Andrián Pertout
Three Microtonal Compositions: The Utilization of Tuning Systems
in Modern Composition
Volume 1
Submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy
Produced on acid-free paper
Faculty of Music
The University of Melbourne
March, 2007
Abstract
Three Microtonal Compositions: The Utilization of Tuning Systems in
Modern Composition encompasses the work undertaken by Lou
Harrison (widely regarded as one of America’s most influential and
original composers) with regards to just intonation, and tuning and
scale systems from around the globe – also taking into account the
influential work of Alain Daniélou (Introduction to the Study of
Musical Scales), Harry Partch (Genesis of a Music), and Ben
Johnston (Scalar Order as a Compositional Resource). The
essence of the project being to reveal the compositional
applications of a selection of Persian, Indonesian, and Japanese
musical scales utilized in three very distinct systems: theory versus
performance practice and the ‘Scale of Fifths’, or cyclic division of
the octave; the equally-tempered division of the octave; and the
‘Scale of Proportions’, or harmonic division of the octave
championed by Harrison, among others – outlining their theoretical
and aesthetic rationale, as well as their historical foundations. The
project begins with the creation of three new microtonal works
tailored to address some of the compositional issues of each
system, and ending with an articulated exposition; obtained via the
investigation of written sources, disclosure of compositional
technique, mathematical analysis of relevant tuning systems,
spectrum analysis of recordings, and face-to-face discussions with
relevant key figures.
THE UNIVERSITY OF MELBOURNE Faculty of Music
TO WHOM IT MAY CONCERN
This is to certify that
(i) the thesis comprises only my original work towards the PhD except where indicated in the Preface*,
(ii) due acknowledgement has been made in the text to all other material used,
(iii) the thesis is less than 80,000 words in length, exclusive of tables, maps, bibliographies and
appendices or the thesis is [number of words] as approved by the RHD Committee.
Signature:
Name in Full: Andrián Pertout
Date: 2 March, 2007
Dedicated to my father,
the late Aleksander Herman Pertout
(b. Slovenia, 1926; d. Australia, 2000)
Acknowledgements
A special thanks to the supervisors:
Professor Brenton Broadstock (Coordinator of Composition, Faculty of Music, University of Melbourne)
and Associate Professor Neil McLachlan (School of Behavioural Science, Faculty of Medicine, Dentistry
and Health Sciences, University of Melbourne). Brenton Broadstock should be especially thanked for
being an inspirational force not only during the last four years of the PhD candidature, but throughout the
last ten years of my composition studies at the University of Melbourne. His encouragement, support,
and direction have exceeded well beyond his duties as supervisor and composition teacher, and
consequently remain forever grateful.
Professor Andrew Schultz (Dean of the Faculty of Creative Arts, University of Wollongong, NSW, Australia)
also deserving a mention with regards to L’assaut sur la raison for Symphony Orchestra (2003) and
Bénédiction d’un conquérant for Symphony Orchestra (2004), which were especially composed for
ACOF 2003 and 2004 (Australian Composers’ Orchestral Forum – Composition workshops with Brenton
Broadstock, Andrew Schultz, and the Tasmanian Symphony Orchestra).
A special thanks also to Dr. Julian Yu who was the official mentor for the 2003 and 2004 ACOF project.
A special thanks to the following people for their direct assistance to the composition folio:
Stephen Adams (Presenter, ABC Classic FM) for producing an excellent program featuring La flor en la
colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) on his ABC Classic FM radio
program.
Susan Batten (Presenter, 3MBS FM) for producing two excellent programs featuring L’assaut sur la raison
for Symphony Orchestra (2003), Navigating the Labyrinth for String Orchestra (2002), and Aristotle’s
Rhetoric, Suite for Orchestra (2001-02, Rev. 2005) – together with an interview with the composer – on
her 3MBS FM Radio ‘Music in Melbourne’ program, in celebration of the Betty Amsden Award – 2005
3MBS FM National Composer Awards.
APRA (Australasian Performing Right Association) for recognizing L’assaut sur la raison for Symphony
Orchestra (2003) with the APRA Encouragement Award – 2004 3MBS FM National Composer Awards.
Andrew Blackburn (Artistic Director, 2007 Melbourne Town Hall Organ Project), Jean Penny and the
Melbourne City Council for commissioning Symétrie intégrante for Flute, Organ and Electronics (2005-06)
for the upcoming 2007 Melbourne Town Hall Organ Project, Melbourne, Australia.
Enmanuel Blanco (Executive Director, Festival Internacional de Música Electroacústica) for selecting
Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to be performed at the XI Festival
Internacional de Música Electroacústica ‘Primavera en la Habana’ 2006, 6-12 March, 2006, Habana, Cuba.
Associate Professor Jack Body (Artistic Director, 2007 Asia Pacific Festival, 26th Asian Composers League
Festival & Conference, and Associate Professor of Composition, New Zealand School of Music, Victoria
University, Wellington, New Zealand) and the festival organizers for selecting Àzàdeh for Santär and Tape
(2004, Rev. 2005) to be performed at the 2007 Asia Pacific Festival (26th Asian Composers League
Festival & Conference), 8-16 February, 2007, Wellington, New Zealand. Also, for selecting the
conference paper Theory Versus Performance Practice: Àzàdeh for Santär and Tape to be presented at
the 2007 Asia Pacific Festival ‘Tradition/Transformation’ Conference.
Warren Burt (Wollongong, NSW, Australia) for his generous support and contribution to the direction of
the PhD research, and especially with regards to Exposiciones for Sampled Microtonal Schoenhut Toy
Piano (2005).
Ao Changqun (Organizing Committee Chairman, 2005 Second Sun River Student New Composition
Competition, and President, Sichuan Conservatory of Music, Chengdu, People’s Republic of China) for
recognizing La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) with the
Third Prize in the 2005 Second Sun River Student New Composition Competition (Chengdu, People’s
Republic of China).
Phyllis Chen for requesting a Toy Piano and Tape arrangement of Exposiciones for Sampled Microtonal
Schoenhut Toy Piano (2005), and for her incredible talent, evident in her virtuosic interpretations of the
work in Bloomington, Indiana and Chicago, Illinois, USA, as well as at the 2007 International Gaudeamus
Interpreters Competition & Chamber Music Week in Amsterdam, The Netherlands.
David Claman (Assistant Professor, Music Department, College of the Holy Cross, Worcester,
Massachusetts, USA) and Matt Malsky (Associate Professor of Music, Department of Visual and Performing
Arts, Clark University, Worcester, Massachusetts, USA) for selecting Exposiciones for Sampled Microtonal
Schoenhut Toy Piano (2005) to be part of the Extensible Toy Piano Project, 5-6 November, 2005,
Department of Visual and Performing Arts, Clark University, Worcester, Massachusetts, USA; and the
÷××× Acknowledgements
Extensible Toy Piano Festival, 4 March, 2007, Performing Arts Center, Department of Music, State University
of New York, Albany, New York, USA.
Barry Cockroft (tenor saxophone) and Adam Pinto (pianoforte) for commissioning and performing
Digressioni modali for Tenor Saxophone and Pianoforte (2003) at the Melbourne International Festival of
Single Reeds, 26-29 March, 2005, Victorian College of the Arts, Southbank, Melbourne, Australia. Also,
for recording the work for the ‘rompduo’ Crazy Logic CD release. Barry Cockroft (tenor saxophone) and
Marc Ryser (pianoforte) for performing the work at The Banff Centre, Banff, Alberta, Canada, and finally
Barry Cockroft (Reed Music) for publishing the work with Reed Music.
Professor Barry Conyngham (former Emeritus Professor of the University of Wollongong and Southern
Cross University, Lismore, NSW, Australia) for his compositional direction during his residency at the
University of Melbourne in 2005.
David Collins (Technical Officer, Faculty of Music, University of Melbourne) for technical assistance
throughout the PhD candidature, as well as invaluable advice with regards to sound diffusion concepts.
David B. Doty (Author of The Just Intonation Primer, Founder of the Just Intonation Network, and Editor of
the Network’s Journal, 1/1, San Francisco, California, USA) for making time for me during my 2004 visit to
San Francisco, California, USA, and for his compositional guidance with regards to just intonation
concepts.
Ensamble Contemporáneo (Aliocha Solovera [artistic director], Cristián Gonzáles [flute], Dante Burotto
[bass clarinet], Alexandros Jusakos [pianoforte], Davor Miric [violin], and Celso López [violoncello]) for
performing La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) at the XV
Festival de Música Contemporánea Chilena (15th Chilean Festival of Contemporary Music), 21-27
November, 2003, Santiago, Chile.
Ivano Ercole (Presenter, Rete Italia) for producing an excellent program featuring L’assaut sur la raison for
Symphony Orchestra (2003), Navigating the Labyrinth for String Orchestra (2002), Gèrëémeler for
Amplified Èrhú, Sampled Harmonium, Cajón and Bombo (2001), Bénédiction d’un conquérant for
Symphony Orchestra (2004), An Honourable Silence for Solo Santär (2001), Renascence for Violin,
Violoncello, Piano and Percussion (2001, Rev. 2006), and Seeds of Passion for Amplified Violoncello
(1999) – together with an interview with the composer – on his Rete Italia radio program.
Acknowledgements ×Ø
The Ónix Ensamble (Alejandro Escuer [flute], Fernando Domínguez [clarinet], Abel Romero [violin],
Edgardo Espinosa [violoncello], and Krisztina Deli [pianoforte]) for selecting La flor en la colina for Flute,
Clarinet, Violin, Violoncello and Pianoforte (2003-04) to be performed at the XXIX Foro Internacional de
Música Nueva (29th International Forum of New Music), “Manuel Enríquez”, 2007, May-June, 2007,
México City, México.
Isabel Ettenauer (St. Poelten, Austria) and Goska Isphording (Eindhoven, The Netherlands) for inspiring the
arrangement of Exposiciones for Toy Piano and Spinet (2005), and for performing the work at the 2006
BMIC Cutting Edge Series, London, UK, and at Axes/Jazzpower, Eindhoven, The Netherlands.
Rodolfo Fischer (Conductor, Basel, Switzerland) for selecting Bénédiction d’un conquérant for Symphony
Orchestra (2004) to be performed by the Orquestra Petrobras Sinfônica at the Theatro Municipal do Rio
de Janeiro, in Rio de Janeiro, Brazil as part of the Orquestra Petrobras Sinfônica ‘Série Ouro Negro’ 2006
concert series, and also for his excellent direction during the rehearsals and final performance.
Robert Franz (Associate Conductor, Louisville Orchestra, Louisville, Kentucky, USA) and the Louisville
Orchestra for recognizing L’assaut sur la raison for Symphony Orchestra (2003) as the winner of the First
Prize in the 2004 ISU Contemporary Music Festival/Louisville Orchestra Composition Competition, and also
for presenting a memorable performance of the work at the Indiana State University 38th Annual
Contemporary Music Festival, ‘Plugged In: Music With an Electric Edge’, November 10-12, 2004, Terre
Haute, Indiana, USA. Indiana State University for sponsoring the award, and providing an opportunity to
conduct a lecture at the festival.
Professor Don Freund (Professor of Music Composition, Indiana University School of Music, Bloomington,
Indiana, USA) and Sandra Freund for their hospitality and enormous generosity during my weekend stay in
2004 with the Freunds in Bloomington, Indiana, USA. A further warm thanks to Don Freund for his
contribution to the development of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte
(2003-04).
Kyle Gann (Associate Professor of Music, Faculty, Bard College, Annandale-on-Hudson, New York, USA)
for his support during my 2004 visit to Bard College (Annandale-on-Hudson, New York, USA), and for his
compositional guidance with regards to just intonation concepts.
Dr Noah Getz (Instructor of Saxophone, American University, and Jazz Saxophone Instructor, Levine
School of Music, Washington, DC, USA) and the judges of the 2005 American University Saxophone
Ø Acknowledgements
Symposium Composition Contest for recognizing Digressioni modali for Tenor Saxophone and Pianoforte
(2003) as the winner of the Third Prize in the international composition competition. Noah Getz (tenor
saxophone) and John Kilkenny (marimba) for inspiring the arrangement of Digressioni modali for Tenor
Saxophone and Marimba (2003), and for performing the work in Alexandria, Virginia and Washington, DC,
USA. Noah Getz (tenor saxophone) and Laurence Gingold (pianoforte) for performing the work in
Lancaster, Pennsylvania, USA, and finally, Noah Getz (tenor saxophone) and Jeffrey Chappell (pianoforte)
for recording the work for CD release.
Brooke Green (Presenter, ABC Classic FM) for producing an excellent program featuring L’assaut sur la
raison for Symphony Orchestra (2003) on her ABC Classic FM ‘Composers Emerging’ program, together
with an interview with the composer, as part of ACOF 2003 (Australian Composers’ Orchestral Forum –
Composition workshops with Brenton Broadstock, Andrew Schultz, and the Tasmanian Symphony
Orchestra).
Dr. Stuart Greenbaum (Lecturer in Composition, Faculty of Music, University of Melbourne) for his
incredible support throughout the PhD candidature.
Alejandro Guarello (Artistic Director, XV Festival de Música Contemporánea Chilena, Instituto de Música,
Facultad de Artes, Pontificia Universidad Católica de Chile, Santiago, Chile) for selecting La flor en la colina
for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) to be performed at the XV Festival de
Música Contemporánea Chilena (15th Chilean Festival of Contemporary Music), 21-27 November, 2003,
Santiago, Chile.
Christian Haines (Lecturer and Unit Coordinator, Electronic Music Unit, Elder Conservatorium of Music,
University of Adelaide) for selecting Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) to
be part of the Medi(t)ations: Computers, Music and Intermedia, Australasian Computer Music Association
Conference 2006, 11-13 July, 2006, Adelaide, Australia.
Michael Harrison (New York, NY, USA) for his demonstration of the ‘harmonic piano’ – a modified seven-
foot Schimmel grand piano – during my visit to New York, NY, USA in 2004.
The international jury of the ISCM (consisting of Stanko Horvat [Croatia], Zygmunt Krauze [Poland],
Giampaolo Coral [Italy], Frank Corcoran [Ireland/Germany], Arne Nordheim [Norway], and Berislav Šipuš
[Croatia]) for selecting La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) to
Acknowledgements Ø×
be performed at the International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd
Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia.
Jerome Kitzke (New York, NY, USA) for his compositional direction during his McGeorge Fellowship
residency at the University of Melbourne in 2005.
Jennifer Logan (Co-Artistic Director, Los Angeles Sonic Odyssey, Electronic and Computer Music Concert
Series 2006, Los Angeles, California, USA) for selecting Paåc hazàr chakêà kaâ andar for Prepared Multi-
tracked Disklavier (2000), Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005), Àzàdeh for
Tape (2004, Rev. 2005), La Homa Kanto for Tape (2005), and Sonic Junk Yard for Tape (1999) to be part
of the Los Angeles Sonic Odyssey Electronic and Computer Music Concert Series 2005, 2006, and 2007
Los Angeles and Pasadena, California, USA.
Jana Haluza Lucic (Producer, HRT, Hrvatska Radio, Zagreb, Croatia) for producing an excellent program
featuring La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-04) – together with
an interview with the composer – on her HRT, Hrvatska Radio (Croatian Radio) ‘World of Music’ program
in Zagreb, Croatia.
Dr. Susan McDonald (Department of Fine Arts, Philadelphia, Pennsylvania, USA) for selecting Exposiciones
for Sampled Microtonal Schoenhut Toy Piano (2005) to be performed at the ‘La Salle University:
Electroacoustic Works Inspired by Popular Music’ concert in November, 2005, Philadelphia, Pennsylvania,
USA.
Marshall McGuire (Artistic Director, Sonic Art Ensemble, Sydney, NSW, Australia) and the Sonic Art
Ensemble (Christine Draeger [flute], Margery Smith [bass clarinet], Rowan Martin [violin], Adrian Wallis
[violoncello], and Bernadette Balkus [pianoforte]) for programming the Australian premier of La flor en la
colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) within the 2006 ‘Southern Stars’
concert in Sydney, Australia. Marshall McGuire for also inspiring and presenting the world premier of
Zambalogy for Harp (2004) in Sydney.
Pavel Mihelœiœ (Artistic Director, Ensemble MD7, and Dean of the Ljubljana Academy of Music, Ljubljana,
Slovenia) and Ensemble MD7 (Steven Loy [conductor], Anamarija Tomac [flute], Jože Kotar [clarinet], Katja
Krajnik [viola], Igor Mitrovic [violoncello], Uroš Polanc [trombone], Luca Ferrini [pianoforte], and Franci
Krevh [percussion]) for commissioning and performing Aequilibrium for Flute, Clarinet, Viola, Cello,
Ø×× Acknowledgements
Trombone, Piano and Percussion (2006) at the Ljubljana Festival 2006, 19 June – 31 August, 2006,
Ljubljana, Slovenia.
Adam Muller (Associate Professor of Saxophone, Florida International University, Miami, Florida, USA) and
Matthew Van Hoose (Accompanist in Residence, Department of Performing Arts, College of Arts and
Sciences, American University, Washington, DC, USA) for performing Digressioni modali for Tenor
Saxophone and Pianoforte at the First American University Saxophone Symposium, 26 March, 2005,
Washington, DC, USA.
Anne Norman (shakuhachi) and Peter Hagen (harpsichord) for assisting in the development of Tres
Imágenes Norteñas for Shakuhachi and Harpsichord (2006), and for performing the work at the
Melbourne Composers’ League ‘From a Silence Well’ concert as part of the 2006 Australia-Japan Year of
Exchange Celebrations.
Juan Miranda (Presenter, SBS Radio, ‘Spanish Radio’ Program) for producing an excellent program
featuring Navigating the Labyrinth for String Orchestra (2002), Seeds of Passion for Amplified Violoncello
(1999), and Bénédiction d’un conquérant for Symphony Orchestra (2004) – together with an interview
with the composer – on his SBS Radio, ‘Spanish Radio’ program.
Peter Neville (Head of Percussion, School of Music, Victorian College of the Arts) for his incredible
enthusiasm for contemporary music and Australian composition, as well as for his insight into
polyrhythmic science.
John D. Nugent (Music Editor, Oregon Literary Review: An Online Collection of Literature, Hypertext, Art,
Music, and Hypermedia, Portland, Oregon, USA) for publishing Exposiciones for Sampled Microtonal
Schoenhut Toy Piano (2005) in the Winter/Spring 2006, Vol. 1, No. 1 edition of the Oregon Literary
Review.
The Omni Ensemble (David Wechsler [flute], Paul Garment [bass clarinet], Olivier Fluchaire [violin], Deborah
Sepe [violoncello], and Jim Lahti [pianoforte]) for presenting the American premier, as well as a follow-up
performance of La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-2004) in
Brooklyn and New York, NY, USA during their 2006 concert series.
José Oplustil Acevedo (Presenter, Radio Beethoven [Radioemisoras], Siglo XX, Santiago, Chile) for
producing an excellent program featuring Bénédiction d’un conquérant for Symphony Orchestra (2004),
Acknowledgements Ø×××
L’assaut sur la raison for Symphony Orchestra (2003), Görüsmeler for Amplified Èrhú, Sampled
Harmonium, Cajón and Bombo (2001), and Pañc hazar chakra kai andar for Prepared Disklavier (2000) –
together with an interview with the composer – on his Radio Beethoven (Radioemisoras) ‘Programa Siglo
XX’ program in Santiago, Chile.
Alex Pertout (Head of Improvisation, School of Music, Victorian College of the Arts) for his invaluable
advice with regards to Afro-Latin percussion, rhythm and improvisation.
Katija Farac-Pertout, my wife, for her amazing belief and understanding not only during the last four years
of the PhD degree, but throughout the last ten years of my composition studies at the University of
Melbourne.
Maritza Pertout (Library Technician, State Library of Victoria) for her assistance with Spanish grammar, as
well as countless other aspects of music publishing dilemmas.
Qmars Piraglu (formerly Siamak Noory) for his great inspiration and dedication to the realization of
Àzàdeh for Santär and Tape (2004, Rev. 2005), as well as for the performance of the work at the 2007
Asia Pacific Festival (26th Asian Composers League Festival & Conference), 8-16 February, 2007,
Wellington, New Zealand.
Glen Riddle (Coordinator, Foundation Program, Music Performance, School of Music, Victorian College of
the Arts) for the French lessons.
Hans Roels (Concert Program Manager and Producer, Logos Foundation, Ghent, Belgium) for presenting
the European premier of Exposiciones for Sampled Microtonal Schoenhut Toy Piano (2005) at the Logos
Foundation 2006 ‘Tape Tum & Heleen Van Haegenborgh’ concert in Ghent, Belgium.
Johanna Selleck for her incredible support throughout the PhD candidature.
Berislav Šipuš (Artistic Director, International Society for Contemporary Music (ISCM) World Music Days
2005 / 23rd Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia) for his hospitality during the
International Society for Contemporary Music (ISCM) World Music Days 2005 / 23rd Music Biennale
Zagreb, Croatia.
Ø×÷ Acknowledgements
The Sonemus Ensemble [Bosnia-Herzegovina] (Ališer Sijaric [Artistic Director], Boris Previšic [flute], Vedran
Tuce [bass clarinet], Julia Gubaidulina [pianoforte], Petar Haluza [violin], and Conradin Brodbek
[violoncello]) for the performance of La flor en la colina for Flute, Clarinet, Violin, Violoncello and
Pianoforte (2003-2004) at the International Society for Contemporary Music (ISCM) World Music Days
2005 / 23rd Music Biennale Zagreb, 15-24 April, 2005, Zagreb, Croatia.
Dr. Todd E. Sullivan (Chairperson, Department of Music, Indiana State University, Terre Haute, Indiana,
USA) for his incredible hospitality during the Indiana State University 38th Annual Contemporary Music
Festival, ‘Plugged In: Music With an Electric Edge’, November 10-12, 2004. A further warm thanks for
driving me all the way from Terre Haute to Bloomington, Indiana.
Natasha Talmacs (Presenter, SBS Radio, ‘Croatian Radio’ Program, Sydney, Australia) and Silvio Rivier
(Presenter, Narrator and Series Producer, Global Village, SBS TV, Sydney, Australia) for producing an
excellent program featuring La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte (2003-
2004), L’assaut sur la raison for Symphony Orchestra (2003), and Seeds of Passion for Amplified
Violoncello (1999) – together with an interview with the composer – on her SBS Radio, ‘Croatian Radio’
program.
Antonio Tenace for his incredible support throughout the PhD candidature, and more importantly, for
fixing my scientific calculator.
Kenneth Young (Conductor, Wellington, New Zealand) and The Tasmanian Symphony Orchestra for the
performance of L’assaut sur la raison for Symphony Orchestra (2003) and Bénédiction d’un conquérant
for Symphony Orchestra (2004), which were especially composed for ACOF 2003 and 2004 (Australian
Composers’ Orchestral Forum – Composition workshops with Brenton Broadstock, Andrew Schultz, and
the Tasmanian Symphony Orchestra).
A special thanks to the following people for their general assistance, advice and support:
Betty Amsden (OAM), Celia Anderson, Dr. Jeri-Mae Astolfi (Assistant Professor, Department of Music,
Henderson State University, Arkadelphia, Arkansas, USA), Rachel Atkinson (Trio Erytheia), Peter Aviss
(Conductor and Musical Director, Oare String Orchestra, Faversham, UK), Laura Baker-Goldsmith, Pip Barry,
Natasha Bennett, Jennifer Bird (New Audience Ensemble), David Black (Rarescale, London, UK), Ellen
Bottorff (Orenunn Trio, Kansas City, Missouri, USA), Julianne Boren (Orenunn Trio, Kansas City, Missouri,
USA), Mark Boren (Orenunn Trio, Kansas City, Missouri, USA), James Bradley (Doubling Up Trio), Le Brass
Acknowledgements Ø÷
Band du Nord-Pas de Calais (Calais, France), Stuart Brownley (Doubling Up Trio), Gary Robert Buchanan
(Conductor and Musical Director, The Foundation Orchestra, Reno, Nevada, USA), David C. Bugli
(Conductor and Musical Director, Carson City Symphony, Carson City, Nevada, USA), Stuart Byrne
(Doubling Up Trio), Isin Cakmakcioglu (Trio Erytheia), José Miguel Candela (Coordinator, Comunidad
Electroacústica de Chile [CECh], Santiago, Chile), Erik Carlson (New York Miniaturist Ensemble, New York,
NY, USA), Robert Casteels (Dean of the Faculty of Performing Arts, LASALLE-SIA College of the Arts,
Singapore), Robert Chamberlain (Trio Erytheia), La Chapelle Musicale de Tournai (Tournai, Belgium),
Radiance Chen (New Audience Ensemble), Penelope Clarke (Thunder Bay, Ontario, Canada), Dr.
Christopher Coleman (Radio Television Hong Kong Radio 4, Hong Kong), Andrew Conley, Rolando Cori
(Associate Professor of Music, Facultad de Artes, Departamento de Música, Universidad de Chile, and
President, Asociacion Nacional de Compositores de Chile, Santiago, Chile), Nicholas Cowall (Conductor,
Victorian Youth Symphony Orchestra), Patricia Da Dalt (Quinteto CEAMC, Buenos Aires, Argentina), Lerida
Delbridge (The Tin Alley String Quartet), Madonna Douglas (Thunder Bay, Ontario, Canada), Eve Duncan
(President, The Melbourne Composers’ League), Shannon Ebeling, Mark Engebretson (Conference Chair,
2005 Society of Composers [SCI] National Conference, School of Music, University of North Carolina at
Greensboro, Greensboro, North Carolina, USA), Ed Ferris (New Audience Ensemble), Barbara Finch
(Thunder Bay, Ontario, Canada), The Foundation Orchestra (Reno, Nevada, USA), Johannes Fritzsch
(Nürnberg, Germany), Steve Gibson (Open Space Art Society, Victoria, British Columbia, Canada), Yves
Gigon (Canadian Electroacoustic Community [CEC], Montréal, Québec, Canada), Ian Godfrey (Lecturer in
Music and Education, Faculty of Music, University of Melbourne), Ben Goudy, Alejandro Guarello (Artistic
Director, XIII Festival de Música Contemporánea Chilena, Instituto de Música, Facultad de Artes, Pontificia
Universidad Católica de Chile, Santiago, Chile), Elías Gurevich (Quinteto CEAMC, Buenos Aires, Argentina),
Steven Heyman (The Syracuse Ensemble, Syracuse, New York, USA), Nancy Hosking, Luke Howard,
Ashley Hribar (Speak Percussion), Frédéric Inigo (Artistic Director, 3èmes Rencontres Musiques Nouvelles,
Lunel, France), Jason Kenner, Danae Killian, Victoria Jacono (3 Lines String Trio, Sydney), Jérôme Joy
(Coordinator, Locus Sonus – Audio in Art, École Nationale Supérieure d’Art de Nice-Villa Arson, Nice,
France), Stijn Kuppens (Artistic Director, Violoncello 2005, Brussels, Belgium), Laura Lentz (Crossroads Trio,
New York, NY, USA), Jennifer Logan (Co-Artistic Director, Los Angeles Sonic Odyssey, Electronic and
Computer Music Concert Series 2005, Los Angeles, California, USA), Phillipe Lorthios (Conductor, Le Brass
Band du Nord-Pas de Calais, Calais, France), Eric Lyon (Assistant Professor, Dartmouth College, Hanover,
New Hampshire, USA), George Macero (The Syracuse Ensemble, Syracuse New York, USA), Briony
Mackenzie (New Audience Ensemble), Marco Antonio Mazzini (Duo Dicto and Diversity, Ghent, Belgium),
John McMurtery (Doctoral Fellow, The Juilliard School of Music, New York, NY, USA), Nyssa McPhail, The
Melbourne University Orchestra, Natsuko Mineghishi, Patrick Murphy (3 Lines String Trio, Sydney), Simona
Musiani (Crossroads Trio, Rome, Italy), Tom Nelson (Southhampton, UK), Cliff Ojala (Thunder Bay, Ontario,
Ø÷× Acknowledgements
Canada), Jorge Pérez (Quinteto CEAMC, Buenos Aires, Argentina), Sonni Petrovski (Musical Director, The
Alea Contemporary Music Ensemble, Skopje, Republic of Macedonia), Marina Phillips (3 Lines String Trio,
Sydney), Timothy Phillips (Speak Percussion), Judy Pile, Vladimir Pritsker (The Syracuse Ensemble, Syracuse
New York, USA), Aleksander Pusz, Ryszard Pusz, Sabina Rakcheyeva (Diversity, Ghent, Belgium), Carla Rees
(Rarescale, London, UK), Darlene Chepil Reid (President, New Music North, Thunder Bay, Ontario,
Canada), Dr. James Romig (Co-Musical Director, The Society for Chromatic Art, New York, NY, USA),
Joelene Rzepisko, Guillermo Sánchez (Quinteto CEAMC, Buenos Aires, Argentina), Naomi Sato (The
Netherlands), Ginevra Schiassi (Ensemble Octandre, Bologna, Italy), José Schiller (Rádio MEC ‘Concerto
das Américas’, Rio de Janeiro, Brazil), Sam Schmetterer (New Audience Ensemble), Phillip Schroeder
(Associate Professor, Department of Music, Henderson State University, Arkadelphia, Arkansas, USA),
Haydée Schvartz (Quinteto CEAMC, Buenos Aires, Argentina), Johanna Selleck, Gemma Sherry, Tarko
Sibbel, Robert Sipos-Ori, Frank Sita (Plenty Valley FM), Emma Skillington (The Tin Alley String Quartet), Laura
Sullivan, Gabriella Swallow (Rarescale, London, UK), Matt Tait, Gaspare Tirincanti (Ensemble Octandre,
Bologna, Italy), Jo To, Eugene Ughetti (Speak Percussion), Josephine Vains, Amy Valent, Carlos Vera
(Santiago, Chile), Lauren Van Der Werff, Orchestra Victoria, The Victorian Youth Symphony Orchestra,
Professor Cirilo Vila Castro (Facultad de Artes, Departamento de Música, Universidad de Chile), Ward de
Vleeschhouwer (Duo Dicto, Ghent, Belgium), Carina Voly (Crossroads Trio, Buenos Aires, Argentina), Cory
Wagstaff, Koen Walraevens (Diversity, Ghent, Belgium), Russell Ward, Anneliese Weibel (Artistic Director,
2004 Society of Composers [SCI] Region II Conference, University of New York, School of Performing
Arts, Geneseo, New York, USA), Larissa Weller (New Audience Ensemble), Justin Williams (The Tin Alley
String Quartet), Elissa Wilson, Michelle Wood (The Tin Alley String Quartet), Larry Zimmerman (Minneapolis,
Minnesota, USA).
Acknowledgements Ø÷××
Table of Contents
Volume 1
Introduction ............................................................................................................................................................ 1
Microtonality ..................................................................................................................................................... 1
Pitch Audibility and Discrimination ................................................................................................................ 3
Three Microtonal Compositions .................................................................................................................... 4
Folio of Compositions ..................................................................................................................................... 5
Methodology ................................................................................................................................................... 6
Interval Nomenclature and Notation System ............................................................................................... 9
1. Theory Versus Performance Practice: Àzàdeh for Santñr and Tape ..................................... 11
A Brief History of Persian Classical Music .................................................................................................... 11
The Seventeen-Note Gamut ........................................................................................................................ 12
Persian Musical Scholarship in the Twentieth Century .............................................................................. 15
The Twenty-Four Equally-Tempered Quarter-Tone Scale ........................................................................ 16
The Pythagorean Division of the Octave ..................................................................................................... 18
Alain Daniélou’s Scale of Fifths ..................................................................................................................... 22
The Twenty-Two Note Division of the Octave .......................................................................................... 26
The Theory of Flexible Intervals .................................................................................................................... 27
Àzàdeh for Santñr and Tape ....................................................................................................................... 30
The Artist ......................................................................................................................................................... 30
The Instrument ................................................................................................................................................ 31
The Persian Modal System ............................................................................................................................ 32
Tuning Analysis Protocols ............................................................................................................................. 36
Tuning of the Santñr ...................................................................................................................................... 38
Spectrum Analysis Results............................................................................................................................. 42
Analysis of Variance ...................................................................................................................................... 49
Tuning System Comparison .......................................................................................................................... 52
Performance Practice and Tuning ................................................................................................................ 53
The Piano Tuner’s Octave and Inharmonicity ............................................................................................. 55
The Tuning of Unisons ................................................................................................................................... 56
Climate and Tuning ........................................................................................................................................ 58
Gušes of Dastgàh-e Segàh ........................................................................................................................... 60
Sampling of the Santñr and Vocals ............................................................................................................. 63
2. The Equally-Tempered Archetype: Exposiciones for Sampled Microtonal Schoenhut
Toy Piano ....................................................................................................................................................... 67
Equal Temperaments .................................................................................................................................... 67
Studies of Microtonal Equal Temperaments .............................................................................................. 68
Nicolas Mercator’s Fifty-Three-Tone Equally-Tempered Division of the Octave .................................. 71
Pietro Aron’s Quarter-Comma Meantone Tempered Division of the Octave ....................................... 74
Joseph Sauveur’s Forty-Three-Tone Equally-Tempered Division of the Octave ................................... 80
Origins of Equal Temperament .................................................................................................................... 83
The Twelve-Tone Equally-Tempered Division of the Octave .................................................................. 87
Exposiciones for Sampled Microtonal Schoenhut Toy Piano ................................................................. 92
A Brief History of the Toy Piano ................................................................................................................... 92
The Schoenhut Toy Piano Sample .............................................................................................................. 94
Sound Diffusion ............................................................................................................................................. 95
Polyrhythmic Theory ...................................................................................................................................... 96
Alain Daniélou’s Scale of Proportions ......................................................................................................... 99
Notation for the Twenty-Four Equal Temperaments .............................................................................. 102
Sléndro and Pélog Scales .......................................................................................................................... 104
One-Tone Equal Temperament ................................................................................................................. 107
Two-Tone Equal Temperament ................................................................................................................. 109
Three-Tone Equal Temperament ............................................................................................................... 111
Four-Tone Equal Temperament ................................................................................................................. 113
Five-Tone Equal Temperament .................................................................................................................. 115
Six-Tone Equal Temperament .................................................................................................................... 117
Seven-Tone Equal Temperament .............................................................................................................. 119
Eight-Tone Equal Temperament ................................................................................................................ 121
Nine-Tone Equal Temperament................................................................................................................. 123
Ten-Tone Equal Temperament .................................................................................................................. 127
Eleven-Tone Equal Temperament ............................................................................................................. 129
Twelve-Tone Equal Temperament ............................................................................................................ 132
Thirteen-Tone Equal Temperament ........................................................................................................... 135
ØØ Table of Contents
Fourteen-Tone Equal Temperament......................................................................................................... 139
Fifteen-Tone Equal Temperament ............................................................................................................. 144
Sixteen-Tone Equal Temperament ........................................................................................................... 150
Seventeen-Tone Equal Temperament ..................................................................................................... 153
Eighteen-Tone Equal Temperament ......................................................................................................... 157
Nineteen-Tone Equal Temperament ........................................................................................................ 162
Twenty-Tone Equal Temperament ........................................................................................................... 168
Twenty-One-Tone Equal Temperament .................................................................................................. 171
Twenty-Two-Tone Equal Temperament .................................................................................................. 174
Twenty-Three-Tone Equal Temperament ................................................................................................ 179
Twenty-Four-Tone Equal Temperament .................................................................................................. 184
Blackwood’s Dictum .................................................................................................................................. 188
3. The Harmonic Consideration: La Homa Kanto for Harmonically Tuned Synthesizer
Quartet ........................................................................................................................................................ 189
Just Intonation ............................................................................................................................................. 189
The Harmonic and Subharmonic Series .................................................................................................. 190
The Monochord ......................................................................................................................................... 198
Combinational Tones ................................................................................................................................. 200
Periodicity Pitch ........................................................................................................................................... 201
Prime Numbers, Primary Intervals, and Prime Limits ................................................................................ 202
The Just Diatonic Scale .............................................................................................................................. 202
The Just Chromatic Scale ........................................................................................................................... 205
Ben Johnston’s Fifty-Three-Tone Just Intonation Scale .......................................................................... 209
Harry Partch’s Forty-Three-Tone Just Intonation Scale ........................................................................... 212
Adriaan Daniël Fokker’s Thirty-One-Tone Equally-Tempered Division of the Octave ........................ 216
La Homa Kanto for Harmonically Tuned Synthesizer Quartet ............................................................... 218
The Harpsichord Sample ........................................................................................................................... 221
Ben Johnston’s System of Notation ......................................................................................................... 222
Compositional Strategy .............................................................................................................................. 223
Composing With Melodicles ..................................................................................................................... 224
Three-Limit Just Intonation ......................................................................................................................... 231
Five-Limit Just Intonation ............................................................................................................................ 236
Seven-Limit Just Intonation ........................................................................................................................ 243
Table of Contents ØØ×
Eleven-Limit Just Intonation ........................................................................................................................ 252
Thirteen-Limit Just Intonation ...................................................................................................................... 259
Seventeen-Limit Just Intonation ................................................................................................................. 266
Nineteen-Limit Just Intonation .................................................................................................................... 272
Twenty-Three-Limit Just Intonation ............................................................................................................ 278
Twenty-Nine-Limit Just Intonation.............................................................................................................. 284
Thirty-One-Limit Just Intonation ................................................................................................................. 289
Johnston’s Dictum ....................................................................................................................................... 296
Conclusion ........................................................................................................................................................... 297
‘Manual’ of Microtonal Composition ........................................................................................................ 297
A Vast Universe of Subtle Intervallic Relationships .................................................................................. 297
Bibliography ....................................................................................................................................................... 301
Appendices ........................................................................................................................................................ 311
Appendix A: Comparative Table of Musical Intervals ............................................................................ 311
Appendix B: Microtonal Notation Font .................................................................................................... 345
Volume 2
Recordings – Folio of Compositions 2003-2007: Volume 2 .......................................................................... vii
1. Àzàdeh for Santär and Tape, no. 389 (2004, Rev. 2005) .................................................................. 1
2. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, no. 392 (2005) ........................... 47
3. La Homa Kanto for Harmonically Tuned Synthesizer Quartet, no. 393 (2005) .......................... 91
4. Symétrie intégrante for Flute, Organ and Electronics, no. 394 (2005-2006) ......................... 153
5. Tres Imágenes Norteñas for Shakuhachi and Harpsichord, no. 396 (2006) .......................... 203
ØØ×× Table of Contents
Volume 3
Recordings – Folio of Compositions 2003-2007: Volume 3 .......................................................................... vii
1. L’assaut sur la raison for Symphony Orchestra, no. 386 (2003) .................................................... 1
2. Digressioni modali for Tenor Saxophone and Pianoforte, no. 387 (2003) ................................ 71
3. La flor en la colina for Flute, Clarinet, Violin, Violoncello and Pianoforte, no. 388 .................. 97
(2003, Rev. 2004)
4. Bénédiction d’un conquérant for Symphony Orchestra, no. 390 (2004) ........................... 175
5. Zambalogy for Harp, no. 391 (2004) ............................................................................................... 245
6. Aequilibrium for Flute, Clarinet, Viola, Cello, Trombone, Piano and Percussion, no. 395 ..... 257
(2006)
Table of Contents ØØ×××
Introduction
Microtonality
In a Perspectives of New Music article, Douglas Keislar states that the term microtonality “conjures up
images of impossibly minute intervals, daunting instruments with hundreds of notes per octave, and wildly
impractical performance instructions,” but that “such difficulties in fact characterize only a small
percentage of the music that uses tunings other than standard twelve-note equal temperament.” Keislar
then suggests that American composer Ivor Darreg’s proposal of the Greek term ‘xenharmonic’ or
‘unfamiliar modes’ is perhaps better suited to music utilizing “radically different tunings.”1 Alternative
language for the term ‘microtonal’ is presented by Lydia Ayers in Exploring Microtonal Tunings: A
Kaleidoscope of Extended Just Tunings and their Compositional Applications, with the following list of
expressions: “tuning; microintervals; macrointervals or macrotones, such as 5-tone, 7-tone, and 10-tone
equal temperaments; omnitonal; omnisonics; neoharmonic; xenharmonic; ‘exploring the sonic spectrum’;
and non-twelve.” Although in spite of Ayers’s general attraction to the broadness of ‘omnitonal’,
‘microtonal’ is nevertheless espoused for its universality.2
The actual term ‘microtonal’ is generally reserved for music utilizing “scalar and harmonic resources”
outside of Western traditional twelve-tone equal temperament, with “music which can be performed in
twelve-tone equal temperament without significant loss of its identity” not considered “truly microtonal”
by some theorists. Most non-western musical traditions (intonationally disengaged from contemporary
Western musical practice) almost certainly accommodate this description. In the online Encyclopedia of
Microtonal Music Theory, Joe Monzo provides the following discussion about the etymology of
‘microtonal’:
“Strictly speaking, as can be inferred by its etymology, ‘microtonal’ refers to small intervals. Some theorists
hold this to designate only intervals smaller than a semitone (using other terms, such as ‘macrotonal’, to
describe other kinds of non-12-edo intervals), while many others use it to refer to any intervals that deviate
from the familiar 12-edo scale, even those which are larger than the semitone – the extreme case being
exemplified by Johnny Reinhard, who states that all tunings are to be considered microtonal.”3
In the West, the concept of microtonality was notably given prominence to during the Renaissance by
Italian composer and theorist Nicola Vicentino (1511-1576), in response to “theoretical concepts and
1 Douglas Keislar, “Introduction,” Perspectives of New Music 29.1 (Winter, 1991): 173.
2 Lydia Ayers, “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional
Applications,” (DMA diss., U. of Illinois, Urbana-Champaign, 1994, PA 9512292) 1-2.
3 Joe Monzo, “Encyclopedia of Microtonal Music Theory,” Microtonal, Just Intonation Electronic Music Software,
2005, Tonalsoft, 17 Nov. 2006, <http://www.tonalsoft.com/>.
materials of ancient Greek music,”4 and later, by music theorists R. H. M. Bosanquet (1841-1912), as well
as Hermann L. F. Helmholtz (1821-1894), and his “translator and annotator” Alexander John Ellis (1814-
1890).5 With regards to the adoption of microtonality by composers in more recent times, according to
The New Harvard Dictionary of Music:
“The modern resurgence of interest in microtonal scales coincided with the search for expanded tonal
resources in much 19th-century music. Jacques Fromental Halévy was the first modern composer to
subdivide the semitone, in his cantata Prométhée enchâiné (1847). The first microtonal piece to use
Western instrumental forms is a string quartet by John Foulds (1897); and the earliest known published
quarter-tone composition, Richard Stein’s Zwei Konzertstücke, op. 26 (1906), is for cello and piano.”6
Gardner Read offers the following historical perspective:
“The history of microtonal speculation during the first half of the twentieth century displays six names above
all others: Julián Carrillo, Adriaan Fokker, Alois Hába, Harry Partch, Ivan Wyschnegradsky, and Joseph Yasser.
All six contributed extensive studies on microtones – historical, technical, and philosophical – and all but
Yasser composed a significant body of music based on their individual explorations into microtonal
fragmentation of the traditional twelve-tone chromatic scale. Later theorist-composers – notably Easley
Blackwood, Ben Johnston, Rudolf Rasch, and Ezra Sims – have extended those explorations into various
tuning systems and temperaments, and each has devised a personal notation for various unorthodox
divisions of the octave.”
Read identifies five essential strategies for the procurement of microtonal intervals, which include: quarter-
and three-quarter-tones, or the division of the octave into twenty-four equal intervals; eighth- and
sixteenth-tones, or forty-eight and ninety-six equal intervals; third-, sixth-, and twelfth-tones, or eighteen,
thirty-six, and seventy-two equal intervals; and fifth-tones, or thirty-one equal intervals; as well as
“extended and compressed microtonal scales” with forty-three, fifty-three, sixty, seventy-two, or more
equal or unequal intervals in the octave.7 J. Murray Barbour on the other hand pronounces Pythagorean
(“excellent for melody, unsatisfactory for harmony”), just intonation (“better for harmony than for
melody”), meantone (“a practical substitute for just intonation, with usable triads all equally distorted”),
and equal temperament (“good for melody, excellent for chromatic harmony”) as the “four leading tuning
4 Accounts of the arcicembalo (a two-manual harpsichord capable of producing thirty-six distinct pitches per
octave) and arciorgano (organ adaptation) were presented by Nicola Vicentino in his treatises L’antica musica ridotta a la
moderna prattica of 1555 and Descrizione dell’ arciorgano (1561). For a further discussion, see Don Michael Randel, ed.,
The New Harvard Dictionary of Music (Cambridge, Mass.: Belknap Press of Harvard U Press, 1986) 47.
5 John H. Chalmers, Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales (Hanover,
NH: Frog Peak Music, 1993) 1-2.
6 Randel, ed., The New Harvard Dictionary of Music 492.
7 Gardner Read, 20th-Century Microtonal Notation (Westport, CT: Greenwood Press, 1990) 2-127.
2 Introduction
systems,” or the “Big Four.” Barbour also makes mention of the “more than twenty varieties of just
intonation,” and “six to eight varieties of the meantone temperament,” as well as the “geometric,
mechanical, and linear divisions of the line” for the mathematical approximation of equal temperament.8
According to Barbour, tuning systems may be classified into two distinct classes: the first being ‘regular’,
where all fifths but one are equal in size; and the second, ‘irregular’, where more than one fifth is unequal
in size. The former includes Pythagorean, meantone, and equal temperament, while the latter (as
classified by Barbour) excludes just intonation.9 Pitch Audibility and Discrimination Although it may be stated that the human ear has a general capacity to hear frequencies between the
ranges of 16Hz and 16,000Hz (equal to 16 to 16,000 cycles per seconds, and approximately C0 and
B9), it must be noted that numerous factors influence the actual outcomes. The 16Hz lower limit is
dependent on two principal factors, being wave intensity and shape; with the inclusion and exclusion of
pure tones displacing the figures for the lower limit to anywhere between 12Hz and 100Hz
(approximately Gþ0 and G2). The 16,000Hz upper limit is generally reserved for a healthy population
under the age of forty, with adolescent capacity as high as 25,000Hz (approximately G10); a supposed
‘normal hearing’ population in some cases not surpassing a 5,000Hz (approximately DÚ8) upper limit; and
another probable large percentage incapable of hearing beyond 10,000Hz (approximately DÚ9).10 The
frequency range of the 88-key pianoforte is between 27.5Hz and 4,186Hz, or A0 to C8, and therefore
encompasses pitch material with a range of over seven octaves. The seven-octave range additionally
represents the range embodied within the collection of instruments that constitute the traditional
symphony orchestra.11
The pitch discrimination threshold for an average adult is around 3Hz at 435Hz, which is
approximately one seventeenth of an equal tone, or 11.899 cents, although a “very sensitive ear can hear
as small a difference as 0.5Hz or less” (approximately a hundredth of a tone, or 1.989 cents). Tests
conducted in 1908 by Norbert Stücker (Zeitschrift für Sinnesphysiologie 42: 392-408) of sixteen
professional musicians in the Viennese Royal Opera conclude a pitch discrimination threshold between
one five-hundred-and-fortieth (0.1Hz) and one forty-ninth of a tone (1.1Hz), or 0.370 and 4.082 cents,
8 J. Murray Barbour, “Irregular Systems of Temperament,” Journal of the American Musicological Society 1.3
(Autumn, 1948): 20. 9 J. Murray Barbour, Tuning and Temperament: A Historical Survey (New York: Dover Publications, 2004) x-xi
10 Carl E. Seashore, Psychology of Music (New York: Dover Publications, 1967) 54-55. 11 Harry F. Olson, Music, Physics and Engineering, 2nd ed. (New York: Dover Publications, 1967) 123.
Introduction 3
with an average of 0.556Hz (approximately a hundredth of a tone), or 2.060 cents.12 In Tuning, Timbre,
Spectrum, Scale William A. Sethares adds the following to the discussion:
“The Just Noticeable Difference (JND) for frequency is the smallest change in frequency that a listener can
detect. Careful testing such as that of E. Zwicker and H. Fastl (Psychoacoustics, Springer-Verlag, Berlin
[1990]) has shown that the JND can be as small as two or three cents, although actual abilities vary with
frequency, duration and intensity of the tones, training of the listener, and the way in which JND is
measured.”13
Three Microtonal Compositions Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition encompasses
the work undertaken by Lou Harrison (widely regarded as one of America’s most influential and original
composers) with regards to just intonation, and tuning and scale systems from around the globe – also
taking into account the influential work of Alain Daniélou (Introduction to the Study of Musical Scales),
Harry Partch (Genesis of a Music), and Ben Johnston (Scalar Order as a Compositional Resource). The
essence of the project being to reveal the compositional applications of a selection of Persian,
Indonesian, and Japanese musical scales utilized in three very distinct systems: theory versus performance
practice and the ‘Scale of Fifths’, or cyclic division of the octave; the equally-tempered division of the
octave; and the ‘Scale of Proportions’, or harmonic division of the octave championed by Harrison,
among others – outlining their theoretical and aesthetic rationale, as well as their historical foundations.
The project begins with the creation of three new microtonal works tailored to address some of the
compositional issues of each system, and ending with an articulated exposition; obtained via the
investigation of written sources, disclosure of compositional technique, mathematical analysis of relevant
tuning systems, spectrum analysis of recordings, and face-to-face discussions with relevant key figures.
The three microtonal works discussed in the thesis include Àzàdeh for santñr and tape, no 389
(2004, Rev. 2005) – composed for Iranian santñrist Qmars Piraglu (formerly Siamak Noory) – which
features the Persian santär (72-string box zither), and serves as a practical study of Persian tuning systems,
with its presentation of both ‘theoretical’ and ‘performance practice’ tunings; an ‘acousmatic’ work
entitled Exposiciones for sampled microtonal Schoenhut toy piano, no. 392 (2005), which attempts to
explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut
model 6625, 25-key toy piano, a complex polyrhythmic scheme, and sequential tuning modulations
12 “Pitch discrimination is measured by sounding two pure tones in quick succession and gradually reducing the
difference in frequency until the observer is unable to tell which of the two tones is higher. The steps usually employed in
such a series are 30, 23, 17, 12, 8, 5, 3, 2, 1, and 0.5Hz, at the level of international (standard) pitch.” For a further
discussion, see Seashore, Psychology of Music 56-57.
13 William A. Sethares, Tuning, Timbre, Spectrum, Scale, 2nd ed. (London: Springer-Verlag, 2005) 44.
4 Introduction
featuring the first twenty-four equally-tempered divisions of the octave; and La Homa Kanto (or ‘The
Human Song’ in Esperanto) for harmonically tuned synthesizer quartet, which derives its pitch material
from Lou Harrison’s five-tone scales (presented in Lou Harrison’s Music Primer: Various Items About Music
to 1970) and features ten distinct tuning modulations: 3-limit through to 31-limit just intonation systems
based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth, twenty-third, twenty-ninth,
and thirty-first partials of the harmonic series.
The aim of the dissertation is to present an articulated exposition of three ‘original’ and unique
microtonal composition models individually exploring the expanded tonal resources of Pythagorean
intonation, equal temperament, and just intonation. It is also proposed that the thesis outlines their
theoretical and aesthetic rationale, as well as their historical foundations, with mathematical analysis of
relevant tuning systems, and spectrum analysis of recordings providing further substance to the project.
Theory versus performance is also taken into account, and the collaboration with an actual performer is
intended to deliver the corporeal perspective. It is anticipated that the thesis will not represent current
acoustic and psychoacoustic research at any great depth, and therefore should not be seen to serve as a
comprehensive study of physics and music. It will nevertheless provide a foundation for the exploration
of tuning systems, and additionally, present a composer’s perspective – as opposed to a musicological
or ethnomusicological study – of microtonal music composition. Folio of Compositions Other works incorporated into volume two and three of ‘Folio of Compositions 2003-07’ include: Symétrie
intégrante for Flute, Organ and Electronics, no. 394 (2005-06); Aequilibrium for flute, clarinet, viola, cello,
trombone, piano and percussion, no. 395 (2006); Tres imágenes norteñas for shakuhachi and harpsichord,
no. 396 (2006); L’assaut sur la raison for symphony orchestra, no. 386 (2003); Digressioni modali for tenor
saxophone and pianoforte, no. 387 (2003); La flor en la colina for flute, clarinet, violin, violoncello and
pianoforte, no. 388 (2003-04); Bénédiction d’un conquérant for symphony orchestra, no. 390 (2004);
and Zambalogy for harp, no. 391 (2004). These works do not represent the microtonal models of the first
three compositions, yet certainly adhere to an exploration of alternative scalar and harmonic materials,
and their application in contemporary compositional practice. Pitch material for these works has been
generated via a selection of methods such as multi-octave grouping (pitch material based on multi-
octave scales constructed of dissimilar tetrachords), modality (modes generated by the major, in,
hirajoshi and kumoijoshi scales), aleatoric formation (pitch material generated via indeterminate means),
pitch class set theory (pitch material derived from the 208 basic pitch-class sets of set theory), synthetic
symmetry (hexatonic and octatonic major and minor scales), cluster generation (pitch material derived
from five-note chords and inversions), physical and psychological concepts of consonance and
Introduction 5
dissonance (the harmonic language of the twelve primary intervals), polymodal and polytonal
juxtaposition (multiple scales and tonalities), as well as cross-cultural abstraction (non-Western music
theoretical concepts).
Methodology Chapter one (theory versus performance practice) begins with a brief history of Persian music, and is
followed by the presentation of Éafå al-Dån Urmawå’s seventeen-note gamut and division of the whole-
tone, and an explanation of the significance of the tetrachord in the construction of melodic and
harmonic structures. A discussion of Persian musical scholarship in the twentieth century then introduces
the three separate theories on intervals and scales of Persian music proposed in the twentieth century:
the twenty-four equally-tempered quarter-tone scale proposed by Ali Naqi Vaziri in the 1920s, the
alternative twenty-two-note scale proposed by Mehdi Barkešli in the 1940s based on Pythagorean
principles, as well as the theory of the five primary intervals of performance practice presented by
Hormoz Farhat in the 1990 publication of his doctoral thesis The Dastgàh Concept in Persian Music.14 The
division of the octave into twenty-four equally-tempered quarter-tones is given a historical perspective,
as well as a mathematical exposition, while the concept of Pythagorean intonation is firstly illustrated via
the construction of a twenty-seven-note Pythagorean scale with the necessary intervals to facilitate the
general modulations of Western tonal music; and secondly, via Daniélou’s ascending ‘scale of fifths’, or
cyclic division of the octave, which presents a series of fifty-nine consecutive fifths, or sixty lü. The BCE
Chinese origins of Pythagoreanism and its philosophical significance according to theorist King Fâng are
also subsequently discussed.15 The development of the seventeen-note gamut by Mehdi Barkešli into a
twenty-two-note Pythagorean scale is then presented, which is followed by Farhat’s theory of flexible
intervals, or of the five primary intervals of performance practice – advocated by Farhat in opposition to
both twenty-four-tone equally-tempered, and twenty-two-note Pythagorean scales of Vaziri and
Barkešli.16
The work, Àzàdeh for santñr and tape, is then introduced, together with a brief biography of the
artist, Qmars Piraglu; a description of instrument, the Persian santär (a 72-string [or 18 quadruple-stringed]
box zither); and a discussion of the essence of the Persian modal system. Following the establishment of
the tuning analysis protocols, a detailed exposition of the tuning process of the santñr for dastgàh-e
segàh (on F) is presented. Spectrum analysis results collected on three separate occasions (with a
14 Hormoz Farhat, The Dastgàh Concept in Persian Music (New York: Cambridge U. Press, 1990) 7.
15 Alain Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness
(Rochester, VT: Inner Traditions, 1995) 35-37.
16 Farhat, The Dastgàh Concept in Persian Music 15-16.
6 Introduction
periodicity of 3-6 months) for each of the twenty-seven sets of strings are then analyzed with regards to
the intervallic size of octaves, perfect fifths, perfect fourths, tempered perfect fourths, and neutral thirds.
An analysis of variance is then conducted with the data collected, which in turn produces mean
measurements with the capacity to characterize tuning characteristics. A tuning system comparison then
concludes a relationship between Farhat’s and Piraglu’s division of the whole-tone, with Farhat’s theory of
flexible intervals accorded as the most plausible hypothesis.
In view of the fact that stretched, as well as compressed octaves are a common occurrence in
Piraglu’s tuning of the santñr, the theory of the ‘piano tuner’s octave’ is discussed, along with the natural
phenomenon of inharmonicity – a factor especially affecting plucked and struck strings (along with other
musical sounds with a short decay).17 A comparison is also made between the tuning of a triple-string
unison of a piano and a quadruple-string unison of a santñr. Climate and its effects on tuning are then
considered, and especially in order to substantiate Piraglu’s claims of the climatic conditions of
Melbourne, Australia being “unsatisfactory” for the tuning of the santär in comparison to Tehran, Iran. The
twenty-four gušes for dastgàh-e segàh according to a prominent radif associated with Mñsà Marñfi are
then presented, followed by the pitch organization of the adopted six most prominent elements of the
radif of dastgàh-e segàh. Finally, the structural scheme of the work and its basis on ‘golden mean’
proportions are explained, as well as the sampling process of the santär and vocals, and digital
processing that culminates in the tape element of Àzàdeh for santñr and tape.
Chapter two (the equally-tempered archetype) begins with a discussion about Partch’s notion of
two distinct classes of equal temperaments: those that produce equal third-tones, quarter-tones, fifth-
tones, sixth-tones, eighth-tones, twelfth-tones, and sixteenth-tones; as opposed to those that divide the
octave into nineteen, thirty-one, forty-three, and fifty-three equally-tempered intervals.18 This is followed
by a brief history of some important studies of the equally-tempered paradigm, namely by Julián Carrillo
Trujillo, Ferruccio Busoni, Ramon Fuller, and Easley Blackwood, with the latter two serving as benchmarks
for the establishment of the criteria to properly assess the musical virtues of a particular equal
temperament. The deviation of basic equally-tempered intervals from just intonation, Fuller’s eight best
equal temperaments, and Blackwood’s concept of ‘recognizable diatonic tunings’ are then discussed.
Nicolas Mercator’s fifty-three-tone equally-tempered division of the octave, which is Fuller’s
recommendation for a temperament with the capacity to approximate just intervals, is consequently
presented, along with an opposing view by Dirk de Klerk.
In order to illustrate the principal evolutionary markers leading up to the adoption of equal
temperament in the West – from Pythagorean intonation, meantone and well temperament, to equal
17 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 166-67.
18 Harry Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments, 2nd ed. (New York:
Da Capo, 1974) 425.
Introduction 7
temperament – Pietro Aron’s quarter-comma meantone temperament is introduced, as well as Joseph
Sauveur’s forty-three-tone equal temperament, which approximates fifth-comma meantone temperament.
The origins of equal temperament are then traced back to 1584 China, and Prince Chu Tsai-yü’s
monochord. What follows is a discussion of the geometrical and numerical approximations of Marin
Mersenne and Simon Stevin, which culminate in Johann Faulhaber’s monochord, and the first printed
numerical solution to equal temperament based on the theory of logarithmic computation.19 The
mathematical formula for twelve-tone equal temperament, the equally-tempered monochord, and
beating characteristics of the twelve-tone equally-tempered major and minor triads are then sequentially
presented, which are followed by the equal thirds, sixths, fifths, and fourths in piano tuning.
The work, Exposiciones for Sampled Microtonal Schoenhut Toy Piano, is then introduced, together
with a brief history of the toy piano, the Schoenhut toy piano sample, as well as concepts of sound
diffusion and polyrhythmic theory utilized in the composition. In order to illustrate the design of the
proposed notation for the twenty-four equal temperaments, Daniélou’s ‘scale of proportions’, or
harmonic division of the octave, which presents a series of sixty-six unique intervals is introduced. Paul
Rapoport’s Pythagorean notation then provides an alternative to the system of notation based on
Daniélou’s subdivision of the whole-tone. Sléndro and pélog scales are then discussed from a historical
perspective, with the gamelan gedhé sléndro and pélog tunings from Sri Wedhari theatre auditorium in
Solo, Central Java serving as the ‘performance practice’ model. The harmonic characteristics of the
sléndro and pélog scales are then presented in accordance to five-limit intonation principles. What
follows is a systematic exposition of the compositional application of each equal temperament between
one and twenty-four.
Chapter three (the harmonic consideration) begins with a basic outline of just intonation and ‘extended
just intonation’, or the incorporation of partials beyond the sixth harmonic.20 A historical and scientific
perspective of the harmonic series is then presented, together with examples of the beating characteristics of
the first eight partials of the harmonic series, as well as of the mistuned and properly tuned unison, and
mistuned and properly tuned octave. Dissonance, with special reference to the theory of beats, is defined
according to James Tenney, Helmholtz, Bosanquet, and Johnston. The complement or mirror image of the
harmonics series, or the ‘subharmonic series’, is also discussed, together with Partch’s theory of ‘otonalities’
(pitches derived from the ascending series) and ‘utonalities’ (pitches derived from the descending series).21 A
comparative table of intonation then provides interval, ratio, and cents data for the twelve basic intervals of just
intonation, Pythagorean intonation, meantone temperament, and equal temperament.
19 Barbour, Tuning and Temperament: A Historical Survey 78.
20 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 106-07.
21 David D. Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation, 3rd ed.
(San Francisco: Other Music, 2002) 28-30.
8 Introduction
In order to illustrate the basic principles of proportions and string lengths, the traditional structure and
function of the monochord is explained, with the generation of simple octaves and fifths utilized to
demonstrate the theoretical basis for the Pythagorean monochord. A table depicting all the intervals of
the harmonic series from the first partial through to the one-hundred-and-twenty-eighth partial is then
presented. Combinational tones, or differential and summation tones, are also subsequently explained,
together with their implications on the intervals of the octave, just perfect fifth, just perfect fourth, just
major third, just minor sixth, just minor third, and just major sixth. This is followed by a discussion of
periodicity pitch, and its theoretical significance in relation to JND, or Just Noticeable Difference. The
relationship of prime numbers, primary intervals, and prime limits to just intonation principles is
subsequently explained.
The concept of just intonation is then illustrated via the construction of a seven-note just diatonic
scale, and the presentation of the beating characteristics of the just major triad. This is followed by the
construction of a twenty-five-note just enharmonic scale, and its development into Johnston’s fifty-three-
tone just intonation scale. Harry Partch’s forty-three-tone just intonation scale, and his rationale for the
consequential harmonic expansion to eleven-limit is then explained. The twenty unique triads, fifteen
unique tetrads, and six unique pentads made possible via the inclusion of the eleven-limit intervals are
additionally presented. The final octave division discussed in the chapter is Adriaan Daniël Fokker’s thirty-
one-tone equally-tempered division of the octave, and in view of its capability to approximate the tonal
resources of seven-limit just intonation.
The work, La Homa Kanto for Harmonically Tuned Synthesizer Quartet, is then introduced, together
with a presentation of Harrison’s five pentatonic scales, which serve as the pitch material, the ‘1967
William Dowd French Double Harpsichord’ sample, and Johnston’s system of notation, which serves as
the system of notation utilized in the score. Compositional strategy is then discussed, together with
Harrison’s concept of composing with melodicles, or neumes, which is adopted and developed into a
system incorporating three categories of motivic manipulation: melodic transformation of motive, rhythmic
transformation of motive, and harmonic transformation of motive. What follows is a systematic exposition
of the compositional application of each just intonation limit between three and thirty-one. Interval Nomenclature and Notation System Intervals based on Pythagorean intonation have been simply named according to their cyclical position,
and therefore follow an either ascending 3/2 incremental progression from natural, sharp, double sharp,
to triple sharp; or a descending 4/3 incremental progression from natural, flat, double flat, to triple flat.
The procedure is exemplified via the twenty-seven-note Pythagorean scale, which incorporates fifteen
intervals generated by an ascending series of fifths, or the pitches C, G, D, A, E, B, F!, C!, G!, D!, A!, E!,
Introduction 9
B!, F#, C#, and G#; and another eleven intervals, by a descending series, or C, F, B", E", A", D", G", C", F",
B$, E$, and A$. The method adopted in equal temperament on the other hand is a nomenclature
based on the comma approximations to Daniélou’s ‘scale of proportions’, or sixty-six-note just intonation
scale, with every interval not characterized by the equal semitones and quarter-tones of 12-et and 24-et
further indentified via its origin (for example: 5-et supermajor second, 7-et grave or small tone, and 9-et
great limma, or large half-tone). Exceptions to this rule include 31-et, 43-et, and 53-et, which because
are not discussed in the thesis with relation to other intervals, do not require a differential prefix with the
same conditions. Adriaan Daniël Fokker’s thirty-one-tone equally-tempered division of the octave
introduces a further element to intervallic nomenclature. The system, which was developed by David C.
Keenan, involves the prefixes: double diminished, subdiminished, diminished, sub, perfect, super,
augmented, superaugmented, and double augmented for unisons, fourths, fifths, and octaves; while
subdiminished, diminished, subminor, minor, neutral, major, supermajor, augmented, and
superaugmented for seconds, thirds, sixths, sevenths, and ninths. Perfect and major, or “the ones implied
when there is no prefix,” represent the central position of a range based on comma or diesis increments
from ß4 to +4 (for example: diminished third, subminor third, minor third, neutral third, major third,
supermajor third, and augmented third).22 For intervals beyond five-limit intonation, James B. Peterson’s
recommendations for the naming of bases has been adopted, which results in the following additional
prefixes for seven-, eleven-, thirteen-, seventeen-, nineteen-, twenty-three-, twenty-nine-, and thirty-one-
limit: septimal, undecimal, tridecimal, septendecimal, nonadecimal, trivigesimal, nonavigesimal, and
untrigesimal (for example: septimal superfifth, undecimal subfifth, tridecimal subfifth, septendecimal
superfifth, nonadecimal superfifth, trivigesimal superfifth, nonavigesimal subfifth, and untrigesimal
superfifth).23 The classification of 724 unique intervals incorporated into the comparative table of musical
intervals (see Appendix A) includes all the intervals cited in the current study.
The notation symbols utilized in the thesis include the five standard accidental signs of Western
music; four common quarter-tone and three-quarter-tone symbols; twenty-three unique symbols based
on Daniélou’s division of the whole-tone; Ali Naqi Vaziri’s notation system, or four accidentals of Persian
music; Johnston’s system of notation, which contains twenty-three unique symbols for the notation of just
intonation up to the thirty-first harmonic; as well as Fokker’s nine symbols for the notation of thirty-one
equal temperament. All these symbols have been incorporated into a 238-character microtonal notation
PostScript Type 1 font (see Appendix B), which was created via the modification of a selection of
symbols in the Coda Music Finale’s Maestro font utilizing CorelDraw 13.0 and FontMonger 1.0.8.
22 David C. Keenan, “A Note on the Naming of Musical Intervals,” David Keenan’s Home Page, 3 Nov. 2001, 22 Nov.
2006, <http://users.bigpond.net.au/d.keenan/Music/IntervalNaming.htm>. 23 James B. Peterson, “Names of Bases,” The Math Forum: Ask Dr. Math, 15 Apr. 2002, Drexel U., Philadelphia, PA,
22 Nov. 2006, <http://mathforum.org/library/drmath/view/60405.html>.
10 Introduction
1. Theory Versus Performance Practice: Àzàdeh for Santñr and Tape
A Brief History of Persian Classical Music Modern Persian scholarship on the theory of intervals and scales may be mainly attributed to the
theoretical writings of medieval music scholars Éafå al-Dån ‘Abd al-Mu’min al-Urmawå’ (d. 1294) and Quðb
al-Dån ‘Maämñd ibn Mas’ñd al-Shåràzå’ (1236-1311). “The latter half of the thirteenth century constitutes
one of the most important periods in the history of Arab and Persian musical theory,” notes Owen Wright.
“It witnessed the emergence of a corpus of theoretical writings that not only demonstrate a considerable
degree of originality, but also provided the framework within which all the major theorists of the following
two centuries were to operate.” Éafå al-Dån in particular is acknowledged for founding the ‘Systematist
school’ with his two influential treatises: Kitàb al-adwàr of 1252 (‘Book of Cycles’) and Risàla al-sharafiyya fi
al-nisàb al-ta’lifiyya of 1267 (‘Sharafian Treatise on Intervallic Relations’), while Quðb al-Dån for his further
contribution to the theory within a section about music contained in his encyclopedia Durrat al-tàj (‘Pearl
of the Crown’), published circa 1300.24 In the spirit of their predecessors – Al-Kindå (d. 873), and
celebrated author of Kitàb al-mñsåqå al-kabår (‘Great Book on Music’), Abu Nasr Fàràbå (872-950); as well
as Ibn Sånà (980-1037) – their findings were essentially based on the musical theories of the classical
Greeks; from Pythagoras of Samos (fl. 530 B.C.) to Aristoxenus of Tarentum (fl. 400 B.C.).25 Cultural links
between Persia and Ancient Greece existed between 500 B.C. and 300 A.D., and were further infused
by Alexander the Great’s conquest of the Achaemenid Empire in 330 B.C. that generated the hundred
years of Greek rule in Persia. As a consequence, “the works of Euclid, Aristoxenus, Ptolemy, and others
translated into Arabic at Baghdad during the ninth century,26 served as models for the great Islamic
theorists,” notes Ella Zones.27
24 Owen Wright, The Modal System of Arab and Persian Music A.D. 1250-1300, London Oriental Series, vol. 28
(Oxford: Oxford U. Press, 1978) 1-20. 25 Hormoz Farhat, “Iran: Classical Traditions,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie
and John Tyrrell, 2nd ed., vol. 12 (London: Macmillan Reference, 2001) 531.
26 “During the Abbasid period (750-1258) many branches of Islamic scholarship developed rapidly, among them
medicine, astronomy, alchemy, geography, mathematics, and also music theory. This development was stimulated by
contact with ancient Greek writings which became available to Islamic scholars through translations done in the Bait al-
Äikma (House of Wisdom), a library, astronomical observatory, and translation institute established in Baghdad by caliph al
Ma’mñn.” For a further discussion, see by Josef M. Pacholczyk, “Secular Classical Music in the Arabic Near East,” Musics of
Many Cultures (Berkeley, CA: U. of California Press, 1980) 255. 27 Ella Zonis, “Contemporary Art Music in Persia,” The Music Quarterly 51.4 (Oct., 1965): 636-37.
The Seventeen-Note Gamut In Kitbag al-adware, Éafå al-Dån proposes that a Pythagorean whole-tone (equal to the frequency ratio of
9/8, or 203.910 cents) should only be subdivided into either one Pythagorean limma (256/243, or
90.225 cents), or two Pythagorean limmas (equal to a Pythagorean diminished third, 65536/59049, or
180.450 cents). This in effect generates a theoretical basis for a whole-tone constructed from the sum of
two limmas and a Pythagorean comma (531441/524288, or 23.460 cents), and a tetrachord made up of
two whole-tones and a limma that is implemented in an octave as two conjunct tetrachords, plus an
additional whole-tone. The result is a seventeen-note scale,28 and modality based on two conjunct
tetrachords, which may be theoretically referred to as a bitetrachordal system.29
Table 1. Éafå al-Dån Urmawå’s division of the whole-tone
PATTERN L L + L L + L + C
INTERVAL Pythagorean
limma
Pythagorean
diminished third
just
major tone
RATIO 256/243 65536/59049 9/8
CENTS 90.225 180.450 203.910
The tetrachord (a four-note series enclosed within the range of a perfect fourth) is a concept borrowed
from ancient Greek music, where as part of the ‘Greater Perfect System’ – a two-octave system made up
of four conjunct and disjunct tetrachords (Hypaton, Meson, Diezeugmeson and Hyperbolaeon), as well
as an additional whole-tone (Proslambanomenos) to complete the lower part of the range – was “the
basic building block of Greek music,” and therefore at the core of Greek theory on intervals and scales.30
The essence of Ancient Greek music and its proponents is summarized by R. P. Winnington-Ingram thus:
Ancient Greek music was purely or predominantly melodic; and in such music subtleties of intonation
count for much. If our sources of information about the intervals used in Greek music are not always easy
to interpret, they are at any rate fairly voluminous. On the one hand we have Aristoxenus, by whom
musical intervals were regarded spatially and combined and subdivided by the processes of addition and
subtraction; for him the octave consisted of six tones, and the tone was exactly divisible into fractions such
28 Owen Wright, “Arab Music: Art Music,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie and
John Tyrrell, 2nd ed., vol. 12 (London: Macmillan Reference, 2001) 806.
29 Dariush Talai, “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System,” ed. Virginia
Danielson, Scott Marcus and Dwight Reynolds, The Garland Encyclopedia of World Music: The Middle East, vol. 6 (New
York: Routledge, 2002) 871.
30 André C. Barbera, “Greece,” ed. Don Michael Randel, The New Harvard Dictionary of Music (Cambridge, Mass.:
Belknap Press of Harvard U Press, 1986) 347-49.
12 Theory Versus Performance Practice
as the half and quarter, so that the fourth was equal to two tones and a half, the fifth to three tones and a
half, and so on. On the other hand we have preserved for us in Ptolemy’s Harmonics the computations of
a number of mathematicians, who realized correctly that intervals could only be expressed as ratios (e.g. of
string-lengths), that the octave was less than the sum of six whole tones and that this tone could not be
divided into equal parts. These authorities are Archytas, the Pythagorean of the early fourth century,
Eratosthenes (third century), Didymus (first century), and Ptolemy himself (second century A.D.). To these
we must add the scale of Plato’s Timaeus (35B) and, closely related to it, the computations of the pseudo-
Philolaus (ap. Boethium, Mus. III, 8) and of Boethius himself (IV, 6).31
With regards to the function of tetrachords in the construction of melodic and harmonic structures, John
H. Chalmers presents the following discussion:
“Tetrachords are modules from which more complex scalar and harmonic structures may be built. These
structures range from the simple heptatonic scales known to the classical civilizations of the eastern
Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the
world’s music, including that of Europe, the Near East, the Catholic and Orthodox churches, Iran, and India,
are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world’s
music.”
Chalmers then further expands on the issue with the subsequent definition:
“The tetrachord is the interval of a perfect fourth, the diatessaron of the Greeks, divided into three
subintervals by the interposition of two additional notes. The four notes, or strings, of the tetrachord were
named hypate, parhypate, lichanos, and mese in ascending order from 1/1 to 4/3 in the first tetrachord of
the central octave of the ‘Greater Perfect System’, the region of the scale of most concern to theorists.
Ascending through the second tetrachord, they were called paramese, trite, paranete, and nete.”32
Stringed instruments are recognized as a major factor in the design of tuning and scale systems. “The
fretting and tuning of stringed instruments was directly connected to the development of modes. We
can deduce this from the fact that stringed instruments have been used to study intervals and tetrachords
from antiquity, and from the fact that in the past, musicians were poet-singers first of all, accompanying
their poetry and song with stringed instruments,” explains Dariush Talai. It is interesting to note that “the
tetrachord also corresponds to a physical area on the neck of instruments such as the ‘ñd, tàr, and setàr,
where the fingers can reach the notes without changing position.”33
31 R. P. Winnington-Ingram, “Aristoxenus and the Intervals of Greek Music,” The Classical Quarterly 26.3/4 (Jul.-Oct.,
1932): 195.
32 Chalmers, Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales 4.
33 Talai, “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System,” The Garland
Encyclopedia of World Music: The Middle East 868-69.
Theory Versus Performance Practice 13
Table 2. The seventeen-note gamut.34
TRADITIONAL
PERSIAN SYSTEM
PHONETIC
TRANSCRIPTION
PYTHAGOREAN
NOTATION
CONTEMPORARY
NOTATION
RATIO
(FRACTION)
CENTS
� � � � � � � � ��
� �� �� �� �� ��
��
A
b
j
d
h
v
z
ä
ë
y
yà
yeb
yej
yed
yeh
yu
yez
yeä
C
D"
E$
D
E"
F"
E
F
G"
A$
G
A"
B$
A
B"
C"
B
C
C
D"
Dî
D
E"
Eî
E
F
G"
Gî
G
A"
Aî
A
B"
Bî
B
C
1/1
256/243
65536/59049
9/8
32/27
8192/6561
81/64
4/3
1024/729
262144/177147
3/2
128/81
32768/19683
27/16
16/9
4096/2187
243/128
2/1
0.000
90.225
180.450
203.910
294.135
384.360
407.820
498.045
588.270
678.495
701.955
792.180
882.405
905.865
996.090
1086.315
1109.775
1200.000
Illustrated in following music example is the scale of Éafå al-Dån, which is referred to as the seventeen-note
gamut and based on Al-Fàràbå’s first tetrachord division on the Khorasan long-necked lute, or ðunbñr
khuràsànå. The seventeen-note gamut (utilizing G as the tonic) is further characterized by the following
intervallic pattern: L, L, C – L, L, C – L – L, L, C – L, L, C – L + L, L, C. The transposition to C of the intervallic
pattern reorganized thus: L, L, C – L, L, C – L + L, L, C – L, L, C – L, L, C – L.35
Ex. 1. Subdivision of the tetrachord and the seventeen-note gamut (Éafå al-Dån Urmawå, 13th century)
34 Taqi Bineš, The Short History of Persian Music (Tehran: Àrvin Publication, 1995) 106.
35 Wright, “Arab Music: Art Music,” The New Grove Dictionary of Music and Musicians 806.
14 Theory Versus Performance Practice
Although the theoretical notion of a seventeen-note division of the octave became widely adopted by
the Islamic world for centuries to come, there is much speculation amongst musicologists today about
the actual implementation of this synthetic scale by musicians of the time in the context of performance
practice, as the employment of an intuitive system of flexible intervals seems more probable. It should
be noted that no mode has ever contained all seventeen notes (heptatonic being the dominant
character), and that the Pythagorean comma was not utilized as an actual successive pitch in a scale
formation but rather as an addition to another intervallic value to form a composite interval.36 From a
historical perspective, with regards to the adoption of a theoretical scale by the Arabs and Persians in the
fourteenth century and the “so-called messel or octave of seventeen third tones,”37 A. J. Hipkins writes:
“The arithmetical reasonings of philosophers who sought to explain the musical scale could never have
been, excepting in the larger intervals, the practical art of the musicians.”38
Persian Musical Scholarship in the Twentieth Century In the sixteenth century, with Shi’a rule imposed by the Sadavid dynasty (1501-1722) generating a certain
hostility towards music, Persian musical scholarship enters its period of decline, and it is not until the
beginning of the twentieth century during the Pahlavi dynasty (1925-1979) that we begin to see the
resurgence of theoretical research into the Persian theory of intervals and scales.39
“In the beginning of the twentieth century, Iran entered a period during which the Western world was
idealized. Western music had already been taught in Iran for over three decades; thus, Iranian musicians
were thoroughly educated in it,” explains Talai. “In fact, some people began to question the value of
Iranian music as serious and dignified, doubting that its instruction should be institutionalized. Others,
motivated in part by a spirit of nationalism, tried to revive Persian music. They wanted to create a theory
relating it to the perspective of what they regarded as the more progressive, Western classical music.”40
36 Farhat, The Dastgàh Concept in Persian Music 12-13.
37 “La Borde, Villoteau, and Kiesewetter, studying Arab music from obscure treatises of medieval Arab philosophers
such as Khalil, Al-Kindå, Ibn Khaldñn, and Al-Fàràbå, maintained that Arab music is based on the so-called messel or octave
of seventeen third tones, and such was purely Oriental and fundamentally different from the Greek diatonic system, which
is based on tones and semitones. They proceed, therefore, to see the origin of Arab music in Persia, where we find a
scale of semitones, demi-semitones, and even semidemi-semitones, their theory being that Persia, after being conquered
by the Arabs, had imposed its music upon the conquerors.” For a further discussion, see Joseph Reider, “Jewish and
Arabic Music,” The Jewish Quarterly Review 7.4 (Apr., 1917): 640.
38 Francesco Salvador-Daniel, and Henry George Farmer, The Music and Musical Instruments of the Arab: With
Introduction on How to Appreciate Arab Music (Portland, ME: Longwood Press, 1976) 182.
39 Farhat, The Dastgàh Concept in Persian Music 5.
40 Dariush Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah, trans. Manoochehr Sadeghi (Coasta Mesa,
CA: Mazda, 2000) 7-8.
Theory Versus Performance Practice 15
This ideological shift in effect being the impetus for three separate theories on intervals and scales of
Persian music proposed in the twentieth century: the twenty-four equally-tempered quarter-tone scale
proposed by Ali Naqi Vaziri in the 1920s, the alternative twenty-two-note scale proposed by Mehdi
Barkešli in the 1940s based on Pythagorean principles, as well as the theory of the five primary intervals of
performance practice presented by Hormoz Farhat in the 1990 publication of his doctoral thesis The
Dastgàh Concept in Persian Music.41 In spite of the almost universal rejection of the proposal for a twenty-
four-tone equally-tempered scale, one cannot underestimate Vaziri’s influence to Persian musical
scholarship in the twentieth century, as some of the reforms implemented to the art form by the theorist
did gain wide acceptance.
Vaziri was born in Tehran, in 1887, and following his traditional studies in classical Persian music,
went on to study piano, harmony, and voice at the École Supérieure de Musique in Paris, and
counterpoint and composition at the Hochschule für Musik in Berlin. Upon his return to Iran in 1923, he
founded his conservatory, the Madresse-ye Ali-ye Musiqi, and amongst his many achievements is today
credited for the first published transcriptions of Persian music in Western staff notation.42 As theorist, his
most notable writings on the twenty-four equally-tempered quarter-tone scale and the Persian modal
system are contained within his 1934 publication Musiqi-ye Nazari.43 The Twenty-Four Equally-Tempered Quarter-Tone Scale The concept of the division of the octave into twenty-four equally-tempered quarter-tones is not a new
phenomena to the region, with Lebanese theorist Måkhà’ål Måshàqàh (1800-1889)44 proposing an Arabian
quarter-tone system in the previous century.45 Earlier examples include Zalzal (d. after 842) – a prominent
ñd teacher of the ninth century who reformed Pythagorean intonation of the time with his introduction of
eleven-limit ratios to the fretting of the lute (an intonation system limited to the eleventh harmonic with
intervals closely resembling those of equal temperament). The Zalzal intervals included the undecimal
grave or small neutral second (12/11, or 150.637 cents), undecimal subfifth (16/11, or 648.682 cents),
and undecimal grave or small neutral sixth (18/11, or 852.592 cents); and represents a system subscribed
41 Farhat, The Dastgàh Concept in Persian Music 10.
42 Ella Zonis, “Classical Iranian Music,” Musics of Many Cultures (Berkeley, CA: U. of California Press, 1980) 186-87.
43 Farhat, The Dastgàh Concept in Persian Music 9.
44 Habib Hassan Touma, The Music of the Arabs, trans. Laurie Schwartz (Portland, OR: Amadeus Press, 1957) 19.
45 Måshàqàh’s treatise proposing the twenty-four equally-tempered division of the octave for Turko-Arabian music
translated into English by American protestant missionary and scholar Eli Smith (1801-57), and then consequently
published in the Journal of the American Oriental Society (1849). For a further discussion, see Alexander J. Ellis, and
Alfred J. Hipkins, “Tonometrical Observations on Some Existing Non-Harmonic Musical Scales,” Proceedings of the Royal
Society of London 37 (1884): 371.
16 Theory Versus Performance Practice
to for seven centuries prior to the reforms of fourteenth century theorists Quðb al-Dån and Abd al Qàdir
al-Maràghå (d 1435).46
The twenty-four-tone equally-tempered division of the octave is based on the Western system of
twelve-tone equal temperament – the division of the octave into twelve equal intervals, technically
referred to as tempered half-tones, while the division of the octave into twenty-four equal intervals,
technically referred to as tempered quarter-tones; the frequency ratio of each quarter-tone therefore
mathematically representing the twenty-fourth root of two, or in different terms, the distance between
any two tones representing twenty-four times the logarithm on the base of two of the frequency ratio.47
The ratio of the equally-tempered quarter-tone may be expressed in mathematical terms as
1: 24 2 =1:1.029302237, or the fraction approximations: 527/512 (17×31/2ù) and 35/34 (5×7/2×17).48
The unit of a cent being the method to further subdivide the semitone, and equal to the twelve-
hundredth root of two; with 12 semitones per octave, 1200 cents per octave, and hence, 100 cents per
semitone. The ratio of the cent may be expressed in mathematical terms as 1: 1200 2 =1:1.000577789, or
approximately 1731/1730. The calculation of cents (a system of measurement devised by Alexander J.
Ellis [1814-90]) is obtained via the formula: 1200ïlog2Ïlogf=cents.49
The methodology involved in the construction of a twenty-four equally-tempered quarter-tone
scale requires the establishment of twenty-four degrees via the equation f= 24 2 ; the equation producing
the figure 1.029302237 (the frequency ratio of one equally-tempered quarter-tone), which when
multiplied by the powers of 2, 3, 4, etc., generates the ratios for the remainder of the scale degrees.50 It
should be noted that the multiplication by the power of 18 ( 1824 ]2[ ) generates the frequency ratio of
46 “In view of the fact that the division into ‘quarter-tones’ gives two of the ten ratios of 11 with a falsity of only 0.6
cent (12/11, or 150.637 cents; and 11/6, or 1049.363 cents), two with a falsity of only 1.3 cents (11/8, or 551.318
cents; and 16/11, or 648.682 cents), and two with a falsity of 2.6 cents (11/9, or 347.408 cents; and 18/11, or 852.592
cents), the probabilities are that Måshàqàh’s claims for Arabic folk melodies are valid. But the fate of the other ratios of 11,
and of the 5 and 7 identities (the 5th and 7th partials of the harmonic series) in ‘quarter-tones’ is another story.” For a
further discussion see, Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 426.
47 Olson, Music, Physics and Engineering 46-47.
48 Alain Daniélou, Tableau Comparatif des Intervalles Musicaux (Pondichéry, India: Institut Français d’Indologie,
1958) 14.
49 Hermann L. F. Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music, 2nd ed.
(New York: Dover Publications, 1954) 431.
50 “‘Quarter-tones’ are obtained by a simple duplication of the twelve-tone tempered scale a ‘quarter-tone’ higher.
The original, since it is the usual scale, has good 3 identities (the 3rd partial of the harmonic series) and two fairly good 9
identities (the 9th partial of the harmonic series), but poor 5 identities (13.7 and 15.6 false). The doubling, a ‘quarter-
tone’ higher, gives six of the ten 11 identities (the 11th partial of the harmonic series) almost perfectly, but fails to
represent the other four at all; it comes closer to the 7 identities than the original, but not close enough, since they show
a maximum falsity of 18.8 cents; it improves the 5s not at all.” For a further discussion see, Partch, Genesis of a Music: An
Account of a Creative Work, its Roots and its Fulfilments 428.
Theory Versus Performance Practice 17
A=440Hz (1.68179283), or equal major sixth ( 34 ]2[ , approximately 37/22, or 900.000 cents),51 while
440ï1.68179283 presents the relative frequency of middle C (261.6255654Hz), or 1/1. Once data is
available for all frequency ratios, it simply becomes a matter of multiplying every ratio by 261.6255654Hz
(1/1) in order to obtain the frequency values in hertz (Hz) of every scale degree. The mathematical
formula for cents is then utilized to generate the cent values of every scale degree, although unnecessary
in equal temperament because of the obvious outcomes.
The Pythagorean Division of the Octave
‘Pythagorean’ intonation, and the generation of a scale based on a series of ‘pure’ fifths (3/2s), is in
reference to Greek philosopher Pythagoras (570-504 B.C.), who according to Llewelyn Southworth
Lloyd, and Hugh Boyle is “usually credited with the discovery that a vibrating string, stopped at two thirds
or one-half of its length, sounds the fifth (3/2) or octave (2/1) of the note it produces when vibrating
freely (1/1).”52 The system of intonation “prevailed in China and Arabia almost to the exclusion of anything
else,” notes Partch, and “was for many centuries the only system in ecclesiastical Roman and Byzantine
music.”53 According to Fritz A. Kuttner, the Pythagorean conceptual basis for the construction of a scale
of twelve semitones via the generation of twelve consecutive fifths was recognized in China “several
centuries before Pythagoras, probably in the ninth century B.C., or even earlier.” Kuttner explains:
“The twelve semitones (lüs) received pitch names which are mentioned for the first time in the Kuo Yü, a
work usually dated towards the end of the Chou Dynasty (c. 1030-722 B.C). However, the Kuo Yü is
believed to contain a good deal of much ‘earlier material from ancient written sources,’ so the origin of the
pitch names must be assigned to about 900 B.C. or earlier.”54
In order to construct a Pythagorean scale with the necessary intervals to facilitate the general modulations
of Western tonal music one must produce twenty-seven distinct pitches.55 Fifteen intervals are generated
by an ascending series of fifths, or the pitches C, G, D, A, E, B, F!, C!, G!, D!, A!, E!, B!, F#, C#, and G#;
while the other eleven intervals, by a descending series, or C, F, B", E", A", D", G", C", F", B$, E$, and A$.
The mathematical procedure thus begins with the generation of the first ascending fifth, G; which has a
ratio of 3/2 (the just and Pythagorean perfect fifth, measuring 701.955 cents) in relation to the
51 Daniélou, Tableau Comparatif des Intervalles Musicaux 79.
52 Llewelyn Southworth Lloyd, and Hugh Boyle, Intervals, Scales and Temperaments: An Introduction to the Study
of Musical Intonation (London: McDonald and Jane’s, 1978) 2.
53 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 399.
54 Fritz A. Kuttner, “A Musicological Interpretation of the Twelve Lüs in China’s Traditional Tone System,”
Ethnomusicology 9.1 (Jan., 1965): 22.
55 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 433.
18 Theory Versus Performance Practice
Tabl
e 3.
The
twen
ty-fo
ur-to
ne e
qua
lly-te
mp
ered
div
ision
of t
he o
ctav
e Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
idd
le C
)=26
1.62
5565
4Hz
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L FA
CTO
R RA
TIO
(D
ECIM
AL)
FR
EQUE
NCY
(H
ERTZ
) CE
NTS
TU
NIN
G
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ ÐÑ
CCË
/ Dì
CÚ
/ DÝ
CÍ
/ Dí
D
DË /
Eì
DÚ /
EÝ
DÍ /
Eí
E EË
/ Fí
F
FË / G
ì FÚ
/ GÝ
FÍ
/ Gí
G
GË /
Aì
GÚ /
AÝ
GÍ /
Aí
A AË
/ Bì
AÚ
/ BÝ
AÍ
/ Bí
B
BË /
Cí
C
uniso
n eq
ual q
uarte
r-ton
e eq
ual s
emito
ne, o
r min
or s
econ
d
equa
l thr
ee-q
uarte
r-ton
e eq
ual t
one
five
equa
l qua
rter-
tone
s eq
ual a
ugm
ente
d s
econ
d, o
r min
or th
ird
seve
n eq
ual q
uarte
r-ton
es
equa
l maj
or th
ird
nine
eq
ual q
uarte
r-ton
es
equa
l per
fect
four
th
elev
en e
qua
l qua
rter-t
ones
eq
ual t
riton
e, a
ugm
ente
d fo
urth
, or d
imin
ished
fifth
thirt
een
equa
l qua
rter-t
ones
eq
ual p
erfe
ct fi
fth
fifte
en e
qua
l qua
rter-t
ones
eq
ual a
ugm
ente
d fi
fth, o
r min
or s
ixth
se
vent
een
equa
l qua
rter-t
ones
eq
ual m
ajor
six
th
nine
teen
eq
ual q
uarte
r-to
nes
equa
l aug
men
ted
six
th, o
r min
or s
even
th
twen
ty-o
ne e
qua
l qua
rter-t
ones
eq
ual m
ajor
sev
enth
tw
enty
-thre
e eq
ual q
uarte
r-ton
es
oc
tave
1/1
242
12
2
82
6
2 5
24)
2(
4
2 7
24)
2(
3
2 3
8)
2(
5
12)
2(
11
24)
2(
2
2 13
24)
2(
7
12)
2(
5
8)
2(
2
3)
2(
17
24)
2(
3
4)
2(
19
24)
2(
5
6)
2(
7
8)
2(
11
12)
2(
23
24)
2(
2/1
1.00
0000
1.
0293
02
1.05
9463
1.
0905
08
1.12
2462
1.
1553
53
1.18
9207
1.
2240
54
1.25
9921
1.
2968
40
1.33
4840
1.
3739
54
1.41
4214
1.
4556
53
1.49
8307
1.
5422
11
1.58
7401
1.
6339
15
1.68
1793
1.
7310
73
1.78
1797
1.
8340
08
1.88
7749
1.
9430
64
2.
0000
00
261.
626
269.
292
277.
183
285.
305
293.
665
302.
270
311.
127
320.
244
329.
628
339.
286
349.
228
359.
461
369.
994
380.
836
391.
995
403.
482
415.
305
427.
474
440.
000
452.
893
466.
164
479.
823
493.
883
508.
355
52
3.25
1
0.00
0 50
.000
10
0.00
0 15
0.00
0 20
0.00
0 25
0.00
0 30
0.00
0 35
0.00
0 40
0.00
0 45
0.00
0 50
0.00
0 55
0.00
0 60
0.00
0 65
0.00
0 70
0.00
0 75
0.00
0 80
0.00
0 85
0.00
0 90
0.00
0 95
0.00
0 10
00.0
00
1050
.000
11
00.0
00
1150
.000
1200
.000
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+00
+
50
+
00
Theory Versus Performance Practice 19
fundamental C (1/1). 2/1 (measuring 1200.000 cents) is consequently subtracted from intervals
exceeding the 2/1 ratio in order to bring the interval within the octave. These procedures are repeated
consecutively, producing the following series of fifteen ascending fifths:
i. G (just and Pythagorean perfect fifth) generated by ratio 3/2
ii. D (just major tone) by the equation (3/2×3/2)/2=9/8
iii. A (Pythagorean major sixth) by 9/8×3/2=27/16
iv. E (Pythagorean major third, or ditone) by (27/16×3/2)/2=81/64
v. B (Pythagorean major seventh) by 81/64×3/2=243/128
vi. F! (Pythagorean tritone, or augmented fourth) by (243/128×3/2)/2=729/512
vii. C! (Pythagorean apotome) by (729/512×3/2)/2=2187/2048
viii. G! (augmented fifth) by 2187/2048×3/2=6561/4096
ix. D! (augmented second) by (6561/4096×3/2)/2=19683/16384
x. A! (augmented sixth) by 19683/16384×3/2=59049/32768
xi. E! (augmented third) by (59049/32768×3/2)/2=177147/131072
xii. B! (Pythagorean comma) by (177147/131072×3/2)/2=531441/524288
xiii. F# (double augmented fourth) by 531441/524288×3/2=1594323/1048576
xiv. C# (double augmented octave) by (1594323/1048576×3/2)/2=4782969/4194304
xv. G# (double augmented fifth) by 4782969/4194304×3/2=14348907/8388608
The process is then reversed, with the generation of the first descending fifth (F) by the mathematical
equation of 4/3 (the just and Pythagorean perfect fourth, measuring 498.045 cents). These procedures
are repeated consecutively, producing the following concluding series of eleven descending fifths:
i. F (just and Pythagorean perfect fourth) generated by the ratio 4/3, or 2/3×2=4/3
ii. B" (just minor seventh) by the equation (4/3×2/3)×2=16/9
iii. E" (Pythagorean minor third, or trihemitone) by 16/9×2/3=32/27
iv. A" (Pythagorean minor sixth) by (32/27×2/3)×2=128/81
v. D" (Pythagorean limma) by 128/81×2/3=256/243
vi. G" (diminished fifth) by (256/243×2/3)×2=1024/729
vii. C" (diminished octave) by (1024/729×2/3)×2=4096/2187
viii. F" (diminished fourth) by 4096/2187×2/3=8192/6561
ix. B$ (diminished sixth) by (8192/6561×2/3)×2=32768/19683
x. E$ (diminished third) by 32768/19683×2/3=65536/59049
xi. A$ (diminished sixth) by (65536/59049×2/3)×2=262144/177147
20 Theory Versus Performance Practice
Tabl
e 4.
The
Pyt
hago
rean
div
ision
of t
he o
ctav
e Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
iddl
e C)
=26
1.62
5565
4Hz
DE
GRE
E N
UMBE
R N
OTE
IN
TERV
AL
RATI
O
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
Ò× ÐÑ
C BÚ
DÝ
CÚ
EÞ
D CÛ
EÝ
DÚ
FÝ E F EÚ
GÝ
FÚ
AÞ
G FÛ
AÝ
GÚ
BÞ
A GÛ
BÝ
AÚ
CÝ B C
uniso
n Py
thag
orea
n co
mm
a (A
) XI
I
Pyth
agor
ean
limm
a, o
r dia
toni
c se
mito
ne (D
) V
Pyth
agor
ean
apot
ome,
or c
hrom
atic
sem
itone
(A)
VII
Pyth
agor
ean
dim
inish
ed th
ird (D
) X
just
maj
or to
ne (A
) II (
9th
harm
onic
) Py
thag
orea
n d
oubl
e au
gmen
ted
oct
ave
(A)
XIV
Pyth
agor
ean
min
or th
ird, o
r trih
emito
ne (D
) III
Pyth
agor
ean
augm
ente
d s
econ
d (A
) IX
Pyth
agor
ean
dim
inish
ed fo
urth
(D)
VIII
Pyth
agor
ean
maj
or th
ird, o
r dito
ne (A
) IV
(81s
t har
mon
ic)
just
and
Pyt
hago
rean
per
fect
four
th (D
) I
Pyth
agor
ean
augm
ente
d th
ird (A
) XI
Pyth
agor
ean
dim
inish
ed fi
fth (D
) V
I Py
thag
orea
n tri
tone
, or a
ugm
ente
d fo
urth
(A)
VI
Pyth
agor
ean
dim
inish
ed s
ixth
(D)
XI
just
and
Pyt
hago
rean
per
fect
fifth
(A)
I (3r
d h
arm
onic
) Py
thag
orea
n d
oubl
e au
gmen
ted
four
th (A
) XI
II
Pyth
agor
ean
min
or s
ixth
(D)
IV
Pyth
agor
ean
augm
ente
d fi
fth (A
) V
III
Pyth
agor
ean
dim
inish
ed s
even
th (D
) IX
Pyth
agor
ean
maj
or s
ixth
(A)
III (2
7th
harm
onic
) Py
thag
orea
n d
oubl
e au
gmen
ted
fifth
(A)
XV
Pyth
agor
ean
min
or s
even
th (D
) II
Pyth
agor
ean
augm
ente
d s
ixth
(A)
X
Pyth
agor
ean
dim
inish
ed o
ctav
e (D
) V
II
Pyth
agor
ean
maj
or s
even
th (A
) V
oc
tave
1/1
5314
41/5
2428
8 25
6/24
3 21
87/2
048
6553
6/59
049
9/8
4782
969/
4194
304
32/2
7 19
683/
1638
4 81
92/6
561
81/6
4 4/
3 17
7147
/131
072
1024
/729
72
9/51
2 26
2144
/177
147
3/2
1594
323/
1048
576
128/
81
6561
/409
6 32
768/
1968
3 27
/16
1434
8907
/838
8608
16
/9
5904
9/32
768
4096
/218
7 24
3/12
8
2/1
1.00
0000
1.
0136
43
1.05
3498
1.
0678
71
1.10
9858
1.
1250
00
1.14
0349
1.
1851
85
1.20
1355
1.
2485
90
1.26
5625
1.
3333
33
1.35
1524
1.
4046
64
1.42
3828
1.
4798
11
1.50
0000
1.
5204
65
1.58
0247
1.
6018
07
1.66
4787
1.
6875
00
1.71
0523
1.
7777
78
1.80
2032
1.
8728
85
1.89
8438
2.00
0000
261.
626
265.
195
275.
622
279.
382
290.
367
294.
329
298.
344
310.
075
314.
305
326.
663
331.
120
348.
834
353.
593
367.
496
372.
510
387.
156
392.
438
397.
792
413.
433
419.
074
435.
551
441.
493
447.
517
465.
112
471.
458
489.
995
496.
680
52
3.25
1
0.00
0 23
.460
90
.225
11
3.68
5 18
0.45
0 20
3.91
0 22
7.37
0 29
4.13
5 31
7.59
5 38
4.36
0 40
7.82
0 49
8.04
5 52
1.50
5 58
8.27
0 61
1.73
0 67
8.49
5 70
1.95
5 72
5.41
5 79
2.18
0 81
5.64
0 88
2.40
5 90
5.86
5 92
9.32
5 99
6.09
0 10
19.5
50
1086
.315
11
09.7
75
12
00.0
00
+00
+
23
ß10
+
14
ß20
+
04
+27
ß
06
+18
ß
16
+08
ß
02
+22
ß
12
+12
ß
22
+02
+
25
ß08
+
16
ß18
+
06
+29
ß
04
+20
ß
14
+10
+00
Theory Versus Performance Practice 21
Alain Daniélou’s Scale of Fifths An alternative method for deriving a Pythagorean scale is Alain Daniélou’s ‘scale of fifths’ (the cyclic
division of the octave), as presented by Daniélou in Music and the Power of Sound: The Influence of
Tuning and Interval on Consciousness (1995) – the current edition of his 1943 monumental work
Introduction to the Study of Musical Scales. The theoretical basis for the ‘cyclic division of the octave’ is
the ascending scale of fifths.56
The theory states that when a length of bamboo (called ‘81 parts’ for symbolic reasons) is
reduced by a third (2/3), it will produce the perfect fifth (3/2, or 54 parts). If the process is continued
by increasing the length of bamboo by a third (4/3), and then alternated between the two ratios, it will
result in the following five-note series of pipe lengths and frequency ratios: 81 (1/1), 54 (3/2), 72 (9/8),
48 (27/16), and 64 (81/64). The juxtaposition of two sets of six lü, or pitch pipes a Pythagorean major
half-tone apart generates a Pythagorean twelve-tone scale, or twelve lü (alternatively achieved via the
simple expansion of the series of consecutive fifths initially forming a heptatonic scale). Han Dynasty
historian Ssu-ma Chi’en (145-86 B.C.) attributes the “mathematical formula for the pentatonic scale to Ling
Lun, minister or court musician under Emperor Huang-Ti, of the twenty-seventh century B.C.”57 Chinese
theorist King Fâng (c. 45 B.C.) followed the series up to the sixtieth sound (equal to fifty-nine consecutive
fifths), although mainly for symbolic reasons. The fifty-third fifth is the cyclic octave (3õó/2øô, or 3.615
cents), but the series being of an infinitive nature, it is musically impractical to follow it beyond this given
point, so it is rounded off to the ratio of 2/1. Fang based his system of the sixty lü on the “analogy of the
eight gua or mystical trigrams of the I ching, which united two by two, form sixty-four distinct
combinations.” Sixty-four hexagrams allow for the “representation of all aspects of existence.” The
twelve original lü, multiplied by five (the number of elements), also equal to sixty lü. Daniélou makes the
following statement with regards to the infinite nature of the series:
“However, after the 52nd fifth, the octave is filled up and the 53rd fifth (note 54) comes out of the octave
and inserts itself between the octave C (2/1) and the twelfth fifth C¢ (531441/524288), thus forming,
above C (1/1), a small interval of 0.84 savarts (3.349 cents [1 savart = 3.986313725 cents, or 1 cent =
56 “Alain Daniélou (1907-94), the founder of the International Institute for Comparative Musicology in Berlin,
elucidated for tens of thousands of readers the meanings of the arts and religious traditions of both East and West. He
was an accomplished player on the vånà and taught in the music department at the University of Benares. His numerous
books, the product of a career spanning six decades, include The Myths and Gods of India; Gods of Love and Ecstasy;
While the Gods Play; Virtue, Success, Pleasure, and Liberation; The Phallus; Mastering the Secrets of Matter and the
Universe; and The Complete Kàma Sñtra.” For a further discussion, see Daniélou, Music and the Power of Sound: The
Influence of Tuning and Interval on Consciousness 35.
57 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 362.
22 Theory Versus Performance Practice
0.250858329 savarts]).58 Thus begins a new cycle, which in its turn, with a period of 53 fifths, will divide
the octave into small intervals of 0.84 savarts. The next cycle will appear during the seventh series of 53,
when the 359th fifth (note 360) comes out of the octave and forms, with C (1/1), an interval of 0.47
savarts (1.874 cents). The next cycle is of 666 notes, with a basic interval of 0.035 savarts (0.140 cents).
Then comes a cycle of 25,524 notes with a basic interval of 0.0021174 savarts (0.008 cents). This cycle is
very near to that of the precession of the equinoxes, or the Pythagorean great year, which is of 25,920
solar years. The small difference between the twelfth fifth and octave, similar to that of the lunar and the
solar year, leaves the door open for further cycles.”59
Table 5. Some correspondences of the first twelve lü NOTE NAME OF
LÜ
SEVEN
BEGINNINGS
FUNDAMENTAL
FOR SACRIFICES
ANNUAL
CYCLE
GENDER MOON
C
D¸
D
EÝ
E¢
F¢
F¥
G
AÝ
A¢
BÝ
B¢
huang zhong (I)
da lü (VIII)
tai cu (male form
of jia zhong (III)
jia zhong or
yuan zhong (X)
gu xian (V)
zhong lü (XIII)
rui bin (VII)
lin zhong or
han zhong (II)
yi tze (IX)
nan lü (IV)
wu yi (XI)
ying zhong (VI)
heaven
man
spring
summer
earth
autumn
winter
to the yellow emperor,
to heaven
to the white emperor,
to ancestors
to the green emperor,
to the first agriculturists
to the red emperor,
to earth
to the black emperor, to
the empress, to the moon
winter
solstice
spring
equinox
summer
solstice
autumn
equinox
masculine
feminine
masculine
feminine
masculine
feminine
masculine
feminine
masculine
feminine
masculine
feminine
11th (December)
12th (January)
1st (February)
2nd (March)
3rd (April)
4th (May)
5th (June)
6th (July)
7th (August)
8th (September)
9th (October)
10th (November)
58 “An alternative method (to cents) of interval measurement, but one used far less frequently, is the savart. Named
after the French scientist Félix Savart (1791-1841), this method assigns a total of 25 savarts to each semitone (or
25.08583297), one savart being approximately equal to four cents.” For a further discussion, see Read, 20th-Century
Microtonal Notation 7.
59 Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness 20-50.
Theory Versus Performance Practice 23
Tabl
e 6.
Ala
in D
anié
lou’
s sc
ale
of fi
fths
(the
cyc
lic d
ivisi
on o
f the
oct
ave)
Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
iddl
e C)
=26
1.62
5565
4Hz
DE
GRE
E N
UMBE
R N
OTE
IN
TERV
AL
RATI
O
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
C C¢
C£
CÚ
D¹
D¸
DÝ
D³
D²
D D¢
D£
DÚ
E¸
EÝ
E³
E² E E¢ E£
F³
F² F F¢
F£
FÚ
uniso
n cy
clic
oct
ave
(A)
LIII
Pyth
agor
ean
com
ma
(A)
XII
grea
t die
sis (A
) XX
IV
cycl
ic g
rave
or s
mal
l chr
omat
ic s
emito
ne, o
r min
or h
alf-t
one
(A)
XXXV
I
cycl
ic P
ytha
gore
an li
mm
a (A
) XL
VIII
Pyth
agor
ean
apot
ome,
dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
(A)
VII
cycl
ic g
reat
lim
ma,
acu
te o
r lar
ge h
alf-t
one
(A)
XIX
cycl
ic g
rave
or s
mal
l ton
e (A
) XX
XI
cycl
ic m
inor
tone
(A)
XLIII
just
maj
or to
ne (A
) II (
9th
harm
onic
) 55
th c
yclic
fifth
(A)
LV
acut
e or
larg
e to
ne (A
) XI
V
cycl
ic s
uper
maj
or s
econ
d (A
) XX
VI
cycl
ic a
ugm
ente
d se
cond
(A)
XXXV
III
cycl
ic P
ytha
gore
an m
inor
third
, or t
rihem
itone
(A)
L
cycl
ic m
inor
third
(A)
IX
cycl
ic n
eutra
l thi
rd (A
) XX
I
cycl
ic g
rave
or s
mal
l maj
or th
ird (A
) XX
XIII
cycl
ic m
ajor
third
(A)
XLV
Pyth
agor
ean
maj
or th
ird, o
r dito
ne (A
) IV
(81
st h
arm
onic
) 57
th c
yclic
fifth
(A)
LVII
cycl
ic a
cute
or l
arge
maj
or th
ird (A
) XV
I
cycl
ic s
ubfo
urth
(A)
XXV
III
cycl
ic g
rave
or s
mal
l fou
rth (A
) XL
cycl
ic p
erfe
ct fo
urth
(A)
LII
cycl
ic a
cute
or l
arge
four
th (A
) XI
cycl
ic s
uper
four
th (A
) XX
III
cycl
ic g
rave
or s
mal
l aug
men
ted
four
th (A
) XX
XV
1 ñ/1
ñ3õ
ó/2øô
3ñò/2
ñù 3ò
ô/2óø
3óö/2
õ÷ 3ô
ø/2÷ö
3÷/2
ññ
3 ñù/2
óð 3 ó
ñ/2ôù
3ôó/2
öø 3ò
/2ó
3õõ/2
ø÷ 3ñ
ô/2òò
3òö/2
ôñ 3ó
ø/2öð
3õð/2
÷ù 3ù
/2ñô
3ò
ñ/2óó
3óó/2
õò 3ô
õ/2÷ñ
3ô/2
ö 3õ
÷/2ùð
3ñö/2
òõ 3ò
ø/2ôô
3ôð/2
öó 3õ
ò/2øò
3ññ/2
ñ÷ 3ò
ó/2óö
3óõ/2
õõ
1.00
0000
1.
0020
90
1.01
3634
1.
0274
73
1.04
1491
1.
0557
00
1.06
7871
1.
0824
40
1.09
7208
1.
1121
78
1.12
5000
1.
1273
52
1.14
0349
1.
1559
07
1.17
1677
1.
1876
63
1.20
1355
1.
2177
45
1.23
4359
1.
2512
00
1.26
5625
1.
2682
71
1.28
2892
1.
3003
95
1.31
8137
1.
3361
20
1.35
1524
1.
3699
64
1.38
8654
261.
626
262.
172
265.
195
268.
813
272.
481
276.
198
279.
382
283.
194
287.
058
290.
974
294.
329
294.
944
298.
344
302.
415
306.
541
310.
723
314.
305
318.
593
322.
940
327.
346
331.
120
331.
812
335.
637
340.
217
344.
858
349.
563
353.
593
358.
417
363.
307
0.00
0 3.
615
23.4
60
46.9
20
70.3
80
93.8
40
113.
685
137.
145
160.
605
184.
065
203.
910
207.
525
227.
370
250.
830
274.
290
297.
750
317.
595
341.
055
364.
515
387.
975
407.
820
411.
435
431.
280
454.
740
478.
200
501.
660
521.
505
544.
965
568.
425
+00
+
04
+23
+
47
ß30
ß
06
+14
+
37
ß39
ß
16
+04
+
08
+27
+
51
ß26
ß
02
+18
ß
59
ß35
ß
12
+08
+
11
+31
ß
45
ß22
+
02
+22
+
45
ß32
24 Theory Versus Performance Practice
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(F
RACT
ION
) RA
TIO
(D
ECIM
AL)
FR
EQUE
NCY
(H
ERTZ
) CE
NTS
TU
NIN
G
Ò×
ÒØ ÒÙ
ÓÐ
ÓÑ
ÓÒ ÓÓ
ÓÔ
ÓÕ
ÓÖ
Ó×
ÓØ
ÓÙ
ÔÐ
ÔÑ ÔÒ
ÔÓ
ÔÔ
ÔÕ
ÔÖ
Ô×
ÔØ
ÔÙ
ÕÐ ÕÑ
ÕÒ
ÕÓ ÐÑ
F ¤ F¥ GÝ
G³
G²
G G¢
G£
GÚ
A¸
AÝ
A³
A²
A A¢ A£
AÚ
B¹
B¸
BÝ
B³
B² B B¢ B£
C³
C² C
cycl
ic tr
itone
, or a
ugm
ente
d fo
urth
(A)
XLV
II
Pyth
agor
ean
trito
ne, o
r aug
men
ted
four
th (A
) V
I
59th
cyc
lic fi
fth (A
) LIX
cycl
ic a
cute
or l
arge
dim
inish
ed fi
fth (A
) XV
III
cycl
ic s
ubfif
th (A
) XX
X
cycl
ic g
rave
or s
mal
l fift
h (A
) XL
II
just
and
Pyt
hago
rean
per
fect
fifth
(A)
I (3r
d h
arm
onic
) 54
th c
yclic
fifth
(A)
LIV
cycl
ic a
cute
or l
arge
fifth
(A)
XIII
cycl
ic s
uper
fifth
(A)
XXV
cycl
ic a
ugm
ente
d fi
fth (A
) XX
XVII
cycl
ic m
inor
six
th (A
) XL
IX
Pyth
agor
ean
min
or s
ixth
(A)
VIII
cycl
ic n
eutra
l six
th (A
) XX
cycl
ic g
rave
or s
mal
l maj
or s
ixth
(A)
XXXI
I
cycl
ic m
ajor
six
th (A
) XL
IV
Pyth
agor
ean
maj
or s
ixth
(A)
III (2
7th
harm
onic
) 56
th c
yclic
fifth
(A)
LVI
cycl
ic a
cute
or l
arge
maj
or s
ixth
(A)
XV
cycl
ic a
ugm
ente
d six
th (A
) XX
VII
cycl
ic g
rave
or s
mal
l min
or s
even
th (A
) XX
XIX
cycl
ic m
inor
sev
enth
(A)
LI
cycl
ic a
cute
or l
arge
min
or s
even
th (A
) X
cycl
ic n
eutra
l sev
enth
(A)
XXII
cycl
ic g
rave
or s
mal
l maj
or s
even
th (A
) XX
XIV
cycl
ic d
iato
nic
maj
or s
even
th (A
) XL
VI
Pyth
agor
ean
maj
or s
even
th (A
) V
58th
cyc
lic fi
fth (A
) LV
III
cycl
ic a
cute
or l
arge
maj
or s
even
th (A
) XV
II
cycl
ic s
uboc
tave
(A)
XXIX
cycl
ic g
rave
or s
mal
l oct
ave
(A)
XLI
oc
tave
3 ô÷/2
÷ô 3 ö
/2ù
3õù/2
ùó 3ñ
ø/2òø
3óð/2
ô÷ 3ô
ò/2öö
3ñ/2
ñ 3õ
ô/2øõ
3ñó/2
òð 3ò
õ/2óù
3ó÷/2
õø 3ô
ù/2÷÷
3ø/2
ñò
3òð/2
óñ 3ó
ò/2õð
3ôô/2
öù 3ó
/2ô
3õö/2
øø 3ñ
õ/2òó
3ò÷/2
ôò 3ó
ù/2öñ
3õñ/2
øð 3ñ
ð/2ñõ
3òò/2
óô 3ó
ô/2õó
3ôö/2
÷ò 3õ
/2÷
3õø/2
ùñ 3ñ
÷/2òö
3òù/2
ôõ 3ô
ñ/2öô
2ñ/1
ñ
1.40
7600
1.
4238
28
1.42
6804
1.
4432
54
1.46
2944
1.
4829
04
1.50
0000
1.
5031
35
1.52
0465
1.
5412
09
1.56
2236
1.
5835
50
1.60
1807
1.
6236
61
1.64
5813
1.
6682
67
1.68
7500
1.
6910
27
1.71
0523
1.
7338
60
1.75
7516
1.
7814
94
1.80
2032
1.
8266
18
1.85
1539
1.
8768
00
1.89
8480
1.
9024
06
1.92
4338
1.
9505
93
1.97
7205
2.00
0000
368.
264
372.
510
373.
289
377.
592
382.
744
387.
966
392.
438
393.
259
397.
792
403.
220
408.
721
414.
297
419.
074
424.
791
430.
587
436.
461
441.
493
442.
416
447.
517
453.
622
459.
811
466.
084
471.
458
477.
890
484.
410
491.
019
496.
680
497.
718
503.
456
510.
325
517.
287
52
3.25
1
591.
885
611.
730
615.
345
635.
190
658.
650
682.
110
701.
955
705.
570
725.
415
748.
875
772.
335
795.
795
815.
640
839.
100
862.
560
886.
020
905.
865
909.
480
929.
325
952.
785
976.
245
999.
705
1019
.550
10
43.0
10
1066
.470
10
89.9
30
1109
.775
11
13.3
90
1133
.235
11
56.6
95
1180
.155
1200
.000
ß08
+
12
+15
+
35
ß41
ß
18
+02
+
06
+25
+
49
ß28
ß
04
+16
ß
61
ß37
ß
14
+06
+
09
+29
ß
47
ß24
+
00
+20
ß
57
ß34
ß
10
+10
+
13
+33
ß
43
ß20
+00
Theory Versus Performance Practice 25
The Twenty-Two Note Division of the Octave The twenty-two note division of the octave was conceptualized by Persian scholar and physicist Barkešli
in the 1940s, following his personal analysis of contemporary tradition, and is based on the Pythagorean
cycle of fifths, as well as the writings of medieval theorists Éafå al-Dån Urmawå and Abu Nasr Fàràbå. After
scientific analysis of intervals in recordings of vocal music by five respected traditional musicians, Barkešli
concluded that the intervallic structure of the whole-tone and semitone in Persian music were reasonably
fixed, and closely resembled the first tetrachord of the Pythagorean diatonic model with the figures:
206+204+89=499 cents. In exact Pythagorean terms this is expressed as 203.910 (two limmas and a
comma) + 203.910 + 90.225 (limma) = 498.045, or 9/8×9/8×256/243=4/3. In other tetrachords he
also identified a “120-cent interval as the characteristic interval of Persian music”, which is close to a
Pythagorean apotome (2187/2048, or 113.685 cents), as well another less popular interval measuring
181 cents similar to a Pythagorean diminished third (65536/59049, or 180.450 cents).
Éafå al-Dån’s seventeen-note gamut, with its whole-tone pattern of L, L + L, and L + L + C is
therefore extended by Barkešli into one exhibiting the following intervallic pattern: L, L + C, L + L, and L
+ L + C; or in more elaborate terms as: 256/243, 256/243×531441/524288=2187/2048 (Pythagorean
apotome), 256/243×256/243=65536/59049 (Pythagorean diminished third), and L + L + C = 9/8, or
just major tone.
Table 7. Mehdi Barkešli’s division of the whole-tone
PATTERN L L + C L + L L + L + C
INTERVAL Pythagorean
limma
Pythagorean
apotome
Pythagorean
diminished third
just
major tone
RATIO 256/243 2187/2048 65536/59049 9/8
CENTS 90.225 113.685 180.450 203.910
It may therefore be summarized that the first eight intervals of the twenty-two-note division of the octave
are derived from the construction of a Pythagorean C mixolydian scale (the pattern of a ratio of 3/2, or
ascending fifths generating the pitches G, D, A, E, and B, while the ratio 4/3, or descending fifths
generating the pitches F and B"). The basic design essentially allows for the formation of both major (C,
D, E, and F) and minor (G, A, B", and C) tetrachords. Another fourteen intervals are then derived from the
construction of a whole-tone based on the four-microtone model of L, L + L, and L + L + C. The
twenty-two-note division of the octave is characterized by the following intervallic pattern: L, C, DDT, C –
L, C, DDT, C – L + L, C, DDT, C – L, C, DDT, C – L, C, DDT, C – L. ‘DDT’, or Pythagorean double diminished
26 Theory Versus Performance Practice
third (134217728/129140163, or 66.765 cents) represents the interval between the Pythagorean
diminished third (65536/59049) and apotome (2187/2048).
Ex. 2. Subdivision of the tetrachord and the 22-note division of the octave (Mehdi Barkešli, 20th century)
The Theory of Flexible Intervals The third proposition for a theory on intervals and scales of Persian music – the theory of flexible intervals,
or of the five primary intervals of performance practice – advocated by Farhat in opposition to both
twenty-four-tone equally-tempered, and twenty-two-note Pythagorean scales of Vaziri and Barkešli.
Farhat offers the following critical assessment:
“Both theories suffer equally from a tendency to accommodate certain Western concepts. Each theory, by
suggesting very exact intervals, remains oblivious to the fluidity and flexibility of Persian intervals. Vaziri did
not take account of this instability, as his apparent objective was to make Persian music adhere to a process
of equal temperament so that it can be harmonized. Barkešli, on the other hand, was not interested in the
Westernisation of Persian music but was committed to prove that today’s music is still rooted in the
medieval system. He has taken the exactness of the medieval theory very seriously – as have many others –
and has proposed a system vested with even greater precision.”
Utilizing a stroboconn and a melograph as measurement devices, and fretted instruments (tàrs and setàrs)
as the sources, Farhat identifies five primary intervals, with semitones and whole-tones representative of
relatively fixed intervals, while the rest, of a flexible nature. His classification of Persian intervals include: (1)
the semitone or minor second, measuring approximately 90 cents; (2) the small neutral tone, measuring
between 125 and 145 cents, or the average of 135 cents; (3) the large neutral tone, measuring between
150 and 170 cents, or the average of 160 cents; (4) the whole-tone or major second, measuring
approximately 204 cents; and (5) the plus-tone, measuring approximately 270 cents.60
60 Farhat, The Dastgàh Concept in Persian Music 10-16.
Theory Versus Performance Practice 27
Tabl
e 8.
Meh
di B
arke
šli’s
twen
ty-tw
o-no
te d
ivisi
on o
f the
oct
ave
Rela
tive
Pitc
h: A
4=44
0Hz
/ C4 (
mid
dle
C)=
261.
6255
654H
z
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ ÐÑ
C CÎ
CÚ
Dî
D DÎ
DÚ
Eî E F FÎ
FÚ
Gî
G GÎ
GÚ
Aî
A BÝ
AÚ
Bî B C
uniso
n
sup
eroc
tave
[6] (
D)
V
Pyth
agor
ean
apot
ome,
or c
hrom
atic
sem
itone
[7] (
A)
VII
neut
ral s
econ
d [8
] (D
) X
just
maj
or to
ne (A
) II (9
th h
arm
onic
)
sup
erm
ajor
sec
ond
[9] (
D) II
I
Pyth
agor
ean
augm
ente
d s
econ
d [1
0] (
A)
IX
neut
ral t
hird
[11]
(D)
VIII
Pyth
agor
ean
maj
or th
ird, o
r dito
ne (A
) IV
(81s
t har
mon
ic)
just
and
Pyt
hago
rean
per
fect
four
th (D
) I
sup
erfo
urth
[12]
(D
) V
I
Pyth
agor
ean
trito
ne, o
r aug
men
ted
four
th [1
3] (
A)
VI
subf
ifth
[14]
(D)
XI
just
and
Pyt
hago
rean
per
fect
fifth
(A)
I (3r
d h
arm
onic
)
sup
erfif
th [1
] (D
) IV
Pyth
agor
ean
augm
ente
d fi
fth [2
] (A
) V
III
neut
ral s
ixth
[3] (
D) IX
Pyth
agor
ean
maj
or s
ixth
(A)
III (2
7th
harm
onic
)
Pyth
agor
ean
min
or s
even
th (
D)
II
Pyth
agor
ean
augm
ente
d s
ixth
[4] (
A)
X
neut
ral s
even
th [5
] (D
) V
II
Pyth
agor
ean
maj
or s
even
th (A
) V
octa
ve
1/1
256/
243
2187
/204
8
6553
6/59
049
9/8
32/2
7
1968
3/16
384
8192
/656
1
81/6
4
4/3
1024
/729
729/
512
2621
44/1
7714
7
3/2
128/
81
6561
/409
6
3276
8/19
683
27/1
6
16/9
5904
9/32
768
4096
/218
7
243/
128
2/1
1.00
0000
1.05
3498
1.06
7871
1.10
9858
1.12
5000
1.18
5185
1.20
1355
1.24
8590
1.26
5625
1.33
3333
1.40
4664
1.42
3828
1.47
9811
1.50
0000
1.58
0247
1.60
1807
1.66
4787
1.68
7500
1.77
7778
1.80
2032
1.87
2885
1.89
8438
2.00
0000
261.
626
275.
622
279.
382
290.
367
294.
329
310.
075
314.
305
326.
663
331.
120
348.
834
367.
496
372.
510
387.
156
392.
438
413.
433
419.
074
435.
551
441.
493
465.
112
471.
458
489.
995
496.
680
523.
251
0.00
0
90.2
25
113.
685
180.
450
203.
910
294.
135
317.
595
384.
360
407.
820
498.
045
588.
270
611.
730
678.
495
701.
955
792.
180
815.
640
882.
405
905.
865
996.
090
1019
.550
1086
.315
1109
.775
1200
.000
+00
ß10
+14
ß20
+04
ß06
+18
ß16
+08
ß02
ß12
+12
ß22
+02
ß08
+16
ß18
+06
ß04
+20
ß14
+10
+00
28 Theory Versus Performance Practice
Table 9. Five-limit approximation of Hormoz Farhat’s division of the whole-tone
PATTERN L L + TSC L + GJCS L + L + C
INTERVAL Pythagorean
limma
great limma, acute or
large half-tone
grave or
small tone
just
major tone
RATIO 256/243 27/25 800/729 9/8
CENTS 90.225 133.238 160.897 203.910
The five-limit approximation of Farhat’s division of the whole-tone exhibits the following intervallic pattern:
L, L + TSC, L + GJCS, and L + L + C. ‘TSC’, or two syntonic commas (also referred to as the Mathieu
superdiesis, and produced by the factors 0.58081 and 6561/6400, or 43.013 cents), and ‘GJCS’, or grave
just chromatic semitone (25/24, or 70.672 cents) represent the intervals between the Pythagorean great
limma (27/25, or 133.238 cents) and limma (256/243), and grave or small tone (800/729) and limma
(256/243) respectively. The interval between the great limma (27/25) and limma (256/243) is
recognized by Manuel Op de Coul as the minimal diesis (20000/19683, or 27.660 cents).61
Farhat’s research concludes that “in Persian music, intervals are often unstable: they tend to
fluctuate, within a certain latitude, depending on the mode and according to the performer’s tastes and
inclination.” Farhat also acknowledges that the organization of melodic material beyond a tetrachord or
pentachord, and the actual notion of a ‘Persian scale’ has “no practical applications” in Persian music, and
is therefore misleading.62 The statement that, “these theories were written by eminent scholars who, in
most cases, were not practicing musicians,” additionally suggestive of the divide between theorists and
practitioners; a point further marked by Farhat’s insistence that “today’s musical traditions do not support
the exactitude of those theories.”63 With regards to the relationship between frequency ratios, string
lengths, or cents and the performer, it is interesting to note that as Scott Marcus suggests:
“The term ‘comma’ is not meant to signify an interval of any specific size and is thus never defined in terms
of frequency ratios, string lengths, or cents. Rather, like the term æruti as it occurs in North Indian music
today, kñmà is used to refer to slight changes in the pitch of a given note. Thus when a specific note is said
to be ‘minus a comma’, it means only that the note is slightly lowered from what is perceived to be the
note’s standard position.”64
61 Manuel Op de Coul, “Huygens-Fokker: List of intervals,” Huygens-Fokker Foundation: Centre for Microtonal Music,
2006, Huygens-Fokker Foundation, 22 Nov. 2006, <http://www.xs4all.nl/~huygensf/doc/intervals.html>.
62 Farhat, “Iran: Classical Traditions,” The New Grove Dictionary of Music and Musicians 532.
63 Farhat, The Dastgàh Concept in Persian Music 15.
64 Scott Marcus, “The Interface Between Theory and Practice: Intonation in Arab Music,” Asian Music 24.2 (Spring-
Summer, 1993): 41.
Theory Versus Performance Practice 29
Àzàdeh for Santñr and Tape Àzàdeh for santñr and tape – composed for Iranian santñrist Qmars Piraglu – features the Persian santär (a
72-string [or 18 quadruple-stringed] box zither), and will hereby serve as a practical study of Persian
tuning systems, with its presentation of both ‘theoretical’ and ‘performance practice’ tunings. Within the
context of an analysis of the tuning methodology of performer Qmars Piraglu, an attempt will be made to
make comparisons not just with the three propositions outlined previously, but also with other existing
alternative tuning systems. This discussion presented in order to generate a greater understanding of the
musical processes that govern the theory of Persian intervals, at the same time disclosing some of the
strategies employed in the general contemporary ‘cross-cultural’ compositional process. The enquiry into
the basic structure of the Persian modal system reveals some of this music’s essential characteristics, while
also providing the necessary organisational framework for the composition.
The Artist Iranian artist Qmars Piraglu (formerly Siamak Noory) was born in Tehran in 1961, and initiated his musical
education at the age of five via Orff instruments (children’s instruments designed by German composer
Carl Orff [1895-1982] such as the xilophon, metalophon, and vibraphon).65 Following a seven-year
period, he then adopted the Persian santär (a 72-string [or 18 quadruple-stringed] box zither), initially
learning the instrument under the guidance of Ms Farzaneh Noshad (a student at the Persian Traditional
Music Conservatory in Tehran). Three years later, he is accepted as a student of “one of Iran’s greatest
contemporary composers and master santärists,” Ostàd Faràmarz Pàyvar (1933-). For the next eight years
Qmars develops his knowledge and understanding of the complete Persian classical music repertoire. As
well as this, he synchronically studies Western music – the pianoforte with Taher Djalili (a pianist from the
Music Conservatory), and for a year, the bassoon with Khosrow Soltani.
In 1979, the Islamic revolution in Iran bestowed the study of music with an illegal status,
consequently bringing about the premature closure of the Music Conservatory. “Because of the Islamic
revolution in 1979, which caused an undemocratic situation, more than three million Iranians became
obliged to escape from the country. Of these three million nearly sixty thousand came to Sweden during
the 1980s. More than a million people fled to USA, and the rest mostly went to Germany, France, and
other European countries,” explains Qmars. After the fact, for his own ‘anti-establishment’ beliefs in
65 German composer Carl Off (1895-1982) developed Orff-Schilwerk, “a system of music education intended for
groups of children singing and playing together,” which emphasized “the development of creativity and the ability to
improvise.” For a further discussion, see Randel, ed., The New Harvard Dictionary of Music 577-78.
30 Theory Versus Performance Practice
artistic freedom, he too becomes a victim of the ongoing institutionalised persecution, and in 1981 is
consequently arrested and imprisoned for a period of two years.
Qmars ultimately escaped from Iran in 1986 (during the Iran-Iraq war of the eighties), arriving in
Turkey as a refugee, to then temporarily settle in Belgium, where he studied the piano at the Music
Academy in Antwerp with Hedvig Vanvarenberg, before establishing a permanent base in Sweden in
1989. Here he studied the piano for two years at the Birkagårdens Folkhögskolan in Stockholm with Stella
Tchaikowsky, and went on to graduate from the University of Göteborg with a Bachelor of Science in
Musicology. Since 2001, Qmars has been living in Melbourne, Australia. In 2003, he completed a Master
of Music (MMus) degree at the University of Melbourne, and is currently undertaking a Doctor of
Philosophy (PhD) degree at Monash University.66
Fig. 1. Andrián Pertout, Qmars Piraglu.67
The Instrument The Persian or Iranian santär is an integral part of the traditional orchestra, sharing the lute repertoire of the
tàr and setàr. It is also utilized in the motrebi ‘entertainment music’ genre, but folk styles are excluded.
An article by Jean During, Scheherazade Q. Hassan, and Alastair Dick describes the santär’s construction
thus:
66 Andrián Pertout, “Siamak Noory: The Santurist – Part 1,” interview with Siamak Noory (Qmars Piraglu) Mixdown
97 (May 2002): 25.
67 Qmars Piraglu, personal photograph of Andrián Pertout, 22 Oct. 2006.
Theory Versus Performance Practice 31
“The santär consists of a trapeziform case made of walnut wood, approximately 90cm wide at the broad
end, 35cm wide at the narrow end and 6cm deep. The sides form an angle of 45 degrees to the wider
end. The strings are fixed to hitch-pins along the left-hand side and wound round metal wrest-pins on the
right by means of which they are tuned with a tuning-key. Each quadruple set of strings rests on a movable
bridge of hardwood (kharak). These bridges are aligned almost parallel with the sides of the case. The
right-hand rank corresponds to the bass strings and that on the left to the treble strings. In the centre of the
santär the low-pitched strings on the right cross the high-pitched strings on the left.”
Adding to this, the instrument features three courses of strings (the bass strings, made of brass, while the
treble ones, steel), with a total of 72 strings, or 18 groups of strings, capable of producing 27 different
pitches, and is played “by striking the strings with two hammers (mezràb) held in three fingers of each
hand.”68
Ex. 3. Range of the santñr
Persian 72-string (or 18 quadruple-stringed) box zither
The Persian Modal System An important aspect of understanding the basic concept behind the Persian modal system is the radif
(row, series), being the term used to describe the complete collection of melodies that constitute the
repertoire of Persian traditional music, as well as the separate issue of melodic patterns associated with
each individual mode represented within the subsystem of the twelve dastgàhs.69 According to Talai:
“To understand the radif, we must first understand that it is something different from the modal system. This
repertory is not like Western art music, which is composed and intended to be played exactly as written.
The radif consists of traditional melodies, many of which are derived from popular and folk sources, and
whose origins have been obscured with the passage of time.”70
On the other hand, the notion of dastgàh (organisation, system) – the subsystem of the radif – further
delineates the twelve groupings of modes, with their own collection of associated melodies, or guêes
(corner, section, piece). The introductory section of each mode referred to as the daràmad (opening,
introduction), while the conclusive, as the forñd (descent, cadence). Another factor of note includes 68 Jean During, Scheherazade Q Hassan, and Alastair Dick, “Santñr,” The New Grove Dictionary of Musical
Instruments, ed. Stanley Sadie, vol. 3 (London: Macmillan Reference, 1984) 291-92.
69 Farhat, The Dastgàh Concept in Persian Music 21.
70 Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah 4.
32 Theory Versus Performance Practice
the specific role of certain tones within a mode, which include the ist, or ‘stop’ note used to conclude
phrases; the šàhed (witness), or prominent note; as well as the àqàz (beginning), or introductory note in
improvisation. The moteqayyer (changeable), or variable note further represents a tone reintroduced to
a mode in an altered form. It is quite common for example for a mode to utilize A koron (a microtonal
inflection on the tone of A) in the bottom registers, while A flat, in the top; or for there to be two
microtonal possibilities for a note within the same register, with the use of one or both pitches governed
by attributes of a particular guše.71 Habib Hassan Touma offers the following discussion with regards to
the compositional and improvisational nature of the dastgàh system in the music of the Middle East:
“The realization of a truly convincing and original maqam requires a creative faculty like that of a composer
of genius. Nevertheless, this phenomenon can be considered only partly as a composed form, because
no maqam, makam, mugam, or dastgàh can be identical to any other. Each time it is re-created as a new
composition. The compositional aspect is demonstrated in the pre-determined tonal-spatial organization
of a fixed number of tone-levels without repetitions, while the improvisation aspect unfolds itself freely in
the rhythmic-temporal scheme. Thus, the interplay of composition and improvisation is one of the most
distinctive features of the maqam phenomenon.”72
The presentation of the twelve dastgàhs that follows (the basic pitch material of the modes) adheres to
the range of the santñr, and utilizes Vaziri’s system of accidentals, which was originally intended for the
microtonal sharpening and flattening of tones in equal quarter-tone ( 24 2 ) increments, but consequently
adopted as a general standard for Persian music notation.73
Fig. 2. The Ali Naqi Vaziri notation system (the accidentals of Persian music)
sori – raised by one Pythagorean limma (90.225 cents)
sharp – raised by one limma and one Pythagorean comma (113.685 cents)
koron – lowered by one Pythagorean limma (90.225 cents)
flat – lowered by one limma and one Pythagorean comma (113.685 cents)
71 Farhat, The Dastgàh Concept in Persian Music 19-26.
72 Habib Hassan Touma, “The Maqam Phenomenon: An Improvisation Technique in the Music of the Middle East,”
Ethnomusicology 15.1 (Jan., 1971): 47.
73 Zonis, Classical Persian Music: An Introduction 56-57.
Theory Versus Performance Practice 33
Ex. 4. The twelve dastgàhs i. Šhñr
ii. Dašhtå
iii. Abñ atà
iv. Màhñr
v. Ràst-Panjgàh
34 Theory Versus Performance Practice
vi. Homàyñn
vii. Bayàt-e Eéfahàn
viii. Bayàt-e Tork
ix. Segàh
x. Áahàrgàh
Theory Versus Performance Practice 35
xi. Afšhàrå
xii. Navà
Tuning Analysis Protocols The obvious consequential procedure in the compositional process now being the selection of the
appropriate dastgàh, which is an exercise delivering an outcome based on definite instinctive principles,
although assisted somewhat by the audition of modes in workshops organized with Piraglu. Upon the
decision to select dastgàh-e segàh, it simply becomes a matter of tuning the santñr to concert
performance standard so as to acquire twenty-seven naturally decaying (approximately six seconds
each) samples. The sound of every string therefore captured in the digital domain (in this case utilizing an
Audio-Technica AT4050/CM5 condenser microphone [with switchable cardioid, omnidirectional, or
figure-8 operation, and a frequency response from 20Hz-20kHz], Mackie 32×8×2 8-Bus Mixing Console,
and a Fostex D-160 Digital Multitrack Recorder), to be then digitally transferred (via optical cable) to a
digital audio editor PC software environment (Sony Sound Forge 8.0). The strategy adopted in order to
obtain the frequencies (in hertz) of each string with the aid of spectrum analyzing software, which will
provide an invaluable source of data for later extensive analysis. This data (collected on three separate
occasions), once subjected to an analysis of variance (utilizing average and standard deviation criteria)
will then form a ‘mean’ scale, which in essence will represent the intervallic nature of dastgàh-e segàh
according to Persian performer Qmars Piraglu. The first collection of samples will additionally serve as the
source material for the creation of a multi-sampled santñr for the Akai S3000XL Midi Stereo Digital
Sampler.
In view of the fact that Sony Sound Forge 8.0 is unable to provide accurate enough frequency
readings, a series of tests is then implemented in order to locate the appropriate software package.
These tests include the multiplatform readings of 1Hz, 10Hz, 100Hz, 1KHz, 10KHz, and 440Hz sine wave
36 Theory Versus Performance Practice
sync tones, with the results nominating Adobe Audition 2.0 as the most error free vehicle. Utilizing the
frequency analysis window function of the program it is now possible to generate FFT (Fast Fourier
Transform) computations that translate frequency and amplitude analysis into a simple display of the mean
frequency of the most prominent frequency peak, or the fundamental.74 The Adobe Audition manual
describes the FFT process thus: “Fourier Theory states that any waveform consists of an infinite sum of sin
and cos functions, allowing frequency and amplitude to be quickly analyzed.” The program uses an
algorithm based on the Fourier Theory for filtering, spectral view, and frequency analysis. The FFT size
menu includes 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, and 65536 sample
options; while the FFT type menu provides eight type of FFT windows – Triangular, Hanning, Hamming,
Blackmann, Welch (Gaussian), Blackmann-Harris, Kaiser (120dB), and Kaiser (180dB). According to the
manual:
“The Triangular window gives a more precise frequency estimate, but it’s also the noisiest, meaning that
other frequencies will be shown as present, even though they may be much lower in volume. At the other
extreme, the Blackmann-Harris window has a broader frequency band, which isn’t as precise, but the
sidelobes are very low, making it easier to pick out the major frequency components.”
An analysis of the test tone readings (all one hundred percent accurate apart from the 0.94257Hz result
for 1Hz) reveal that a FFT size of 65536 and a Hamming75 type is required. At 44.1KHz, or 44,100
samples per second, a 65536 FFT sample window length allows for the equivalent of 673 milliseconds of
sample time. One further test procedure is nevertheless implemented in order to confirm the above
results, and this involves obtaining data (utilizing a smaller 4096 FFT sample window length, or 93
milliseconds) from all Cs (C3, C4, C5, and C6) at multiple start points (in 50 millisecond intervals beginning
from 0 and ending on 500 milliseconds). The vibrating strings are shown to be extremely stable over
time, with approximately ß99.923564% and +100.122296% deviations from the 65536 FFT sample
window length reading of the first sample (C3), which represents fluctuations within 0.26Hz, or 3.440
cents.
74 “Frequency, which is a measurement expressed in hertz (Hz), is derived from German physicist Heinrich Rudolf
Hertz (1857-94), and represents the rate of vibrations per second of a sound wave, with each cycle a sonic structure
incorporating positive and negative amplitudes.” For a further discussion, see Arthur H. Benade, Fundamentals of Musical
Acoustics, 2nd ed. (New York: Dover Publications, 1990) 159.
75 The Hamming window is “a very popular compromise between simplicity and effectiveness. It’s named after
Richard W. Hamming, a pioneer in the application of computers to practical computation. Mathematically, it consists of a
single cycle of a cosine, raised and weighted so that it drops to 0.08 at the end-points and has a peak value of one:
ht=0.54ß0.46cos(2�t/nß1), nt <0 � .” For a further discussion, see Ken Steiglitz, A DSP Primer: with Applications to
Digital Audio and Computer Music (Menlo Park, CA: Addison-Wesley Pub., 1995) 206.
Theory Versus Performance Practice 37
Tuning of the Santñr
The tuning of the santñr for dastgàh-e segàh (on F) presents the following series of pitches: C3, F3, G3,
Aù3, B"3, C4, Dù4, E"4, F4 (first position); Eù4, F4, G4, Aù4, B"4, C5, Dù5, E"5, F5 (second position); and Eù5,
F5, G5, A"5, B"5, C6, Dù6, E6, F6 (third position). The ‘position’ is in reference to each of the instrument’s
three courses of strings. The instrument’s design incorporates eighteen quadruple set of strings, with the
division via movable bridges of the second set of nine strings generating the second and third courses
(two individual pitches generally tuned octaves apart), to produce twenty-seven distinct pitches in total.
The santñrist generally commences with the tuning of the first front string, which then becomes the basis
for the generation of unison intervallic relationships (removing beating as much as possible) with the other
three strings of each set. An interesting point of note is that although the primary structure of dastgàh-e
segàh can be essentially stated to be heptatonic, in performance practice the scale is expanded into a
three-octave scale with dissimilar tetrachords. Some points of interest include the low brass string tuned
to C'; the A’s in the first two octaves tuned to Aù, while in the third, tuned to A"; E" and Eù alternatives
provided in two octaves; and the duplication of F pitches in the second and third octaves. The design
of the instrument – presenting a total of nine strings per octave, with two strings, E and F, tunable to either
E', E" or Eù; and F' or F! respectively – allows for the performance of heptatonic scales (inclusive of their
auxiliary pitches) from the complete Persian modal system.76
Dastgàh-e segàh is mode number nine of the twelve dastgàhs (according to Piraglu’s own
codification derived from the santñr tradition of Pàyvar), and its name (“literally ‘third place’”) is in
reference to “the positioning of the central pitch relative to other modes.” According to Zonis, “se is the
Persian word for ‘three’; gàh, the word for ‘time’ or ‘place’,” and alludes to the fact that “this kind of
designation was formerly used both in Persia and in the Arab countries to indicate the degrees of the
scale.”77 The guêe nomenclature and design of the dastgàh is extensively interconnected with dastgàh-e
áahàrgàh (“literally ‘fourth place’”).78 “Segàh and áahàrgàh allied to a degree unparalleled in any other
two dastgàhs,” notes Farhat. “Virtually every piece performed in segàh can be performed in áahàrgàh,
although áahàrgàh includes a few guêes peculiar to that dastgàh and not performed in segàh.”79 The
following three examples present dastgàh-e segàh’s basic structure, as well as the tuning of the santñr,
and harmonic characteristics (in Pythagorean terms).
76 Ella Zonis, Classical Persian Music: An Introduction (Cambridge, MA: Harvard U. Press, 1973) 57-58.
77 Zonis, Classical Persian Music: An Introduction 88.
78 Laudan Nooshin, “The Song of the Nightingale: Processes of Improvisation in Dastgàh Segàh (Iranian Classical
Music),” British Journal of Ethnomusicology 7 (1998): 76.
79 Farhat, The Dastgàh Concept in Persian Music 56.
38 Theory Versus Performance Practice
Ex. 5. Dastgàh-e segàh i. Basic mode
ii. Tuning of the santñr for dastgàh-e segàh (on F)
iii. The harmonic characteristics of dastgàh-e segàh
The tuning of the santñr for dastgàh-e segàh (on F) involves a specific tuning scheme, which begins with
the acquisition of a fundamental C' in second position (C5) calibrated to A=440Hz. The notion of
standard pitch only idealistic as in actuality on one occasion demonstrating a 10.223 cent deviation from
A=440Hz (or A=442.606Hz), while on other occasions, 10.387 cents (or A=442.648Hz), and 1.584
cents (or A=440.403Hz). In spite of Piraglu’s insistence of possessing perfect pitch and therefore the
ability to tune to a sung high C at standard pitch,80 something that becomes clearly evident via this
experience is that in this particular context the solo performer is rarely that concerned about the
procurement of A=440Hz in literal terms, hence it is a musical pitch certainly not in the domain of
obligatory criteria. Although it must be noted that in a performance setting involving the santñr in
80 “Absolute pitch, while extremely rare in the general population (its incidence has been estimated as 1 in
10,000), is in reality common among professional musicians, and to a limited extent may serve as a marker for musicality.
But as N. Slonimsky, in his autobiography, Perfect Pitch, writes: ‘The lack of it does not exclude musical talent, or even
genius. Neither Wagner nor Tchaikovsky had absolute pitch, while a legion of mediocre composers possessed it to the
highest degree.’ There is a greatly heightened occurrence of absolute pitch in some other populations: among the autistic
the incidence may be about one in 20; and among those with savant syndrome, more than a third have musical gifts –
and all musical savants, apparently, have absolute pitch.” For a further discussion, see Oliver Sacks, G. Schlaug, L. Jäncke,
Y. Huang, and H. Steinmetz, “Musical Ability,” Science 268.5211 (May 5, 1995): 621.
Theory Versus Performance Practice 39
conjunction with Western traditional instruments, some form of pitch calibration will be required in order
to accommodate the limitations of some of the instruments. A performance setting involving a Persian
ensemble on the other hand will require the whole group to calibrate their pitch to the santñr, for the
simple reason of practicality – the instrument requiring the tuning of seventy-two strings. It is interesting to
note that following a discussion with the performer about the rationale of the last measurement of
A=440.403Hz, and its obvious discrepancy in comparison with the mean of the first two (442.627Hz,
and therefore a difference of almost nine cents), it is revealed that a duet performance with a flautist had
required the santñr to be calibrated down to A=440Hz.
Stage one of the tuning process encapsulates the following procedures:
I. i. C5 is tuned to A=440Hz
ii. C4 to C5
iii. C3 to C4
iv. C6 to C5
II. i. F4 (2nd position) is tuned to C5, generating a perfect fifth
ii. F3 to F4 (2nd position)
iii. F5 (3rd position) to F4 (2nd position)
iv. F5 (2nd position) to F5 (3rd position)
v. F6 to F5 (2nd position)
vi. F4 (1st position) to F4 (2nd position)
III. i. G4 is tuned to C5, generating a perfect fourth
ii. G3 to G4
iii. G5 to G4
IV. i. B"4 is tuned to F4, generating a tempered perfect fourth
ii. B"3 to B"4
iii. B"5 to B"4
V. i. Aù4 is tuned to first tetrachord melodic patterns (F4, G4, Aù4 and B"4)
ii. Aù3 to Aù4
VI. i. Dù5 is tuned to Aù4, generating a tempered perfect fourth
ii. Dù4 to Dù5
iii. Dù6 to Dù5
VII. i. E"5 is tuned to B"4, generating a tempered perfect fourth
ii. E"4 to E"5
iii. E"6 to E"5
VIII. i. Eù4 is tuned to Aù4, generating a tempered perfect fourth
40 Theory Versus Performance Practice
ii. Eù5 to Eù4
IX. i. A"5 is tuned to second tetrachord melodic patterns (Eù5, F5, G5 and A"5)
It may be further noted that although the technique applied to the tuning of Aù4 has a basis of first
tetrachord melodic patterns (F4, G4, Aù4, and B"4), the interval is additionally calculated according to the
evaluation of the perfect fourth simultaneous sonorities of F4 (2nd position) and B"4, as well as F4 (2nd
position) and Aù4. As a matter of interest, it must be stated that the perfect fourth is “the smallest
invariable interval” in Persian music, and recognized as the most important.81 The duplication of Fs (F4 in
first and second positions, and F5 in second and third positions) serve merely as alternatives for the
equivalent pitches, although in performance practice, F4 (first position) and F5 (second position) are
generally avoided in dastgàh-e segàh (on F) due to their inferior timbral quality (the strings, shorter in this
region of the instrument, hence dynamically weaker). In the context of the pitch material requirements of
other dastgàhs, this additional pitch serves to provide a mandatory chromatic alternation. The
synchronous utilization of Eù and E" in dastgàh-e segàh (on F) illustrates the application of that principle.
The fact that the integrity of second and third position pitches are determined by the placement of
individual bridges dividing the string into specific ratios, these bridges may also have to be manually
adjusted in order to produce the desired sonorities. Third position A"5 is produced via this method,
with the adjustment of the bridge dividing the relevant string essentially simultaneously producing second
position Aù4 and third position A"5. This also holds true for other pitches acquired during the dipartite
tuning process not conforming to the aesthetics of dastgàh-e segàh (on F), which may have to be
alternatively fine-tuned via the adjustments of the bridge. The final pitch of interest in need of some
explanation is C3 in first position, which according to Piraglu, is a relatively modern phenomenon –
adopted post 1980 by a new generation of santñr players such as Parviz Meshkàtiàn (1955-) and
Pashang Kàmkàr (1951-). The practice favours the tuning of the low string to C or D (depending on the
dastgàh), as opposed to the earlier schools of Abol-Äasan Éabà (1902-1957) and Faràmarz Pàyvar
(1933-) subscribing to the acquisition of E or Eù. The C3 in first position of course provides the facility for
the instrument to produce both perfect fourth (between C3 and F3) and perfect fifth (between C3 and
G3) dyads in the low tessitura of the instrument.82
Stage two of the tuning process – the evaluation of simultaneous sonorities (perfect fifths, perfect
fourths, tempered perfect fourths, and neutral thirds) – then encapsulates the following procedures:
I. i. F4 (2nd position) is evaluated with C5, generating a perfect fifth
81 Talai, Traditional Persian Art Music: The Radif of Mirza Abdollah 10.
82 Qmars Piraglu, “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties,” diss. Göteborg U., Swed.,
1998, 33.
Theory Versus Performance Practice 41
ii. C4 with F4 (2nd position), generating a perfect fourth
iii. C4 with F4 (1st position), generating a perfect fourth
iv. C5 with F5 (3rd position), generating a perfect fourth
v. C5 with F5 (2nd position), generating a perfect fourth
vi. F5 (3rd position) with C6, generating a perfect fifth
vii. F3 with C4, generating a perfect fifth
viii. C3 with F3, generating a perfect fourth
ix. C6 with F6, generating a perfect fourth
II. i. C4 is evaluated with G4, generating a perfect fifth
ii. G4 with C5, generating a perfect fourth
iii. C5 with G5, generating a perfect fifth
iv. G5 with C6, generating a perfect fourth
v. G3 with C4, generating a perfect fourth
vi. C3 with G3, generating a perfect fifth
III. i. F4 (2nd position) is evaluated with B"4, generating a tempered perfect fourth
ii. F4 (2nd position) with Aù4, generating a neutral third
iii. Aù4 with Dù5, generating a tempered perfect fourth
iv. F3 with B"3, generating a tempered perfect fourth
v. F3 with Aù3, generating a neutral third
vi. Aù3 with Dù4, generating a tempered perfect fourth
vii. F5 (3rd position) with B"5, generating a tempered perfect fourth
IV. i. B"4 is evaluated with E"5, generating a tempered perfect fourth
ii. B"3 with E"4, generating a tempered perfect fourth
iii. B"5 with E"6, generating a tempered perfect fourth
V. i. Eù4 is evaluated with Aù4, generating a tempered perfect fourth Spectrum Analysis Results The following three tables present the spectrum analysis results collected on three separate occasions
(with a periodicity of 3-6 months) for each of the twenty-seven sets of strings, and therefore denotes all
data for string set, pitch, order, frequency (Hz), ratio (decimal), and cents. The calculation of frequency
represents prime (not normalized) data, and therefore ratio and cents are the only two comparative
frames of reference. Ratios have been calculated from the relationship of frequencies to the base pitch
of the second position C5 (the reference pitch, or 526.350Hz, 526.400Hz, and 523.730Hz respectively
for each of the three tunings), while cents are a derivative of ratio data.
42 Theory Versus Performance Practice
Table 10. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 1 1st position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ
C3
F3
G3
Aî3
BÝ3
C4
Dî4
EÝ4
F4
3 (I-iii)
6 (II-ii)
12 (III-ii)
18 (V-ii)
15 (IV-ii)
2 (I-ii)
20 (VI-ii)
23 (VII-ii)
10 (II-vi)
130.830
174.990
196.480
213.800
232.060
261.390
286.780
310.080
351.200
1.988487
1.329838
1.493151
1.624774
1.763541
1.986435
1.089693
1.178227
1.334473
1190.005
493.500
694.032
840.287
982.171
1188.218
148.706
283.942
499.524
2nd position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ
Eî4
F4
G4
Aî4
BÝ4
C5
Dî5
EÝ5
F5
25 (VIII-i)
5 (II-i)
11 (III-i)
17 (V-i)
14 (IV-i)
1 (I-i)
19 (VI-i)
22 (VII-i)
8 (II-iv)
322.540
350.090
394.970
429.610
465.950
526.350
578.200
620.720
703.100
1.225572
1.330256
1.500788
1.632412
1.770495
1.000000
1.098509
1.179291
1.335803
352.147
494.044
702.865
848.406
988.983
0.000
162.655
285.504
501.249
3rd position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×
Eî5
F5
G5
AÝ5
BÝ5
C6
Dî6
EÝ6
F6
26 (VIII-ii)
7 (II-iii)
13 (III-iii)
27 (IX-i)
16 (IV-iii)
4 (I-iv)
21 (VI-iii)
24 (VII-iii)
9 (II-v)
645.220
697.240
788.260
827.400
930.310
1049.300
1144.400
1246.000
1406.200
1.225838
1.324670
1.497597
1.571958
1.767474
1.993540
1.087109
1.183623
1.335803
352.522
486.759
699.179
783.075
986.027
1194.399
144.596
291.852
501.249
Theory Versus Performance Practice 43
Table 11. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 2 1st position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ
C3
F3
G3
Aî3
BÝ3
C4
Dî4
EÝ4
F4
3 (I-iii)
6 (II-ii)
12 (III-ii)
18 (V-ii)
15 (IV-ii)
2 (I-ii)
20 (VI-ii)
23 (VII-ii)
10 (II-vi)
130.790
174.260
197.400
213.740
231.990
260.930
286.690
310.050
351.170
1.987690
1.324164
1.500000
1.624164
1.762842
1.982751
1.089248
1.178002
1.334233
1189.311
486.098
701.955
839.637
981.484
1185.004
147.999
283.610
499.212
2nd position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ
Eî4
F4
G4
Aî4
BÝ4
C5
Dî5
EÝ5
F5
25 (VIII-i)
5 (II-i)
11 (III-i)
17 (V-i)
14 (IV-i)
1 (I-i)
19 (VI-i)
22 (VII-i)
8 (II-iv)
322.490
348.840
394.670
428.470
465.940
526.400
578.150
620.620
703.090
1.225266
1.325380
1.499506
1.627926
1.770289
1.000000
1.098309
1.178989
1.335657
351.714
487.687
701.385
843.642
988.782
0.000
162.341
285.061
501.060
3rd position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×
Eî5
F5
G5
AÝ5
BÝ5
C6
Dî6
EÝ6
F6
26 (VIII-ii)
7 (II-iii)
13 (III-iii)
27 (IX-i)
16 (IV-iii)
4 (I-iv)
21 (VI-iii)
24 (VII-iii)
9 (II-v)
645.160
697.280
788.220
829.380
930.230
1048.700
1144.200
1254.800
1406.000
1.225608
1.324620
1.497378
1.575570
1.767154
1.992211
1.086816
1.191869
1.335486
352.197
486.694
698.927
787.049
985.714
1193.245
144.129
303.871
500.838
44 Theory Versus Performance Practice
Table 12. Tuning of the santñr for dastgàh-e segàh (on F) – tuning no. 3 1st position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ
C3
F3
G3
Aî3
BÝ3
C4
Dî4
EÝ4
F4
3 (I-iii)
6 (II-ii)
12 (III-ii)
18 (V-ii)
15 (IV-ii)
2 (I-ii)
20 (VI-ii)
23 (VII-ii)
10 (II-vi)
131.160
174.570
197.130
212.300
231.460
262.250
285.400
311.140
346.900
1.001738
1.333282
1.505585
1.621446
1.767781
1.001470
1.089875
1.188169
1.336185
3.005
497.979
708.389
836.737
986.328
2.543
148.995
298.489
501.743
2nd position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ
Eî4
F4
G4
Aî4
BÝ4
C5
Dî5
EÝ5
F5
25 (VIII-i)
5 (II-i)
11 (III-i)
17 (V-i)
14 (IV-i)
1 (I-i)
19 (VI-i)
22 (VII-i)
8 (II-iv)
320.500
348.970
394.930
427.390
462.800
523.730
569.000
617.650
698.550
1.223913
1.332633
1.508144
1.632101
1.767323
1.000000
1.086438
1.179329
1.333798
349.801
497.136
711.328
848.076
985.879
0.000
143.526
285.560
498.648
3rd position
STRING SET NOTE ORDER FREQUENCY (HERTZ) RATIO (DECIMAL) CENTS
ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×
Eî5
F5
G5
AÝ5
BÝ5
C6
Dî6
EÝ6
F6
26 (VIII-ii)
7 (II-iii)
13 (III-iii)
27 (IX-i)
16 (IV-iii)
4 (I-iv)
21 (VI-iii)
24 (VII-iii)
9 (II-v)
638.520
695.250
791.640
824.560
925.770
1051.000
1141.700
1242.500
1396.800
1.219178
1.327497
1.511542
1.574399
1.767647
1.003380
1.089970
1.186203
1.333512
343.090
490.450
715.226
785.761
986.197
5.84102
149.146
295.621
498.276
Theory Versus Performance Practice 45
An analysis of simultaneous sonorities based on stage one of the tuning process for the three tunings
reveal octaves with a mean value of 1201.019 cents, a range between 1181.941 and 1218.810 cents,
and a standard deviation of 7.771183. These results effectively pronounce the octave as being in the
vicinity of the just perfect octave (2/1), although in the range from the double tritone (2025/1024, or
1180.447 cents) and acute or large octave (81/40, or 1221.506 cents) with a falsity of +1.494 and
ß2.696 cents on each count. In comparison with the just perfect octave (2/1), the octave is within the
range of ß18.059 and +18.810 cents.83
Table 13. Stage one of tuning process – octaves
NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
C3 and C4
F3 and F4(2)
G3 and G4
Aî3 and Aî4
BÝ3 and BÝ4
C4 and C5
Dî4 and Dî5
EÝ4 and EÝ5
Eî4 and Eî5
F4(2) and F5(3)
G4 and G5
BÝ4 and BÝ5
C5 and C6
Dî5 and Dî6
EÝ5 and EÝ6
F5(2) and F6
1195.693 � 1199.538
1199.157 � 1201.589
1199.430 � 1208.833
1204.005 � 1211.339
1199.551 � 1207.298
1197.457 � 1214.996
1194.532 � 1214.343
1187.071 � 1201.563
1193.289 � 1200.483
1192.715 � 1199.007
1196.314 � 1203.897
1197.044 � 1200.318
1193.245 � 1205.841
1181.941 � 1205.620
1206.347 � 1218.810
1199.628 � 1200.000
1197.814
1200.430
1203.734
1207.821
1204.554
1208.078
1207.608
1196.695
1198.049
1195.012
1199.251
1198.098
1197.828
1189.783
1211.740
1199.802
1.953313
1.220009
4.751527
3.675986
4.339307
9.337949
11.325798
8.334857
4.122885
3.472527
4.070141
1.923469
6.963106
13.715187
6.398768
0.187035
Simultaneous sonorities based on stage two of the tuning process for the three tunings on the other hand
reveal perfect fifths with a mean value of 706.894 cents, a range between 694.718 and 716.381 cents,
and a standard deviation of 6.537279. The primary perfect fifths, or those unique and sequentially
superior are represented by F4 (2nd position) and C5, C4 and G4, and the mean values of 707.044 and
713.271 cents respectively. The range of these fifths being between 698.927 and 715.226 cents
indicates an inclination to represent the meantone perfect fifth (3/2× 48180 , approximately 154/103, or
696.578 cents), just perfect fifth (3/2), and equal perfect fifth ( 712 ]2[ , approximately 767/512, or
700.000 cents) with a falsity of +2.274, ß3.028, and ß1.073 cents at the lower end of the scale; while
83 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation ix.
46 Theory Versus Performance Practice
the sléndro acute or large fifth ( 35 ]2[ , approximately 97/64, or 720.000 cents), just perfect fifth (3/2),
and equal perfect fifth ( 712 ]2[ ) with a falsity of ß4.774, +13.271, and +15.226 cents at the higher
end.84 The sléndro acute or large fifth is the identical interval found in five-tone equal temperament, and
is produced by the ratio 1: 35 )2( =1:1.515717.85
Table 14. Stage two of tuning process – perfect fifths
NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
C3 and G3
F3 and C4
C4 and G4
F4(2) and C5
C5 and G5
F5(3) and C6
704.027 � 712.644
694.718 � 704.565
708.785 � 716.381
702.864 � 712.313
698.927 � 715.226
706.550 � 715.391
707.351
699.396
713.271
707.044
704.444
709.860
4.633233
4.941669
3.980471
4.817387
9.338180
4.820301
An analysis of simultaneous sonorities based on stage two of the tuning process for the three tunings
reveal perfect fourths with a mean value of 497.436 cents, a range between 483.049 and 514.208 cents,
and a standard deviation of 7.640715. The primary perfect fourths are represented by C4 and F5 (2nd
position), and G4 and C5, and the mean values of 501.034 and 494.807 cents respectively. The range of
these fourths being between 488.672 and 505.826 indicates an inclination to represent the sléndro
grave or small fourth ( 25 ]2[ , approximately 128/97, or 480.000 cents), just perfect fourth (4/3), and
equal perfect fourth ( 512 ]2[ , approximately 1024/767, or 500.000 cents) with a falsity of +8.672,
ß9.373, and ß11.328 cents at the lower end of the scale; while the meantone perfect fourth
(4/3× 48081 , approximately 103/77, or 503.422 cents), just perfect fourth (4/3), and equal perfect fourth
( 512 ]2[ ) with a falsity of +2.380, +7.781, and +5.826 cents at the higher end.86 The sléndro grave or
small fourth is the identical interval found in the five-tone equal temperament, and is produced by the
ratio 1: 25 )2( =1:1.319508. The sléndro acute or large fifth and grave or small fourth may also be
referred to as the 5-et grave or small fourth and 5-et acute or large fifth, or alternatively as the quintal
equal subfourth and quintal equal superfifth.87
84 Daniélou, Tableau Comparatif des Intervalles Musicaux 119-25.
85 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 455.
86 Daniélou, Tableau Comparatif des Intervalles Musicaux 118-24.
87 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 455.
Theory Versus Performance Practice 47
Table 15. Stage two of tuning process – perfect fourths
NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
C3 and F3
G3 and C4
C4 and F4(2)
C4 and F4(1)
G4 and C5
C5 and F5(3)
C5 and F5(2)
G5 and C6
C6 and F6
494.973 � 503.495
483.049 � 494.186
494.592 � 505.826
499.200 � 514.208
488.672 � 498.615
486.694 � 490.450
498.648 � 501.249
490.616 � 495.220
492.435 � 507.593
498.418
490.463
501.034
508.238
494.807
487.968
500.319
493.385
502.293
4.488880
6.420814
5.795747
7.960749
5.365009
2.149959
1.450061
2.440269
8.544883
Simultaneous sonorities based on stage two of the tuning process for the three tunings reveal tempered
perfect fourths with a mean value of 501.373 cents, a range between 488.349 and 518.700 cents, and a
standard deviation of 8.744854. The primary tempered fourths are represented by F4 (2nd position) and
BÝ4, Aî4 and Dî4, BÝ4 and EÝ5, Eî4 and Aî4, and the mean values of 494.926, 509.466, 497.494, and
495.487 cents respectively. The range of these fourths being between 488.743 and 518.700 indicates
an inclination to again (as in the primary perfect fourths) represent the sléndro grave or small fourth
( 25 ]2[ ), just perfect fourth (4/3), and equal perfect fourth ( 512 ]2[ ), but this time with a falsity of +8.098,
ß9.302, and ß11.257 cents at the lower end of the scale, while the acute or large fourth (27/20, or
519.551 cents), just perfect fourth (4/3), and equal perfect fourth ( 512 ]2[ ) with a falsity of ß0.852,
+20.655, and +18.700 cents at the higher end.
Table 16. Stage two of tuning process – tempered perfect fourths
NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
F3 and BÝ3
Aî3 and Dî4
BÝ3 and EÝ4
Eî4 and Aî4
F4(2) and BÝ4
Aî4 and Dî5
BÝ4 and EÝ5
F5(2) and BÝ5
BÝ5 and EÝ6
488.349 � 495.385
508.362 � 512.257
501.771 � 512.161
491.928 � 498.275
488.743 � 501.094
495.451 � 518.700
496.279 � 499.681
495.746 � 499.267
505.825 � 518.158
490.802
509.679
505.353
495.487
494.926
509.466
497.494
498.011
511.136
3.973019
2.232698
5.898947
3.243106
6.175730
12.340378
1.897984
1.965164
6.342067
48 Theory Versus Performance Practice
An analysis of simultaneous sonorities based on stage two of the tuning process for the three tunings on
the other hand reveal neutral thirds with a mean value of 350.057 cents, a range between 338.758 and
355.954 cents, and a standard deviation of 6.396299. The primary neutral third is represented by F4
(2nd position) and Aî4, and the mean value of 353.752 cents. The range of these thirds being between
350.940 and 355.954 indicates an inclination to represent seven equal quarter-tones (approximately
60/49, or 350.00 cents), the just major third (5/4, or 386.314 cents), and equal major third ( 3 2 ,
approximately 63/50, or 400.000 cents)88 with a falsity of +0.940, ß35.374, and ß49.060 cents at the
lower end of the scale; while an extreme grave or small major third (27/22, or 354.547 cents), just major
third (5/4), and equal major third ( 3 2 ) with a falsity of +1.407, ß30.359, and ß44.046 cents at the
higher end. Seven equal quarter-tones is the identical interval found in the twenty-four-tone equally-
tempered division of the octave, and is produced by the ratio 1: 724 )2( =1:224054.
Table 17. Stage two of tuning process – neutral thirds
NUMBER NOTES (POSITION) RANGE (CENTS) AVERAGE (CENTS) STANDARD DEVIATION
ÐÑ
ÐÒ
F3 and Aî3
F4(2) and Aî4
338.758 � 353.539
350.940 � 355.954
346.361
353.752
7.399249
2.562172
Analysis of Variance The fourth table presents an ‘analysis of variance’ with regards to the three tunings, and therefore
provides a platform for the evaluation of the tuning of the seventy-two strings of the santñr for dastgàh-e
segàh (on F) utilizing average and standard deviation criterion.89 In this case standard deviation has been
derived from cents data, and frequencies normalized to A=440Hz, although in direct relation to ratio
data, which has been obtained via the mathematical equation of 1200ïcentsÏ 24 2 =ratio.
88 Daniélou, Tableau Comparatif des Intervalles Musicaux 90-102.
89 “A computational procedure frequently used to analyze the data from an experimental study employs a
statistical procedure known as the analysis of variance. For a single-factor experiment, this procedure uses a hypothesis
test concerning equality of treatment means to determine if the factor has a statistically significant effect on the response
variable. For experimental designs involving multiple factors, a test for the significance of each individual factor as well as
interaction effects caused by one or more factors acting jointly can be made. A variety of numerical measures are used to
summarize data. The proportion, or percentage, of data values in each category is the primary numerical measure for
qualitative data. The mean, median, mode, percentiles, range, variance, and standard deviation are the most commonly
used numerical measures for quantitative data. The mean, often called the average, is computed by adding all the data
values for a variable and dividing the sum by the number of data values, while standard deviation is a measure of the
variability (dispersion or spread) of any set of numerical values about their arithmetic mean. It is specifically defined as the
square root of the arithmetic mean of the squared deviations.” For a further discussion, see Encyclopaedia Britannica,
“Statistics,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM, Chicago: Encyclopaedia Britannica, 2001.
Theory Versus Performance Practice 49
Table 18. Tuning of the santñr for dastgàh-e segàh (on F) – analysis of variance 1st position
STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ
C3
F3
G3
Aî3
BÝ3
C4
Dî4
EÝ4
F4
130.368
173.862
196.163
212.369
230.848
260.408
285.069
309.099
349.261
1.993204
1.329089
1.499570
1.623461
1.764720
1.990689
1.089605
1.181457
1.334963
1194.107
492.526
701.459
838.887
983.327
1191.922
148.566
288.680
500.160
7.713846
5.999891
7.191309
1.889998
2.620925
9.337949
0.512523
8.496235
1.380069
2nd position
STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION
ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ
Eî4
F4
G4
Aî4
BÝ4
C5
Dî5
EÝ5
F5
320.470
347.810
393.173
426.662
462.912
523.251
572.642
617.019
698.585
1.224917
1.329419
1.502808
1.630811
1.769368
1.000000
1.094404
1.179203
1.335086
351.221
492.956
705.193
846.708
987.881
0.000
156.174
285.375
500.319
1.248094
4.817387
5.365009
2.660578
1.737114
0.000000
10.954514
0.273364
1.450061
3rd position
STRING SET NOTE FREQUENCY (HERTZ) RATIO (DECIMAL) AVERAGE (CENTS) STANDARD DEVIATION
ÑÙ ÒÐ ÒÑ ÒÒ ÒÓ ÒÔ ÒÕ ÒÖ Ò×
Eî5
F5
G5
AÝ5
BÝ5
C6
Dî6
EÝ6
F6
640.217
693.619
786.006
823.584
924.807
1048.400
1138.557
1242.435
1397.011
1.223537
1.325595
1.502158
1.573975
1.767425
1.001813
1.087964
1.187227
1.334933
349.270
487.968
704.444
785.295
985.979
1197.828
145.957
297.115
500.121
5.354195
2.149959
9.338180
2.027411
0.245081
6.963106
2.771441
6.147445
1.610794
50 Theory Versus Performance Practice
Utilizing the mean ratios from the ‘analysis of variance’ data, and considering solely the base pitch of
second position C5, as well as all primary derivative pitches (second position F4, G4, B"4, Aù4, Dù5, E"5,
Eù4, and third position A"5), it is possible to arrive at the ‘tuning characteristics’ personified by the three
tunings, and therefore what may be stated as being a ‘performance practice’ tuning obtained via the
intervallic analysis of Persian performer Qmars Piraglu’s instrument, which was tuned to dastgàh-e segàh by
ear, and therefore not artificially influenced by tuning devices in order to adhere to strict theoretical
schemes. Piraglu’s tuning methodology involved the obtainment of a fundamental C note calibrated
approximately to A=440Hz, which would then serve as the reference for the generation of all required
perfect fifths and fourths, as well as tempered fourths, with problem intervals such as A koron and A flat
left to the discretion of the ear and the perceived musicality of performed extracts from associated gušes
(individual pieces which make up the repertoire of a particular dastgàh).
Table 19. Dastgàh-e segàh (on F) – tuning characteristics Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE APPROXIMATE INTERVAL FREQUENCY
(HERTZ)
RATIO
(DECIMAL)
AVERAGE
(CENTS)
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ
ÐÑ
C4
D4
EÝ4
Eî4
F4
G4
Aî4
BÝ4
Bî4
C5
unison
just major tone (9th harmonic)
Pythagorean minor third, or trihemitone
neutral third
just and Pythagorean perfect fourth
just and Pythagorean perfect fifth (3rd harmonic)
grave or small major sixth
Pythagorean minor seventh
grave or small major seventh
octave
261.626
295.748
309.753
320.939
348.206
393.594
430.750
464.127
482.120
523.251
1.000000
1.130424
1.183957
1.226709
1.330933
1.504416
1.646439
1.774012
1.842785
2.000000
0.000
212.237
292.339
353.752
494.926
707.044
863.219
992.419
1058.265
1200.000
The abovementioned process consequently produces intervals approximating the just major tone (9/8),
Pythagorean minor third (32/27), neutral third (11/9), just perfect fourth (4/3), just perfect fifth (3/2), grave
or small major sixth (400/243), Pythagorean minor seventh (16/9), and grave or small major seventh
(50/27). The findings proclaim the obvious Pythagorean connection with the resulting ‘mean’
measurements for the major second (equal to 212.237 cents, with a range between and 208.821 and
214.193 cents, and a standard deviation of 2.968984); minor third (equal to 292.339 cents, with a range
between 288.626 and 299.361 cents, and a standard deviation of 6.084603); perfect fourth (equal to
494.926 cents, with a range between 488.743 and 501.094 cents, and a standard deviation of
Theory Versus Performance Practice 51
6.175730); perfect fifth (equal to 707.044 cents, with a range between 702.864 and 712.313 cents,
and a standard deviation of 4.817387); and minor seventh (equal to 992.419 cents, with a range
between 988.424 and 997.374 cents, and a standard deviation of 4.551339). The research interestingly
also suggests a distinction between the neutral orientation of microtonal inflections such as Eù4 (equal to
353.752 cents, with a range between 350.940 and 355.954 cents, and a standard deviation of
2.562172); Aù5 (equal to 863.219 cents, with a range between 846.391 and 874.654 cents, and a
standard deviation of 14.883326); and Bù4 (equal to 1058.265 cents, with a range between 1052.666
and 1064.027 cents, and a standard deviation of 5.682304); which reveal a ‘grave or small major’
intervallic orientation.
Tuning System Comparison
A tuning system comparison table incorporating data from the twenty-four equally-tempered quarter-
tone scale of Vaziri, twenty-two-note Pythagorean scale of Barkešli, and Farhat’s theory of flexible
intervals, or of the five primary intervals of performance practice; as well as the performance practice
tuning of Piraglu, reveals a close link between the latter two. Farhat and Piraglu share similarities with all
intervals (a falsity of +8.237, ß1.661, ß10.248, ß3.074, +5.044, +1.219, ß3.581, and ß7.735 cents
on each count) but for the neutral third (Eî4), which has a closer association with the equally-tempered
quarter-tone of Vaziri (a falsity of +3.752, as opposed to ß10.248). Interestingly, at 384.360 cents,
Barkešli’s neutral third represents the Pythagorean diminished fourth (8192/6561), which is 34.360,
24.360, and 30.608 cents larger than the neutral thirds of Vaziri, Farhat, and Piraglu.
Table 20. Dastgàh-e segàh (on F) – tuning system comparison
DEGREE
NUMBER
NOTE ALI NAQI VAZIRI
(CENTS)
MEHDI BARKEÊLI
(CENTS)
HORMOZ FARHAT
(CENTS)
QMARS PIRAGLU
(CENTS)
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ
ÐÑ
C4
D4
EÝ4
Eî4
F4
G4
Aî4
BÝ4
Bî4
C5
0.000
200.000
300.000
350.000
500.000
700.000
850.000
1000.000
1050.000
1200.000
0.000
203.910
294.135
384.360
498.045
701.955
882.405
996.090
1086.315
1200.000
0.000
204.000
294.000
364.000
498.000
702.000
862.000
996.000
1066.000
1200.000
0.000
212.237
292.339
353.752
494.926
707.044
863.219
992.419
1058.265
1200.000
52 Theory Versus Performance Practice
Utilizing Farhat’s theory of the division of the whole-tone and nomenclature for Persian intervals, it is then
possible to conduct an analysis of all minor second (D and E"), small neutral tone (Eù and F, Bù and C),
large neutral tone (D and Eù, G and Aù), and major second (C and D, E" and F, F and G, B" and C)
intervals (encountered within the framework established via the base and primary derivative pitches in the
performance practice tuning of Piraglu); and consequently produce a comparison table outlining range,
as well as the average for each interval in the two supportive propositions.
Table 21. Dastgàh-e segàh (on F) – Hormoz Farhat’s and Qmars Piraglu’s division of the whole-tone
DEGREE
NUMBER
INTERVAL HORMOZ FARHAT
RANGE (CENTS)
HORMOZ FARHAT
AVERAGE (CENTS)
QMARS PIRAGLU
RANGE (CENTS)
QMARS PIRAGLU
AVERAGE (CENTS)
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ
unison
minor second
small neutral tone
large neutral tone
major second
0.000
90.000
125.000 � 145.000
150.000 � 170.000
204.000
0.000
90.000
135.000
160.000
204.000
0.000
74.433 � 85.664
135.973 � 147.334
136.747 � 162.655
200.117 � 214.193
0.000
80.102
141.454
148.845
208.631
Performance Practice and Tuning
In conclusion, it must be stated that Farhat’s theory of flexible intervals or of the five primary intervals of
performance practice certainly holds true in the final analysis, and especially in view of the fact that
although general assumptions may be reached with regards to tuning practice, there is no doubt that
standard deviation data is so conflicting in some instances (the three tunings producing an overall
standard deviation average of 3.258156, with a range between 0.245081 and 10.954514; while octaves
between 0.187035 and 13.715187; perfect fifths between 3.980471 and 9.338180; perfect fourths
between 1.450061 and 7.960749; tempered fourths between 1.897984 and 12.340378; and neutral
thirds between 2.562172 and 7.399249) that the results cannot be stated as being the axiom. Ayers’s
own tuning research into the performance practice of Persian music – utilizing “thirteen improvised
recorded examples from the mode áahàrgàh and two recorded examples from the mode êhñr” –
declare the following proposition:
“The analysis of actual performance practice in Persian music raises the issue of accuracy in acoustic
performance of any microtonal music. If pitch variation in a mode of as much as 30 cents, which is a sixth
of a tone, can be acceptable, then what place do theoretical systems have in this music? If we are
composing computer music, then we don’t have to depend on performers and we can tune the
Theory Versus Performance Practice 53
computer as accurately as we wish. If we chose a precise tuning, will it sound realistic if the pitches never
vary? Will perfect music sound natural?”90
N. A. Jairazbhoy, and A. W. Stone, in their studies of intonation in present-day North Indian classical music
found that, “It would appear that within each performance the intonation does vary, and that a variation
of as much as fifteen cents, in seconds and thirds at least, could easily pass unnoticed. This is particularly
noticeable in the series of notes taken from the upper register which suggests that there may be a
tendency towards sharpening intervals in this register.” With regards to the notion of the universal
systemization of a theoretical tuning, the following conclusions are reached:
“With this divergence between musicians (the maximum divergence noticed in the interval of the third was
between Pannalal Gosh (439 cents) and Ustàd Umaro Khan (375 cents), a difference of 64 cents, or more
than a quarter-tone), it would appear that intonation is a matter of personal choice, perhaps influenced by
the teacher’s intonation, but not bound to it, and that any intonation within certain limits (perhaps within 25
or 30 cents in either side of the tempered intonation) can be acceptable. Under these circumstances it
would seem pointless to consider applying the ancient 22 æruti system, or for that matter, any system of
exact intonation to North Indian classical music.”91
In support of these findings, in Intervals, Scales and Temperaments: An Introduction to the Study of
Musical Intonation Boyle makes the appropriate observation that “sounds should be pitched according to
the dictates of the ear. For this certain notes in the scale must be free to move about a comma
(approximately 21.506 cents), which is perfectly possible on all but keyboard instruments.” It is further
relevant to note that according to foreword contributor Kenneth Van Barthold, in the piano tuning
practice of the latter part of the twentieth century, octaves began to be often “stretched for added
brilliance,” and therefore “theoretically accurate equal temperament” has never existed in piano tuning,
with stretched octaves often induced in the highest and lowest octaves.92 Theoretically accurate octaves
or the ratio 2/1 is defined in musical acoustics by a “beat-free condition between all the partials of the
upper musical tone and the even-numbered partials of the lower tone.” Stretched, as well as
compressed octaves a common occurrence in Piraglu’s tuning of the santñr.
90 Ayers, “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional
Applications,” 197-98.
91 N. A. Jairazbhoy, and A. W. Stone, “Intonation in Present-Day North Indian Classical Music,” Bulletin of the School
of Oriental and African Studies, University of London 26.1 (1963): 130-31.
92 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation ix.
54 Theory Versus Performance Practice
The Piano Tuner’s Octave and Inharmonicity The ‘piano tuner’s octave’ may be is stated as being an octave with a falsity of around three cents, or in
other words, when (for example) “the fundamental component of C5 on a good piano is set about three
cents higher than twice the fundamental component of C4.”93
Table 22. Octaves with a falsity of 3 cents
C4 (PARTIAL) FREQUENCY (HERTZ) C5 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)
1
2
3
4
5
6
7
8
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
– – – –
1
– – – –
2
– – – –
3
– – – –
4
– – – –
524.159
– – – –
1048.317
– – – –
1572.476
– – – –
2096.635
– – – –
0.908
– – – –
1.815
– – – –
2.723
– – – –
3.630
In a chapter concerned with tunings and temperaments Boyle adds to the discussion of theoretically
accurate octaves with the following comments:
“The tuner, who tunes to eliminate beats between the fundamental of the upper note and the second
partial tone of the lower note, will naturally stretch the octave physically, and the musical ear, not
surprisingly, seems to prefer them that way. This is simply a case of the ear asserting its right to judge –
within the limits of choice allowed to it – when such an instrument is mostly nearly in tune with itself
musically, and provides a good example of the danger of leaving out the ear and regarding any particular
musical interval as being defined primarily and absolutely by a mathematical ratio.”
The present discussion coincides with the natural phenomenon of inharmonicity, which is a factor
especially affecting plucked and struck strings (along with other musical sounds with a short decay), and
one that displaces the upper partials of a vibrating string with the harmonic series.94 The upwards slope
of inharmonicity on the pianoforte is encountered at around middle C in both directions. A reduction of
93 Benade, Fundamentals of Musical Acoustics 319.
94 “In a grand piano, the tension in each string is over 100 pounds, creating a total force on the frame of between
40,000 and 60,000 pounds. A large variation in tension between the lower and the higher strings could lead to warping
of the piano frame, so that, in order to apply even tension throughout, the higher strings are shorter and smaller in
diameter while the bass strings are constructed of heavy wire wound with additional thin wire. This construction makes
the wires stiff, causing the overtones to be higher in frequency than the ideal harmonics and leading to the slight
inharmonicity that plays an important part in the characteristic piano tone.” For a further discussion, see Encyclopaedia
Britannica, “Sound,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM, Chicago: Encyclopaedia Britannica, 2001.
Theory Versus Performance Practice 55
inharmonicity in a string may be brought about via an increase of physical length, and a decrease of
diameter, which is the reasoning behind the double and triple stringing (unison twos and threes, or
bichords and trichords) of the pianoforte. Physical length is therefore synonymous with flexibility, and
directly responsible for the consequential decrease in inharmonicity, and hence, a more well-defined
musical tone.95 The Tuning of Unisons A survey incorporating both musically trained and untrained participants conducted in 1959 by Roger Kirk
of the Baldwin Piano Company revealed the general preference for one or two cents deviation among
the strings of a triple-string unison of a piano. Although the upper threshold could have been lessened
for the musically trained, unisons with as much as eight cents deviation were accorded with wide
acceptance. Unisons with a falsity of two cents are described by Arthur H. Benade as “reasonably
smooth”, while ones with a falsity of eight cents as a “rather brighter sound, but is not yet the sort of
jangle one gets with a spread of 15 to 20 cents.” It should be pointed out that in general performance
practice it can be extremely difficult or sometimes even impossible to tune unisons to a “true zero-beat
condition,” and one must also consider the important fact that theoretical unisons will produce reduced
decay times.
Table 23. Unisons with a falsity of 2 cents
C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)
1
2
3
4
5
6
7
8
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
1
2
3
4
5
6
7
8
261.928
523.856
785.784
1047.712
1309.640
1571.568
1833.496
2095.424
0.302
0.605
0.907
1.210
1.512
1.814
2.117
2.419
95 “This reduction is directly proportional to the fourth power of the length and to the square of the diameter,”
explains Lloyd and Boyle. “Thus, increasing the length by a 51 th, i.e. from 1 to 5
6 , reduces the inharmonicity by
( 56 ) 4 =2.07, just over two times, i.e. more than halves its previous value. Again, decreasing the diameter by a 2
1 , i.e. from
1 to 21 , reduces its inharmonicity by ( 2
1 ) 2 = 41 , i.e. quarters its previous values.” For a further discussion, see Lloyd, and
Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 166-67.
56 Theory Versus Performance Practice
Table 24. Unisons with a falsity of 8 cents
C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)
1
2
3
4
5
6
7
8
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
1
2
3
4
5
6
7
8
262.837
525.675
788.512
1051.349
1314.187
1577.024
1840.496
2102.699
1.212
2.424
3.635
4.847
6.059
7.271
9.117
9.694
The Persian santär, essentially 72 strings organized into 18 sets of quadruple-string unisons, is an
interesting case for comparison. In order to access deviation among the strings of a quadruple-string
unison of a santär, analysis of a separate performance practice tuning by Piraglu restricted to the primary
pitches of the second position strings is conducted. The results reveal unisons with a mean falsity of
4.442 cents, which is a pronouncement that coincides with a median perspective of Benade’s
“reasonably smooth” and “rather brighter” unisons with falsities of two and eight cents respectively.96
Table 25. Unisons with a falsity of 4.442 cents
C4 (PARTIAL) FREQUENCY (HERTZ) C4 (PARTIAL) FREQUENCY (HERTZ) BEATS (HERTZ)
1
2
3
4
5
6
7
8
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
1
2
3
4
5
6
7
8
262.298
524.595
786.893
1049.191
1311.489
1573.786
1836.084
2098.382
0.672
1.344
2.016
2.689
3.361
4.033
4.705
5.377
The ‘unison’ experiment consists of the spectrum analysis of second position pitches Eî4, F4, G4, Aî4,
BÝ4, C5, Dî5, and EÝ5, which have all been tuned in relationship to the first string of each set. In other
words, in stage one of the tuning process of the santñr for dastgàh-e segàh (on F), second position F4 is
acquired by tuning the first ‘outside’ F4 string to the first ‘outside’ C5 string (generating a perfect fifth). The
tuning of F4 is then concluded with the generation of unisons by the three other strings of the set (2nd,
3rd, and 4th strings of the quadruple-string unison of the santär) in relationship to the first string. The
96 Benade, Fundamentals of Musical Acoustics 334-36.
Theory Versus Performance Practice 57
analysis of the eight pitches surveyed conclude the characterization of unisons with a mean value of
4.442 cents, a range between ß9.381 and +9.130 cents, and a standard deviation of 2.416688. In
consideration of beats, the results present a mean value of 1.201Hz, a range between ß1.280 and
+1.283Hz, and a standard deviation of 0.749103. Climate and Tuning Specific climatic conditions no doubt play a significant part in tuning, and therefore the important factor
of relative humidity must be taken into account.97 Piraglu makes a point of stating that he has found the
climatic conditions of Melbourne, Australia, in comparison to Tehran, Iran, as “unsatisfactory” for the tuning
of the santär. The questions therefore arise as to what are the desirable climatic conditions for the tuning
of an instrument, and how can official meteorological statistical data for climatic dissimilarities between
the two regions support Piraglu’s observations? The climate of Tehran, Iran (latitude: 35.41°N, longitude:
51.25°E, elevation: 1191m) features average minimum and maximum temperatures of 10.3°C and 22.8°C;
record minimum and maximum temperatures ranging between ß21.0°C and 43.0°C (averaging ß1.0°C
and 31.9°C); an average monthly total precipitation of 20.8mm; and average relative humidity ranging
between 59.1% in the morning to 53.3% in the afternoon (averaging 56.2%, with a total range of 5.8%).
Melbourne, Australia on the other hand (latitude: ß37.49°S, longitude: 144.58°E, elevation: 35m) features
average minimum and maximum temperatures of 9.9°C and 19.8°C; record minimum and maximum
temperatures ranging between ß3.0°C and 46.0°C (averaging 1.1°C and 34.8°C); an average monthly
total precipitation of 54.2mm; and average relative humidity ranging between 68.7% in the morning to
55.8% in the afternoon (averaging 62.2%, with a total range of 12.9%).98
The tuning of a pianoforte for example, can be affected severely by significant changes in relative
humidity over a one- or two-day period, with perpetual and excessive precipitation detrimental to the
intonation of the instrument. The ideal level of humidity is 42%, or a level within the range of 35% and
97 According to the Encyclopaedia Britannica humidity is “the amount of water vapour in the air. It is the most
variable characteristic of the atmosphere and constitutes a major factor in climate and weather. Atmospheric water
vapour is an important factor in weather for several reasons. It regulates air temperature by absorbing thermal radiation
both from the Sun and the Earth. Moreover, the higher the vapour content of the atmosphere, the more latent energy is
available for the generation of storms. In addition, water vapour is the ultimate source of all forms of condensation and
precipitation.” The article making the further point of importance that “care must be taken to distinguish between the
relative humidity of the air and its moisture content or density, known as absolute humidity.” For a further discussion, see
Encyclopaedia Britannica, “Humidity,” Encyclopaedia Britannica 2001, deluxe ed., CD-ROM (Chicago: Encyclopaedia
Britannica, 2001) n. pag.
98 “BBC Weather Centre: World Weather,” BBC Home Page, 6 Apr. 2006, British Broadcasting Corporation, 6 Apr.
2006, <http://212.58.224.86/weather/>
58 Theory Versus Performance Practice
Table 26. Tuning of the santñr for dastgàh-e segàh (on F) – relationship of unisons to first strings
PITCH STRING FREQUENCY
(HERTZ)
RATIO
(DECIMAL)
BEATS
(HERTZ)
FALSITY
(CENTS)
Eî4
F4
G4
Aî4
BÝ4
C5
Dî5
EÝ5
Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô Ñ Ò Ó Ô
322.490
323.260
322.860
321.740
351.100
351.600
351.830
350.360
396.020
395.740
394.630
393.880
428.820
429.530
429.410
430.750
466.700
467.430
464.670
464.760
529.180
527.870
527.300
528.530
572.060
574.090
572.470
573.470
622.200
625.490
623.430
621.190
1.000000
1.002388
1.001147
0.997674
1.000000
1.001424
1.002079
0.997892
1.000000
0.999293
0.996490
0.994596
1.000000
1.001656
1.001376
1.004501
1.000000
1.001564
0.995650
0.995843
1.000000
0.997524
0.996447
0.998772
1.000000
1.003549
1.000717
1.002465
1.000000
1.005288
1.001977
0.998377
0.000
0.770
0.370
0.750
0.000
0.500
0.730
0.740
0.000
0.280
1.390
2.140
0.000
0.710
0.590
1.930
0.000
0.730
2.030
1.940
0.000
1.310
1.880
0.650
0.000
2.030
0.410
1.410
0.000
3.290
1.230
1.010
+0.000
+4.129
+1.985
ß4.031
+0.000
+2.464
+3.596
ß3.653
+0.000
ß1.224
ß6.087
ß9.381
+0.000
+2.864
+2.380
+7.774
+0.000
+2.706
ß7.547
ß7.211
+0.000
ß4.291
ß6.161
ß2.128
+0.000
+6.133
+1.240
+4.262
+0.000
+9.130
+3.419
ß2.813
Theory Versus Performance Practice 59
55%. High levels of humidity, 60% and over, produce an expansion of the soundboard, greater string
tension, and therefore an increase in overall pitch. “Sticking keys, sluggish action, and rusting strings/tuning
pins are other consequences of continued high humidity,” points out Martha Beth Lewis. Low levels of
humidity, 34% and under, on the other hand produces a contraction of the soundboard, lesser string
tension, and therefore a decrease in overall pitch. “Other effects of low humidity include rattling (loose)
keys, slipping tuning pins, and cracks in the soundboard,” explains Lewis. Instruments constructed of a
soundboard of reduced density and deficient of the structural support provided by the iron plate of the
pianoforte are even more predisposed to the adverse effects of humidity.99
Meteorological statistical data reveal Melbourne as having no climactic period with the ideal level
of relative humidity within the range of 35% and 55%, while Tehran, in striking contrast, a period of seven
months between April and October. High levels of humidity, 60% and over, are prevalent in Melbourne
for ten months of the year, between February and November, while in Tehran, only for five months of the
year, between November and March. Neither region suffers from low levels of humidity of 34% and
under. The average monthly total precipitation of Melbourne exceeds that of Tehran by 33.4mm, over
two and a half times the level, and is therefore a significant divergent factor. All these factors substantiate
Piraglu’s claims about the “ideal” conditions for the tuning of the santñr in Tehran. Gušes of Dastgàh-e Segàh A prominent radif associated with Mñsà Marñfi (1889-1965) has been utilized in the organization of
modal material for the work, which subscribes to a collection of the following twenty-four gušes for
dastgàh-e segàh:
i. Mogadameh ii. Daràmad-e Avva
iii. Daràmad-e Dovvom iv. Daràmad-e Sevvom
v. Piš Zangñleh vi. Zangñleh
vii. Zangeh Šotor viii. Zàbol
ix. Zàbol (Qesmat-e Dovvom) x. Zangñleh
xi. Panjeh Mñye xii. Àvàz-e Mñye
xiii. Forñd-e Mñye xiv. Bagiye-e Zàbol
xv. Hesàr xvi. Hesàr (Qesmat-e Dovvom)
xvii. Nagmeh xviii. Hesàr (Qesmat-e Sevvom)
xix. Kerešmeh xx. Forñd-e Hesàr
99 Martha Beth Lewis, “Tuning Your Piano: Why Pianos Go Out of Tune,” Martha Beth Lewis’ Home Page, 1999, 8
Apr. 2006, <http://www.serve.com/marbeth/tune_piano.html>.
60 Theory Versus Performance Practice
xxi. Hozzàn xxii. Pas Hesàr
xxiii. Moarbad xxiv. Moxàlef
Àzàdeh for santñr and tape will incorporate six of the most prominent elements of the radif of the
dastgàh-e segàh belonging to the school of Marñfi. Pitch material from the main gušes, which include
guše-ye zàbol, mñye, moxàlef and maqlub, as well as the daràmad and forñd have been accessed and
categorized (according to Piraglu) to be then utilized in adherence to Western contemporary
compositional practices, and therefore resulting in a work that has no intention of representing Persian
classical music tradition.100 The selection of the six guše-ye of dastgàh-e segàh are represented within the
compositional framework as individual demarcations (dividing the work methodically), with composer
improvisation (sequentially exploring each guše) utilized to generate all the notated material of the work.
Ex. 6. Six of the most prominent elements of the radif of the dastgàh-e segàh First tetrachord i. Daràmad-e segàh
ii. Guêe-ye zabçl
iii. Guêe-ye mñye
Second tetrachord iv. Guêe-ye moxàlef
100 Piraglu, “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties,” 41-43.
Theory Versus Performance Practice 61
v. Guêe-ye maqlub
vi. Segàh forñd
A structural scheme based on ‘golden mean’ or ‘golden section’ proportions has been incorporated in
the linear plan. The aesthetic notion of these proportions being a technique directly borrowed from art
and architecture, with its conceptual basis stating that “if the proportion of ‘ab’ to ‘bc’ is the same as the
proportion of ‘bc’ to the whole line, then ‘ac’ is segmented according to the golden mean.” The ratio
represented by this ideology is approximately 1:1.618 (or the relationship acbc
bcab � ), and manifests itself
in the work at guêe-ye moxàlef with its introduction of melodic material based on the second tetrachord
of dastgàh-e segàh (720 seconds Ï .618 = 444.96 seconds, or approximately 7:25).101
Table 27. Structural scheme
NUMBER EVENT CLOCK COUNTER
(SECONDS)
DURATION GOLDEN
MEAN
1
2
3
4
5
6
7
8
9
10
11
12
13
14
daràmad-e segàh (first tetrachord)
taknavàzi-e santñr
guêe-ye zabçl
taknavàzi-e santñr
guêe-ye mñye
taknavàzi-e santñr
guêe-ye moxàlef (second tetrachord)
taknavàzi-e santñr
guêe-ye maqlub
taknavàzi-e santñr
forñd-e segàh
taknavàzi-e santñr
fade out
end
0:00
2:50
4:35
5:40
6:20
7:00
7:25
9:10
10:15
10:55
11:20
11:45
12:00
13:00
0
170
275
340
380
420
445
550
615
655
680
705
720
780
2:50
1:45
1:05
0:40
0:40
0:25
1:45
1:05
0:40
0:25
0:25
0:15
1:00
0:00
– – – –
3 (3-1)
2 (7-1)
5 (5-3)
4 (7-3)
6 (7-5)
1 (13-1)
8 (9-7)
7 (13-7)
10 (11-9)
9 (13-9)
11 (13-11)
– – – –
– – – –
101 Stefan Kostka, Materials and Techniques of Twentieth-Century Music, 2nd ed. (Upper Saddle River, NJ: Prentice-
Hall, 1999) 150-51.
62 Theory Versus Performance Practice
Sampling of the Santñr and Vocals The following stage of the compositional process involves the assembly of the tape element of the work,
which begins with the transferral of the complete set of santñr samples (the twenty-seven individual
samples collected during the first recording session with Piraglu) into the memory of an Akai S3000XL
Midi Stereo Digital Sampler. This will provide a platform for the retuning (to a hundredth of a cent) of
each individual sample in the edit sample ‘pitch offset’ parameter window of the Akai S3000XL (to firstly
adhere to equal temperament), in order to be then readjusted (in cent increments) in the edit program
‘tune’ parameter window (enabling the capture of the frequency ratios of Barkešli’s twenty-two-note
division of the octave). The frequencies (in hertz), obtained via the spectrum analyzing module of
Adobe Audition 2.0, once converted into cents will provide all the necessary data required to offset
pitch. The alternative tuning system will service the accompanying sampled santñr – one of the two
principal tape elements of the work – and the ‘call and response’ component that will effectuate the
‘theoretical’ and ‘performance practice’ tuning comparison proposed by Àzàdeh for santñr and tape.
Performances of the ‘sampled santñr’ notated phrases of the work have been recorded on a midi
sequencer (Roland MC-500 Mark II Micro Composer), utilizing the Akai S3000XL as a sound source, with
the product finally digitally transferred to Sony Sound Forge 8.0 via optical cable (routed through the
Fostex D-160 Digital Multitrack Recorder). A modified ‘white noise’ preset of the Waves X-Noise 5.2 DX
plug-in (“an audio plug-in that intelligently learns from a section of noise, and then applies a broadband
noise reduction to eliminate background noise from any source”) is utilized to remove unwanted white
noise (threshold: ß20dB, reduction: 70%, [dynamics] attack: 30ms, release: 220ms; [high shelf]
frequency: 1415Hz, gain: +0.0dB).
Table 28. Mehdi Barkešli’s twenty-two-note division of the octave tuning matrix ‘key of F’ (Program 01)
NOTE C – – Dî E" Eî F – – G A" Aî B" – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 ß20 ß06 ß16 ß02 +00 +02 ß08 ß18 ß04 +00
Stage two of the assembly requires the recording of the recitation of poetry by Piraglu (an original poem
entitled Àzàdeh), as well as the individual syllables constituting the Farsi (Modern Persian) vocalizations of
Àzàdeh,102 which include À, zà, and deh – the three syllables sung at two distinct pitches (348.834Hz
102 “The official language of Iran is Iranian or Modern Persian (Iranian/Indo-European), which is written in Arabic
script.” For a further discussion, see Akira Nakanishi, Writing Systems of the World: Alphabets, Syllabaries, Pictograms
(Rutland, VT: Charles E. Tuttle Co., 1980) 30.
Theory Versus Performance Practice 63
and 174.417Hz, or F4 and F3). Àzàdeh includes the following four lines of text, which are presented
below in Farsi (Arabic and Roman script), together with the English translation:
Kas nadànad darde bi darmàne mà,
Nobody knows the pain inside me,
Jore Leili dar dele por êñre mà.
The pain that emanates, being away from my beloved Àzàdeh.
Hazar kardim ze haráe xalgo donyà
Because my soul is with her, I am oblivious to everybody and everything
Kajàst Àzàdeh in mahpeykare mà?
Where is my beloved Àzàdeh, who reflects the beauty of moonlight?
Àzàdeh, Qmars Piraglu
Àzàdeh, Qmars Piraglu
All the samples are recorded on the Fostex D-160 Digital Multitrack Recorder, and then processed within
Sony Sound Forge 8.0 utilizing various audio plug-ins: Waves L3 Multimaximizer 5.2 (“a multi-band peak
limiting audio plug-in”)103 is utilized for general compression (left and right input: +0.0dB, threshold:
ß2.3dB, out ceiling: ß0.3dB, release: 10.00ms); dB-L Audioware Mastering Limiter 1.05 (“an intelligent
loudness maximizer and brickwall limiter audio plug-in with dithering”),104 for general limiting (gain: 3.0dB,
release: 50ms, output: ß0.3dB, attack: hard, dither: off); while Sony Noise Reduction 2.0 (“a collection of
four professional-level sound restoration audio plug-ins: vinyl restoration, noise reduction, clipped peak
103 Waves: The World’s Leading Developer of Audio Signal Processing Software, 2006, Waves Incorporated, 15
Dec. 2006, <http://www.waves.com/>.
104 dB Audioware: Professional Audio Software, 2006, dB Audioware Limited, 15 Dec. 2006, <http://www.db-
audioware.com/>.
64 Theory Versus Performance Practice
restoration, and click and crackle removal”),105 to remove unwanted tape hiss (reduction type: mode 3,
reduce noise by: 30.0dB, noise bias: +0.0dB, attack speed: 90ms, release speed: 50ms, windowing FFT
size: 1024, windowing overlap: 67%).
The two sets of three syllables are then arranged within the digital domain as a stereo configuration
sounding a low F3-set panned left, left-right, and left-right-left; followed by a high F4-set panned right,
right-left, and right-left-right – the time points selected in accordance to the ‘golden mean’ structural
scheme and therefore acting as structural markers. An Antares Auto-tune 4.31 DX plug-in (“an audio
plug-in that corrects intonation problems in vocals or solo instruments”)106 is utilized to tune the samples
(input type: low male voice, key: F, scale: chromatic, remove: F! and E), while a Spin Audio 3D Delays
1.1 DX plug-in (“a multi-tap dimensional space delay audio plug-in that provides six independent stereo
delay lines which can be freely positioned in 3D sound field”),107 to generate simultaneous quarter-note
and eighth-note triplet delays (delay 1 [mode: ms, tempo: 120bpm, time: 375ms, feedback: 55%, filter:
off, taps: ß6.0dB]; delay 2 [mode: ms, tempo: 120bpm, time: 125ms, feedback: 65%, filter: off, taps:
ß6.0dB]; I/O [in: ß3.0dB, dry: +0.0dB, wet: ß4.0dB]).
The singular 174.417Hz À component is further developed as an F pedal point or drone in the
form of a 500ms and 250ms rhythmic pulse (at Ê=120 equal to simultaneous crotchets and quavers). A
Waves Super Tap 5.0 DX plug-in (“an audio plug-in with six taps with up to six seconds of mono or true
stereo delay, as well as independent Q10-style filtering, rotation [stereo panning], gain, and precision
time control”)108 is utilized to generate a thirteen-minute rhythmic pulse ([tempo: 120bmp, ms: 500,
modulation: off; gain: ß3.6dB]; delay 1 [gain: +0.0dB, rotation: ß45°, delay: 250ms, eq: off]; delay 2
[gain +0.0dB, rotation: ß45°, delay: 500ms, eq: off]; feedback [gain: 99%, rotation: 0°, delay: 500ms]),
while a PSP Audioware Nitro 1.0.2 DX plug-in (“a multimode filter audio plug-in with many filter types
derived from analog prototypes in addition to other useful processing blocks such as phaser, bit-
crusher/downsampler, waveshaper, and interpolated delay blocks”),109 to generate ‘state variable low-
pass’ filter sweeps throughout the sample (op 1 [frequency: 553Hz, resonance: 29.40%, level: +6.0dB];
op 2 [frequency: 790Hz, resonance: 68.97%, level: +6.0dB]; op 3 [attenuation: +0.0dB, level:
maximum]; op 4 [attenuation: +0.0dB, level: maximum]; I/O [in: +0.0dB, mix: 25.00%, out: +1.0dB]).
105 Sony Media Software: Home for Vegas, Sound Forge and Acid, 2006, Sony Corporation of America, 15 Dec.
2006, <http://www.sonymediasoftware.com/>.
106 Antares Audio Technologies, 2006, Antares Audio Technologies, 15 Dec. 2006,
<http://www.antarestech.com/>.
107 Spin Audio Software, 2006, Spin Audio Software, 15 Dec. 2006, <http://www.spinaudio.com/>.
108 Waves: The World’s Leading Developer of Audio Signal Processing Software, 2006, Waves Incorporated, 15
Dec. 2006, <http://www.waves.com/>.
109 PSP Audioware: Audio Processors and Effects Plug-ins, 2006, PSP Audioware, 15 Dec. 2006,
<http://www.pspaudioware.com/>.
Theory Versus Performance Practice 65
A secondary thirteen-minute rhythmic pulse is then created from the sequential layering of a one-
second sample (in order to duplicate sonic parameters generated by the Waves Super Tap 5.0 DX plug-
in) of the primary rhythmic pulse, which is then subjected to thirteen digital fade-ins and therefore every
‘golden mean’ point of the structural scheme (a 500ms fade-out additionally executed to smooth out the
junctions). The abovementioned PSP Audioware Nitro 1.0.2 DX plug-in setting with a slight modification
of the effect level (mix: 75.00%) is also utilized to generate ‘state variable low-pass’ filter sweeps
throughout the sample. The adoption of dissimilar effect levels (25.00% in the primary and 75.00% in
secondary rhythmic pulses), and the fact that the superimposition of the two rhythmic pulses essentially
generates a series of gradual digital cross-fades, adds up to an unbroken rhythmic pulse of continually
transforming timbral colour.
The sequential layering of a one-second sample of the secondary thirteen-minute rhythmic pulse is
then employed to create a tertiary thirteen-minute rhythmic pulse. The PSP Audioware Nitro 1.0.2 DX
plug-in is utilized to generate ‘stereo width balance’ spatial nuances (stereo field modifications) in the
sample (op 1 [width: 100.00%, balance: left/100%, level: maximum]; op 2 [width: 100.00%, balance:
right/100%, level: maximum]; op 3 [feedback: 50.00%, depth: 94.04%, level: maximum]; op 4 [pan left:
left/26.6%, pan right: right/24.6%, level: +0.0dB]; I/O [in: +2.0dB, mix: 100.00%, out: ß8.0dB]). The
resulting .wav file (similarly to the secondary rhythmic pulse) is then also subjected to thirteen digital fade-
ins. The next step involves a DSound Stomp’n Fx DN-SG1 Noise Gate DX plug-in (“a dynamics effects
processor audio plug-in whose function is to remove unwanted audio material below a certain
threshold”),110 which is utilized to suppress the sustaining quality of the sample (tone level: ß32dB,
attack: 170ms, release: 1880ms), while the Spin Audio 3D Delays 1.1 DX plug-in, to generate
simultaneous ‘high cut’ filtered multiple-time delays (delay 1 [mode: ms, tempo: 120bpm, time: 1250ms,
feedback: 3%]; delay 2 [mode: ms, tempo: 120bpm, time: 1000ms, feedback: 3%]; filter 1 and 2
[routing: out, type: low pass, frequency: 400Hz, gain: +0.0dB, q/filter slope steepness: 1.0]; delay 3
[mode: ms, tempo: 120bpm, time: 1500ms, feedback: 3%]; delay 4 [mode: ms, tempo: 120bpm, time:
1750ms, feedback: 3%]; filter 3 and 4 [routing: out, type: low pass, frequency: 200Hz, gain: +0.0dB,
q/filter slope steepness: 1.0]; taps: ß1.9dB; I/O [in: ß1.0dB, dry: +0.0dB, wet: +0.0dB]).
The final recordings are further subjected to digital sound processing modifications (namely
equalization and reverberation) at the final stages of mixing and mastering, where all the tracks (including
the principal solo santñr recording [recorded separately on the Fostex D-160 Digital Multitrack Recorder],
as well as the accompanying ‘sampled’ santñr, primary pulse, secondary pulse, tertiary pulse, vocal
samples, and recitation of poetry, which constitute the tape element of the work) are conclusively
assembled. A separate mix omitting the principal solo santñr is also produced in order to reproduce the
work in a live performance context.
110 DSound, 2005, DSound, 15 Dec. 2006, <http://www.dsound1.com/>.
66 Theory Versus Performance Practice
2. The Equally-Tempered Archetype: Exposiciones for Sampled Microtonal Schoenhut Toy Piano
Equal Temperaments
According to American composer and theorist Harry Partch,111 there are two distinct classes of equal
temperaments, with the first including “those which divide the already equal tone into further equal
parts,” while the second including “those which compress one of the Pythagorean cycles into the 2/1 to
obtain such divisions as nineteen equal degrees and fifty-three equal degrees.”112 The results of the
former being third-tones (as in eighteen-tone equal temperament), quarter-tones (as in twenty-four-tone
equal temperament), fifth-tones (as in thirty-tone equal temperament), sixth-tones (as in thirty-six-tone
equal temperament), eighth-tones (as in forty-eight-tone equal temperament), twelfth-tones (as in
seventy-two-tone equal temperament), and sixteenth-tones (as in ninety-six-tone equal temperament);
while of the latter, namely nineteen-tone, thirty-one-tone, forty-three-tone, and fifty-three-tone equal
temperaments. The superimposition of a second twelve-tone equally-tempered division of the octave a
sixth of a tone, or 33.333 cents higher, as in thirty-six-tone equal temperament, is termed by Partch as a
polypythagoreanism, due to its correlation to the Pythagorean concept of the juxtaposition of perfect
fifths.113 The general accepted abbreviation for equal temperament, or tuning system of “logarithmically
equal intervals” is ‘ET’, although some theorists subscribe to the lesser ambiguity of ‘EDO’, or equally-
divided octave, which better defines the probability, or improbability of the ‘extended’ perimeter. Dan
Streams makes the following observations: “Usually, but not always, equal temperaments assume octave-
equivalence, of which the usual 12-edo is the most obvious example. For many theorists the preferred
abbreviation or these types of temperaments is EDO, for which some other theorists substitute ED2; both
of these specify that it is the 2:1 ratio which is to be equally divided.”114
111 “Visionary composer, theorist, and creator of musical instruments, Harry Partch (1901-1974) was a leading figure
in the development of an indigenously American contemporary music. A pioneer in his explorations of new instruments
and new tunings, Partch created multimedia theatre works that combine sight and sound in a compelling synthesis. He is
acknowledged as a major inspiration to postwar experimental composers as diverse as Gyõrgy Ligeti, Lou Harrison, Philip
Glass, and Laurie Anderson, and his book Genesis of a Music, first published in 1949, is now considered a classic.” For a
further discussion, see Bob Gilmore, Harry Partch: A Biography (New Haven, CT: Yale U. Press, 1998) n. pag.
112 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 425. 113 Read, 20th-Century Microtonal Notation 13-145.
114 Dan Stearns, “Some Thoughts on an Alternative Definition of Equal Temperament,” Kronosonic, 2006, The
International Society for Creative Guitar and String Music, 1 Aug. 2006,
<http://kronoson.station185.com/kronosonic/forum/index>.
Studies of Microtonal Equal Temperaments Mexican composer Julián Carrillo Trujillo (1875-1965) utilized quarter-tones, eighth-tones, and sixteenth-
tones in his compositions; publishing his theories under the heading of Sonido Trece or the ‘Thirteenth
Sound’ in a series of writings published in Mexico and the United States between 1927 and 1957 – the
number thirteen merely representing a system beyond that of the traditional twelve tones, and essentially
ninety-six-tone equal temperament ( 96 2 ). The pitch resources of such a system provide for 780 distinct
intervals as opposed to the 12 of twelve-tone equal temperament. “The reason for stopping with this
intervallic division was not dictated by theoretical considerations, but by a purely physical one: Carrillo
felt that the human ear could not distinguish clearly beyond 1/16th of a tone,” explains Gerald R.
Benjamin. Carrillo also introduced new or adapted instruments to produce quarter-, eighth-, and
sixteenth-tones, and devised a numerical notation system based on “relative number notation.” The
quarter-tone or “quarters of tone” system is represented thus: 0, 1/4, 2/4, 3/4, 5/4, 6/4, 7/4, 8/4, 9/4,
10/4, 11/4, 12/4, 13/4, 14/4, 15/4, 16/4, 17/4, 18/4, 19/4, 20/4, 21/4, 22/4, 23/4, and 0.115
Italian composer Ferruccio Busoni (1866-1957) on the other hand, subscribed to the theorem
pronouncing the third-tone as a product of the division of the octave into six equal tones, and the further
threefold subdivision of each equal tone, which results in an eighteen-tone scale and therefore eighteen-
tone equal temperament ( 18 2 ). In his 1911 publication of Sketch of a New Aesthetic of Music Busoni
states, “Tripartite tones are wholly independent intervals with a pronounced character, and not to be
confounded with ill-tuned semitones. They form a refinement in chromatics based, as at present
appears, on the whole-tone scale.” Other tripartite systems investigated by Busoni include sixth-tones, or
the threefold subdivision of each equal semitone, which results in a thirty-six-tone scale and therefore
thirty-six-tone equal temperament ( 36 2 ). Busoni devised a notation system for sixth-tones consisting of a
six-line staff designating “open noteheads” to the spaces and “solid noteheads” to the lines, with each
representing the tripartite divisions of diatonic and chromatic semitones respectively.116
If one were to entertain the notion that the “most important characteristic sought in a temperament
is its ability, in principle, to support both traditional tonality and atonality with intervals of greater purity
than are found in twelve-tone equal temperament,” Ramon Fuller’s A Study of Microtonal Equal
Temperaments presents a set of sound guidelines. According to Fuller, octaves classify as the most
important interval, because of their ability “to maintain audibly stable musical structures”, and therefore
‘pure’ or ‘just’ 2/1 ratios are of the outmost importance. Second in line come fifths, or 3/2s (the third
harmonic), “because of their role in musical structure and the ear’s sensitivity to mistuning”, which should
be represented within a temperament with a size ranging from 699.500 cents to 704.400 cents, or a just 115 Gerald R. Benjamin, “Julian Carrillo and ‘Sonido Trece’ (Dedicated to the Memory of Nabor Carrillo),” Anuario 3
(1967): 33-68. 116 Read, 20th-Century Microtonal Notation 95-96.
68 The Equally-Tempered Archetype
perfect fifth (3/2) with a falsity of around two cents; the ideology representing a tempered cycle of fifths
as opposed to a “mathematically exact” one, and therefore providing a link between the perfect fifth and
the octave. Major and minor thirds (resembling the just major third, with a ratio of 5/4, or 386.314; and
just minor third, with a ratio of 6/5, or 315.641 cents) become the next consideration. It is interesting to
note that although twelve-tone equal temperament qualifies on the first two counts (representing the just
perfect octave [2/1] correctly, while the just perfect fifth [3/2] with a falsity of ß1.955 cents), the
common temperament features major and minor thirds with a falsity of +13.686 and ß15.641 cents
respectively; the mean of the falsity equal to the value of 9.776 cents, and the maximum amount of error,
17.596 cents (represented by the equal minor seventh).
Table 29. Deviation of basic equally-tempered intervals from just intonation
COMPARATIVE TABLE JUST INTONATION EQUAL TEMPERAMENT
INTERVAL RATIO CENTS RATIO CENTS FALSITY (CENTS)
unison
diatonic semitone
major tone
minor third
major third
perfect fourth
tritone
perfect fifth
minor sixth
major sixth
minor seventh
major seventh
octave
1/1
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1
0.000
111.731
203.910
315.641
386.314
498.045
590.224
701.955
813.686
884.359
1017.596
1088.269
1200.000
1.000000
1.059463
1.122462
1.189207
1.259921
1.334840
1.414214
1.498307
1.587401
1.681793
1.781797
1.887749
2.000000
0.000
100.000
200.000
300.000
400.000
500.000
600.000
700.000
800.000
900.000
1000.000
1100.000
1200.000
+0.000
ß11.731
ß3.910
ß15.641
+13.686
+1.955
+9.776
ß1.955
ß13.686
+15.641
ß17.596
+11.731
+0.000
Fuller also recommends that major thirds “should be no more than seven cents sharp in a good
microtonal temperament,” adding that “to accommodate the ear’s bias, we will require a major third to
be no more than four cents flat, for a total acceptable range from about 382.000 cents and 393.000
cents;” while allocating the rule for minor thirds that they be “no more than eight cents flat or four cents
sharp, for a total acceptable range from 308.000 cents to 320.000 cents.” In conclusion, Fuller’s eight
best temperaments (selected from all equal temperaments from 1 to 144) in ascending order include the
following: fifty-three-tone, sixty-five-tone, eighty-seven-tone, ninety-nine-tone, one hundred and six-tone,
one hundred and eighteen-tone, one hundred and thirty-tone, and one hundred and forty-tone equal
temperaments (53-et, 65-et, 87-et, 99-et, 106-et, 118-et, 130-et, and 140-et).
The Equally-Tempered Archetype 69
Table 30. Fuller’s eight best equal temperaments
TEMPERAMENT MINOR THIRD
PC NUMBER
MINOR THIRD
(CENTS)
MAJOR THIRD
PC NUMBER
MAJOR THIRD
(CENTS)
PERFECT FIFTH
PC NUMBER
PERFECT FIFTH
(CENTS)
53-et
65-et
87-et
99-et
106-et
118-et
130-et
140-et
14
17
23
26
28
31
34
37
316.981
313.846
317.241
315.152
316.981
315.254
313.846
317.143
17
21
28
32
34
38
42
45
384.906
387.692
386.207
387.879
384.906
386.441
387.692
385.714
31
38
51
58
62
69
76
82
701.887
701.538
703.448
703.030
701.887
701.695
701.538
702.857
In The Structure of Recognizable Diatonic Tunings Easley Blackwood117 presents the concept of equal
temperaments and ‘recognizable diatonic tunings’ – “those in which the perfect fifths ultimately form a
closed circle” – proposing that “in order for any array of notes to contain recognizable diatonic scales, it
is both necessary and sufficient that the array should contain seven adjacent intervals that are the same
size, and are perfect fifths within the range of recognizability.” The theorem in effect pronouncing that for
a tuning to be capable of generating recognizable diatonic scales, its intervallic boundaries must adhere
to the formula: ava 53
74 << , or 685.714 << v 720.000. The criteria for a “perfect fifth within the range
of recognizability” therefore stipulates that the interval be not smaller than four sevenths of an octave, and
not larger than three fifths of an octave, hence within the range of +18.045 cents and ß16.241 cents
from a just perfect fifth (3/2). Tunings accommodating Blackwood’s ideals include 12, 17, 19, 22, 24, 26,
27, 29, 31, 33, 34, and 36 or more equal divisions of the octave.118 Dutch musicologist Rudolf Rasch
subscribes to the modification of the formula: ava 53
74 �� (‘less than’ replaced by ‘less than or equal
to’), which has the capacity to indicate “which equal temperaments have no recognizable v, as a by-
product of showing which have more than one.”119 117 American composer-theorist Easley Blackwood, a significant proponent of the equally-tempered modus
operandi, subscribes to the notion that “extended microtonal systems based on equal-temperament tunings are as valid
acoustically and musically, and as technically challenging as fractional divisions of the octave adhering to just or mean-tone
intonational principles.” For a further discussion, see Read, 20th-Century Microtonal Notation 5.
118 Easley Blackwood, The Structure of Recognizable Diatonic Tunings (Princeton, NJ: Princeton U. Press, 1985) 221-
54.
119 Paul Rapoport, “The Structural Relationships of Fifths and Thirds in Equal Temperaments,” Journal of Music Theory
37.2 (Autumn, 1993): 359.
70 The Equally-Tempered Archetype
Nicolas Mercator’s Fifty-Three-Tone Equally-Tempered Division of the Octave According to Fuller, fifty-three-tone equal temperament ( 53 2 ) is “the most nearly ideal of all
temperaments for working with pure intervals and just scales,” noting that due to the fact that it has a
prime structure, there are no “embedded subtemperaments,” and “any one of the intervals of T53 can
be arranged in a cycle that will generate the complete set of T53 pitch-classes.”120 Nicolas Mercator
(1620-87) is acknowledged for discovering “that if the octave is divided into 53 equal intervals, 31 of
them give a very perfect fifth and 17 a very good major third.”121 Daniel James Wolf offers the following
criticism of fifty-three-tone equal temperament:
“While 53tet does provide excellent approximations of 5-limit intervals, it is problematic in at least two
ways. For one, the temperament is awkward, if not unsuitable, for the performance of existing repertoire.
If Western classical triadic tonality can be heard as premised upon having the best major third present in
the tuning system equivalent to the (octave-equivalent) sum of four consecutive perfect fifths – a properly
equally present in each of the major tuning or temperament environments used in common practice
counterpoint and harmony (meantone, well-temperaments, 12tet) – then this premise is unfulfilled by
53tet. In 53tet, which might be thought of a scale of 53 modestly tempered syntonic commas, the best
major third remains one scale step distant from the sum of four perfect fifths; as a consequence,
realizations of existing repertoire may well tend to ‘drift’ in pitch from an initial tonic by the approximate
comma interval of 1/53 octave. But, perhaps more critically in this speculative context, given the
extravagant resources required to implement 53tet in notation or instruments, it does not offer significantly
better and consistent representations of intervals beyond the 5-limit.”122
A contrasting ideology of the equally-tempered paradigm is presented by Dirk de Klerk in his study of
equal temperaments (subscribing to thirty-four and forty-six equal divisions of the octave), which
presents the following analytical commentary:
“If we ignore the seventh harmonic, the tunings with 34 and 46 divisions in the octave give results that are
feasible as in the Mercator temperament (53-et). They are also a good deal more perfect than our 12-
semitone system and also better than those of Wesley Woolhouse (19-et), Christian Huygens (31-et), and
Von Janko (41-et). There is no point preferring Mercator’s system to 34 and 46 and Janko’s to 34. If we
also include the seventh harmonic it appears that it is as feasible in the temperaments of Janko and
Mercator and in 46 as it is in that of Huygens (and Fokker), whereas in the cases of 46 and Mercator the
fifth and the minor third are better.”123
120 Ramon Fuller, “A Study of Microtonal Equal Temperaments,” Journal of Music Theory 35.1/2 (Spring-Autumn,
1991): 212-20. 121 Dirk de Klerk, “Equal Temperament,” Acta Musicologica 51.1 (Jan.-Jun., 1979): 140. 122 Daniel James Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 22.1-2 (2003): 4. 123 Klerk, “Equal Temperament,” 150.
The Equally-Tempered Archetype 71
Tabl
e 31
. N
icol
as M
erca
tor’s
fifty
-thre
e-to
ne e
qua
lly-te
mp
ered
div
ision
of t
he o
ctav
e Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
iddl
e C)
=26
1.62
5565
4Hz
DE
GRE
E N
UMBE
R N
OTE
IN
TERV
AL
FACT
OR
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
C C¢
C£
CÚ
D¹
D¸
DÝ
D³
D²
D D¢
D£
DÚ
E¸
EÝ
E³
E² E E¢
E£
F³
F² F F¢
F£
FÚ
uniso
n 53
-et s
ynto
nic
com
ma
53-e
t gre
at d
iesis
53
-et g
rave
or s
mal
l jus
t chr
omat
ic s
emito
ne, o
r min
or h
alf-t
one
53-e
t Pyt
hago
rean
lim
ma
53-e
t jus
t dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
53-e
t gre
at li
mm
a, a
cute
or l
arge
hal
f-ton
e 53
-et g
rave
or s
mal
l ton
e 53
-et j
ust m
inor
tone
53
-et j
ust m
ajor
tone
53
-et a
cute
or l
arge
tone
53
-et s
uper
maj
or s
econ
d
53-e
t aug
men
ted
seco
nd
53-e
t Pyt
hago
rean
min
or th
ird, o
r trih
emito
ne
53-e
t jus
t min
or th
ird
53-e
t neu
tral t
hird
53
-et g
rave
or s
mal
l maj
or th
ird
53-e
t jus
t maj
or th
ird
53-e
t Pyt
hago
rean
maj
or th
ird, o
r dito
ne
53-e
t acu
te o
r lar
ge m
ajor
third
53
-et s
ubfo
urth
53
-et g
rave
or s
mal
l fou
rth
53-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
53-e
t acu
te o
r lar
ge fo
urth
53
-et s
uper
four
th
53-e
t gra
ve o
r sm
all a
ugm
ente
d fo
urth
1/1
532
253
)2
(
353
)2
(
453
)2
(
553
)2
(
653
)2
(
753
)2
(
853
)2
(
953
)2
(
1053
)2
(
1153
)2
(
1253
)2
(
1353
)2
(
1453
)2
(
1553
)2
(
1653
)2
(
1753
)2
(
1853
)2
(
1953
)2
(
2053
)2
(
2153
)2
(
2253
)2
(
2353
)2
(
2453
)2
(
2553
)2
(
1.00
0000
1.
0131
64
1.02
6502
1.
0400
15
1.05
3705
1.
0675
77
1.08
1630
1.
0958
69
1.11
0295
1.
1249
11
1.13
9720
1.
1547
23
1.16
9924
1.
1853
25
1.20
0929
1.
2167
38
1.23
2756
1.
2489
84
1.26
5426
1.
2820
84
1.29
8961
1.
3160
61
1.33
3386
1.
3509
39
1.36
8723
1.
3867
41
261.
626
265.
070
268.
559
272.
094
275.
676
279.
305
282.
897
286.
707
290.
482
294.
306
298.
180
302.
105
306.
082
310.
111
314.
194
318.
330
322.
520
326.
766
331.
068
335.
426
339.
841
344.
315
348.
848
353.
440
358.
093
362.
807
0.00
0 22
.642
45
.283
67
.925
90
.566
11
3.20
8 13
5.84
9 15
8.49
1 18
1.13
2 20
3.77
4 22
6.41
5 24
9.05
7 27
1.69
8 29
4.34
0 31
6.98
1 33
9.62
3 36
2.26
4 38
4.90
6 40
7.54
7 43
0.18
9 45
2.83
0 47
5.47
2 49
8.11
3 52
0.75
5 54
3.39
6 56
6.03
8
+00
+
23
+45
ß
32
ß09
+
13
+36
ß
42
ß19
+
04
+26
+
49
ß28
ß
06
+17
+
40
ß38
ß
15
+08
+
30
ß47
ß
25
ß02
+
21
+43
ß
34
72 The Equally-Tempered Archetype
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L FA
CTO
R RA
TIO
(D
ECIM
AL)
FR
EQUE
NCY
(H
ERTZ
) CE
NTS
TU
NIN
G
Ò×
ÒØ
ÒÙ
ÓÐ
ÓÑ
ÓÒ
ÓÓ
ÓÔ
ÓÕ
ÓÖ
Ó×
ÓØ
ÓÙ
ÔÐ
ÔÑ
ÔÒ
ÔÓ
ÔÔ
ÔÕ
ÔÖ
Ô×
ÔØ
ÔÙ
ÕÐ
ÕÑ
ÕÒ
ÕÓ ÐÑ
F¤ F¥
GÝ
G³
G²
G G¢
G£
GÚ
A¸
AÝ
A³
A²
A A¢
A£
AÚ
B¹
B¸
BÝ
B³
B² B B¢
B£
C³
C² C
53-e
t jus
t trit
one,
or a
ugm
ente
d fo
urth
53
-et a
cute
or l
arge
trito
ne, o
r aug
men
ted
four
th
53-e
t acu
te o
r lar
ge d
imin
ished
fifth
53
-et s
ubfif
th
53-e
t gra
ve o
r sm
all f
ifth
53-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
53
-et a
cute
or l
arge
fifth
53
-et s
uper
fifth
53
-et a
ugm
ente
d fi
fth
53-e
t Pyt
hago
rean
min
or s
ixth
53
-et j
ust m
inor
six
th
53-e
t neu
tral s
ixth
53
-et g
rave
or s
mal
l maj
or s
ixth
53
-et j
ust m
ajor
six
th
53-e
t Pyt
hago
rean
maj
or s
ixth
53
-et a
cute
or l
arge
maj
or s
ixth
53
-et a
ugm
ente
d s
ixth
53
-et g
rave
or s
mal
l min
or s
even
th
53-e
t Pyt
hago
rean
min
or s
even
th
53-e
t acu
te o
r lar
ge m
inor
sev
enth
53
-et n
eutra
l sev
enth
53
-et g
rave
or s
mal
l maj
or s
even
th
53-e
t jus
t dia
toni
c m
ajor
sev
enth
53
-et P
ytha
gore
an m
ajor
sev
enth
53
-et a
cute
or l
arge
maj
or s
even
th
53-e
t sub
octa
ve
53-e
t gra
ve o
r sm
all o
ctav
e
octa
ve
2653
)2
(
2753
)2
(
2853
)2
(
2953
)2
(
3053
)2
(
3153
)2
(
3253
)2
(
3353
)2
(
3453
)2
(
3553
)2
(
3653
)2
(
3753
)2
(
3853
)2
(
3953
)2
(
4053
)2
(
4153
)2
(
4253
)2
(
4353
)2
(
4453
)2
(
4553
)2
(
4653
)2
(
4753
)2
(
4853
)2
(
4953
)2
(
5053
)2
(
5153
)2
(
5253
)2
(
2/
1
1.40
4996
1.
4234
92
1.44
2231
1.
4612
16
1.48
0452
1.
4999
41
1.51
9686
1.
5396
92
1.55
9960
1.
5804
96
1.60
1302
1.
6223
82
1.64
3739
1.
6653
77
1.68
7301
1.
7095
12
1.73
2017
1.
7548
17
1.77
7918
1.
8013
23
1.82
5036
1.
8490
61
1.87
3402
1.
8980
64
1.92
3050
1.
9483
65
1.97
4014
2.00
0000
367.
583
372.
422
377.
324
382.
292
387.
324
392.
423
397.
589
402.
823
408.
126
413.
498
418.
942
424.
457
430.
044
435.
705
441.
441
447.
252
453.
140
459.
105
465.
149
471.
272
477.
476
483.
762
490.
130
496.
582
503.
119
509.
742
516.
452
52
3.25
1
588.
679
611.
321
633.
962
656.
604
679.
245
701.
887
724.
528
747.
170
769.
811
792.
453
815.
094
837.
736
860.
377
883.
019
905.
660
928.
302
950.
943
973.
585
996.
226
1018
.868
10
41.5
09
1064
.151
10
86.7
92
1109
.434
11
32.0
75
1154
.717
11
77.3
58
12
00.0
00
ß11
+
11
+34
ß
43
ß21
+
02
+25
+
47
ß30
ß
08
+15
+
42
ß40
ß
17
+06
+
28
ß49
ß
26
ß04
+
19
+42
ß
36
ß13
+
09
+32
ß
45
ß23
+00
The Equally-Tempered Archetype 73
Pietro Aron’s Quarter-Comma Meantone Tempered Division of the Octave
Before the advent of twelve-tone equal temperament in the West, Pythagorean intonation was the
predominant doctrine (“presumably from the Dark Ages to the 1200s”), although musicians eventually
became dissatisfied with the extreme sharpness of the Pythagorean thirds (the just major third [5/4] with a
falsity of +21.506 cents), and began to experiment by altering the fifths with an “indefinite amount of
diminution called temperament,” or participate.124 Historically, the first theoretical account of
temperament in the West (or quasi-Pythagorean intonation) comes during the Renaissance from Italian
music theorist and composer Franchinus Gaffurius’s (1451-1522) Practica musica, which was published in
Milan, in 1496. Meantone temperament, or the division of the just major third (5/4) into two mean tones
(equal to the meantone major tone, 9/8× 28180 , or 193.157 cents), may be attributed to Italian music
theorist and composer Pietro Aron (1489-1555), who in his 1523 treatise entitled Toscanello in musica
described a method for tempering the fifths that would in time serve as the foundation for numerous
systems of tempered intonation.125 It should be noted that as Mark Lindley rightly points out, “When
meantone systems gradually went out of fashion on keyboard instruments after c. 1650, they yielded, not
to equal temperament, but to a type of irregular system.” Irregular systems were prevalent between
1680 and c. 1800, and include ‘well temperaments’ prescribed by composers Andreas Werckmeister
(1645-1706), Jean-Philippe Rameau (1683-1764), Francesco Antonio Vallotti (1697-1780), and Muzio
Clementi (1752-1832); as well as by scientists and mathematicians Jacopo Francesco Riccati (1676-
1754), Johann Heinrich Lambert (1728-77), and Thomas Young (1773-1829). In A Venerable
Temperament Rediscovered, Douglas Leedy provides an insightful account of the rationale behind the
half-millennium domination of meantone temperament.126
“In the history of Western music, the keyboard temperament with the longest run was meantone, which
was the nearly universal standard from 1350 or so until it was almost totally eclipsed by twelve-tone equal
temperament in the first half of the nineteenth century. Meantone and the development (and ultimate
success) of keyboard instruments are inextricably intertwined: because its sonorities are close to those of
just tuning, meantone was able to emulate the sweetness of post-Pythagorean, pure triadic intonation
124 “In the visual arts, the mid- to late-15th century was a time of mathematically governed precision and elegance.
Geometrically planned perspective was introduced into painting and intarsia, and the first great masters (Brunelleschi,
Alberti, Francesco di Giorgio Martini) were architects who made remarkably sensitive use of very simple ratios in their
buildings. It was at this time that musicians, evidently alert to the distinctive sound of the nearly pure thirds in a quasi-
Pythagorean system, began to temper the fifths among the naturals so as to have unequivocally consonant thirds there as
well.” For a further discussion, see Mark Lindley, Mathematical Models of Musical Scales: A New Approach (Bonn: Verlag
für Systematische Musikwissenschaft, 1993) 134-38.
125 Barbour, Tuning and Temperament: A Historical Survey 25-28.
126 Lindley, Mathematical Models of Musical Scales: A New Approach 134-52.
74 The Equally-Tempered Archetype
characteristics of vocal music from around the time of Dunstable – the earliest pieces were in fact
transcriptions of vocal works.”
The author then continues the discussion with this evocative description of Meantone’s intervallic
properties:
“What unfortunately cannot be conveyed in words is the warm and serene beauty of meantone’s
consonant harmonies, its arresting dissonances, or the vivid colour, kaleidophonic variety, and expressive
strength of its melodic intervals. These need to be heard and savoured over time, for time is necessary to
accustom oneself to meantone’s richness and to becomes attuned to its subtlety and shading - much as
one would need time to become accustomed to the replacement of a diet of uniform blandness with the
variety of flavour, colour, and piquancy afforded by a Lucullan cuisine.”127
Pietro Aron’s quarter-comma meantone tempered division of the octave is based on the premise that the
just major tone (9/8) and just minor tone (10/9), or the sum of the two intervals (the just major third [5/4])
is reduced to the mean of these two intervals via the cumulative diminution of just perfect fifths (3/2s) by
the amount of a quarter of a syntonic comma. The syntonic comma, or comma of Didymus,128 is
represented by the ratio 81/80, or 21.506 cents (equal to the subtraction of 10/9 from 9/8, or the
difference between the two tones), and a quarter of this measurement is equal to 48081 , 1:1.003110, or
5.377 cents.129 81/80, and its reciprocal, 80/81, may also be theoretically referred to as the acute, or
comma, and grave, or hypocomma.130 The consequence of this process (the diminution of just perfect
fifths) also naturally leads to the augmentation of just perfect fourths (4/3s) by the amount of a quarter of
a syntonic comma, and ultimately to ‘true’ just major thirds (5/4s). Lloyd and Boyle offer the following
explanation: “The true major third (ratio 5/4) is the sum of a major and minor tone. Basic mean-tone
temperament therefore made all the whole tones the mean of these two intervals, half a comma smaller
than the major tone, half a comma larger than the minor tone, leaving each of the diatonic semitones (EF
and BC) too sharp by quarter comma.”131
In order to generate a meantone scale, it is first necessary to construct a Pythagorean scale with
twenty-seven distinct pitches, which are the necessary intervals to facilitate the general modulations of
127 Douglas Leedy, “A Venerable Temperament Rediscovered,” Perspectives of New Music 29.2 (Summer, 1991):
202-03.
128 “Didymus, academic philosopher of Nero’s time (ruled 54-68 A.D.) gave his name,” to the syntonic comma
(81/80), “the difference between two between-degree relationships – 9/8 and 10/9 – in his diatonic tetrachord.” For a
further discussion, see Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 368.
129 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 168.
130 Alexander J. Ellis, “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments wit Fixed
Tones,” Proceedings of the Royal Society of London 13 (1863-64): 95.
131 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 168.
The Equally-Tempered Archetype 75
Western tonal music.132 The fifteen intervals generated by an ascending series of fifths, or the pitches C, G,
D, A, E, B, F!, C!, G!, D!, A!, E!, B!, F#, C#, and G# (beginning with G (3/2) are then cumulatively reduced
by the amount of a quarter of a syntonic comma, or by ß¼, ß½, ß¾, ß1, ß1¼, ß1½, ß1¾, ß2,
ß2¼, ß2½, ß2¾, ß3, ß3¼, ß3½, and ß3¾ of a comma in each case; while the other eleven
intervals, generated by a descending series, or C, F, B", E", A", D", G", C", F", B$, E$, and A$, are
cumulatively increased by the amount of a quarter of a syntonic comma, or by +¼, +½, +¾, +1,
+1¼, +1½, +1¾, +2, +2¼, +2½, and +2¾ of a comma in each case. The abovementioned pattern
demonstrates that by a subtraction of a syntonic comma (81/64×80/81=5/4), the Pythagorean major
third (81/64) will be suitably reduced to a just major third (5/4). It may be therefore summarized that the
calculation of the factor for each meantone interval firstly involves the calculation of the reciprocal, or
multiplicative inverse of a series of one-quarter increments (¼, ½, ¾, 1, 1¼, 1½, 1¾, 2, 2¼, 2½, 2¾, 3,
3¼, 3½, and 3¾ equal to 4, 2, 1.333333, 1, 0.8, 0.666666, 0.571429, 0.5, 0.444444, 0.4, 0.363636,
0.307692, 0.285714, and 0.266666), which is then be utilized to divide the syntonic comma (81/80)
into the appropriate quarter-comma additions or subtractions that are cumulatively applied to each
Pythagorean ratio. For example: the meantone perfect fifth is equal to the first ascending Pythagorean fifth
(3/2) minus one quarter of a comma, so 3/2 must be multiplied by the fourth root of 80/81 (four being
the reciprocal of 0.25, or ¼ï1=4), which is mathematically expressed as the factor 3/2× 48180 , or the
decimal ratio 1.495349. The meantone perfect fifth (3/2× 48180 ) at 696.578 cents therefore represents
the just perfect fifth (3/2) tempered by the subtraction of a quarter of a syntonic comma (81/80), or
1:3/2× 48180 =1:1.495349 (equivalent to 696.578+5.377=701.955); while the meantone perfect fourth
(4/3× 48081 ) at 503.422 cents represents the just perfect fourth (4/3) tempered by the addition of a
quarter of a syntonic comma, or 1:4/3× 48081 =1:1.337481 (equivalent to 503.422ß5.377=498.045
cents).
The system reduced to twelve notes to the octave (being the tuning convention applied to
standard keyboard instruments) produces the following series of pitches: C, C!, D, E", E', F, F!, G, G!, A,
B", B', and C, or eight ascending fifths and three descending fifths modified by the quarter-comma
meantone process. The selection of sharps and flats represent “the chromatically altered notes used in
modal music,” and results in F!, G!, and C!, as just major thirds (5/4s) above D, E, and A respectively;
while B", and E", as just major thirds (5/4s) below D and G. “The result of tuning the black notes in this
way was to produce ‘wolves’ in the keys E" major and E major, and in all the major keys more remote,
while the minor keys of C and E were faulty,” notes Lloyd and Boyle. Major keys represented well within
this temperament include C, G, D, A, F, and B"; as well as minor keys G, D, and A,133 and hence “any keys
132 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 434.
133 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 102-
68.
76 The Equally-Tempered Archetype
beyond the six which can be got with only three sharps and two flats are very seriously out of tune, and
full of wolves.”134 The so-called ‘wolves’ are in reference to the A" major (G!, C, and E") and F minor
triads (F, G!, and C), which are severely compromised in twelve-tone meantone temperament by the
replacement of a major third, perfect fifth, and minor third with diminished fourth, diminished sixth, and
augmented second intervals. The major third, perfect fifth, and minor third each at 427.373, 737.637,
and 269.205 cents therefore represent just intervals with a falsity of +41.059, +35.683, and ß46.436
cents on each count. “That is why we find all the early music written in simple keys – they were the only
ones available on keyboard instruments,” notes N. Lindsay Norden.135 The ‘wolf’ fifth (or meantone
diminished sixth) at 737.637 cents, approximates the interval of a superfifth (192/125, or 743.014 cents),
which is significantly higher than an acute or large fifth (243/160, or 723.014 cents), and way beyond any
recognizable form of Fuller’s or Blackwood’s definition of a perfect fifth with a 720.000 cent upper limit.
The interval further represents the eleventh descending Pythagorean fifth (or Pythagorean diminished sixth)
tempered by an addition of two and three-quarter commas, or 262144/177147× 0.3636368081 =1.531237.
“The horrible effect was familiarly compared to the howling of ‘wolves’,” notes Helmholtz. “Similarly for B,
D!, and F!, it was necessary to use B, E", and F!, E" being a great diesis (128/125, or 41.059 cents) too
sharp, with similar excruciating effects.”136
It is interesting to note that according to Edward Dunne, and Mark McConnell, “the syntonic
comma is actually a much greater problem in Western music than the Pythagorean comma. To encounter
a Pythagorean comma, a piece would have to modulate through all twelve keys of the circle of fifth, but
only a few modulations bring you to the syntonic comma.” Also, the fact that the syntonic comma
(81/80) represents an equation involving three primes (81/80=1, or 2ô.5=3ô), just like 2x=3y (tripling
fifths and doubling octaves), which “will never be a scale in which all the fifths, or a complete set of fifths
and thirds, are correct,” the mathematical verity becomes apparent that “any method of constructing a
twelve-tone scale by rational numbers is doomed to inconsistency.”137
134 James Swinburne, “The Ideal Scale: Its AEtiology, Lysis and SequelAE,” Proceedings of the Musical Association,
63rd sess. (1936-1937): 39-64.
135 N. Lindsay Norden, “A New Theory of Untempered Music: A Few Important Features with Special Reference to
‘A Capella’ Music,” The Musical Quarterly 22.2 (Apr., 1936): 221.
136 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 434-55.
137 Edward Dunne, and Mark McConnell, “Pianos and Continued Fractions,” Mathematics Magazine 72.2 (Apr.,
1999): 107-09.
The Equally-Tempered Archetype 77
Tabl
e 32
. Pi
etro
Aro
n’s
qua
rter-
com
ma
mea
nton
e te
mp
ered
div
ision
of t
he o
ctav
e Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
iddl
e C)
=26
1.62
5565
4Hz
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
C CÚ
DÝ
CÛ
D EÞ
DÚ
EÝ E FÝ
EÚ F FÚ
GÝ
FÛ
uniso
n
mea
nton
e ch
rom
atic
sem
itone
, or m
inor
hal
f-ton
e (A
) V
II4
3 1ß
mea
nton
e m
inor
sec
ond
(D)
V41
1�
mea
nton
e d
oubl
e au
gmen
ted
oct
ave
(A)
XIV
213
ß
mea
nton
e m
ajor
tone
(A)
II21
ß
mea
nton
e d
imin
ished
third
(D)
X21
2�
mea
nton
e au
gmen
ted
sec
ond
(A)
IX41
2ß
mea
nton
e m
inor
third
(D)
III4
33�
just
maj
or th
ird (A
) IV
1ß (
5th
harm
onic
)
mea
nton
e d
imin
ished
four
th (D
) V
III2�
mea
nton
e au
gmen
ted
third
(A) X
I4
32
ß
mea
nton
e p
erfe
ct fo
urth
(D)
I41
�
mea
nton
e tri
tone
, or a
ugm
ente
d fo
urth
(A) V
I21
1ß
mea
nton
e d
imin
ished
fifth
(D) V
I21
1�
mea
nton
e d
oubl
e au
gmen
ted
four
th (A
) XI
II41
3ß
1/1
2187
/204
8×0.
5714
298180
256/
243×
0.8
8081
4782
969/
4194
304×
0.28
5714
8180
9/8×
28180
6553
6/59
049×
0.4
8081
1968
3/16
384×
0.44
4444
8180
32/2
7×1.
3333
338081
81/6
4×80
/81,
or 5
/4
8192
/656
1×0.
58081
, or 3
2/25
1771
47/1
3107
2×0.
3636
368180
4/3×
48081
729/
512×
0.66
6667
8180
1024
/729
×0.
6666
678081
1594
323/
1048
576×
0.30
7692
8180
1.00
0000
1.04
4907
1.06
9984
1.09
1830
1.11
8034
1.14
4867
1.16
8241
1.19
6279
1.25
0000
1.28
0000
1.30
6133
1.33
7481
1.39
7542
1.43
1084
1.46
0302
261.
626
273.
374
279.
935
285.
651
292.
506
299.
526
305.
642
312.
977
327.
032
334.
881
341.
718
349.
919
365.
633
374.
408
382.
052
0.00
0
76.0
49
117.
108
152.
098
193.
157
234.
216
269.
206
310.
265
386.
314
427.
373
462.
363
503.
422
579.
471
620.
529
655.
536
+00
ß24
+17
ß48
ß07
+34
ß31
+10
ß14
+27
ß38
+03
ß21
+21
ß44
78 The Equally-Tempered Archetype
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
Ò× ÐÑ
G AÞ
GÚ
AÝ
GÛ
A BÞ
AÚ
BÝ B CÝ
BÚ C
mea
nton
e p
erfe
ct fi
fth (A
) I
41ß
mea
nton
e d
imin
ished
six
th (D
) XI
43
2�
mea
nton
e au
gmen
ted
fifth
(A)
VIII
2ß
(25
th h
arm
onic
)
just
min
or s
ixth
(D)
IV1�
mea
nton
e d
oubl
e au
gmen
ted
fifth
(A) X
V4
33
ß
mea
nton
e m
ajor
six
th (A
) III
43
ß
mea
nton
e d
imin
ished
sev
enth
(D)
IX41
2�
mea
nton
e au
gmen
ted
six
th (A
) X21
2ß
mea
nton
e m
inor
sev
enth
(D)
II21
�
mea
nton
e m
ajor
sev
enth
(A) V
411ß
mea
nton
e d
imin
ished
oct
ave
(D) V
II4
3 1�
mea
nton
e au
gmen
ted
sev
enth
(A)
XII
3ß
octa
ve
3/2×
48180
2621
44/1
7714
7×0.
3636
368081
6561
/409
6×0.
58180
, or 2
5/16
128/
81×
81/8
0, o
r 8/5
1434
8907
/838
8608
×0.
2666
678180
27/1
6×1.
3333
338180
3276
8/19
683×
0.44
4444
8081
5904
9/32
768×
0.4
8180
16/9
×2
8081
243/
128×
0.8
8180
4096
/218
7×0.
5714
298081
5314
41/5
2428
8×0.
3333
338180
2/1
1.49
5349
1.53
1237
1.56
2500
1.60
0000
1.63
2667
1.67
1851
1.71
1975
1.74
6928
1.78
8854
1.86
9186
1.91
4046
1.95
3125
2.00
0000
391.
221
400.
611
408.
790
418.
601
427.
147
437.
399
447.
896
457.
041
468.
010
489.
027
500.
763
510.
987
523.
251
696.
578
737.
637
772.
627
813.
686
848.
676
889.
735
930.
794
965.
784
1006
.843
1082
.892
1123
.951
1158
.941
1200
.000
ß03
+38
ß27
+14
ß51
ß10
+31
ß34
+07
ß17
+24
ß41
+00
The Equally-Tempered Archetype 79
Joseph Sauveur’s Forty-Three-Tone Equally-Tempered Division of the Octave There are a variety of other meantone temperaments that require special mention, and namely the 2/7
comma temperament of Italian music theorist and composer Gioseffo Zarlino (1517-90), 1/3 comma
temperament of Spanish music theorist and organist Francisco de Salinas (1513-90), 1/5 comma
temperament of Dutch organist Abraham Verheijen (fl. 1600), 2/9 comma temperament of Italian music
theorist Lemme Rossi (1602-73), and 1/6 comma temperament of German organ builder and instrument
maker Gottfried Silbermann (1683-1753). Additionally, 1/11 comma meantone temperament serves as an
excellent approximation of twelve-tone equal temperament. 1/11of a syntonic comma (81/80) is equal to
118081 , 1:1.001130, or 1.955 cents, and results in a fifth and fourth equal to 1:3/2× 11
8180 =1:1.498307, or
700.000 cents, and 1:4/3× 118081 =1:1.334840, or 500.000 cents respectively.
Fifth-comma meantone temperament may be expressed as the mathematical equation that
produces the “first ratio for the fifth” as the “fifth root of 15:2,” or 52
15 . The fifth of a syntonic comma
(81/80) is equal to 58081 , 1:1.002488, or 4.301 cents. The fifth-comma meantone perfect fifth
(3/2× 58180 ) at 697.654 cents represents the just perfect fifth (3/2) tempered by the subtraction of a fifth
of a syntonic comma (81/80), or 1:3/2× 58180 =1:1.496278 (equivalent to 697.654+4.301=701.955);
while the fifth-comma meantone perfect fourth (4/3× 58081 ) at 502.346 cents represents the just perfect
fourth (4/3) tempered by the addition of a fifth of a syntonic comma (81/80), or
1:4/3× 58081 =1:1.336650 (equivalent to 502.346ß4.301=498.045 cents). The temperament (which
incidentally approximates the forty-three-tone equally-tempered division of the octave [ 43 2 ]) features
the “equal distortion of the fifths and the major thirds (equal to 390.615 cents); the former being one-fifth
comma flat, while the latter, sharp by the same amount.”138
French acoustician Joseph Sauveur (1653-1716) proposed a forty-three-tone equally-tempered
division of the octave (which closely approximates fifth-comma meantone temperament) in 1701, and
consequently devised a highly complex notation system based on seven diatonic notes and syllables.
Read explains: “Ut to si, were divided into six parts, which he termed mérides. Furthermore, each
mérides was divided into seven eptamérides and these in turn into ten decamérides each, resulting in an
octave comprised of 43 mérides, 301 eptamérides, and 3010 decamérides.”139 The fact that the
common logarithm of two is equal to 0.301030, means that the decamérides “unit of musical measure”
may be derived directly from this source. The 43-et perfect fifth ( 2543 ]2[ ) measures 697.674 cents; the
43-et perfect fourth ( 1843 ]2[ ), 502.326 cents; while the 43-et major third ( 1443 ]2[ ), 390.698 cents.
These measurements represent fifth-comma meantone temperament intervals with a falsity of +0.020,
ß0.020, and +0.083 cents on each count.140 138 Barbour, Tuning and Temperament: A Historical Survey 31-83.
139 Read, 20th-Century Microtonal Notation 133-34.
140 Barbour, Tuning and Temperament: A Historical Survey 122.
80 The Equally-Tempered Archetype
Tabl
e 33
. Jo
sep
h Sa
uveu
r’s fo
rty-th
ree-
tone
eq
ually
-tem
per
ed d
ivisi
on o
f the
oct
ave
Rela
tive
Pitc
h: A
4=44
0Hz
/ C4 (
mid
dle
C)=
261.
6255
654H
z
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
C D$
B#
C!
D"
E&
C#
D E$
C%
D!
E"
F$
D#
E F"
G&
E! F G$
E#
uniso
n
43-e
t dim
inish
ed s
econ
d
43-e
t dou
ble
augm
ente
d s
even
th
43-e
t chr
omat
ic s
emito
ne, o
r min
or h
alf-t
one
43-e
t min
or s
econ
d
43-e
t dou
ble
dim
inish
ed th
ird
43-e
t dou
ble
augm
ente
d o
ctav
e
43-e
t maj
or to
ne
43-e
t dim
inish
ed th
ird
43-e
t trip
le a
ugm
ente
d o
ctav
e
43-e
t aug
men
ted
seco
nd
43-e
t min
or th
ird
43-e
t dou
ble
dim
inish
ed fo
urth
43-e
t dou
ble
augm
ente
d s
econ
d
43-e
t maj
or th
ird
43-e
t dim
inish
ed fo
urth
43-e
t trip
le d
imin
ished
fifth
43-e
t aug
men
ted
third
43-e
t per
fect
four
th
43-e
t dou
ble
dim
inish
ed fi
fth
43-e
t dou
ble
augm
ente
d th
ird
1/1
432
243
)2
(
343
)2
(
443
)2
(
543
)2
(
643
)2
(
743
)2
(
843
)2
(
943
)2
(
1043
)2
(
1143
)2
(
1243
)2
(
1343
)2
(
1443
)2
(
1543
)2
(
1643
)2
(
1743
)2
(
1843
)2
(
1943
)2
(
2043
)2
(
1.00
0000
1.01
6250
1.03
2765
1.04
9547
1.06
6603
1.08
3936
1.10
1550
1.11
9450
1.13
7642
1.15
6129
1.17
4916
1.19
4009
1.21
3412
1.23
3131
1.25
3169
1.27
3534
1.29
4229
1.31
5261
1.33
6634
1.35
8355
1.38
0429
261.
626
265.
877
270.
198
274.
588
279.
051
283.
585
288.
194
292.
877
297.
636
302.
473
307.
388
312.
383
317.
460
322.
618
327.
861
333.
189
338.
603
344.
106
349.
698
355.
380
361.
155
0.00
0
27.9
07
55.8
14
83.7
21
111.
628
139.
535
167.
442
195.
349
223.
256
251.
163
279.
070
306.
977
334.
884
362.
791
390.
698
418.
605
446.
512
474.
419
502.
326
530.
233
558.
140
+00
+28
ß44
ß16
+12
+40
ß33
ß05
+23
ß49
ß21
+07
+35
ß37
ß09
+19
+47
ß26
+02
+30
ß42
The Equally-Tempered Archetype 81
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
Ò×
ÒØ
ÒÙ
ÓÐ
ÓÑ
ÓÒ
ÓÓ
ÓÔ
ÓÕ
ÓÖ
Ó×
ÓØ
ÓÙ
ÔÐ
ÔÑ
ÔÒ
ÔÓ ÐÑ
F!
G"
A&
F #
G
A$
F %
G!
A"
B &
G# A
B $
C &
A! B "
C $
A# B C "
D &
B ! C
43-e
t trit
one,
or a
ugm
ente
d fo
urth
43-e
t dim
inish
ed fi
fth
43-e
t dou
ble
dim
inish
ed s
ixth
43-e
t dou
ble
augm
ente
d fo
urth
43-e
t per
fect
fifth
43-e
t dim
inish
ed s
ixth
43-e
t trip
le a
ugm
ente
d fo
urth
43-e
t aug
men
ted
fifth
43-e
t min
or s
ixth
43-e
t dou
ble
dim
inish
ed s
even
th
43-e
t dou
ble
augm
ente
d fi
fth
43-e
t maj
or s
ixth
43-e
t dim
inish
ed s
even
th
43-e
t trip
le d
imin
ished
oct
ave
43-e
t aug
men
ted
six
th
43-e
t min
or s
even
th
43-e
t dou
ble
dim
inish
ed o
ctav
e
43-e
t dou
ble
augm
ente
d s
ixth
43-e
t maj
or s
even
th
43-e
t dim
inish
ed o
ctav
e
43-e
t dou
ble
dim
inish
ed s
econ
d
43-e
t aug
men
ted
sev
enth
octa
ve
2143
)2
(
2243
)2
(
2343
)2
(
2443
)2
(
2543
)2
(
2643
)2
(
2743
)2
(
2843
)2
(
2943
)2
(
3043
)2
(
3143
)2
(
3243
)2
(
3343
)2
(
3443
)2
(
3543
)2
(
3643
)2
(
3743
)2
(
3843
)2
(
3943
)2
(
4043
)2
(
4143
)2
(
4243
)2
(
2/1
1.40
2861
1.42
5658
1.44
8825
1.47
2369
1.49
6296
1.52
0611
1.54
5321
1.57
0433
1.59
5953
1.62
1888
1.64
8244
1.67
5029
1.70
2249
1.72
9911
1.75
8022
1.78
6591
1.81
5624
1.84
5128
1.87
5112
1.90
5583
1.93
6549
1.96
8019
2.00
0000
367.
024
372.
989
379.
050
385.
209
391.
469
397.
831
404.
296
410.
866
417.
542
424.
327
431.
223
438.
230
445.
352
452.
589
459.
944
467.
418
475.
014
482.
733
490.
577
498.
549
506.
651
514.
884
523.
251
586.
047
613.
953
641.
860
669.
767
697.
674
725.
581
753.
488
781.
395
809.
302
837.
209
865.
116
893.
023
920.
930
948.
837
976.
744
1004
.651
1032
.558
1060
.465
1088
.372
1116
.279
1144
.186
1172
.093
1200
.000
ß14
+14
+42
ß30
ß02
+26
ß47
ß19
+09
+37
ß35
ß07
+21
+49
ß23
+05
+33
ß40
ß12
+16
+44
ß28
+00
82 The Equally-Tempered Archetype
Origins of Equal Temperament Equal temperament, or the division of the octave into equal parts may be accredited to Prince Chu Tsai-
yü, “a sixth-generation descendant of Hung Hsi, the fourth emperor of the Ming dynasty,” who in his 1584
treatise entitled A New Account of the Science of the Pitch Pipes presented the string lengths for twelve-
tone equal temperament.141 According to Kuttner:
“Lü Hsüeh Hsin Shuo (A New Account of the Science of the Pitch Pipes) contains a complete nine-digit
monochord of equal temperament with the octave ratio 10:5 for what appears to be a calculation of string
lengths; for the lengths of pitch-pipes there monochords based on the octave ratios 100:50 and 90:45,
with four decimals, as well as tabulations of pipe diameters and circumferences on the base 100:50 with
two decimals.”
The roots of two for the string lengths in a monochord are then published in Chu Tsai-yü’s 1595-96
follow-up treatise entitled Lü Lü Ching I, and although correct to nine places, have been calculated with
the absence of logarithms. It should be noted that although “the twelfth root of two is numerically
implied as quantitative definition of the semitone in equal temperament,” in Tsai-yü’s second theoretical
work, “it is never stated explicitly as a mathematical expression.”142 Partch offers the following
commentary on the nature of Prince Chu Tsai-yü’s accomplishment:
“Of the prince’s accomplishment a contemporary modern theorist reminds us that ‘the computation would
have to begin, for certain tones, with numbers containing 108 zeros, of which the 12th root would have to
be extracted, as (Marin) Mersenne did, by taking the square root twice and then the cube root. This
lengthy and laborious procedure was followed without error.’”143
The nine-digit string lengths of Tsai-yü’s monochord depicted in the following table are not a result of
rounded off values of a ten-digit calculation, and therefore a reduction to lower terms, but the true values
of a calculation based on the hundredth millionth.144
141 Stuart Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization
(New York: Vintage, 2003) 163-66. 142 Chu Tsai-yü’s 1595-96 treatise entitled Lü Lü Ching I “contains an enormously detailed mathematical investigation
of all conceivable parameters involved in the definition of pitches in equal temperament tuning, including string lengths
and pitch-pipe dimensions, such as tube lengths, inner and outer diameters, circular surface areas, bore and volume of
pipes through three octave ranges in 9- and 10-digit tabulations,” states Fritz A. Kuttner. For a further discussion, see Fritz
A. Kuttner, “Prince Chu Tsai-Yu’s Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory,”
Ethnomusicology 19.2 (May, 1975): 166-67.
143 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 380-81.
144 Barbour, Tuning and Temperament: A Historical Survey 77-78.
The Equally-Tempered Archetype 83
Table 34. Chu Tsai-yü’s monochord
DEGREE
NUMBER
NOTE LENGTHS RATIO
(DECIMAL)
CENTS FALSITY
(CENTS)
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ
ÐÑ
C4
C!4
D4
D!4
E4
F4
F!4
G4
G!4
A4
A!4
B4
C5
1,000,000,000
943,874,312
890,898,718
840,896,415
793,700,525
749,153,538
707,106,781
667,419,927
629,960,524
594,603,557
561,231,024
529,731,547
500,000,000
1.000000
1.059463
1.122462
1.189207
1.259921
1.334840
1.414214
1.498307
1.587401
1.681793
1.781797
1.887749
2.000000
0.000
100.000
200.000
300.000
400.000
500.000
600.000
700.000
800.000
900.000
1000.000
1100.000
1200.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
The mathematical solution to the dilemma of the cycle of fifths, or the irresolvable succession of just
perfect fifths or 3/2s rests on the unequivocal formula, equivalent to the frequency ratio 1.059463094, or
the numerical approximation of the proportion 749:500 (derived from 750:500, which is equal to the
ratio 3/2, and refined in chapter one of Lü Lü Ching I to the proportion 749.153.538:500.000.000). The
explanation for the infinite nature of a 3/2 series is that octaves (or the ratio 2/1) are based on multiples of
two, while fifths (or the ratio 3/2), on multiples of three; and because two and three are prime numbers
(divisible only by themselves and one), presented is the mathematical verity that only powers of identical
prime numbers can be equal.145 Calculus illustrates that a series of twelve 3/2s yields the following values
in cents: 0.000, 113.685, 203.910, 317.595, 407.820, 521.505, 611.730, 701.955, 815.640, 905.865,
1019.550, 1109.775, and 1223.460. The intonational anomaly generated by the twelfth ascending fifth,
which has a ratio of 531441/524288 (1:1.013643) and measures 23.460 cents, is generally referred to as
the Pythagorean comma. In other words, “the sum of twelve 3/2s (3ñò/2ñù) is greater than seven octaves
(2÷/1ñ) by a Pythagorean comma.”146 In striking contrast, a series of twelve 749:500s yields: 0.000,
97.516, 199.290, 296.806, 398.581, 496.097, 597.871, 699.645, 797.161, 898.935, 996.451,
1098.226, and 1195.742; while a series of twelve 749.153.538:500.000.000s yields twelve equal
semitones.
145 Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization 40-170.
146 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 34.
84 The Equally-Tempered Archetype
In the West, French monk, mathematician and physicist Marin Mersenne (1588-1648) is not only
acknowledged for “the discovery of the overtone series in the natural sounds of the trumpet,” but also
for being “the first Westerner to give the correct mathematical solution for equal temperament.”147
Mersenne’s most accurate geometrical approximations are characterised by Barbour as follows:
“By the familiar Euclidean method he found the mean proportional between a line and its double,
subtracted the original line from the mean, and then subtracted this difference from the doubled line. The
length thus found was the larger of the desired means – that is, the string length for the major third.”
In mathematical terms, Mersenne’s solution is depicted via the equation (3ß 2 2 ):2, which is equal to the
string length of 0.7928930, or the decimal ratio 1.261204 for the major third. The division of the major
third into four equal semitones or “mean proportionals” produce a semitone equal to 100.440 cents in
this region, while the subsequent division of the remaining major sixth (the distance between the major
third and the octave) into eight equal semitones produce a dissimilar semitone equal to 99.780 cents.
The mean falsity of the semitones in Mersenne’s geometrical approximation for equal temperament,
therefore have a value of 0.881 cents, with a maximum error of 1.762 cents.
Table 35. Marin Mersenne’s geometrical approximation for equal temperament
DEGREE
NUMBER
NOTE LENGTHS RATIO CENTS FALSITY
(CENTS)
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C4
C!4
D4
D!4
E4
F4
F!4
G4
G!4
A4
A!4
B4
C5
1.000000
0.943634
0.890446
0.844670
0.792893
0.748487
0.706567
0.666996
0.629640
0.594377
0.561088
0.529664
0.500000
1.000000
1.059733
1.123033
1.190115
1.261204
1.336029
1.415293
1.499260
1.588209
1.682435
1.782251
1.887989
2.000000
0.000
100.440
200.881
301.321
401.762
501.542
601.321
701.101
800.881
900.661
1000.440
1100.220
1200.000
+0.000
+0.440
+0.881
+1.321
+1.762
+1.542
+1.321
+1.101
+0.881
+0.661
+0.440
+0.220
+0.000
147 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 381-82.
The Equally-Tempered Archetype 85
It is then, circa 1596, that Flemish mathematician and inventor Simon Stevin (1548-1620) recognizes that
the solution to equal temperament lies in finding “eleven mean proportional parts between two and
one,”148 and hence produces “an essay containing the mathematical formulation of equal temperament as 12 2 for the first time in Western musical theory.” The essay – rediscovered and edited in 1884 by Dutch
mathematician Dr. David Bierens de Haan (1822-95), to be then published for the very first time –
presents the calculation of a monochord defining “twelve semitone values, correct to four decimal
places, as the 12 successive powers of the twelfth root of two.”149 The method adopted by Stevin
involves the calculation of degree numbers 7, 4, and 5 (or F!, D!, and E) – each subordinate to cubic
and quartic levels of mathematical complexity – with the next step requiring the strategy of proportion, or
“the rule of three”, and therefore the division of the fifth degree (7937) by the fourth (8408) to produce
the second degree (9440). “This method is much easier than to extract the roots for each individual
note, which runs into difficulties with the roots of the prime numbers, as for the notes 2, 6, 8, and 12 (C!,
F, G, and B), where the 12th root itself must be extracted,” explains Barbour. The method delivers a
mean falsity of 0.199 cents, with a maximum error of 0.391 cents.
Table 36. Simon Stevin’s monochord
DEGREE
NUMBER
NOTE LENGTHS RATIO CENTS FALSITY
(CENTS)
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ
ÐÑ
C4
C!4
D4
D!4
E4
F4
F!4
G4
G!4
A4
A!4
B4
C5
10000
9440
8911
8408
7937
7493
7071
6675
6301
5945
5612
5298
5000
1.000000
1.059322
1.122209
1.189343
1.259922
1.334579
1.414227
1.498127
1.587050
1.682086
1.781896
1.887505
2.000000
0.000
99.769
199.609
300.199
400.001
499.662
600.017
699.792
799.617
900.302
1000.096
1099.776
1200.000
+0.000
ß0.231
ß0.391
+0.199
+0.001
ß0.338
+0.017
ß0.208
ß0.383
+0.302
+0.096
ß0.224
+0.000
148 Barbour, Tuning and Temperament: A Historical Survey 54-55.
149 Kuttner, “Prince Chu Tsai-Yu’s Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory,”
Ethnomusicology 167-68.
86 The Equally-Tempered Archetype
It must be noted that in spite of the geometrical and numerical approximations of Mersenne, Stevin, and
other European theorists of the sixteenth and seventeenth centuries, it is not until 1630 that the first
printed numerical solution to equal temperament based on the theory of logarithmic computation
appears; German mathematician Johann Faulhaber (1580-1635) establishing lengths derived from the
division of a monochord with a length of the required 20,000 units in order to generate a series
legitimately expressing twelve equally-tempered semitones. The mean falsity of 0.100 cents (with a
maximum error of 0.181 cents) depicted in Faulhaber’s monochord is a direct result of the expected
error obtained via the utilization of logarithmic tables.150
Table 37. Johann Faulhaber’s monochord
DEGREE
NUMBER
NOTE LENGTHS RATIO CENTS FALSITY
(CENTS)
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C4
C!4
D4
D!4
E4
F4
F!4
G4
G!4
A4
A!4
B4
C5
20000
18877
17817
16817
15874
14982
14141
13347
12598
11891
11224
10594
10000
1.000000
1.059490
1.122523
1.189273
1.259922
1.334935
1.414327
1.498464
1.587554
1.681944
1.781896
1.887861
2.000000
0.000
100.045
200.095
300.096
400.001
500.124
600.139
700.181
800.166
900.156
1000.096
1100.103
1200.000
+0.000
+0.045
+0.095
+0.096
+0.001
+0.124
+0.139
+0.181
+0.166
+0.156
+0.096
+0.103
+0.000
The Twelve-Tone Equally-Tempered Division of the Octave The twelve-tone equally-tempered division of the octave is the division of the octave into twelve equal
intervals, technically referred to as tempered half-tones; the frequency ratio of each semitone therefore
mathematically representing the twelfth root of two, or in different terms, the distance between any two
tones representing twelve times the logarithm on the base of two of the frequency ratio.151 The ratio of
150 Barbour, Tuning and Temperament: A Historical Survey 54-78.
151 Olson, Music, Physics and Engineering 46-47.
The Equally-Tempered Archetype 87
the equally-tempered semitone may be expressed in mathematical terms as 1: 12 2 =1:1.059463094,152
or the fraction approximations: 1024/967 (2ñð/967) and 512/483 (2ù×3×7×23). The correct
measurements for the string lengths (based on 2,000,000 units) of the equally-tempered monochord are
depicted in the following table.153
Table 38. The equally-tempered monochord
DEGREE
NUMBER
NOTE LENGTHS RATIO CENTS FALSITY
(CENTS)
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C4
C!4
D4
D!4
E4
F4
F!4
G4
G!4
A4
A!4
B4
C5
2000000
1887749
1781797
1681793
1587401
1498307
1414214
1334840
1259921
1189207
1122462
1059463
1000000
1.000000
1.059463
1.122462
1.189207
1.259921
1.334840
1.414214
1.498307
1.587401
1.681793
1.781797
1.887749
2.000000
0.000
100.045
200.095
300.096
400.001
500.124
600.139
700.181
800.166
900.156
1000.096
1100.103
1200.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
The methodology involved in the construction of a twelve-tone equally-tempered scale requires the
establishment of twelve degrees via the equation f= 12 2 ; the equation producing the figure
1.059463094 (the frequency ratio of one tempered semitone), which when multiplied by the powers of
2, 3, 4, etc., generates the ratios for the remainder of the scale degrees. The multiplication by the power
of 9 ( 912 ]2[ ) generates the frequency ratio of A=440Hz (1.68179283), or equal major sixth ( 34 ]2[ ),
while the mathematical equation of 440ï1.68179283 presents the relative frequency of middle C
(261.6255654Hz), or 1/1. Once data is available for all frequency ratios it simply becomes a matter of
the multiplication of every ratio by 261.6255654Hz (the frequency of middle C) in order to obtain the
frequency values in hertz (Hz) of every scale degree. The mathematical formula for cents is then utilized
to generate the cent values of every scale degree, although unnecessary in equal temperament because
of the obvious outcomes.
152 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 431.
153 Daniélou, Tableau Comparatif des Intervalles Musicaux 28.
88 The Equally-Tempered Archetype
The acoustical realities of equal temperament are that although equally-tempered fifths approximate just
perfect fifths, the system delivers major thirds and minor thirds tempered by seven and eight times as
much as fifths respectively. “Lustrous and calm in their pure form, they were now slightly rough and
somewhat bland,” notes Stuart Isacoff with regards to the intonational transformation of just major thirds
(5/4s) into equal major thirds ( 3 2 ).154 The root position twelve-tone equally-tempered major triad is
represented by the ratio 1.000000:1.259921:1.498307, and the pitches C, E¢, and G; and presents the
just major third (5/4) and just perfect fifth (3/2) with a falsity of ß13.686 and +1.955 cents on each
count. G4 (391.995Hz), or the equal perfect fifth ( 712 ]2[ ), presents 0.886 beats between the third
harmonic of C4 (784.877Hz) and the second harmonic of G4 (783.991Hz), and 1.772 beats between
the sixth harmonic of C4 (1569.753Hz) and the fourth harmonic of G4 (1567.982Hz); while E¢4
(329.628Hz), or the equal major third ( 3 2 ), 10.382 beats between the fifth harmonic of C4 (1308.128Hz) and the fourth harmonic of E¢4 (1318.510Hz).
Table 39. The beating characteristics of the twelve-tone equally-tempered major triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E¢4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
329.628
– – – –
– – – –
659.255
– – – –
988.883
– – – –
– – – –
1318.510
– – – –
1648.138
– – – –
1977.765
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
10.382
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
391.995
– – – –
– – – –
783.991
– – – –
– – – –
1175.986
– – – –
1567.982
– – – –
– – – –
1959.977
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.886
– – – –
– – – –
– – – –
– – – –
1.772
– – – –
– – – –
– – – –
– – – –
The root position twelve-tone equally-tempered minor triad is represented by the ratio 1.000000:
1.189207:1.498307, and the pitches C, E¸, and G; and presents the just minor third (6/5) and just
perfect fifth (3/2) with a falsity of ß15.641 and +1.955 cents on each count. G4, or the equal perfect
154 Isacoff, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization 118.
The Equally-Tempered Archetype 89
fifth ( 712 ]2[ ), presents 0.886 beats between the third harmonic of C4 and the second harmonic of G4,
and 1.772 beats between the sixth harmonic of C4 and the fourth harmonic of G4; while E¸4
(311.127Hz), or the equal minor third ( 4 2 ), 14.118 beats between the sixth harmonic of G4 and the
fifth harmonic of EÝ4 (1555.635Hz).
Table 40. The beating characteristics of the twelve-tone equally-tempered minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E¸4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
311.127
– – – –
– – – –
622.254
– – – –
933.381
– – – –
– – – –
1244.508
– – – –
1555.635
– – – –
1866.762
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
14.118
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
391.995
– – – –
– – – –
783.991
– – – –
– – – –
1175.986
– – – –
– – – –
1567.982
– – – –
– – – –
1959.977
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.886
– – – –
– – – –
– – – –– –
– –
– – – –
1.772
– – – –
– – – –
– – – –
– – – –
The following series of tables depict the beating characteristics of the thirds, sixths, fifths, and fourths of
twelve-tone equal temperament with reference to the tuning scheme of the pianoforte. The ‘tuning
beats’ column represents the approximation of beats per second that a piano tuner must consider. The
tuning methodology begins with the tuning of the first fifth, or F3 and C4 (middle C), with 0.591 (or 0.6)
beats; and is followed by the tuning of the first fourth, G3 and C4, with 0.886 (or 0.9) beats. “Leave the F
slightly sharp, so that it sounds as a very slow wow – wow trailing off to nothing,” explains Ian McCombie.
“This interval should beat just over one beat in two seconds. It is the only note in the tempered scale
which is sharp. The G is left slightly flat, and should beat about beat per second flat.” The process
continues, to include a selection of thirds, sixths, fifths, and fourths, as well as relevant octave (2/1)
relationships.155
155 Ian McCombie, The Piano Handbook (London: David & Charles, 1980) 89-95.
90 The Equally-Tempered Archetype
Table 41. Piano tuning – twelve-tone equal temperament (thirds)
NUMBER LOWER NOTE 5TH PARTIAL (HERTZ) UPPER NOTE 4TH PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
F3
FÚ3
G3
GÚ3
A3
AÚ3
B3
C4
873.071
924.986
979.989
1038.262
1100.000
1165.409
1234.708
1308.128
A3
AÚ3
B3
C4
CÚ4
D4
DÚ4
E4
880.000
932.328
987.767
1046.502
1108.731
1174.659
1244.508
1318.510
6.929
7.341
7.778
8.241
8.731
9.250
9.800
10.382
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5 Table 42. Piano tuning – twelve-tone equal temperament (sixths)
NUMBER LOWER NOTE 5TH PARTIAL (HERTZ) UPPER NOTE 3RD PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
F3
FÚ3
G3
GÚ3
873.071
924.986
979.989
1038.262
D4
DÚ4
E4
F4
880.994
933.381
988.883
1047.685
7.924
8.395
8.894
9.423
8.0
8.5
9.0
9.5 Table 43. Piano tuning – twelve-tone equal temperament (fifths)
NUMBER LOWER NOTE 3RD PARTIAL (HERTZ) UPPER NOTE 2ND PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
F3
G3
A3
FÚ3
GÚ3
AÚ3
523.842
587.993
660.000
554.992
622.957
699.246
C4
D4
E4
CÚ4
DÚ4
F4
523.251
587.330
659.255
554.365
622.254
698.456
0.591
0.664
0.745
0.626
0.703
0.789
0.6
0.7
0.8
0.6
0.7
0.8 Table 44. Piano tuning – twelve-tone equal temperament (fourths)
NUMBER LOWER NOTE 4TH PARTIAL (HERTZ) UPPER NOTE 3RD PARTIAL (HERTZ) BEATS (HERTZ) TUNING BEATS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
G3
A3
B3
FÚ3
GÚ3
AÚ3
783.991
880.000
987.767
739.989
830.609
932.328
C4
D4
E4
B3
CÚ4
DÚ4
784.877
880.994
988.883
740.825
831.548
933.381
0.886
0.994
1.116
0.836
0.938
1.053
0.9
1.0
1.1
0.8
1.0
1.0
The Equally-Tempered Archetype 91
Exposiciones for Sampled Microtonal Schoenhut Toy Piano Exposiciones for Sampled Microtonal Schoenhut Toy Piano is an ‘acousmatic’156 work that attempts to
explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut
model 6625, 25-key toy piano and a complex polyrhythmic scheme. All equal temperaments between
one and twenty-four – essentially functioning as tuning modulations – as well as all polyrhythms (divisible
only by 1 and including their inversions) between the ranges of 2 and 15 are presented. In other words,
polyrhythmic ratios 3:2Ê (2:3Ê�), 5ç:2Ê (2É:5Ê�), 4:3Ê (3:4�), 5:3Ê (3:5�), and so on – fifty-seven polyrhythmic
sets in total, with the last set represented by 15:14Ê (14:15Ê) – alongside two complementary scales
(Indonesian pélog and sléndro forms with primary and secondary scale tones, as well as primary and
secondary auxiliary tones) shaped via microtonal inflections produced by sequential tuning modulations
featuring the first twenty-four equally-tempered divisions of the octave.
The work was especially composed for ‘The Extensible Toy Piano Project’ – a toy piano festival and
symposium hosted by Clark University, Department of Visual and Performing Arts, in Worcester,
Massachusetts, USA between Friday and Saturday, November 4 and 5, 2005.
“It has a deceptively simple mechanism – plastic hammers hitting steel rods. Yet, the toy piano produces a
rich and quirky sound palette. John Cage brought the instrument from a treasured plaything to a bona fide
musical instrument with his Suite for Toy Piano (1948). Our aim is to bring the instrument into the 21st
Century.” 157
A Brief History of the Toy Piano The history of the toy piano (Fr. piano jouet; Ger. Spielzeugklavier; It. pianino-giocattolo) begins as
Troiger’s ‘Stahlklavier’ (Dessau, 1792) and Franz Schuster’s ‘Adiaphonon’ (Vienna, c. 1818), to be later
developed in Philadelphia, USA, in 1872, where German immigrant Albert Schoenhut ultimately conceives
156 According to Francis Dhomont, “The term Acousmatic Music (or Art) designates works that have been
composed for loudspeakers, to be heard in the home – on radio or on CD/tape – or in concert, through the use of
equipment (digital or analog) that allows the projection of sound in 3-dimensional space.” For a further discussion see,
Francis Dhomont, “Acousmatic Update,” Contact! 8.2 Spring, 1995, CEC – Communauté Électroacoustique Canadienne /
Canadian Electroacoustic Community, 27 Jan. 2006, <http://cec.concordia.ca/contact/contact82Dhom.html>. A further
description of the term ‘acousmatic’ is offered in an interview with François Bayle by Sandra Desantos: “In acousmatic
music, one may recognize the sound sources, but one also notices that they are out of their usual context. In the
acousmatic approach, the listener is expected to reconstruct an explanation for a series of sound events, even if this
explanation is provisional.” For a further discussion see, Sandra Desantos, “Acousmatic Morphology: An Interview with
François Bayle,” Computer Music Journal 21.3 (Fall, 1997): 17.
157 The Extensible Toy Piano Project, ed. David Claman and Matt Malsky, 1 Jan. 2005, Clark U., Worcester, MA, 21
August 2005, <http://www.clarku.edu/xtp/xtp.html>.
92 The Equally-Tempered Archetype
the child’s toy that in time will also capture the imagination of the modern composer. The instrument is
usually made out of wood or plastic, and is dimensionally less than fifty centimetres in width, with a range
between two diatonic and three chromatic octaves. It has a simple sounding mechanism (similar to that
of the full-sized keyboard glockenspiel) consisting of plastic hammers operated via a keyboard, which
strike fixed metal plates or steel rods. Traditionally, toy pianos were modelled on uprights, but following
the 1950s grand piano varieties were commonplace.
Fig. 3. Matt Malsky, Schoenhut Model 6625: 25-Key Toy Piano.158
Contemporary works that have incorporated the toy piano include John Cage’s Suite for Toy Piano
(1948) and George Crumb’s Ancient Voices of Children (1970), as well as other works by Renaud
Gagneux, Mauricio Kagel, Louis Roquin, Zygmunt Krauze, and Leonid Aleksandrovich, among many.
Internationally acclaimed concert pianist Margaret Leng Tan made her debut on the toy piano in 1993 at
New York’s Lincoln Centre, and went on to introduce the model 6625, 25-key Schoenhut Traditional
Spinet to Carnegie Hall in 1997; also releasing a compact disc entitled The Art of the Toy Piano in that
same year – a collection of works by Stephen Montague, John Lennon and Paul McCartney, Toby
Twining, Jed Distler, Philip Glass, David Lang, Julia Wolfe, Ludwig van Beethoven, Guy Klucevsek, Raphael
Mostel, and Erik Satie.159
158 Matt Malsky, “Schoenhut Model 6625: 25-Key Toy Piano,” Feb. 2005, The Extensible Toy Piano Project, ed.
David Claman and Matt Malsky, 1 Mar. 2005, Clark U., Worcester, MA, 21 Aug. 2005,
<http://www.clarku.edu/xtp/photo.html>. 159 Hugh Davies, “Toy Piano,” The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie and John Tyrrell,
2nd ed., vol. 12 (London: Macmillan Reference, 2001) 615.
The Equally-Tempered Archetype 93
The Schoenhut Toy Piano Sample The Schoenhut model 6625, 25-key toy piano samples (recorded in dead studio space [96kHz/24bit]
by engineer John Shirley at Clark University, Department of Visual and Performing Arts, in Worcester, MA,
USA, utilizing two Neumann TLM 103s [positioned front-L and back-R] and a Nuendo recorder) include
three sets of twenty-five (two-octave chromatic span) forte ( � ), mezzo forte ( � ), and piano ( � )
samples, as well as one corresponding set of keyboard release clicks. All these sounds multi-sampled on
an Akai S3000XL Midi Stereo Digital Sampler – tuned firstly to standard A=440Hz twelve-tone equal
temperament, modified within thirty velocity cross-faded patches, and then operated via a midi
sequencer; a gong-like detuned middle C (octave down) sounding the tonal centre, as well as marking
the downbeat, while another severely gated alternative providing the rhythmic pulse. The final Schoenhut
Toy Piano sample may therefore be characterized as comprising of six basic sample elements (three
dynamic ranges, keyboard clicks, gong, and counter).160 The musical rationale of the gong and counter
has a close association with that of the gong and kempli in the gamelan music of Indonesia. Michael
Tenzer explains the significance of the gong in gamelan music:
“The most important function of the gongs in gamelan music is to mark structural points in a composition.
The number of gongs employed for this purpose depends on the ensemble. Gong kebyar uses one or
two gong ageng (the largest and deepest) for the beginnings and endings of melodies and other strong
accents. If two, they are used in alternation and never together. Other divisions are the responsibility of
the medium-sized kempur, the small, chiming kemong, and the nearly ever-present kempli, whose clear, dry
sound taps out the steady beat on which all musicians depend when the rhythms get tricky. Other types
of gamelans make use of some of these gongs, plus others like the tiny kelenang, the flat-bossed kajar and
bebende, or a resonant version of the kempli called tawa-tawa.”161
The elements then undergoing further processing within a digital audio editor PC software environment
(Sony Sound Forge 8.0). A modified ‘white noise’ preset of the Waves X-Noise 5.2 plug-in is utilized to
remove unwanted noise from all the samples (threshold: ß20.0dB, reduction: 70%; [dynamics] attack:
30ms, release: 220ms; [high shelf] frequency: 1415Hz, gain: +0.0dB), while the DSound Stomp’n Fx DN-
160 The Schoenhut model 6625, 25-key toy piano features are as follows: “25-key two-octave designer spinet; age
range: 3 and up; chromatically tuned with lovely chime like notes produced by little hammers striking precision ground,
German steel music rods; play-by-colour teaching method makes playing simple and fun; removable colour strip fits
behind the keys to guide small fingers from chord to chord; helps to build your child’s confidence and develop basic
playing skills; comes with a songbook which contains a collection of familiar tunes; available in mahogany, white, or red
finish; dimensions: 19¾” highÏ10¼” deepÏ17” wide; matching bench: 9¼” highÏ6” deepÏ10” wide; weighs 17 lbs.”
For a further discussion, see Schoenhut Toy Piano Company: Over 130 Years, 2005, Schoenhut Toy Piano Company, 21
Aug. 2005, <http://www.toypiano.com/6625.htm>.
161 Michael Tenzer, Balinese Music (Singapore: Periplus Editions, 1991) 37.
94 The Equally-Tempered Archetype
SG1 Noise Gate DX plug-in, to gate any remaining unwanted sonic material from the keyboard clicks and
counter samples (tone level: ß45dB, attack: 0.0ms, release: 185ms). Sound Diffusion The optimum method of performance for Exposiciones for Sampled Microtonal Schoenhut Toy Piano is
7-channel sound diffusion, with a scheme designed around the speaker system available in the
performance space of the Razzo Recital Hall at Clark University, Department of Visual and Performing Arts,
in Worcester, MA, USA, although the work may be performed utilizing a simple two-channel system (CD
player). The system incorporates seven EAW speakers in various dimensions: three on-stage large soffit-
mounted speakers (left front = 1, centre front = 2, right front = 3); plus four other smaller ‘left and right
side sets’ soffit-mounted speakers (right side = 4, left side = 7); as well as left and right rear sets (left rear
= 6, right rear = 5). Sound diffusion organized in such a manner as to sonically delineate the various
elements of the toy piano samples: toy piano (1 = 75.0%, 3 = 100.0%, 4 = 12.5%, 7 = 50.0%); key
clicks (1 = 100.0%, 3 = 75.0%, 4 = 50.0%, 7 = 12.5%); gong (2 = 100.0%); downbeat counter (4 =
50.0%, 5 = 100.0%, 6 = 100.0%, 7 = 50.0%); and upbeat counter (1 = 50.0%, 3 = 50.0%, 4 =
100.0%, 5 = 50.0%, 6 = 50.0%, 7 = 100.0%).
Fig. 4. Sound diffusion matrix (Razzo Recital Hall, Clark University, Worcester, MA, USA)
The Equally-Tempered Archetype 95
Polyrhythmic Theory The New Harvard Dictionary of Music defines a polyrhythm as being “the simultaneous use of two or
more rhythms that are not readily perceived as deriving from one another or as simple manifestations of
the same meter; sometimes also cross-rhythm.” The most common examples of this phenomena is
illustrated via the juxtaposing of triple and duple subdivisions of the beat, or alternatively of 3/4 and 6/8
meters. In classical music, the simple polyrhythmic technique is termed ‘hemiola’.162 According to Kevin
Barrett, a better understanding of polyrhythms can be achieved via their conceptualization as rhythmic
ratios, and therefore “three notes of equal value over a pulse of two notes” interpreted as a “ratio of
three notes to two notes, or 3:2;” or the inverted alternative of “two notes of equal value over a pulse of
three notes” interpreted as a “ratio of two notes to three notes, or 2:3.” Barrett establishes the criteria for
a true polyrhythm as being when the two rhythms in question do not have a common divisor other than
one, and therefore excluding ratios such as 4:2 and 2:4.163
American guitarist Steve Vai provides a thorough discussion on the technique required to perform
complex polyrhythmic ratios in Little Black Dots: Tempo Metal (an online article discussing his 1978
transcriptions for American composer, guitarist, singer, film director, and satirist, Frank Zappa [1940-
93]).164 In the article, 5ç:2Ê (transcribed as five quavers in the time of two crotchets) is utilized as an
example to demonstrate that the “first number (5) shows the number of beats to be superimposed over
the space provided,” while the “second number (2) designates the number of beats upon which the first
number is to be superimposed.” It is understood therefore, that by finding a common denominator for
the two (in this case the number ten, and therefore sixteenth notes), and “subdividing and putting five
units of measurements on both sides of the beat, you can now see that the second beat will fall on the
upstroke of the third eighth note of the quintuplet.” The following example demonstrates how the 5:2Ê polyrhythmic subdivision, which has the common denominator of ten, is subdivided into five groups of
semiquavers in order to delineate crotchet pulses.
162 Randel, ed., The New Harvard Dictionary of Music 646.
163 Kevin Barrett, “Understanding Polyrhythms,” Funkster’s Groove Theory, 23 Mar. 2004, 18 Jan. 2006,
<http://www.funkdrums.com/polyrhythm.html>. 164 “With more than 60 albums to his credit, composer, arranger, guitarist, and bandleader Frank Zappa
demonstrated a mastery of pop idioms ranging from jazz to rock of every conceivable variety, penned electronic and
orchestral works, parlayed controversial satire, and testified in Congress against censorship. As astute an entrepreneur as
he was a musician, he was impatient with any division between popular and high art; he combined scatological humour
with political wit, required of his players (Little Feat founder Lowell George, guitarists Adrian Belew and Steve Vai, and
drummer Terry Bozzio among them) an intimidating skill, and displayed consistent innovation in instrumental and studio
technology.” For a further discussion, see Patricia Romanowski, Holly George-Warren, and Jon Pareles, ed., The New
Rolling Stone Encyclopedia of Rock & Roll (New York: Fireside, 1995) 1111.
96 The Equally-Tempered Archetype
Ex. 7. 5:2Ê Polyrhythmic subdivision
The second example demonstrates how the 7ç:3Ê polyrhythmic subdivision (transcribed as seven
quavers in the time of three crotches), which has the common denominator of twenty-one, is subdivided
into seven groups of semiquaver triplets in order to delineate crotchet pulses.165
Ex. 8. 7ç:3Ê Polyrhythmic subdivision
The organizational framework of the ‘polyrhythm 1-16 set’ scheme utilized in the work is based on the
proposition of including all the fifty-seven polyrhythmic sets available between the ranges of two and
fifteen, and therefore one-hundred-and-fourteen unique polyrhythms. All the polyrhythms have been
organized in ascending order of complexity, as well as ‘golden mean’ arch form defined polyrhythmic
density, with equal temperaments one to four allocated one polyrhythmic set each (3:2Ê [2:3Ê]; 5ç:2Ê
[2É:5Ê]; 4:3Ê� [3:4�]; 5:3Ê� [3:5�]); five to eight, two (7ç:2Ê [2É�:7Ê�] and 5:4Ê [4:5Ê]; 7ç:3Ê [3É�:7Ê] and 9è:2Ê
[2È:9Ê]; 8ç:3Ê [3É:8Ê] and 7:4Ê [4:7Ê]; 6:5Ê [5:6Ê] and 7:5Ê�� [5:7Ê]; nine to twelve, three (11è:2Ê [2È:11Ê],
10ç:3Ê [3É:10Ê], and 9ç:4Ê [4É:9Ê]; 8:5Ê [5:8Ê], 7:6Ê [6:7Ê], and 11ç:3Ê [3É:11Ê]; 9:5Ê [5:9Ê], 13è:2Ê
[2È:13Ê], and 11ç:4Ê [4É:11Ê]; 8:7Ê [7:8Ê], 13è:3Ê [3È:13Ê], and 11ç:5Ê [5É:11Ê]); thirteen to sixteen, four
(9:7Ê [7:9Ê], 15è:2Ê [2È:15Ê], 14è:3Ê [3È:14Ê], and 13ç:4Ê [4É:13Ê]; 12ç:5Ê [5É:12Ê], 11:6Ê [6:11Ê], 10:7Ê
[7:10Ê], and 9:8Ê [8:9Ê]; 13ç:5Ê [5É:13Ê], 11:7Ê [7:11Ê], 15ç:4Ê [4É:15Ê], and 14ç:5Ê [5É:14Ê]; 13ç:6Ê
[6É:13Ê], 12:7Ê [7:12Ê], 11:8Ê [8:11Ê], and 10:9Ê [9:10Ê]); seventeen to nineteen, three (13:7Ê [7:13Ê],
11:9Ê [9:11Ê], and 13:8Ê [8:13Ê]; 11:10Ê [10:11Ê], 15ç:7Ê [7É:15Ê], and 13:9Ê [9:13Ê]; 15:8Ê [8:15Ê], 14:9Ê
[9:14Ê], and 13:10Ê [10:13Ê]); twenty to twenty-two, two (12:11Ê [11:12Ê] and 13:11Ê [11:13Ê]; 14:11Ê
[11:14Ê] and 13:12Ê [12:13Ê]; 15:11Ê [11:15Ê] and 14:13Ê [13:14Ê]); while the final two, twenty-three to
twenty-four, one each (15:13Ê [13:15Ê]; 15:14Ê [14:15Ê]). A further aspect of the tripartite organizational
framework is the lower limit imposed to the units of time: 2, or half note (1-et to 9-et); 4, or quarter note
(10-et to 17-et); and 8, or eighth note (18-et to 24-et).
165 Steve Vai, “Little Black Dots: Tempo Metal,” The Official Steve Vai Website, 1983, 18 Jan. 2006,
<http://www.vai.com/LittleBlackDots/tempomental.html>
The Equally-Tempered Archetype 97
Fig. 5
. Po
lyrh
ythm
1-1
6 se
t 1-
et
2-et
3-et
4-et
5-et
6-et
7-et
8-et
9-et
10-e
t
11-e
t
12-e
t
13-e
t
14-e
t
15-e
t
16-e
t
17-e
t
18-e
t
19-e
t
20-e
t
21-e
t
22-e
t
23-e
t
24-e
t
3:2 Ê
5ç:2
��4:
3��
5:3Ê
��7ç
:2��
7ç:3
Ê��8ç
:3��
6:5Ê
��11
è:2Ê
�8:
5�
9:5Ê
�8:
7�
9:7Ê
12
ç:5Ê
13
ç:5Ê
13
ç:6Ê
13
:7Ê
11:1
0Ê
15:8
Ê 12
:11Ê
14:1
1Ê
15:1
1Ê
15:1
3Ê
15:1
4Ê
(2:3
�) (2
É:5Ê�)
(3:4
�) (3
:5�)
(2É�:7Ê
�) (3
�:7�)
(3É:8
�) (5
:6�)
(2È:1
1Ê)
(5:8
Ê) (5
:9Ê)
(7:8
Ê) (7
:9Ê)
(5É:1
2Ê)
(5É:1
3Ê)
(6É:1
3Ê)
(7:1
3Ê)
(10:
11Ê)
(8:1
5Ê)
(11:
12Ê)
(11:
14Ê)
(11:
15Ê)
(13:
15Ê)
(14:
15Ê)
5:4Ê
��9è
:2��
7:4Ê
��7:
5��
10ç:
3�
7:6Ê
�13
è:2Ê
�13
è:3Ê
�15
è:2Ê
11
:6Ê
11:7
Ê 12
:7Ê
11:9
Ê 15
ç:7Ê
14:9
Ê 13
:11Ê
�13
:12Ê
�14
:13Ê
(4:5
�) (2
È:9Ê�)
(4:7
�) (5
:7�)
(3É:1
0Ê)
(6:7
Ê) (2
È:13Ê
)
(3È:1
3Ê)
(2È:1
5Ê)
(6:1
1Ê)
(7:1
1Ê)
(7:1
2Ê)
(9:1
1Ê)
(7É:1
5Ê)
(9:1
4Ê)
(11:
13Ê)
(12:
13Ê)
(13:
14Ê)
9ç:4
Ê 11
ç:3Ê
11ç:
4Ê
11ç:
5Ê
14è:
3Ê
10:7
Ê 15
ç:4Ê
11:8
Ê 13
:8Ê
13:9
Ê 13
:10Ê
(4É:9
Ê) (3
É:11Ê
)
(4É:1
1Ê)
(5É:1
1Ê)
(3È:1
4Ê)
(7:1
0Ê)
(4É:1
5Ê)
(8:1
1Ê)
(8:1
3Ê)
(9:1
3Ê)
(10:
13Ê)
13ç:
4�
9:8Ê
14ç:
5Ê
10:9
Ê
(4É:1
3Ê)
(8:9
Ê) (5
É:14Ê
)
(9:1
0Ê)
98 The Equally-Tempered Archetype
Alain Daniélou’s Scale of Proportions The intervallic structure of the Indonesian pélog and sléndro pentatonic scales utilized in the work have
been defined by the ratios of just intonation, or the ‘scale of proportions’ (the harmonic division of the
octave), as presented by Daniélou in Music and the Power of Sound: The Influence of Tuning and Interval
on Consciousness. The ‘harmonic division of the octave’ is based on the harmonic series, and is a
division of the octave into fifty-three distinct intervals. It is a scale of just intonation, where the intervals
are called ‘pure’ or ‘just’ because there are no beats between the notes or their harmonics. The quarter-
tone (three-quarter-tone) is a result of the further division of the disjunctions of this scale (just diatonic
semitone, or major half-tone [16/15]), which altogether presents a total of sixty-six unique intervals (the
octave included). In Indian musical theory this system is referred to as the ‘sixty-six ærutis’ – a theoretical
extension to the system of the ‘twenty-two ærutis’ of contemporary performance practice.166 It must be
further noted that the ‘scale of proportions’ (the modal or harmonic division of the octave) is based on
the ‘modal’ musical model, and therefore in principal serves a selection of intervals with primary
relationships to a fixed tonic. Daniélou explains the significance of the ‘modal’ musical application of
intervals:
“Indian deæå music is essentially modal, which means that the intervals on which the musical structure is built
are calculated in relation to a permanent tonic. This does not mean that the relations between sounds
other than the tonic are not considered, but that each note will be established first according to its relation
to the fixed tonic and not, as in the case of the cycle of fifths, by any permutation of the basic note. The
modal structure can thus be compared to the proportional division of a straight line rather than to the
periodic movement of a spiral. According to the symbolism of numbers, the proportional divisions are
connected with certain ideas, forms, and emotions.”
With regards to the nature of harmonic science, Daniélou adds the following:
“The object of harmonic science is to classify these proportions according to their symbolism and the
feelings, images, or symbols they express. Only on this basis can modes be logically constructed and their
expression precisely defined. All the notes obtained in the harmonic system are distinct from those of the
cyclic system, which is based on different data. Yet though the notes are theoretically distinct and their
sequence follows completely different rules, in practice they lead to a similar division of the octave into
fifty-three intervals.”167
166 “Æruti means ‘to hear’ or ‘that which is heard’. Musically, it points to the interval, between notes, which can be
just perceived auditorily. Musically viable pitches in an octave are literally infinite: this was recognized and explicitly stated
so in our ancient texts. But for practical reasons twenty-two have been enumerated and distributed within the span Sa to
Sa’ (C to C’). Further, like the seven notes, they have been given names and divided into five classes based, purportedly,
on their aesthetic connotations.” For a further discussion, see B. Chaitanya Deva, Indian Music (New Delhi: Indian Council
for Cultural Relations, 1974) 29-30.
167 Daniélou, Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness 91-145.
The Equally-Tempered Archetype 99
Tabl
e 45
. A
lain
Dan
iélo
u’s
scal
e of
pro
por
tions
(th
e ha
rmon
ic d
ivisi
on o
f the
oct
ave)
Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
iddl
e C)
=26
1.62
5565
4Hz
DE
GRE
E N
UMBE
R N
OTE
IN
TERV
AL
RATI
O
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ ÐÔ
ÐÕ
ÐÖ
Ð× ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ ÑÓ
ÑÔ
ÑÕ ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÒÕ ÒÖ
C C¢
C£
CË /
Dì
CÚ
D¹
D¸
DÝ
CÍ /
Dí
D³
D²
D D¢
D£
DË /
Eì
DÚ
E¸
EÝ
DÍ /
Eí
E³
E² E E¢
E£
EË /
Fí
F³
F² F F¢
F£
FË / G
ì FÚ
uniso
n sy
nton
ic c
omm
a gr
eat d
iesis
G
reek
enh
arm
onic
or s
eptim
al q
uarte
r-ton
e gr
ave
or s
mal
l jus
t chr
omat
ic s
emito
ne, o
r min
or h
alf-t
one
Pyth
agor
ean
limm
a ju
st d
iato
nic
sem
itone
, or m
ajor
hal
f-ton
e gr
eat l
imm
a, a
cute
or l
arge
hal
f-ton
e th
ree-
qua
rter-t
one
grav
e or
sm
all t
one
just
min
or to
ne
just
maj
or to
ne (
9th
harm
onic
) ac
ute
or la
rge
tone
su
per
maj
or s
econ
d
five
qua
rter-t
ones
au
gmen
ted
seco
nd (7
5th
harm
onic
) Py
thag
orea
n m
inor
third
, or t
rihem
itone
ju
st m
inor
third
se
ven
qua
rter-t
ones
ne
utra
l thi
rd
grav
e or
sm
all m
ajor
third
ju
st m
ajor
third
(5t
h ha
rmon
ic)
Pyth
agor
ean
maj
or th
ird, o
r dito
ne (8
1st h
arm
onic
) ac
ute
or la
rge
maj
or th
ird
nine
qua
rter-t
ones
su
bfou
rth
grav
e or
sm
all f
ourth
ju
st a
nd P
ytha
gore
an p
erfe
ct fo
urth
ac
ute
or la
rge
four
th
sup
erfo
urth
el
even
qua
rter-t
ones
gr
ave
or s
mal
l aug
men
ted
four
th
1/1
81/8
0 12
8/12
5 31
/30
25/2
4 25
6/24
3 16
/15
27/2
5 13
5/12
4 80
0/72
9 10
/9
9/8
256/
225
5904
9/51
300
93/8
0 75
/64
32/2
7 6/
5 75
/62
8000
/656
1 10
0/81
5/
4 81
/64
32/2
5 31
/24
125/
96
320/
243
4/3
27/2
0 51
2/37
5 62
/45
25/1
8
1.00
0000
1.
0125
00
1.02
4000
1.
0333
33
1.04
1667
1.
0534
98
1.06
6667
1.
0800
00
1.08
8710
1.
0973
94
1.11
1111
1.
1250
00
1.13
7778
1.
1510
53
1.16
2500
1.
1718
75
1.18
5185
1.
2000
00
1.20
9677
1.
2193
26
1.23
4568
1.
2500
00
1.26
5625
1.
2800
00
1.29
1667
1.
3020
83
1.31
6872
1.
3333
33
1.35
0000
1.
3653
33
1.37
7778
1.
3888
89
261.
626
264.
896
267.
905
270.
346
272.
527
275.
622
279.
067
282.
556
284.
834
287.
106
290.
695
294.
329
297.
672
301.
145
304.
140
306.
592
310.
075
313.
951
316.
483
319.
007
322.
995
327.
032
331.
120
334.
881
337.
933
340.
658
344.
527
348.
834
353.
195
357.
206
360.
462
363.
369
0.00
0 21
.506
41
.059
56
.767
70
.672
90
.225
11
1.73
1 13
3.23
8 14
7.14
3 16
0.89
7 18
2.40
4 20
3.91
0 22
3.46
3 24
3.54
5 26
0.67
7 27
4.58
2 29
4.13
5 31
5.64
1 32
9.54
7 34
3.30
1 36
4.80
7 38
6.31
4 40
7.82
0 42
7.37
3 44
3.08
1 45
6.98
6 47
6.53
9 49
8.04
5 51
9.55
1 53
9.10
4 55
4.81
2 56
8.71
7
+00
+
22
+41
+
57
ß29
ß
10
+12
+
33
+47
ß
39
ß18
+
04
+23
+
44
+61
ß
25
ß06
+
16
+30
ß
57
ß35
ß
14
+08
+
27
+43
ß
43
ß23
ß
02
+20
+
39
+55
ß
31
100 The Equally-Tempered Archetype
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(F
RACT
ION
) RA
TIO
(D
ECIM
AL)
FR
EQUE
NCY
(H
ERTZ
) CE
NTS
TU
NIN
G
Ò×
ÒØ
ÒÙ ÓÐ
ÓÑ
ÓÒ
ÓÓ
ÓÔ ÓÕ
ÓÖ
Ó× ÓØ
ÓÙ
ÔÐ
ÔÑ
ÔÒ ÔÓ
ÔÔ
ÔÕ
ÔÖ Ô×
ÔØ
ÔÙ
ÕÐ
ÕÑ ÕÒ
ÕÓ ÐÑ
F ¤ F¥
GÝ
FÍ /
Gí
G³
G²
G G¢
G£
GË /
Aì
GÚ
A¸
AÝ
GÍ /
Aí
A³
A²
A A¢
A£
AË /
Bì
AÚ
B¹
B¸
BÝ
AÍ /
Bí
B³
B² B B¢
B£
BË /
Cí
C³
C² C
just
trito
ne, o
r aug
men
ted
four
th (
45th
har
mon
ic)
acut
e or
larg
e tri
tone
, or a
ugm
ente
d fo
urth
ac
ute
or la
rge
dim
inish
ed fi
fth
thirt
een
qua
rter-t
ones
su
bfift
h gr
ave
or s
mal
l fift
h ju
st a
nd P
ytha
gore
an p
erfe
ct fi
fth (
3rd
har
mon
ic)
acut
e or
larg
e fif
th
sup
erfif
th
fifte
en q
uarte
r-ton
es
augm
ente
d fi
fth (
25th
har
mon
ic)
Pyth
agor
ean
min
or s
ixth
ju
st m
inor
six
th
seve
ntee
n q
uarte
r-ton
es
neut
ral s
ixth
gr
ave
or s
mal
l maj
or s
ixth
ju
st m
ajor
six
th
Pyth
agor
ean
maj
or s
ixth
(27t
h ha
rmon
ic)
acut
e or
larg
e m
ajor
six
th
nine
teen
qua
rter-t
ones
au
gmen
ted
six
th
grav
e or
sm
all m
inor
sev
enth
Py
thag
orea
n m
inor
sev
enth
ac
ute
or la
rge
min
or s
even
th
twen
ty-o
ne q
uarte
r-ton
es (
29th
har
mon
ic)
neut
ral s
even
th
grav
e or
sm
all m
ajor
sev
enth
ju
st d
iato
nic
maj
or s
even
th (
15th
har
mon
ic)
Pyth
agor
ean
maj
or s
even
th
acut
e or
larg
e m
ajor
sev
enth
tw
enty
-thre
e q
uarte
r-ton
es
subo
ctav
e (1
25th
har
mon
ic)
grav
e or
sm
all o
ctav
e
octa
ve
45/3
2 64
/45
36/2
5 90
/62
375/
256
40/2
7 3/
2 24
3/16
0 19
2/12
5 31
/20
25/1
6 12
8/81
8/
5 50
/31
81/5
0 40
0/24
3 5/
3 27
/16
128/
75
31/1
8 12
5/72
22
5/12
8 16
/9
9/5
29/1
6 40
00/2
187
50/2
7 15
/8
243/
128
48/2
5 60
/31
125/
64
160/
81
2/
1
1.40
6250
1.
4222
22
1.44
0000
1.
4516
13
1.46
4844
1.
4814
81
1.50
0000
1.
5187
50
1.53
6000
1.
5500
00
1.56
2500
1.
5802
47
1.60
0000
1.
6129
03
1.62
0000
1.
6460
91
1.66
6667
1.
6875
00
1.70
6667
1.
7222
22
1.73
6111
1.
7578
13
1.77
7778
1.
8000
00
1.81
2500
1.
8289
89
1.85
1852
1.
8750
00
1.89
8438
1.
9200
00
1.93
5484
1.
9531
25
1.97
5309
2.00
0000
367.
911
372.
090
376.
741
379.
779
383.
241
387.
593
392.
438
397.
344
401.
857
405.
520
408.
790
413.
433
418.
601
421.
977
423.
833
430.
659
436.
043
441.
493
446.
508
450.
577
454.
211
459.
889
465.
112
470.
926
474.
196
478.
510
484.
492
490.
548
496.
680
502.
321
506.
372
510.
987
516.
791
52
3.25
1
590.
224
609.
776
631.
283
645.
188
660.
896
680.
449
701.
955
723.
014
743.
014
758.
722
772.
627
792.
180
813.
686
827.
592
835.
193
862.
852
884.
359
905.
865
925.
418
941.
126
955.
031
976.
537
996.
090
1017
.596
10
29.5
77
1045
.256
10
66.7
62
1088
.269
11
09.7
75
1129
.328
11
43.2
33
1158
.941
11
78.4
94
12
00.0
00
ß10
+
10
+31
+
45
ß39
ß
20
+02
+
23
+43
+
59
ß27
ß
08
+14
+
28
ß75
ß
37
ß16
+
06
+25
+
41
ß45
ß
23
ß04
+
18
+30
ß
55
ß33
ß
12
+10
+
29
+43
ß
41
ß22
+00
The Equally-Tempered Archetype 101
Notation for the Twenty-Four Equal Temperaments
The notation adopted in Exposiciones for Sampled Microtonal Schoenhut Toy Piano (inspired by
Daniélou’s work) is highly illustrative of the affects of each individual equal temperament on the two
pentatonic scales, and their consequential intervallic deviation from just intonation. It is based on
approximations of the harmonic division of the octave, to the closest syntonic comma (81/80), and
Daniélou’s subdivision of the whole-tone, or just major tone (9/8).
Table 46. Alain Daniélou’s subdivision of the whole-tone
NOTE INTERVAL RATIO (FRACTION) CENTS
C
C¢
C£
CË / Dì
CÚ
D¹
D¸
DÝ
CÍ / Dí
D³
D²
D
unison
syntonic comma
great diesis
Greek enharmonic or septimal quarter-tone
grave or small just chromatic semitone
Pythagorean limma
just diatonic semitone
great limma
three-quarter-tone
small tone
just minor tone
just major tone (9th harmonic)
1/1
81/80
128/125
31/30
25/24
256/243
16/15
27/25
135/124
800/729
10/9
9/8
0.000
21.506
41.059
56.767
70.672
90.225
111.731
133.238
147.143
160.897
182.404
203.910
The system of accidentals utilizes twenty-three unique symbols – three identical to conventional sharp
(Ú), flat (Û), and natural (Ö) accidentals (raising, lowering or neutralizing a tone by 25/24, or 70.672 cents),
with an additional four derived from standard quarter-tone notation in twentieth century contemporary
music practice; the latter representative of the division of the ‘unequal’ major half-tone, or just diatonic
semitone (16/15), and not of the ‘equal’ quarter-tone (1:1.029302237, or 50.000 cents) derived from
twenty-four-tone equal temperament ( 24 2 ). The quarter-tone symbols include (Ë) and (í), raising or
lowering a tone by 31/30 (Greek enharmonic or septimal quarter-tone), or 56.767 cents; and (Í) and
(ì), raising or lowering a tone by 135/124 (three-quarter-tone), or 147.143 cents. Additional symbols
include (¢, ¦, ¤, ¨) and (², ¶, ´, ¸), raising or lowering an unaltered, naturalized, sharpened, or flattened
tone by 81/80 (one syntonic comma), or 21.506 cents; and (£, §, ¥, ©) and (³, ·, μ, ¹), raising or
lowering an unaltered, naturalized, sharpened, or flattened tone by 128/125 (approximately two syntonic
commas, or one great diesis), or 41.059 cents. It should be noted that two syntonic commas is equal to
the ratio 6561/6400, and 43.013 cents.
102 The Equally-Tempered Archetype
Paul Rapoport has devised a viable alternative for notating equal temperaments based on the expansion
of Pythagorean notation, limited to the fifth harmonic. Nevertheless, this approach was not adopted in
the work as it would not have provided a platform for comparative analysis of the twenty-four equal
temperaments. In spite of this very fact, the system of notation was extensively explored and in view of
its virtues, deserves a mention.
In simple terms, Rapoport’s system of notation is based on firstly acknowledging the octave, or the
symbol a, and establishing the number of units and size of an individual unit (in 12-et, twelve units equal
to a hundred cents per unit, or 1200÷12=100). The next task then becomes to identify three key
elements: the fifth, or v closest to just (in 12-et equal to seven units, or 700 cents); major third, or t
closed to just (in 12-et equal to four units, or 400 cents); and the Pythagorean major third, or �,
generated via the formula �=4vß2a (in 12-et also equal to four units, or 400 cents). It is then simply a
matter of further establishing the existence of the following intervals, and setting a unit where appropriate:
i. syntonic comma (81/80), or k (�ßt, or 4vßtß2a)
ii. Pythagorean comma (531441/524288), or p (3�ßa, or 12vß7a)
iii. great diesis (128/125, or 41.059 cents), or d (aß3t, or 3kßp)
iv. skhisma (32805/32768, or 1.955 cents), or s (2�+tßa, or pßk) v. diaskhisma (2048/2025, or 19.553 cents), or q (aß[�ß2t] or 2kßp)
vi. Pythagorean and just major tone (9/8), or �w (2vßa)
vii. Pythagorean limma (256/243), or �h (3aß5v)
viii. Pythagorean apotome (2187/2048), or ! (7vß4a)
ix. just minor third (6/5), or jm3 (vßt)
x. Pythagorean minor third, or trihemitone (32/27, or 294.135 cents), or �m3 (v�)
xi. just diatonic semitone (16/15), or jh (aßvßt)
xii. grave or small just chromatic semitone (25/24), or jc (2tßv)
In 12-et, only �w, �h, !, jm3, �m3, jh, and jc present a positive result, with 2, 1, 1, 3, 3, 1, and 1 units
respectively; and therefore the Pythagorean apotome (2187/2048), or !, is selected as the appropriate
notational device. The paper concludes that the “most important comma for notation is k (syntonic
comma), followed in order by d (great diesis), q (diaskhisma), and s (skhisma).” Rapoport’s system of
notation certainly has its merits when dealing with one single equal temperament at a time, although in the
context of tuning modulations, presented is the dilemma of having to decipher seventy-two unique
symbols for chromas such as the syntonic comma (81/80), Pythagorean comma (531441/524288), great
The Equally-Tempered Archetype 103
diesis (128/125), skhisma (32805/32768), and diaskhisma (2048/2025), along with conventional quarter-
sharp and flat, sharp and flat, and three-quarter-sharp and flat symbols.168
The ‘notation for the twenty-four equal temperaments’ table presents all the possible pitch
allocations within the octave, and represents the scheme utilized in the work for the notation of all equal
temperament between one and twenty-four. The system has been adopted for practicality, although it is
able to approximate the intervals to the closest syntonic comma, and therefore useful when making
comparisons between one equal temperament and another. Sléndro and Pélog Scales Sléndro and pélog scales represent the two genera (the feminine and masculine genus) of Javanese
music. Saléndro or sléndro, derived from Æailendra, being the name of the Indian dynasty in regional rule
from the latter part of the eighth century through to the latter part of the tenth century.169 The tunings of
sléndro and pélog scales from East and Central Java are based on two very distinct non-equidistant
pentatonic and heptatonic tuning systems or laras, and accordingly are represented within the Javanese
orchestra or gamelan via different sets of instruments.170 A gamelan orchestra or ensemble is typically
made up of distinct combinations of gongs, metallophones, xylophones, drums, bowed and plucked
strings, bamboo flutes, small cymbals and singers, with the participation of between “three or four
musicians” and “twenty-five instrumentalists and ten to fifteen singers.”171 A complete gamelan consists of
a double set of instruments; each tuned to sléndro and pélog, and called a gamelan gedhé. The
gamelan situated at the Sri Wedhari theatre auditorium in Solo, Java – belonging to the Central Javanese
gamelan ensemble tradition of Karawitan – serves as an illustration of the two tuning systems, although it
168 Paul Rapoport, “The Notation of Equal Temperaments,” Xenharmonikôn: An Informal Journal of Experimental
Music 16 (Autumn, 1995): 61-84.
169 Laurence Picken, “The Music of Far Eastern Asia: 2. Other Countries,” Ancient and Oriental Music, ed. Egon
Wellesz (London: Oxford U. Press, 1957) 166-67. 170 “Sléndro is a five-tone scale that very roughly approaches equal-size intervals. The intervals vary within a given
scale and across orchestras, but the underlying tuning concept can be considered as a rough five-tone equal
temperament,” explains Braun, while “pélog is a seven-tone scale, whose underlying tuning concept is less obvious. One
possibility is that the tuning approaches a nine-tone equal temperament, using both single steps (133 cents) and double
steps (267 cents) from an imagined nine-tone equal temperament scale, which is a particular element of the historical
tradition of Javanese Music.” Martin Braun, “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic
Musical Scale,” Neuroscience of Music, 11 Mar. 2006, Sweden, 10 Apr. 2006,
<http://web.telia.com/~u57011259/pelog_main.htm>.
171 Benjamin Brinner, Knowing Music, Making Music: Javanese Gamelan and the Theory of Musical Competence and
Interaction (Chicago: U. of Chicago Press, 1995) XVII-XX.
104 The Equally-Tempered Archetype
Table 47. Notation for the twenty-four equal temperaments
NOTATION COMPASS OR RANGE
(CENTS)
NOTATION COMPASS OR RANGE
(CENTS)
C
C¢
C£
CË / Dì
CÚ
D¹
D¸
DÝ
CÍ / Dí
D³
D²
D
D¢
D£
DË / Eì
DÚ
E¸
EÝ
DÍ / Eí
E³
E²
E
E¢
E£
EË / Fí
F³
F²
F
F¢
F£
FË / Gì
FÚ
F¤
1189.247 � 10.752
10.753 � 31.282
31.283 � 48.912
48.913 � 63.719
63.720 � 80.448
80.449 � 100.977
100.978 � 122.483
122.484 � 140.189
140.190 � 154.019
154.020 � 171.650
171.651 � 193.156
193.157 � 213.685
213.686 � 233.503
233.504 � 252.110
252.111 � 267.629
267.630 � 284.358
284.359 � 304.887
304.888 � 322.593
322.594 � 336.423
336.424 � 354.053
354.054 � 375.560
375.561 � 397.066
397.067 � 417.595
417.596 � 435.226
435.227 � 450.032
450.033 � 466.761
466.762 � 487.291
487.292 � 508.797
508.798 � 529.327
529.328 � 546.957
546.958 � 561.764
561.765 � 579.470
579.471 � 599.999
F¥
GÝ
FÍ / Gí
G³
G²
G
G¢
G£
GË / Aì
GÚ
A¸
AÝ
GÍ / Aí
A³
A²
A
A¢
A£
AË / Bì
AÚ
B¹
B¸
BÝ
AÍ / Bí
B³
B²
B
B¢
B£
BË / Cí
C³
C²
C
600.000 � 620.528
620.529 � 638.234
638.235 � 653.041
653.042 � 670.671
670.672 � 691.201
691.202 � 712.483
712.484 � 733.013
733.014 � 750.867
750.868 � 765.674
765.675 � 782.403
782.404 � 802.932
802.933 � 820.638
820.639 � 831.391
831.392 � 849.021
849.022 � 873.605
873.606 � 895.111
895.112 � 915.640
915.641 � 933.271
933.272 � 948.077
948.078 � 965.783
965.784 � 986.313
986.314 � 1006.842
1006.843 � 1023.586
1023.587 � 1037.416
1037.417 � 1056.008
1056.009 � 1077.515
1077.516 � 1099.021
1099.022 � 1119.550
1119.551 � 1136.279
1136.280 � 1151.086
1151.087 � 1168.716
1168.717 � 1189.246
1189.247 � 10.752
The Equally-Tempered Archetype 105
must be noted that absolute pitch, as well as fixed theoretically defined tuning systems, should be
considered anomalies, and hence intonation traditions unique to but one gamelan. The roman numerals
above the pitches represent a form of the Central Javanese cipher notation system of Kepatihan, which is
based on the principles of the undotted numeral representative of the central octave, while dots above
and below, delineating the high and low octaves respectively. The names accorded to each pitch are
based on a system of interchangeable sléndro and pélog repertoire. It must be further noted that in spite
of the heptatonic structure of pélog, the tuning system is essentially pentatonic in traditional
compositions, with additional ‘auxiliary’ pitches serving as temporary substitute neighbour tones.172
Ex. 9. Gamelan gedhé sléndro and pélog tunings (Sri Wedhari theatre auditorium, Solo, Central Java)
In order to ascertain the size of the intervals of this particular gamelan gedhé, the following example
represents a transposition to the key of C of the actual tunings.
Ex. 10. Gamelan gedhé sléndro and pélog tunings in the key of C
The following presentation represents a harmonically tuned ‘just intonation’ reinterpretation of the sléndro
and pélog scales; the intervallic delineations facilitating the display of the harmonic characteristics.
Additional auxiliary tones have been added to the scales in order to generate a heptatonic scale
172 Marc Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory (Berkeley: U. of California
Press, 2004) XV-41.
106 The Equally-Tempered Archetype
structure from the two essentially pentatonic forms. The sléndro tuning may be simplified as consisting of
the intervals: major second, perfect fourth, perfect fifth, and minor seventh; and in the key of C consisting
of the pitches: C, D, F, G, and B"; while pélog: minor second, minor third, perfect fifth, and minor sixth;
and in the key of C consisting of the pitches: C, D", E", G and A".
Ex. 11. The harmonic characteristics of the sléndro and pélog scales i. The sléndro scale
ii. The pélog scale
One-Tone Equal Temperament One-tone equal temperament is generated by the factor 1 2 , and produces the intervals of the octave
(with the simple frequency ratio of 2/1) and the unison (1/1), or what David D. Doty describes as “the
‘non-interval’ between two tones with exactly the same frequency.” 1/1, or the unison represents relative
pitch, and not absolute pitch in general terms, although in this particular context, the frequency relative to
A=440Hz (standard pitch), and equal to 261.626Hz. The octave is a twofold multiplication of 1/1,
hence equal to the frequency of 523.251Hz. The octave may also be stated as being the primary interval
The Equally-Tempered Archetype 107
of the prime number two, and “the only interval just intonation and equal temperament have in
common.”173 Sethares proclaims the octave as “the most consonant interval after the unison.” 2/1 and
1/1 are of course represented by the first two partials of the harmonic series.174 “The proportion of the
whole to its half or of the half to the whole is so natural that it is the first to be understood,” states French
composer and theorist Jean-Philippe Rameau; acknowledging Italian music theorist and composer
Gioseffo Zarlino (1517-90) for recognizing the octave as “the mother, the source, and the origin of all
intervals,” because, “by the division of its two terms all other harmonious chords are generated.”175 The
first three bars of the work highlight the singular application of both intervals.
Ex. 12. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 1-3
The following table depicts the tonal resources of one-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 48. The one-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÑ
C
C
unison
octave
1.000000
2.000000
261.626
523.251
0.000
1200.000
+00
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
173 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 36.
174 Sethares, Tuning, Timbre, Spectrum, Scale 1-3. 175 Jean-Philippe Rameau, Treatise on Harmony, trans. Philip Gossett (New York: Dover Publications, 1971) 8-10.
108 The Equally-Tempered Archetype
Ex. 13. 1-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 49. 1-tone equal temperament tuning matrix no. 1 (Program 01)
NOTE C – – – – – – – – – – – – – – – – – – – – – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00
Two-Tone Equal Temperament Two-tone equal temperament is generated by the factor 2 2 , and introduces the equal tritone, or the
sum of three equal whole-tones, which according to Richard Bobbitt was “proscribed in early
polyphonic music as diabolus in musica (the devil in music),” and “remains even today a ferment among
the family of intervals due to its peculiar characteristics.”176 Interesting among these characteristics is the
fact that the interval of a tritone does not appear in the first sixteen partials of the harmonic series, with
the just tritone, or augmented fourth (45/32, or 590.224 cents) first identified as the forty-fifth harmonic;
the alternative cyclic tritone (3ô÷/2÷ô, or 591.885 cents), as the forty-seventh ascending fifth in the ‘scale
of fifths’, or cyclic division of the octave; while the simpler Pythagorean tritone (729/512, or 611.730
cents), as the sixth ascending fifth. The equal tritone, or augmented fourth ( 2 2 ) is produced by the
176 According to Hans Tischler, “The diabolus in musica, however often it was exorcised by theorists, was
nevertheless as familiar to musicians as was Satan to the people of the age – and of much later ages, witness Luther and
the belief in witches. The avoidance of the tritone was hardly so generally practiced that it can be used as an unfailing
guide to musica ficta.” For a further discussion, see Hans Tischler, “Musica Ficta in the Thirteenth Century,” Music & Letters
54.1 (Jan., 1973): 48.
The Equally-Tempered Archetype 109
division of the octave into two equal parts (or two tones equal to 600.000 cents each), and
approximated with the frequency ratio 181/128.177 It is absolutely symmetrical, and is therefore non-
invertible.178 The notation (F¥) is reflective of the equal tritone fitting within the range of 600.000 and
620.528 cents, and may therefore be prescribed as approximating the acute or large tritone, or
augmented fourth (64/45, or 609.776) with a falsity of ß9.776 cents. It should nevertheless be noted
that theoretically it additionally represents the just tritone (45/32) with a falsity of +9.776 cents. The
pélog scale form scheme of the composition is able to accommodate the interval to produce a two-
tone symmetrical scale form, as well as a simultaneous sonority with an intervallic value of 600.000 cents.
Ex. 14. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 4
The following table depicts the tonal resources of two-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 50. The two-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÑ
C
F¥
C
unison
equal or acute or large tritone, or augmented fourth
octave
1.000000
1.414214
2.000000
261.626
369.994
523.251
0.000
600.000
1200.000
+00
+00
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
177 Daniélou, Tableau Comparatif des Intervalles Musicaux 144-45.
178 Richard Bobbitt, “The Physical Basis of Intervallic Quality and its Application to the Problem of Dissonance,”
Journal of Music Theory 3.2 (Nov., 1959): 190-92.
110 The Equally-Tempered Archetype
Ex. 15. 2-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 51. 2-tone equal temperament tuning matrix no. 1 (Program 02)
NOTE C – – – – – – – – – – F¥ – – – – – – – – – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00
Three-Tone Equal Temperament Three-tone equal temperament is generated by the factor 3 2 , and introduces the equal major third
(approximately 63/50, or 400.000 cents) and its inversion, the equal minor sixth ( 23 ]2[ , approximately
100/63, or 800.000 cents).179 Relative to the just major thirds (5/4s) and just minor sixths (8/5s, or
813.686 cents) of just intonation, three-tone equal temperament produces major thirds ( 3 2 ) and minor
sixths ( 23 ]2[ ) with a falsity of +13.686 and ß13.686 cents on each count. The tuning modulation,
although unable to generate a recognizable sléndro or pélog scale form, is nevertheless able to deliver
the first theoretical triad, with the enharmonic reinterpretation of the pitches C, E¢, and A¸ generating an
augmented triad (C, E¢, and G¤). The triad is nevertheless not included in the work for the very reason
that formulated aesthetic guideless predicate a clear intention to highlight principal tones (of the sléndro
or pélog scale forms generated), with an attempt to exclude secondary ‘auxiliary’ tones from the melodic
framework as much as possible. Simultaneous sonorities introduced in the work with this temperament
include the dyads C and A¸, and E¢ and C, which represent the interval of an equal minor sixth ( 23 ]2[ ).
179 Daniélou, Tableau Comparatif des Intervalles Musicaux 102-03.
The Equally-Tempered Archetype 111
Ex. 16. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 7
The following table depicts the tonal resources of three-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 52. The three-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÑ
C
E¢
A¸
C
unison
equal or Pythagorean major third, or ditone
equal or Pythagorean minor sixth
octave
1.000000
1.259921
1.587401
2.000000
261.626
329.628
415.305
523.251
0.000
400.000
800.000
1200.000
+00
+00
+00
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 17. 3-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
112 The Equally-Tempered Archetype
Table 53. 3-tone equal temperament tuning matrix no. 1 (Program 03)
NOTE C – – – – – – E¢ – – – – – – A¸ – – – – – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00
Four-Tone Equal Temperament Four-tone equal temperament ( 4 2 ) presents a series of four equal tones 300.000 cents in size
(approximately 44/37),180 and therefore the juxtaposition of four equal minor thirds, or two sets of two-
tone equally-tempered intervals interlocked an equal minor third ( 4 2 ) apart. In comparison with the just
minor third (6/5), the equal minor third ( 4 2 ) has a falsity of ß15.641 cents, while its inversion, the equal
major sixth ( 34 ]2[ ), a falsity of +15.641 cents with the just major sixth (5/3, or 884.359 cents). Within
the combined sléndro and pélog scale form schemes, the temperament presents the pitches of the
equally-tempered diminished seventh tetrad, with the pitches C, E¸, F¥, and A¢ enharmonically
reinterpreted as C, E¸, G¸, and A¢. The only simultaneous sonority introduced in the work with this
temperament is the dyad E¸ and C, which represents the interval of an equal major sixth ( 34 ]2[ ).
Ex. 18. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 9
The following table depicts the tonal resources of four-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
180 Daniélou, Tableau Comparatif des Intervalles Musicaux 78.
The Equally-Tempered Archetype 113
Table 54. The four-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÑ
C
E¸
F¥
A¢
C
unison
equal or Pythagorean minor third, or trihemitone
equal or acute or large tritone, or augmented fourth
equal or Pythagorean major sixth
octave
1.000000
1.189207
1.414214
1.681793
2.000000
261.626
311.127
369.994
440.000
523.251
0.000
300.000
600.000
900.000
1200.000
+ 00
+ 00
+ 00
+ 00
+ 00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 19. 4-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 55. 4-tone equal temperament tuning matrix no. 1 (Program 04)
NOTE C – – – – E¸ – – – – F¥ – – – – A¢ – – – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00
114 The Equally-Tempered Archetype
Five-Tone Equal Temperament Five-tone equal temperament is generated by the factor 5 2 , and with its five equal tones 240.00 cents
in size, which represent the 5-et supermajor second (approximately 54/47),181 is not only the first equally-
tempered division of the octave able to produce intervals outside of the domain of twelve-tone equal
temperament, but also the first able to adequately represent at least one of the Javanese scale forms.
The complete sléndro form is outlined via the pitches C, D£, F², G¢, and A!, and subsequently presents
the opportunity to evaluate the common notion proposed by theorists such as Martin Braun, proclaiming
that “the tuning concept (of sléndro) can be considered as a roughly five-tone equal temperament.”182
The sléndro tuning of the gamelan gedhé at the Sri Wedhari theatre auditorium in Solo, Central Java,
analyzed by Marc Perlman presents a platform for comparison with the following series of unequal
intervals: 238.000, 237.000, 252.000, 223.000, and 250.000 cents. The mean of this example (and the
division of the octave into five equal tones) is represented by the figure of 240.000 cents (with a
standard deviation value of 11.683321); proving the theory as somewhat significant, although analysis of
deviation values (ß2.000, ß3.000, +12.000, ß17.000, and +10.000 cents in each case) illustrate a
disproportionate relationship, and make the theoretical position no doubt inconclusive for the very
reasons that Perlman makes clear with the following statement: “There is no absolute pitch, or even a
standardized intonation, for these two laras; each fine gamelan may have its own distinctive realization of
them. The sample tone measurements are therefore only illustrative, not definitive.”183 Simultaneous
sonorities introduced in the work with this temperament include the dyads D£ and A!, F² and C, and C
and G¢, which at 720.000 cents, represents the 5-et acute or large fifth ( 35 ]2[ ). The interval is at the
upper limits of Blackwood’s criteria for a “perfect fifth within the range of recognizability”, and represents
the just perfect fifth (3/2) with a falsity +18.045 cents.184
Ex. 20. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 12
181 Daniélou, Tableau Comparatif des Intervalles Musicaux 64.
182 Braun, “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic Musical Scale,” Neuroscience
of Music n. pag.
183 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41. 184 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
The Equally-Tempered Archetype 115
The following table depicts the tonal resources of five-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 56. The five-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÑ
C
D£
F²
G¢
AÚ
C
unison
5-et supermajor second
5-et grave or small fourth
5-et acute or large fifth
5-et augmented sixth
octave
1.000000
1.148698
1.319508
1.515717
1.741101
2.000000
261.626
300.529
345.217
396.550
455.517
523.251
0.000
240.000
480.000
720.000
960.000
1200.000
+00
+40
ß20
+20
ß40
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 21. 5-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
116 The Equally-Tempered Archetype
Table 57. 5-tone equal temperament tuning matrix no. 1 (Program 05)
NOTE C – – D£ – – – – F² – – G¢ – – – – A! – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +40 +00 +00 ß20 +00 +20 +00 +00 ß40 +00
Six-Tone Equal Temperament Six-tone equal temperament ( 6 2 ), with its return to intervals within the domain of twelve-tone equal
temperament, presents the essential scalar material to produce the equally-tempered whole-tone
hexatonic scale, featuring six equal whole-tones 200.00 cents in size (approximately 55/49).185 The
temperament may alternatively be expressed as the juxtaposition of three sets of two-tone equally-
tempered intervals, or two sets of three-tone equally-tempered intervals interlocked an equal major tone
( 6 2 ) apart. It must be noted that in comparison with the just major tone (9/8) and the just minor tone
(10/9, or 182.404 cents), the equal major tone ( 6 2 ) has a falsity of +3.910 and ß17.596 cents on
each count. Charles Villiers Stanford makes the following statement in opposition to the very notion of an
equally-tempered whole-tone and the scale derived via its juxtaposition – a series of six equally-
tempered whole-tones encompassing the span of an octave:
“It is physically impossible for a scale of whole-tones to reach a half at the octave (9/8, or 203.910×3 =
611.730 cents); a scale of whole greater tones would arrive at an octave which is too sharp (9/8, or
203.910×6 = 1223.460 cents); and any combination of the greater and lesser tones will be found equally
impossible at the octave. The only way to reach the octave by whole-tones is to make each and every
one of the intervals out of tune as the pianoforte is.”186
“The whole-tone hexatonic scale offers a limited basis for extended musical expression,” notes Vincent
Persichetti, and due to the fact that “when the scale is mirrored there is no change except in register.”
The scale also offers only two possible transpositions (minor seconds apart), with no prospects for
modality and hence the generation of unique modes. Persichetti concludes that “its intervallic make-up
deprives the scale of the fundamental intervals, the perfect fourth and fifth, and of the leading tone,” and
that “a real feeling of tonality, therefore, must be established by harmony outside the whole-tone
category.”187 The temperament also introduces the equal minor seventh ( 56 ]2[ , approximately 98/55, or
185 Daniélou, Tableau Comparatif des Intervalles Musicaux 102.
186 Charles Villiers Stanford, “On Some Recent Tendencies in Composition,” Proceedings of the Musical Association,
47th Sess. (1920): 40. 187 Persichetti, Twentieth-Century Harmony: Creative Aspects and Practice 54-55.
The Equally-Tempered Archetype 117
1000.000 cents),188 which represents the acute or large minor seventh (9/5, or 1017.596 cents) with a
falsity of ß17.596 cents. Simultaneous sonorities appearing in the work include the equal major third
( 3 2 ) and major sixth ( 34 ]2[ ), which are depicted via the Pythagorean approximations of A¸ and C, and
C and A¸.
Ex. 22. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 16
The following table depicts the tonal resources of six-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 58. The six-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
ÐÑ
C
D
E¢
F¥
A¸
B¸
C
unison
equal or just major tone
equal or Pythagorean major third, or ditone
equal or acute or large tritone, or augmented fourth
equal or Pythagorean minor sixth
equal or Pythagorean minor seventh
octave
1.000000
1.122462
1.259921
1.414214
1.587401
1.781797
2.000000
261.626
293.665
329.628
369.994
415.305
466.164
523.251
0.000
200.000
400.000
600.000
800.000
1000.000
1200.000
+00
+00
+00
+00
+00
+00
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
188 Daniélou, Tableau Comparatif des Intervalles Musicaux 55.
118 The Equally-Tempered Archetype
Ex. 23. 6-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 59. 6-tone equal temperament tuning matrix no. 1 (Program 06)
NOTE C – – D – – E¢ – – F¥ – – A¸ – – B¸ – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00
Seven-Tone Equal Temperament Seven-tone equal temperament is generated by the factor 7 2 , and with its seven equal tones 171.429
cents in size, which represent the 7-et grave or small tone (approximately 56295/50989),189 make it an
intonation generally associated with the system of tuning in the traditional music of Siam (Thailand). The
equidistant claim is speculative amongst ethnomusicologists, with scientific analysis revealing a much more
complex system of intervallic diversity.190 Terry E. Miller and Sam-ang Sam make the following
observations:
“In the case of Khmer music, there has never been a strong claim for equidistance, and in fact Khmer tuning is
clearly non-equidistant. Certain pitch levels are considered useable, others not. In sum, then, Thai tuning is
functionally equidistant while Khmer is only apparently non-equidistant but variable.”191
189 Daniélou, Tableau Comparatif des Intervalles Musicaux 48.
190 Picken, “The Music of Far Eastern Asia: 2. Other Countries,” Ancient and Oriental Music 163. 191 Terry E. Miller, and Sam-ang Sam, “The Classical Musics of Cambodia and Thailand: A Study of Distinctions,”
Ethnomusicology 39.2 (Spring-Summer, 1995): 237-38.
The Equally-Tempered Archetype 119
The temperament introduces some unique simultaneous sonorities to the work, which include the dyads
F¢ and AÍ, or the 7-et acute or large fourth ( 37 ]2[ , approximately 689/512, or 514.286 cents); C and
G², and D³ and A², or the 7-et grave or small fifth ( 47 ]2[ , approximately 1024/689, or 685.714 cents);
and E³ and C, or the 7-et grave or small major sixth ( 57 ]2[ , approximately 105/64, or 857.143 cents).192
At 685.714 cents, the 7-et grave or small fifth ( 47 ]2[ ) is at the lower limits of Blackwood’s criteria for a
“perfect fifth within the range of recognizability,” and represents the just perfect fifth (3/2) with a falsity of
ß16.241 cents. The 7-et grave or small major sixth ( 57 ]2[ ) and 7-et acute or large fourth ( 37 ]2[ )
represent deviations of just intervals (5/3 and 4/3) by ß27.216 and +16.241 cents on each count.193
Ex. 24. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 18
The following table depicts the tonal resources of seven-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 60. The seven-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐÑ
C
D³
E³
F¢
G²
A²
AÍ / Bí
C
unison
7-et grave or small tone
7-et neutral third
7-et acute or large fourth
7-et grave or small fifth
7-et grave or small major sixth
7-et twenty-one quarter-tones
octave
1.000000
1.104090
1.219014
1.345900
1.485994
1.640671
1.811447
2.000000
261.626
288.858
318.925
352.122
388.774
429.241
473.921
523.251
0.000
171.429
342.857
514.286
685.714
857.143
1028.571
1200.000
+00
ß29
ß57
+14
ß14
ß43
+29
+00
192 Daniélou, Tableau Comparatif des Intervalles Musicaux 89-127.
193 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
120 The Equally-Tempered Archetype
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 25. 7-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 61. 7-tone equal temperament tuning matrix no. 1 (Program 07)
NOTE C – – D³ E³ – – F¢ – – G² – – A² AÍ – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 ß29 +43 +00 +14 +00 ß14 +00 ß43 +29 +00
Eight-Tone Equal Temperament Eight-tone equal temperament ( 8 2 ) presents a series of eight equal tones 150.000 cents in size
(approximately 1024/939), and therefore the juxtaposition of two sets of four-tone equally-tempered
intervals, or four sets of two-tone equally-tempered intervals interlocked an equal three-quarter-tone
( 8 2 ) apart. The exercise produces the equally-tempered three-quarter-tone octatonic scale, which
features eight equal three-quarter-tones, and hence, the addition of the equal three-quarter-tone ( 8 2 ),
nine quarter-tones ( 38 ]2[ , approximately 83/64, or 450.000 cents), fifteen quarter-tones ( 58 ]2[ ,
approximately 128/83, or 750.000 cents), and twenty-one quarter-tones ( 78 ]2[ , approximately
939/512, or 1050.000 cents) to the vocabulary of equal intervals.194 The simultaneous sonorities of E¸
and F¥, CÍ and Fí, Fí and G£, and C and E¸ highlight the utilization of the equal minor third ( 4 2 ) in the
194 Daniélou, Tableau Comparatif des Intervalles Musicaux 42-113.
The Equally-Tempered Archetype 121
work; while C and Fí, E¸ and G£, and F¥ and B³, the nine equal quarter-tones ( 38 ]2[ ), which is one of
these very distinct equal quarter-tone intervals.
Ex. 26. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 22-24
The following table depicts the tonal resources of eight-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 62. The eight-tone equally-tempered division of the octave
Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÑ
C
CÍ / Dí
E¸
EË / Fí
F¥
G£
A¢
B³
C
unison
equal three-quarter-tone
equal or Pythagorean minor third, or trihemitone
nine equal quarter-tones
equal or acute or large tritone, or augmented fourth
fifteen equal quarter-tones
equal or Pythagorean major sixth
twenty-one equal quarter-tones
octave
1.000000
1.090508
1.189207
1.296840
1.414214
1.542211
1.681793
1.834008
2.000000
261.626
285.305
311.127
339.286
369.994
403.482
440.000
479.823
523.251
0.000
150.000
300.000
450.000
600.000
750.000
900.000
1050.000
1200.000
+00
+50
+00
ß50
+00
+50
+00
ß50
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales. Ex. 27. 8-tone equal temperament
122 The Equally-Tempered Archetype
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 63. 8-tone equal temperament tuning matrix no. 1 (Program 08)
NOTE C CÍ – – E¸ – – Fí F¥ G£ – – A¢ – – B³
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +50 +00 +00 +00 ß50 +00 +50 +00 +00 +00 ß50
Nine-Tone Equal Temperament Nine-tone equal temperament ( 9 2 ), with its twofold tripartite division of the octave, presents a
subdivision of the equal major third ( 3 2 ) into three equal tones 133.333 cents in size, which represent
the 9-et great limma (approximately 553/512). The twofold tripartite division of the octave is in reference
to the juxtaposition of three sets of three-tone equally-tempered intervals interlocked an equal 9-et great
limma ( 9 2 ) apart, which delivers not only the first triad in the work, but also the first complete exposition
of the pélog scale with the pitches C, DÝ, Eì, F£, G³, A¸, and B². The triad consists of the pitches C, Eì,
and G³, and presents a minor third 266.667 cents in size ( 29 ]2[ , 9-et five quarter tones, or
approximately 7/6) and a perfect fifth 666.667 cents in size ( 59 ]2[ , 9-et subfifth, or approximately
147/100).195 Eì, although equal to 9-et five quarter tones ( 29 ]2[ ), functions harmonically as a minor third,
and therefore in association with the other pitches forms a harmonic structure that may be regarded as
being essentially a nine-tone equally-tempered minor triad. In comparison with the just minor triad (equal
to the proportions 10:12:15), the nine-tone equally-tempered minor triad features two acutely dissonant
intervals, which form the complex ratio 1.000000:1.166529:1.469734. The fundamental C4 at standard
pitch (A=440Hz) may be utilized to illustrate that the third harmonic of C4 does not correspond with the
second harmonic of G³4 (769.040Hz), or the 9-et subfifth ( 59 ]2[ ). A just perfect fifth with a 3/2
proportion would have presented identical frequencies, and hence a ‘zero beating’ condition between
195 Daniélou, Tableau Comparatif des Intervalles Musicaux 38-102.
The Equally-Tempered Archetype 123
the two intervals. In striking contrast, nine-tone equal temperament presents 15.836 beats, with 31.673
beats between the sixth harmonic of C4 and the fourth harmonic of G³4 (1538.080Hz). Eì4
(305.194Hz), or 9-et five quarter tones ( 29 ]2[ ), presents 43.784 beats between the sixth harmonic of
C4 and the fifth harmonic of Eì4 (1525.969Hz). The level of dissonance is clearly within the sonic realm
of ‘roughness’, and therefore outside of the 20-25Hz threshold that Doty defines as a distinctly audible
rate of beating.196
Table 64. The beating characteristics of the nine-tone equally-tempered minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
Eì4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G³4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
6
– – – –
– – – –
– – – –
– – – –
305.194
– – – –
– – – –
610.388
– – – –
915.581
– – – –
– – – –
1220.775
– – – –
1525.969
1831.163
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
43.784
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
384.520
– – – –
– – – –
769.040
– – – –
– – – –
1153.560
– – – –
– – – –
1538.080
– – – –
– – – –
1922.601
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
15.836
– – – –
– – – –
– – – –
– – – –
– – – –
31.673
– – – –
– – – –
– – – –
– – – –
The pélog scale is outlined with the pitches C, DÝ, Eì, F£, G³, A¸, and B², and equal to 0.000, 133.333,
266.667, 533.333, 666.667, 800.000, and 1066.667 cents. The pélog tuning of the gamelan gedhé
analyzed by Perlman presents the following series of unequal intervals: 116.000, 165.000, 269.000,
119.000, 100.000, 192.000, and 239.000 cents. The mean of this example (and the division of the
octave into seven equal tones) is represented by the figure of 171.429 cents (with a standard deviation
196 “Beats can be perceived clearly when the difference is less than 20-25Hz, but as the difference increases
beyond this point the beats blend together, giving rise to a general sensation of roughness. This roughness gradually
decreases as the difference increases, persisting until the difference exceeds the critical band, which, for most of the
audio range falls between a whole-tone and a minor third,” explains Doty. For a further discussion, see Doty, The Just
Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 20-22.
124 The Equally-Tempered Archetype
value of 65.158415), and therefore with some certainty, it may be stated that with the falsities of
ß17.333, +14.333, +16.667, +2.333, ß31.000, and ß106.667 cents on each count, the intervals of
the gamelan gedhé only remotely resemble those of nine-tone equal temperament. The inclusion of A£
in the pélog system – although far from being representative of any legitimate form of a minor seventh –
replaces the last value of contention (B²) with a falsity of +27.667 cents. In consideration of the
pentatonic outline, or principal tones of the pélog scale (the pitches C, DÝ, Eì, G³, and A¸) – just like the
previous sléndro example – the mean value of 240.00 cents (with a standard deviation of 157.310203)
may be ascertained, but with an even greater degree of inequality.197
Simultaneous sonorities include the triad G³, C, and G³; and the inversion F£, C, and F£; which
may be represented by the 9-et superfourth ( 49 ]2[ , or approximately 200/147) and 9-et subfifth
( 59 ]2[ ).198 The two intervals represent an extremely dissonant just perfect fourth (4/3) and just perfect
fifth (3/2) with a falsity +35.288 and ß35.288 on each count.
Ex. 28. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 27
Other simultaneous sonorities include the dyads Eì and G³, B² and Eì, and G³ and B²; all equal to the
interval of an equal major third ( 3 2 ); with a passage spelling F£ and G³, G³ and A£, and a melodic
descent to A¸, marking the intervals of the 9-et great limma ( 9 2 ), 9-et five quarter tones ( 29 ]2[ ), and 9-
et great limma ( 9 2 ) respectively.
Ex. 29. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 28-29
197 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41.
198 Daniélou, Tableau Comparatif des Intervalles Musicaux 130-131.
The Equally-Tempered Archetype 125
The following table depicts the tonal resources of nine-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 65. The nine-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÐÑ
C
DÝ
DË / Eì
E¢
F£
G³
A¸
AË / Bì
B²
C
unison
9-et great limma, or large half-tone
9-et five quarter-tones
equal or Pythagorean major third, or ditone
9-et superfourth
9-et subfifth
equal or Pythagorean minor sixth
9-et nineteen quarter-tones
9-et grave or small major seventh
octave
1.000000
1.080060
1.166529
1.259921
1.360790
1.469734
1.587401
1.714488
1.851749
2.000000
261.626
282.571
305.194
329.628
356.017
384.520
415.305
448.554
484.465
523.251
0.000
133.333
266.667
400.000
533.333
666.667
800.000
933.333
1066.667
1200.000
+00
+33
ß33
+00
+33
ß33
+00
+33
ß33
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 30. 9-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
126 The Equally-Tempered Archetype
Table 66. 9-tone equal temperament tuning matrix no. 1 (Program 09)
NOTE C DÝ – – Eì E¢ F£ – – G³ A¸ AË – – B²
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +33 +00 ß33 +00 +33 +00 ß33 +00 +33 +00 ß33
Ten-Tone Equal Temperament Ten-tone equal temperament ( 10 2 ) presents a subdivision of the 5-et supermajor second ( 5 2 ) into
two equal tones 120.000 cents in size, which represent the 10-et just diatonic semitone (approximately
15/14). The temperament may be alternatively expressed as the juxtaposition of two sets of five-tone
equally-tempered intervals, or five sets of two-tone equally-tempered intervals interlocked a 10-et just
diatonic semitone ( 10 2 ) apart. The intonation scheme therefore produces the identical pitches for
sléndro as five-tone equal temperament, although additional pitches generate possible auxiliary pitches of
an extended pentatonic form. Sethares makes the following observations: “The 10-tet tuning has no fifth,
no third, no major seconds, and no dominant sevenths. The only interval common to both 10-tet and
12-tet (other than the octave) is the 600-cent interval normally called the tritone, augmented fourth, or
diminished fifth.”199
A variety of simultaneous sonorities are explored in the work within the scope of this temperament,
which include the dyads E² and G¢, and F² and A³, or the 10-et grave or small major third ( 310 ]2[ ,
approximately 16/13, or 360.000 cents), and C and AÚ, or the 5-et augmented sixth ( 45 ]2[ ,
approximately 47/27, or 960.000 cents).200 The two intervals represent the just major third (5/4) and the
acute or large minor seventh (9/5) with a falsity of ß26.314 and ß57.596 cents on each count.
Ex. 31. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 32
199 Sethares, Tuning, Timbre, Spectrum, Scale 291.
200 Daniélou, Tableau Comparatif des Intervalles Musicaux 34-92.
The Equally-Tempered Archetype 127
Other simultaneous sonorities include the dyads D£ and A³, or the equal tritone ( 2 2 ), and C and G¢, or
the 5-et acute or large fifth ( 35 ]2[ ).
Ex. 32. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 34
The following table depicts the tonal resources of ten-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 67. The ten-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÐÑ
C
D¸
D£
E²
F²
F¥
G¢
A³
AÚ
B
C
unison
10-et just diatonic semitone, or major half-tone
5-et supermajor second
10-et grave or small major third
5-et grave or small fourth
equal or acute or large tritone, or augmented fourth
5-et acute or large fifth
10-et neutral sixth
5-et augmented sixth
10-et just diatonic major seventh
octave
1.000000
1.071773
1.148698
1.231144
1.319508
1.414214
1.515717
1.624505
1.741101
1.866066
2.000000
261.626
280.403
300.529
322.099
345.217
369.994
396.550
425.012
455.517
488.211
523.251
0.000
120.000
240.000
360.000
480.000
600.000
720.000
840.000
960.000
1080.000
1200.000
+00
+20
+40
ß40
ß20
+00
+20
ß60
ß40
ß20
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 33. 10-tone equal temperament
128 The Equally-Tempered Archetype
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 68. 10-tone equal temperament tuning matrix no. 1 (Program 10)
NOTE C D¸ D£ – – E² F² F¥ G¢ A³ – – AÚ B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +20 +40 +00 ß40 ß20 +00 +20 +40 +00 ß40 ß20
Eleven-Tone Equal Temperament Eleven-tone equal temperament is generated by the factor 112 , and presents eleven equal intervals
109.091 cents in size, which represent the 11-et just diatonic semitone (approximately 82/77). An
implied triad makes its appearance in the composition in bar 38. This triad (essentially an eleven-tone
equally-tempered major triad) consists of the pitches C, EË, and G³, and presents a major third 436.364
cents in size ( 411 ]2[ , 11-et nine quarter-tones, or approximately 659/512) and a perfect fifth 654.545
cents in size ( 611 ]2[ , 11-et subfifth, or approximately 54/37); hence an extremely dissonant harmonic
construct.201 EË, although equal to 11-et nine quarter-tones ( 411 ]2[ ), functions harmonically as a major
third. In comparison with the just major triad (equal to the proportions 4:5:6), the eleven-tone equally-
tempered major triad forms the complex ratio 1.000000:1.286665:1.459480. The temperament
produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of +50.050 and ß47.410
cents, which is approximately an equal quarter-tone ( 24 2 ) deviation on each count. G³4 (381.837Hz),
or the 11-et subfifth ( 611 ]2[ ), presents 21.202 beats between the third harmonic of C4 and the second
harmonic of G³4 (763.675Hz), and 42.404 beats between the sixth harmonic of C4 and the fourth
harmonic of G³4 (1527.349Hz); while EË4 (336.624Hz), or 11-et nine quarter-tones ( 411 ]2[ ), 38.370
beats between the fifth harmonic of C4 and the fourth harmonic of EË4 (1346.498Hz).
201 Daniélou, Tableau Comparatif des Intervalles Musicaux 30-135.
The Equally-Tempered Archetype 129
Table 69. The beating characteristics of the eleven-tone equally-tempered major triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
EË4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G³4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
336.624
– – – –
– – – –
673.249
– – – –
1009.873
– – – –
– – – –
1346.498
– – – –
1683.122
– – – –
2019.747
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
38.370
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
381.837
– – – –
– – – –
763.675
– – – –
– – – –
1145.512
– – – –
1527.349
– – – –
– – – –
1909.187
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
21.202
– – – –
– – – –
– – – –
– – – –
42.404
– – – –
– – – –
– – – –
– – – –
Simultaneous sonorities introduced in the work with this temperament include the dyads F£ and G³, or
the 11-et just diatonic semitone ( 112 ), and B¹ and C, or the 11-et acute or large tone ( 211 ]2[ ,
approximately 245/216, or 218.182 cents).202
Ex. 34. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 41
Other simultaneous sonorities include the dyads D¢ and F£, EË and Aì, and G³ and B¹, or 11-et seven
quarter-tones ( 311 ]2[ , approximately 29/24, or 327.273 cents), and C and EË, and B¹ and D¢, or 11-et
nine quarter-tones ( 411 ]2[ ).203
202 Daniélou, Tableau Comparatif des Intervalles Musicaux 58.
203 Daniélou, Tableau Comparatif des Intervalles Musicaux 84.
130 The Equally-Tempered Archetype
Ex. 35. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 42
The following table depicts the tonal resources of eleven-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 70. The eleven-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÐÑ
C
D¸
D¢
DÍ / Eí
EË / Fí
F£
G³
GË / Aì
A²
B¹
B
C
unison
11-et just diatonic semitone, or major half-tone
11-et acute or large tone
11-et seven quarter-tones
11-et nine quarter-tones
11-et superfourth
11-et subfifth
11-et fifteen quarter-tones
11-et grave or small major sixth
11-et grave or small minor seventh
11-et just diatonic major seventh
octave
1.000000
1.065041
1.134313
1.208089
1.286665
1.370351
1.459480
1.554406
1.655507
1.763183
1.877862
2.000000
261.626
278.642
296.765
316.067
336.624
358.519
381.837
406.672
433.123
461.294
491.297
523.251
0.000
109.091
218.182
327.273
436.364
545.455
654.545
763.636
872.727
981.818
1090.909
1200.000
+00
+09
+18
+27
+36
+45
ß45
ß36
ß27
ß18
ß09
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 36. 11-tone equal temperament
The Equally-Tempered Archetype 131
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
Table 71. 11-tone equal temperament tuning matrix no. 1 (Program 11)
NOTE C D¸ D¢ DÍ EË F£ – – G³ Aì A² B¹ B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +09 +18 +27 +36 +45 +00 ß45 ß36 ß27 ß18 ß09
Twelve-Tone Equal Temperament Twelve-tone equal temperament ( 12 2 ), with its division of the octave into twelve equally-tempered
semitones 100.000 cents in size, marks the central structural climax of the composition. The
temperament may alternatively be expressed as the juxtaposition of two sets of six-tone equally-
tempered intervals, three sets of four-tone equally-tempered intervals, four sets of three-tone equally-
tempered intervals, or six sets of two-tone equally-tempered intervals interlocked an equal semitone
( 12 2 ) apart. Philippe De Vitry (1291-1361) has the following to say about the semitone: “The semitone,
as Bernardus said, is the sugar and spice of all music, and without it song is corrupted, altered, and
destroyed,” while theorizing the interval of the semitone as “the interval between two unisons, which in
the human voice is incapable of, and will admit of, division or the interpolation of a middle sound.”204
With regards to the rationale behind the adoption of twelve-tone equal temperament as the intonation
standard of Western music, Blackwood offers the following explanation: “There is no doubt in my mind
that of all equal tunings, twelve is the most versatile and most expressive. It’s no accident that we have
twelve instead of thirteen.”205
204 Philippe de Vitry, “Philippe de Vitry’s ‘Ars Nova’: A Translation,” Journal of Music Theory 5.2 (Winter, 1961): 10.
205 Douglas Keislar, Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and William
Schottstaedt, “Six American Composers on Nonstandard Tunings,” Perspectives of New Music 29.1 (Winter, 1991): 180.
132 The Equally-Tempered Archetype
The first simultaneous sonority of interest makes its appearance in the first beat of bar 44, which
incorporates the pitches C, G, F, and C, and may be theoretically explained as two equal perfect fourths
( 512 ]2[ ) juxtaposed an equal minor seventh apart ( 56 ]2[ ), or C(add 11/omit 3). Consequential sonorities
B¢ and F, E¢ and B¸, and F¥ and C (F¥ alternatively expressed via its enharmonic equivalent of G¸), all
equal to 600.000 cents, or the equal tritone ( 2 2 ), while E¢ and C, and C and B¸, the equal minor sixth
( 23 ]2[ ) and equal minor seventh ( 56 ]2[ ).
Ex. 37. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 44-45
Other sonorities include the dyads F and G¸, and B¢ and C, which represent the equal semitone ( 12 2 ).
In comparison with the just diatonic semitone (16/15), the equal semitone ( 12 2 ) has a falsity of ß11.731
cents, while its inversion, the equal major seventh ( 1112 ]2[ , approximately 967/512, or 1100.000
cents),206 a falsity of +11.731 cents with the just diatonic major seventh (15/8, or 884.359 cents)
Twelve-tone equal temperament is the first temperament to contain what Blackwood terms as
recognizable diatonic scales and “perfect fifths within the range of recognizability,”207 and produces the
just perfect fifth (3/2) and just perfect fourth (4/3) with a falsity of +1.955 and ß1955 cents on each
count.
Ex. 38. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 46
The following table depicts the tonal resources of twelve-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
206 Daniélou, Tableau Comparatif des Intervalles Musicaux 29.
207 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
The Equally-Tempered Archetype 133
Table 72. The twelve-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
D¹
D
E¸
E¢
F
F¥
G
A¸
A¢
B¸
B¢
C
unison
equal semitone, or Pythagorean limma
equal or just major tone
equal or Pythagorean minor third, or trihemitone
equal or Pythagorean major third, or ditone
equal, or just and Pythagorean perfect fourth
equal or acute or large tritone, or augmented fourth
equal, or just and Pythagorean perfect fifth
equal or Pythagorean minor sixth
equal or Pythagorean major sixth
equal or Pythagorean minor seventh
equal or Pythagorean major seventh
octave
1.000000
1.059463
1.122462
1.189207
1.259921
1.334840
1.414214
1.498307
1.587401
1.681793
1.781797
1.887749
2.000000
261.626
277.183
293.665
311.127
329.628
349.228
369.994
391.995
415.305
440.000
466.164
493.883
523.251
0.000
100.000
200.000
300.000
400.000
500.000
600.000
700.000
800.000
900.000
1000.000
1100.000
1200.000
+00
+00
+00
+00
+00
+00
+00
+00
+00
+00
+00
+00
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 39. 12-tone equal temperament
The following tuning matrix contains the data required to represent the temperament chromatically within
the program memory of the Akai S3000XL.
134 The Equally-Tempered Archetype
Table 73. 12-tone equal temperament tuning matrix no. 1 (Program 12)
NOTE C D¹ D E¸ E¢ F F¥ G A¸ A¢ B¸ B¢
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00
Thirteen-Tone Equal Temperament Thirteen-tone equal temperament is generated by the factor 13 2 , and presents thirteen equal intervals
92.308 cents in size, which represent the 13-et Pythagorean limma (approximately 77/73), and
approximate the Pythagorean limma (256/243) with a falsity of +2.083 cents. The thirteen-tone equally-
tempered division of the octave now for the first time presents more than one possibility for the
approximation of a particular tone. In this case, that tone being the fifth, with the possible representation
as G£, 13-et thirteen quarter-tones ( 713 ]2[ , approximately 61/42, or 646.154 cents), or the alternative of
Gí, the 13-et superfifth ( 813 ]2[ , approximately 72/47, or 738.462 cents).208 Throughout the work, the
criteria for accessing whether this second tone is incorporated into the scalar scheme as a primary or
secondary ‘auxiliary’ tone will be dependent on the accuracy of the approximation. In thirteen-tone
equal temperament Gí ( 713 ]2[ ), hence accorded as an auxiliary tone, due to the fact that G£ ( 813 ]2[ ) is
36.507 cents higher than the just perfect fifth (3/2), while Gí ( 713 ]2[ ), 55.801 cents lower, or with a
falsity exceeding that of the primary fifth G£ ( 813 ]2[ ) by 19.294 cents. In the case of more than two
possibilities for the approximation of one tone, only the closest two will be taken into account. Melodic
development in bar 51 of the work exploits the intervallic nuances of the two microtonal inflections of
the fifth via the intonational reinterpretation of pitch recapitulation.
Ex. 40. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 51
208 Daniélou, Tableau Comparatif des Intervalles Musicaux 26-137.
The Equally-Tempered Archetype 135
American composer Easley Blackwood (1933-) composed Twelve Microtonal Etudes for Electronic
Music Media in 1979 and 1980 as part of a research project supported by the National Endowment for
the Humanities in association with Webster College in St. Louis, USA. The twelve-movement work
explored the tonal and modal characteristic of all the equal divisions of the octave between thirteen and
twenty-four, stimulating the devising of appropriate notational practices, and culminating in the compact
disc release of Microtonal Compositions by Easley Blackwood, which feature Blackwood on the
polyfusion synthesizer. It is therefore most appropriate to hereon incorporate some of the comments
expressed by the composer in relation to each temperament. According to Blackwood, the thirteen-
tone equally-tempered division of the octave is “the most alien tuning of all; so dissonant that no three-
note combination sounds like major or minor triad.”209 Two new triads are introduced to the work via this
temperament. The first triad (essentially a thirteen-tone equally-tempered major triad) consists of the
pitches C, E², and G£, and presents the 13-et grave or small major third ( 413 ]2[ , approximately 26/21, or
369.231 cents) and the 13-et superfifth ( 813 ]2[ ). The second triad consists of the pitches C, DÚ, and
Gí, and presents the 13-et augmented second ( 313 ]2[ , approximately 601/512, or 276.923 cents) and
13-et thirteen quarter-tones ( 713 ]2[ ). Enharmonically reinterpreted as C, E¹, and Gí, the latter triad
serves as an implied thirteen-tone equally-tempered minor triad.210
Ex. 41. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 55
The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß17.083
and +36.507 cents, or approximate equal sixth-tone ( 36 2 , approximately 261/256, or 33.333 cents)
and third-tone ( 18 2 , 133/128, or 66.667 cents) deviations on each count.211 G£4 (400.802Hz), or the
13-et superfifth ( 813 ]2[ ), presents 16.726 beats between the third harmonic of C4 and the second
harmonic of G£4 (801.603Hz), and 33.453 beats between the sixth harmonic of C4 and the fourth
harmonic of G£4 (1603.206Hz); while E²4 (323.821Hz), or the 13-et grave or small major third ( 413 ]2[ ),
209 Easley Blackwood, liner notes, Microtonal Compositions by Easley Blackwood, perf. Easley Blackwood
(polyfusion synthesizer), and Jeffrey Kust (guitar), rec. 16 Sep. 1990, Cedille, 1994, CDR 90000 018, n. pag.
210 Daniélou, Tableau Comparatif des Intervalles Musicaux 74-94.
211 Daniélou, Tableau Comparatif des Intervalles Musicaux 8-18.
136 The Equally-Tempered Archetype
12.844 beats between the fifth harmonic of C4 and the fourth harmonic of E²4 (1295.283Hz). This triad
is certainly more consonant than the eleven-tone equally-tempered major triad previously presented (C,
EË, and G³), with a major third 436.364 cents in size a perfect fifth 654.545 cents in size, although when
compared with the theoretical major triad of twelve-tone equal temperament (C, E¢, and G), which
presents an equal major third ( 3 2 ) and perfect fifth ( 712 ]2[ ) 400.000 and 700.000 cents in size, it is
quite dissonant in character.
Table 74. The beating characteristics of the thirteen-tone equally-tempered major triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E²4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G£4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
323.821
– – – –
– – – –
647.642
– – – –
971.463
– – – –
– – – –
1295.283
– – – –
1619.104
– – – –
1942.925
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
12.844
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
400.802
– – – –
– – – –
801.603
– – – –
– – – –
1202.405
– – – –
1603.206
– – – –
– – – –
2004.008
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.726
– – – –
– – – –
– – – –
– – – –
33.453
– – – –
– – – –
– – – –
– – – –
The two microtonal inflections of 3/2 are also explored as simultaneous sonorities, with the dyads C and
Gí, and Gì and C, or 13-et thirteen quarter-tones ( 713 ]2[ ), and C and G£, or the 13-et superfifth ( 813 ]2[ ).
Ex. 42. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 50
The Equally-Tempered Archetype 137
Other sonorities include the dyads G£ and B¸, or the 13-et augmented second ( 313 ]2[ ); Gí and B¸, or
the 13-et grave or small major third ( 413 ]2[ ); E² and GÍ, or the 13-et subfourth ( 513 ]2[ , approximately
47/36, or 461.538 cents); DÚ and GÍ, or 13-et eleven quarter-tones ( 613 ]2[ , approximately 705/512, or
553.846 cents); D² and A£, or the 13-et superfifth ( 813 ]2[ ), E² and C, or the 13-et seventeen quarter-
tones ( 913 ]2[ , approximately 21/13, or 830.769 cents); and DÚ and C, or the 13-et acute or large major
sixth ( 1013 ]2[ , approximately 1024/601, or 923.077 cents).212 The following table depicts the tonal
resources of thirteen-tone equal temperament, indicating degree, notation, interval, ratio, frequency,
cents, and tuning.
Table 75. The thirteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÐÑ
C
D¹
D²
DÚ
E²
F³
FË / Gì
FÍ / Gí
G£
GÍ / Aí
A£
B¸
B¢
C
unison
13-et Pythagorean limma
13-et just minor tone
13-et augmented second
13-et grave or small major third
13-et subfourth
13-et eleven quarter-tones
13-et thirteen quarter-tones
13-et superfifth
13-et seventeen quarter-tones
13-et acute or large major sixth
13-et Pythagorean minor seventh
13-et Pythagorean major seventh
octave
1.000000
1.054766
1.112531
1.173460
1.237726
1.305512
1.377009
1.452423
1.531966
1.615866
1.704361
1.797702
1.896155
2.000000
261.626
275.954
291.067
307.007
323.821
341.555
360.261
379.991
400.802
422.752
445.904
470.325
496.083
523.251
0.000
92.308
184.616
276.923
369.231
461.538
553.846
646.154
738.462
830.769
923.077
1015.385
1107.692
1200.000
+00
ß08
ß15
ß23
ß31
ß38
ß46
ß54
+38
+31
+23
+15
+08
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 43. 13-tone equal temperament
212 Daniélou, Tableau Comparatif des Intervalles Musicaux 75-34.
138 The Equally-Tempered Archetype
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 76. 13-tone equal temperament tuning matrix no. 1 (Program 13)
NOTE C D¸ D² DÚ E² F³ Gì G£ GÍ A£ B¸ B¢
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß08 ß15 ß23 ß31 ß38 ß46 +38 +31 +23 +15 +08
Table 77. 13-tone equal temperament tuning matrix no. 2 (Program 14)
NOTE – – – – – – – – – – – – Gí – – – – – – – – – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +46 +00 +00 +00 +00 +00
Fourteen-Tone Equal Temperament Fourteen-tone equal temperament ( 14 2 ) presents a subdivision of the 7-et grave or small tone ( 7 2 )
into two equal tones 85.714 cents in size, which represent the 14-et Pythagorean limma (approximately
269/256). The temperament may alternatively be expressed as the juxtaposition of two sets of seven-
tone equally-tempered intervals, or seven sets of two-tone equally-tempered intervals interlocked a 14-et
Pythagorean limma ( 14 2 ) apart. Within the sléndro scalar scheme two options become available for the
approximation of the major second, being D³ or DË, or the 7-et grave or small tone ( 7 2 ) and 14-et five
quarter-tones ( 314 ]2[ , approximately 297/256, or 257.143 cents); two for the major third, being E³ or
E£, or the 7-et neutral third ( 27 ]2[ , approximately 128/105, or 342.857 cents) and 14-et acute or large
major third ( 514 ]2[ , approximately 16807/13122, or 428.571 cents); as well as two options for the
augmented sixth, being AÚ or AÍ, or the 14-et augmented sixth ( 1114 ]2[ , approximately 512/297, or
942.857 cents) and 7-et twenty-one quarter-tones ( 67 ]2[ , approximately 50989/28147, or 1028.571
The Equally-Tempered Archetype 139
cents).213 The pélog scalar scheme accommodates only the latter set of intervals. Bar 58 of the work
presents D³ and DË as an intonational reinterpretation of a melodic sequence, while bar 59 alternatively
presents C and D³, and C and DË as intonational reinterpretations of a simultaneous sonority.
Ex. 44. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 58-59
A further example of an intonational reinterpretation of a melodic sequence is bar 61, which highlights AÚ
and AÍ.
Ex. 45. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 61
In spite of E£ ( 514 ]2[ ), or the 14-et acute or large major third being technically closer to the just major
third (5/4) – E³ ( 27 ]2[ ), or the 7-et neutral third and E£ ( 514 ]2[ ) represent 5/4 with a falsity of ß43.457
and +42.258 on each count – E³ ( 27 ]2[ ) is relegated with the role of primary tone (and not the
expected secondary ‘auxiliary’ tone) for simple aesthetic considerations; hereby presenting an exception
to the rule where the accorded status of a tone is dependent on the accuracy of the approximation to
just intervals. It must be nevertheless noted that the 1.199 cents difference between the tones is
negligible, and therefore not a point of theoretical contention.
Blackwood offers the following description of fourteen-tone equal temperament: “This very
discordant tuning offers two highly contrasting modal arrangements. One is the division of an octave into
213 Daniélou, Tableau Comparatif des Intervalles Musicaux 24-108.
140 The Equally-Tempered Archetype
seven equal parts, sounding like a diatonic scale with no distinctions of major and minor. The other is a
combination of two differently tuned diminished seventh chords.”214
One new triad (essentially a fourteen-tone equally-tempered major triad) is introduced to the work
via this temperament, which consists of the pitches C, E³, and G², and presents a major third 342.857
cents in size ( 27 ]2[ , or 7-et neutral third) and a perfect fifth 685.714 cents in size ( 47 ]2[ , or 7-et grave
or small fifth). The temperament produces the just major third (5/4) and just perfect fifth (3/2) with a
falsity of ß43.457 and ß16.241 cents on each count. G²4 (388.774Hz), or 7-et grave or small fifth
( 47 ]2[ ), presents 7.329 beats between the third harmonic of C4 and the second harmonic of G²4
(777.548Hz), and 14.658 beats between the sixth harmonic of C4 and the fourth harmonic of G²4
(1555.096Hz); while E³4 (318.925Hz), or the 7-et neutral third ( 27 ]2[ ), 32.427 beats between the fifth
harmonic of C4 and the fourth harmonic of E³4 (1275.700Hz).
Table 78. The beating characteristics of the fourteen-tone equally-tempered major triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E³4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G²4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
318.925
– – – –
– – – –
637.850
– – – –
956.776
– – – –
– – – –
1275.700
– – – –
1594.626
– – – –
1913.551
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
32.427
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
388.774
– – – –
– – – –
777.548
– – – –
– – – –
1166.322
– – – –
1555.096
– – – –
– – – –
1943.870
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
7.329
– – – –
– – – –
– – – –
– – – –
14.658
– – – –
– – – –
– – – –
– – – –
The concept of exploiting the intervallic nuances of two microtonal inflections is developed further in
fourteen-tone equal temperament, with its incorporation into simultaneous sonorities as a structural
compositional device. Bar 62 features a repeat of the harmonic sequence of C and GÚ, or the 14-et
augmented fifth ( 914 ]2[ , approximately 26244/16807, or 771.429 cents); AÚ and G², or the 14-et
214 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
The Equally-Tempered Archetype 141
augmented sixth ( 1114 ]2[ ); A² and F¢, or the 7-et grave or small major sixth ( 57 ]2[ ); and G² and E£, and
F¢ and DË, or the or 14-et augmented sixth ( 1114 ]2[ ); with the two final dyads replaced by G² and E³,
and F¢ and D³, or the 7-et grave or small major sixth ( 57 ]2[ ). The recapitulation of the harmonic
sequence emphasizes intervallic diversity available within the two alternatives for major seconds and
thirds.215
Ex. 46. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 62
The following two bars outline a fourteen-tone equally-tempered sléndro scalar descent with the pitches
C, AÍ, A², G², F¢, E³, D³, and C – equal to 1200.000, 1028.571, 857.143, 685.714, 514.286, 342.857,
and 171.429 cents respectively. The melodic line is further supported via the harmonic progression of
E³ and C, D³ and AÍ, C and A², or the 7-et grave or small major sixth ( 57 ]2[ ); and C and G², or the 7-et
grave or small fifth ( 47 ]2[ ); followed singularly by F¢, or the 7-et acute or large fourth ( 37 ]2[ ); E³, or the
7-et neutral third ( 27 ]2[ ); and D³, or the 7-et grave or small tone ( 7 2 ), to a resolution provided by the
simultaneous sonority of C and E³, or the 7-et neutral third ( 27 ]2[ ), which implies a fourteen-tone
equally-tempered major triad.
Ex. 47. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 63-64
Additional simultaneous sonorities include the pitches C and F¢, and F¢ and AÍ, or the 7-et acute or large
fourth ( 37 ]2[ ); D¹ and GÚ, or the 7-et grave or small fifth ( 47 ]2[ ); and D¹ and AÍ, or the 14-et
215 Daniélou, Tableau Comparatif des Intervalles Musicaux 109.
142 The Equally-Tempered Archetype
augmented sixth ( 1114 ]2[ ). The following table depicts the tonal resources of fourteen-tone equal
temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.
Table 79. The fourteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÐÑ
C
D¹
D³
DË / Eì
E³
E£
F¢
F¥
G²
GÚ
A²
AÚ
AÍ / Bí
B¢
C
unison
14-et Pythagorean limma
7-et grave or small tone
14-et five quarter-tones
7-et neutral third
14-et acute or large major third
7-et acute or large fourth
equal or acute or large tritone, or augmented fourth
7-et grave or small fifth
14-et augmented fifth
7-et grave or small major sixth
14-et augmented sixth
7-et twenty-one quarter-tones
14-et Pythagorean major seventh
octave
1.000000
1.050757
1.104090
1.160129
1.219014
1.280887
1.345900
1.414214
1.485994
1.561418
1.640671
1.723946
1.811447
1.903390
2.000000
261.626
274.905
288.858
303.520
318.925
335.113
352.122
369.994
388.774
408.507
429.241
451.028
473.921
497.976
523.251
0.000
85.714
171.429
257.143
342.857
428.571
514.286
600.000
685.714
771.429
857.143
942.857
1028.571
1114.286
1200.000
+00
ß14
ß29
+57
ß57
+29
+14
+00
ß14
ß29
ß43
ß57
+29
+14
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 48. 14-tone equal temperament
The Equally-Tempered Archetype 143
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 80. 14-tone equal temperament tuning matrix no. 1 (Program 15)
NOTE C D¹ D³ E³ E£ F¢ F¥ G² GÚ A² AÍ B¢
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß14 ß29 +43 +29 +14 +00 ß14 ß29 ß43 +29 +14
Table 81. 14-tone equal temperament tuning matrix no. 2 (Program 16)
NOTE – – – – – – DË – – – – – – – – – – AÚ – – – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 ß43 +00 +00 +00 +00 +00 +43 +00 +00
Fifteen-Tone Equal Temperament Fifteen-tone equal temperament ( 15 2 ), with its twofold tripartite division of the octave, presents a
subdivision of the 5-et supermajor second ( 5 2 ) into three equal tones 80.000 cents in size, which
represent the 15-et grave or small just chromatic semitone (approximately 22/21).216 The temperament
may alternatively be expressed as the juxtaposition of three sets of five-tone equally-tempered intervals,
or five sets of three-tone equally-tempered intervals interlocked a 15-et grave or small just chromatic
semitone ( 15 2 ) apart. Blackwood describes the temperament thus: “One of the most fascinating of the
equal tunings, this contains triads that are sufficiently in tune to serve to serve as the final harmony in
cadences. The major scale, however, is so strange that even the most common diatonic progressions are
disturbing unless precautions are taken.”217
One new triad (essentially a fifteen-tone equally-tempered minor triad) is introduced to the work
via this temperament, which consists of the pitches C, EÝ, and G¢, and presents a minor third 320.000
cents in size ( 415 ]2[ , approximately 77/64, or 15-et just minor third) and a perfect fifth 720.000 cents in
size ( 35 ]2[ , or 5-et acute or large fifth).218 The temperament produces the just minor third (6/5) and just
perfect fifth (3/2) with a falsity of +4.359 and +18.045 cents on each count. G¢4 (396.550Hz), or 5-et
acute or large fifth ( 35 ]2[ ), presents 8.224 beats between the third harmonic of C4 and the second
216 Daniélou, Tableau Comparatif des Intervalles Musicaux 22.
217 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
218 Daniélou, Tableau Comparatif des Intervalles Musicaux 82.
144 The Equally-Tempered Archetype
harmonic of G¢4 (793.100Hz), and 16.447 beats between the sixth harmonic of C4 and the fourth
harmonic of G¢4 (1586.201Hz); while EÝ4 (314.742Hz), or the 15-et just minor third ( 415 ]2[ ), 3.957
beats between the sixth harmonic of C4 and the fifth harmonic of EÝ4 (1573.711Hz).
Ex. 49. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 66
Table 82. The beating characteristics of the fifteen-tone equally-tempered minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
EÝ4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G¢4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
6
– – – –
– – – –
– – – –
– – – –
314.742
– – – –
– – – –
629.484
– – – –
944.226
– – – –
– – – –
1258.968
– – – –
1573.711
1888.453
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
3.957
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
396.550
– – – –
– – – –
793.100
– – – –
– – – –
1189.651
– – – –
– – – –
1586.201
– – – –
– – – –
1982.751
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
8.224
– – – –
– – – –
– – – –
– – – –
– – – –
16.447
– – – –
– – – –
– – – –
– – – –
The closest approximation of the major scale available in fifteen-tone equal temperament in both sléndro
and pélog forms is considered as presenting the series: 0.000, 240.000, 400.000, 480.000, 720.000,
880.000, 1120.000, and 1200.000 cents, and therefore representing the just major tone (9/8), just major
The Equally-Tempered Archetype 145
third (5/4), just perfect fourth (4/3), just perfect fifth (3/2), just major sixth (5/3), and just major seventh
(15/8) with falsity of +36.090, +13.686, ß18.045, +18.045, ß4.359, and +31.731 cents on each
count. Major, minor, and diminished triads produced via this temperament presents major thirds 400.000
cents in size, minor thirds with a range between 240.000 and 320.000 cents, perfect fifths with a range
between 640.000 and 720.000 cents, and a diminished fifth 560.000 cents in size. The most
problematic triad being D minor, or the pitches D£, F², and A, with a minor third and perfect fifth
represented by the 5-et supermajor second ( 5 2 ) and 15-et thirteen quarter-tones ( 815 ]2[ , 741/512, or
640.000 cents),219 and a deviation from just intervals of ß75.641 and ß61.955 cents. In harmonic terms,
the two intervals of the D minor triad essentially approximate a supermajor second (59049/51300, or
243.545 cents) and the division of the disjunction between GÝ and G³, or thirteen quarter-tones (90/62,
or 645.188 cents), and therefore contributing nothing with any possibility of being orally interpreted as
any form of a major triad.
Ex. 50. The fifteen-tone equally-tempered major scale
The closest approximation of the minor scale on the other hand presents the series: 0.000, 240.000,
320.000, 480.000, 720.000, 800.000, 1120.000, and 1200.000 cents, and therefore representing the
just major tone (9/8), just minor third (6/5), just perfect fourth (4/3), just perfect fifth (3/2), just minor sixth
(8/5), and acute or large minor seventh (9/5) with a falsity of +36.090, +4.359, ß18.045, +18.045,
ß13.686, and +2.404 cents on each count.
Ex.51. The fifteen-tone equally-tempered minor scale
The sléndro scalar scheme provides two options for the approximation of the major second, being D³
or D£, or the 15-et grave or small tone ( 215 ]2[ , approximately 34/31, or 160.000 cents) and 5-et
supermajor second ( 5 2 ); while pélog, two options for the tritone, being FË or FÍ, or 15-et eleven
quarter-tones ( 715 ]2[ , 1024/741, or 560.000 cents) and 15-et thirteen quarter-tones ( 815 ]2[ ), as well as
two options for the major seventh, being B³ or B£, or the 15-et neutral seventh ( 1315 ]2[ , approximately
31/17, or 1040.000 cents) and 15-et just diatonic major seventh ( 1415 ]2[ , approximately 489/256, or
219 Daniélou, Tableau Comparatif des Intervalles Musicaux 137.
146 The Equally-Tempered Archetype
1120.000 cents).220 Bar 68 is representative of intonational reinterpretations of a simultaneous sonority
based on the two major seconds, which represent the just major tone (9/8) with a falsity of ß43.013
and +39.635 cents on each count. The latter interval is further represented by the simultaneous
sonorities of D£ and F², and G¢ and AÚ in bar 66. The mean deviation of the two intervals from the just
major tone (9/8) is significantly close to the interval of the great diesis (128/125), or “the defect of three
major thirds from an octave.”221
Ex. 52. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 68
Bars 69 and 70 depicts the two microtonal inflections of the major seventh, or the 15-et neutral seventh
( 1315 ]2[ ) and 15-et just diatonic major ( 1415 ]2[ ).
Ex. 53. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 69-70
Bar 72 on the other hand, depicts the two microtonal inflections of the tritone, 15-et eleven quarter-
tones ( 715 ]2[ ) and 15-et thirteen quarter-tones ( 815 ]2[ ); while the following bar outlines a fifteen-tone
equally-tempered pélog scalar descent (omitting CÚ) with the pitches G¢, FÍ, FË, F², and EÝ – equal to
720.000, 640.000, 560.000, 480.000, and 320.000 cents respectively, and therefore a scalar
progression with equal tones 80.000 cents in size, or the 15-et grave or small just chromatic semitone
220 Daniélou, Tableau Comparatif des Intervalles Musicaux 23-136.
221 The juxtaposition of three just major thirds (5/4), equal to 1158.941 cents is a figure that diminishes the octave
(2/1) by 41.059 cents, and the interval termed as the ‘great diesis’ with a frequency ratio of 128/125. For a further
discussion, see Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 453.
The Equally-Tempered Archetype 147
( 15 2 ). Framed as simultaneous sonorities with repeated C's, the resolution is provided by the
resounding of the fundamental in the downbeat of sixteen-tone equal temperament.
Ex. 54. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 72-73
The following table depicts the tonal resources of fifteen-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
Table 83. The fifteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÐÑ
C
CÚ
D³
D£
EÝ
E¢
F²
FË / Gì
FÍ / Gí
G¢
A¸
A
AÚ
B³
B£
C
unison
15-et grave or small just chromatic semitone
15-et grave or small tone
5-et supermajor second
15-et just minor third
equal or Pythagorean major third, or ditone
5-et grave or small fourth
15-et eleven quarter-tones
15-et thirteen quarter-tones
5-et acute or large fifth
equal or Pythagorean minor sixth
15-et just major sixth
5-et augmented sixth
15-et neutral seventh
15-et just diatonic major seventh
octave
1.000000
1.047294
1.096825
1.148698
1.203025
1.259921
1.319508
1.381913
1.447269
1.515717
1.587401
1.662476
1.741101
1.823445
1.909683
2.000000
261.626
273.999
286.957
300.529
314.742
329.628
345.217
361.544
378.643
396.550
415.305
434.946
455.517
477.060
499.622
523.251
0.000
80.000
160.000
240.000
320.000
400.000
480.000
560.000
640.000
720.000
800.000
880.000
960.000
1040.000
1120.000
1200.000
+00
ß20
ß40
+40
+20
+00
ß20
+60
+40
+20
+00
ß20
ß40
ß60
+20
+00
148 The Equally-Tempered Archetype
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 55. 15-tone equal temperament
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 84. 15-tone equal temperament tuning matrix no. 1 (Program 17)
NOTE C CÚ D£ EÝ E¢ F² FË G¢ A¸ A A! B£
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß20 +40 +20 +00 ß20 ß40 +20 +00 ß20 ß40 +20
Table 85. 15-tone equal temperament tuning matrix no. 2 (Program 18)
NOTE – – – – D³ – – – – – – FÍ – – – – – – B³ – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 ß40 +00 +00 +00 +40 +00 +00 +00 +40 +00
The Equally-Tempered Archetype 149
Sixteen-Tone Equal Temperament Sixteen-tone equal temperament ( 16 2 ) presents a series of sixteen equal tones 75.000 cents in size,
which represent the 16-et grave or small just chromatic semitone (approximately 47/45),222 and is what
may be interpreted as being a scale made up of sixteen equal three-quarter-tone steps. The
temperament may alternatively be expressed as the juxtaposition of two sets of eight-tone equally-
tempered intervals, four sets of four-tone equally-tempered intervals, or eight sets of two-tone equally-
tempered intervals interlocked a 16-et grave or small just chromatic semitone ( 16 2 ) apart. Blackwood
makes the following observations about the temperament: “Triads in 16-note tuning, although
recognizable, are too discordant to serve as the final harmony in cadences. But keys can still be
established by successions of altered subdominant and dominant harmonies.”223
The work outlines a sixteen-tone equally-tempered minor triad, within an added-note chord
incorporating CÚ, and therefore essentially what could be interpreted as being a form of a Cmin(add"9)
tetrad if enharmonically reinterpreted. The triad consists of the pitches C, E¸, and G², and presents a
minor third 300.000 cents in size ( 4 2 , or the equal minor third) and a perfect fifth 675.000 cents in size
( 916 ]2[ , 16-et grave or small fifth, or approximately 189/128).224 The comparison with the just minor third
(6/5) and just perfect fifth (3/2) reveal a falsity of +4.359 and +18.045 cents on each count. G²4
(386.375Hz), or 16-et grave or small fifth ( 916 ]2[ ), presents 12.126 beats between the third harmonic of
C4 and the second harmonic of G²4 (772.751Hz), and 24.251 beats between the sixth harmonic of C4
and the fourth harmonic of G¢4 (1545.502Hz); while E¸4, or the equal minor third ( 4 2 ), 14.118 beats
between the sixth harmonic of C4 and the fifth harmonic of E¸4.
Ex. 56. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 78
222 Daniélou, Tableau Comparatif des Intervalles Musicaux 20.
223 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
224 Daniélou, Tableau Comparatif des Intervalles Musicaux 129.
150 The Equally-Tempered Archetype
Table 86. The beating characteristics of the sixteen-tone equally-tempered minor triad
C4 (PARTIAL)
FREQUENCY
(HERTZ)
E¸4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G²4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
311.127
– – – –
– – – –
622.254
– – – –
933.381
– – – –
– – – –
1244.508
– – – –
1555.635
– – – –
1866.762
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
14.118
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
386.375
– – – –
– – – –
772.751
– – – –
– – – –
1159.126
– – – –
– – – –
1545.502
– – – –
– – – –
1931.877
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
12.126
– – – –
– – – –
– – – –
– – – –
– – – –
24.251
– – – –
– – – –
– – – –
– – – –
The pélog scalar scheme now provides two options for the approximation of the augmented unison,
being CÚ or CÍ, or the 16-et grave or small just chromatic semitone ( 16 2 ) and equal three-quarter-tone
( 8 2 ); two for the fourth, being Fí or F¢, or nine equal quarter-tones ( 38 ]2[ ) and the 16-et acute or
large fourth 716 ]2[ , approximately 256/189, or 525.000 cents); two for the fifth, being G² or G£, or the
16-et grave or small fifth ( 916 ]2[ ) and fifteen equal quarter-tones ( 58 ]2[ ); as well as two options for the
major seventh, being B³ or B£, or twenty-one equal quarter-tones ( 78 ]2[ ) and the 16-et acute or large
major seventh ( 1516 ]2[ , approximately 90/47, or 1125.000 cents). Sléndro accommodates the fourth
and fifth. The microtonal nuances of these pitches are exploited both melodically and harmonically
throughout the section of the work demarcated as ‘16-tone equal temperament’. Simultaneous
sonorities include the dyads F¢ and G², or the equal three-quarter-tone ( 8 2 ); F¢ and G£, or 16-et acute
or large tone ( 316 ]2[ , approximately 41/36, or 225.000 cents); CÚ and F¢, or the equal minor third ( 4 2 ),
CÍ and F¢, or the 16-et grave or small major third ( 316 ]2[ , approximately 77/62, or 375.000 cents); D¢ and G², and CÚ and F¥, or nine equal quarter-tones ( 38 ]2[ ); D¢ and G£, and E¸ and B¹, or the 16-et
acute or large fourth ( 716 ]2[ ); CÚ and GÍ, or the equal tritone ( 2 2 ); C and G², and F¢ and C, or the 16-
et grave or small fifth ( 916 ]2[ ); C and G£, or fifteen equal quarter-tones ( 58 ]2[ ); E¸ and B£, C and GÍ,
B¹ and F¥, and G² and E¸, or 16-et seventeen quarter-tones ( 1116 ]2[ , approximately 124/77, or
825.000 cents); as well as C and B¹, or the 16-et grave or small minor seventh ( 1316 ]2[ , approximately
The Equally-Tempered Archetype 151
72/41, or 975.000 cents).225 The following table depicts the tonal resources of sixteen-tone equal
temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.
Table 87. The sixteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
ÐÑ
C
CÚ
CÍ / Dí
D¢
E¸
E²
EË / Fí
F¢
F¥
G²
G£
GÍ / Aí
A¢
B¹
B³
B£
C
unison
16-et grave or small just chromatic semitone
equal three-quarter-tone
16-et acute or large tone
equal or Pythagorean minor third, or trihemitone
16-et grave or small major third
nine equal quarter-tones
16-et acute or large fourth
equal or acute or large tritone, or augmented fourth
16-et grave or small fifth
fifteen equal quarter-tones, or superfifth
16-et seventeen quarter-tones
equal or Pythagorean major sixth
16-et grave or small minor seventh
twenty-one equal quarter-tones, or neutral seventh
16-et acute or large major seventh
octave
1.000000
1.044274
1.090508
1.138789
1.189207
1.241858
1.296840
1.354256
1.414214
1.476826
1.542211
1.610490
1.681793
1.756252
1.834008
1.915207
2.000000
261.626
273.209
285.305
297.936
311.127
324.902
339.286
354.308
369.994
386.375
403.482
421.345
440.000
459.480
479.823
501.067
523.251
0.000
75.000
150.000
225.000
300.000
375.000
450.000
525.000
600.000
675.000
750.000
825.000
900.000
975.000
1050.000
1125.000
1200.000
+00
ß25
+50
+25
+00
ß25
ß50
+25
+00
ß25
+50
+25
+00
ß25
ß50
+25
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 57. 16-tone equal temperament
225 Daniélou, Tableau Comparatif des Intervalles Musicaux 21-128.
152 The Equally-Tempered Archetype
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 88. 16-tone equal temperament tuning matrix no. 1 (Program 19)
NOTE C CÚ D¢ E¸ E² F¢ F¥ G² GÍ A¢ B¹ B£
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß25 +25 +00 ß25 +25 +00 ß25 +25 +00 ß25 +25
Table 89. 16-tone equal temperament tuning matrix no. 2 (Program 20)
NOTE – – CÍ – – – – – – Fí – – G£ – – – – – – B³
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +50 +00 +00 +00 ß50 +00 +50 +00 +00 +00 ß50
Seventeen-Tone Equal Temperament Seventeen-tone equal temperament is generated by the factor 17 2 , and with its seventeen equal tones
70.588 cents in size, which represent the 17-et grave or small just chromatic semitone, is able to
approximate the important interval of just intonation termed as the grave or small just chromatic semitone
(25/24) with extreme accuracy – a value expressing a falsity of ß0.084 cents.226 This is the ratio in just
intonation “applied to any diatonic note to produce the sharpened form of that note.”227 Blackwood
describes the temperament thus: “17-note triads are very discordant due to the large major third, so the
fundamental consonant harmony of the tuning is a minor triad with an added minor seventh. The scale is
very good due to the relatively small minor second.”228 Blackwood is of course referring to the major
third 423.529 cents in size (E£), which may be referred to as the 17-et acute or large major third
( 617 ]2[ , or approximately 327/256), and deviates from the just major third (5/4) by +37.216 cents. The
present scheme relegates this third to the position of auxiliary tone, due to the better approximation of
226 Daniélou, Tableau Comparatif des Intervalles Musicaux 20.
227 John Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” Perspectives of New Music
29.2 (Summer, 1991): 109.
228 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
The Equally-Tempered Archetype 153
the just major third (5/4) provided by E³ ( 517 ]2[ , or approximately 38/31), the 17-et neutral third, at
352.941 cents.229 The alternative theorization produces a dissonant triad nevertheless, yet seventeen-
tone equal temperament is the second temperament to contain what Blackwood terms as recognizable
diatonic scales and “perfect fifths within the range of recognizability”.230
The seventeen-tone equally-tempered major triad consists of the pitches C, E³, and G, and
presents a major third 352.941 cents in size ( 517 ]2[ , or 17-et neutral third) and a perfect fifth 705.882
cents in size ( 1017 ]2[ , or 17-et just perfect fifth). The temperament produces the just major third (5/4)
and just perfect fifth (3/2) with a falsity of ß33.373 and +3.927 cents on each count. G4 (393.330Hz),
or 17-et just perfect fifth ( 1017 ]2[ ), presents 1.783 beats between the third harmonic of C4 and the
second harmonic of G4 (786.659Hz), and 3.565 beats between the sixth harmonic of C4 and the fourth
harmonic of G4 (1573.318Hz); while E³4 (307.972Hz), or the 17-et neutral third ( 517 ]2[ ), 76.241 beats
between the fifth harmonic of C4 and the fourth harmonic of E³4 (1231.189Hz).
Ex. 58. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 87
229 Daniélou, Tableau Comparatif des Intervalles Musicaux 90-106.
230 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
154 The Equally-Tempered Archetype
Table 90. The beating characteristics of the seventeen-tone equally-tempered major triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E³4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
– – – –
6
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
1308.128
– – – –
1569.753
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
5
– – – –
– – – –
6
– – – –
– – – –
307.972
– – – –
– – – –
615.943
– – – –
923.915
– – – –
– – – –
1231.189
1539.858
– – – –
– – – –
1847.830
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
76.241
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
– – – –
– – – –
393.330
– – – –
– – – –
786.659
– – – –
– – – –
1179.989
– – – –
– – – –
1573.318
– – – –
1966.648
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1.783
– – – –
– – – –
– – – –
– – – –
– – – –
3.565
– – – –
– – – –
– – – –
Blackwood’s consonant minor triad is located in this scheme within the pitches C, DÚ, and G, which when
enharmonically reinterpreted as C, EÝ, and G produces a minor third 282.353 cents in size, and therefore
a just minor third (6/5) with a falsity of ß33.288 cents. The exercise illustrates that the principal minor
triad is no more consonant than the principal major triad in seventeen-tone equal temperament.
Seventeen-tone equal temperament provides two options for the approximation of the
augmented unison, being CÚ or CÍ, or the 17-et grave or small just chromatic semitone ( 17 2 ) and 17-et
three-quarter-tone ( 217 ]2[ , approximately 243/224, or 141.176 cents); two for the major third, being E³
or E£, or the 17-et neutral third ( 517 ]2[ ) and 17-et acute or large major third ( 617 ]2[ ); two for the
tritone, being FÚ or GÝ, or the 17-et grave or small augmented fourth ( 817 ]2[ , approximately 709/512, or
564.706 cents) and 17-et acute or large diminished fifth ( 917 ]2[ , approximately 739/512, or 635.294
cents); two for the major sixth, being A³ or A£, or the 17-et neutral sixth ( 1217 ]2[ , approximately
835/512, or 847.059 cents, and 17-et acute or large major sixth ( 1317 ]2[ , approximately 435/256, or
917.647 cents); as well as two options for the major seventh, being B² or B£, or the 17-et grave or small
major seventh ( 1517 ]2[ , approximately 448/243, or 1058.824 cents) and 17-et grave or small major
seventh ( 1617 ]2[ , approximately 48/25, or 1129.412 cents).
Simultaneous sonorities include the dyads A£ and C, or the 17-et augmented second ( 417 ]2[ ,
approximately 512/435, or 282.353 cents); G and C, and B¸ and DÚ, or the 17-et just perfect fourth
The Equally-Tempered Archetype 155
( 717 ]2[ , approximately 681/512, or 494.118 cents); A³ and C, and C and E³, or the 17-et neutral third
( 517 ]2[ ); C and E£, or the 17-et acute or large major third ( 617 ]2[ ); C and FÚ, F and B², FÚ and B£, GÝ
and C, G and CÚ, and GÚ and CÍ, or the 17-et grave or small augmented fourth ( 817 ]2[ ); C and GÝ, CÚ
and G, and CÍ and GÚ, or the 17-et acute or large diminished fifth ( 917 ]2[ ); C and G, and DÚ and B¸, or
17-et just perfect fifth ( 1017 ]2[ ); DÚ and B£, and C and A³, or the 17-et neutral sixth ( 1217 ]2[ ); C and
A£, or the 17-et acute or large major sixth ( 1317 ]2[ ); CÚ and B², and C and B¸, or the 17-et Pythagorean
minor seventh ( 1417 ]2[ , approximately 3584/2025, or 988.235); as well as C and B², or the 17-et grave
or small major seventh ( 1517 ]2[ ).231 The following table depicts the tonal resources of seventeen-tone
equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.
Table 91. The seventeen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÐÑ
C
CÚ
CÍ / Dí
D
DÚ
E³
E£
F
FÚ
GÝ
G
GÚ
A³
A£
B¸
B²
B£
C
unison
17-et grave or small just chromatic semitone
17-et three-quarter-tone
17-et just major tone
17-et augmented second
17-et neutral third
17-et acute or large major third
17-et just and Pythagorean perfect fourth
17-et grave or small augmented fourth
17-et acute or large diminished fifth
17-et just and Pythagorean perfect fifth
17-et augmented fifth
17-et neutral sixth
17-et acute or large major sixth
17-et Pythagorean minor seventh
17-et grave or small major seventh
17-et acute or large major seventh
octave
1.000000
1.041616
1.084964
1.130116
1.177147
1.226135
1.277162
1.330312
1.385674
1.443341
1.503407
1.565972
1.631142
1.699024
1.769730
1.843379
1.920093
2.000000
261.626
272.513
283.854
295.667
307.972
320.788
334.138
348.044
362.528
377.615
393.330
409.698
426.748
444.508
463.007
482.275
502.346
523.251
0.000
70.588
141.176
211.765
282.353
352.941
423.529
494.118
564.706
635.294
705.882
776.471
847.059
917.647
988.235
1058.824
1129.412
1200.000
+00
ß29
+41
+12
ß18
ß47
+24
ß06
ß35
+35
+06
ß24
ß53
+18
ß12
ß41
+29
+00
231 Daniélou, Tableau Comparatif des Intervalles Musicaux 21-139.
156 The Equally-Tempered Archetype
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 59. 17-tone equal temperament
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 92. 17-tone equal temperament tuning matrix no. 1 (Program 21)
NOTE C CÍ D D! E³ F F! G G! A£ B¸ B²
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +41 +12 ß18 ß47 ß06 ß35 +06 ß24 +18 ß12 ß41
Table 93. 17-tone equal temperament tuning matrix no. 2 (Program 22)
NOTE – – CÚ – – – – E£ – – GÝ – – A³ – – – – B£
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß29 +00 +00 +24 +00 +35 +00 +47 +00 +00 +29
Eighteen-Tone Equal Temperament Eighteen-tone equal temperament ( 18 2 ) presents a subdivision of the equal major third ( 3 2 ) into six
equal tones 66.667 cents in size, or equal third-tones, which represent the 18-et grave or small just
chromatic semitone (approximately 133/128).232 The temperament may alternatively be expressed as the
juxtaposition of two sets of nine-tone equally-tempered intervals, three sets of six-tone equally-tempered
232 Daniélou, Tableau Comparatif des Intervalles Musicaux 18.
The Equally-Tempered Archetype 157
intervals, six sets of three-tone equally-tempered intervals, or nine sets of two-tone equally-tempered
intervals interlocked an 18-et grave or small just chromatic semitone ( 18 2 ) apart. Blackwood makes the
following observations about the temperament: “The perfect fifths are so out of tune that even seventh
chords are disturbingly discordant.”233 Major, minor, and diminished triads produced via the major scale
derived from this temperament presents major thirds with a range between 333.333 and 400.000 cents;
minor thirds between 266.667 and 333.333 cents; perfect fifths between 666.667 and 733.333 cents;
and a diminished fifth 600.000 cents in size. In comparison with the just perfect fifth (3/2), fifths within
the specified range present a deviation of between ß31.378 and +31.378 cents, which is roughly an
equal sixth-tone ( 36 2 ), and approaching two syntonic commas (6561/6400, or 43.013) – clearly
outside of the boundaries set by Blackwood in his theory of “perfect fifths within the range of
recognizability” and its formula: 74 a<v< 5
3 a, or 685.714<v<720.000.234
All triads prove to be problematic, and due to the fact of either producing the just major third
(5/4) and just perfect fifth (3/2) with a falsity of +13.683 and +31.378 cents, as in C major (C, E¢, and
G£) and F major (F², A², and C); or ß52.980 and ß35.288 cents, as in G major (G£, B², and D). Minor
triads produce the just minor third (6/5) and just perfect fifth (3/2) with a falsity of ß48.975 and
ß35.288 cents, as in D minor (D, F², and A²); +17.692 and ß35.288 cents, as in E minor (E¢, G£, and
B²); and +17.692 and +31.378 cents, as in A minor (A², C, and E¢); while the B diminished triad (B², D,
and F²) produces the just minor third (6/5) and just tritone (45/32) with a falsity of +17.692 and +9.776
cents.
Ex. 60. The eighteen-tone equally-tempered major scale
The work outlines an implied eighteen-tone equally-tempered minor triad, within an arpeggiated added-
note chord incorporating B¸, and therefore essentially a Cmin7 tetrad. The triad consists of the pitches C,
Eí and G£, and presents a minor third 333.333 cents in size ( 518 ]2[ , 18-et seven quarter-tones, or
approximately 40/33) and a perfect fifth 733.333 cents in size ( 1118 ]2[ , 18-et superfifth, or
approximately 189/128).235 Eí, although equal to 18-et seven quarter-tones ( 518 ]2[ ), functions
harmonically as a minor third. The comparison with the just minor third (6/5) and just perfect fifth (3/2)
reveal a falsity of +17.692 and +31.378 cents on each count. G£4 (399.616Hz), or the 18-et superfifth
( 1118 ]2[ ), presents 14.355 beats between the third harmonic of C4 and the second harmonic of G£4
233 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
234 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
235 Daniélou, Tableau Comparatif des Intervalles Musicaux 18-129.
158 The Equally-Tempered Archetype
(799.232Hz), and 28.711 beats between the sixth harmonic of C4 and the fourth harmonic of G£4
(1598.464Hz); while Eí4 (317.175Hz), or 18-et seven quarter-tones ( 518 ]2[ ), 16.124 beats between
the sixth harmonic of C4 and the fifth harmonic of Eí4 (1585.877Hz). Bar 88 also serves to highlight the
highly dissonant nature of the two fifths available in eighteen-tone equal temperament.
Ex. 61. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 88
Table 94. The beating characteristics of the eighteen-tone equally-tempered minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
Eí4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G£4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
317.175
– – – –
– – – –
634.351
– – – –
951.526
– – – –
– – – –
1268.702
– – – –
1585.877
– – – –
1903.053
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.124
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
399.616
– – – –
– – – –
799.232
– – – –
– – – –
1198.848
– – – –
– – – –
1598.464
– – – –
– – – –
1998.080
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
14.355
– – – –
– – – –
– – – –
– – – –
– – – –
28.711
– – – –
– – – –
– – – –
– – – –
Eighteen-tone equal temperament provides two options for the approximation of the augmented
unison/minor second, being CÚ or DÝ, or the 18-et grave or small just chromatic semitone ( 18 2 ) and 9-et
The Equally-Tempered Archetype 159
great limma ( 9 2 ); two for the minor third, being Eì or Eí, or 9-et five quarter tones ( 29 ]2[ ) and 18-et
seven quarter-tones ( 518 ]2[ ); two for the fourth, being F² or F£, or the 18-et grave or small fourth
( 718 ]2[ , approximately 512/391, or 466.667 cents) and 9-et superfourth ( 49 ]2[ ); two for the fifth, being
G³ or G£, or the 9-et subfifth ( 59 ]2[ ) and 18-et superfifth ( 1118 ]2[ ); two for the major sixth, being A² or
AË, or the 18-et grave or small major sixth ( 1318 ]2[ , approximately 33/20, or 866.667 cents) and 9-et
nineteen quarter-tones ( 79 ]2[ , approximately 12/7, or 933.333 cents); as well as two options for the
major seventh, being B² or B£, or the 9-et grave or small major seventh ( 89 ]2[ , approximately 1024/553,
or 1066.667 cents) and 18-et acute or large major seventh ( 1718 ]2[ , approximately 256/133, or
1133.333 cents).236 Bars 89 and 90 illustrate the melodic utilization of the two alternative fifths, fourths
and minor thirds, each deviating approximately an equal sixth-tone ( 36 2 ) in either direction of its
associated justly intoned interval.
Ex. 62. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 89-90
The utilization of simultaneous sonorities in this temperament are somewhat limited due to the focus on
melodic development, although includes the dyads C and Eì, or the 9-et five quarter tones ( 29 ]2[ ), C
and A², or the 18-et grave or small major sixth ( 1318 ]2[ ), and C and AË, or 9-et nineteen quarter-tones
( 1418 ]2[ ). The following table depicts the tonal resources of eighteen-tone equal temperament,
indicating degree, notation, interval, ratio, frequency, cents, and tuning.
236 Daniélou, Tableau Comparatif des Intervalles Musicaux 31-116.
160 The Equally-Tempered Archetype
Table 95. The eighteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÐÑ
C
CÚ
DÝ
D
DË / Eì
DÍ / Eí
E¢
F²
F£
F¥
G³
G£
A¸
A²
AË / Bì
B¸
B²
B£
C
unison
18-et grave or small just chromatic semitone
9-et great limma, or large half-tone
equal or just major tone
9-et five quarter-tones
18-et seven quarter-tones
equal or Pythagorean major third, or ditone
18-et grave or small fourth
9-et superfourth
equal or acute or large tritone, or augmented fourth
9-et subfifth
18-et superfifth
equal or Pythagorean minor sixth
18-et grave or small major sixth
9-et nineteen quarter-tones
equal or Pythagorean minor seventh
9-et grave or small major seventh
18-et acute or large major seventh
octave
1.000000
1.039259
1.080060
1.122462
1.166529
1.212326
1.259921
1.309385
1.360790
1.414214
1.469734
1.527435
1.587401
1.649721
1.714488
1.781797
1.851749
1.924448
2.000000
261.626
271.897
282.571
293.665
305.194
317.175
329.628
342.568
356.017
369.994
384.520
399.616
415.305
431.609
448.554
466.164
484.465
503.485
523.251
0.000
66.667
133.333
200.000
266.667
333.333
400.000
466.667
533.333
600.000
666.667
733.333
800.000
866.667
933.333
1000.000
1066.667
1133.333
1200.000
+00
ß33
+33
+00
+67
+33
+00
ß33
+33
+00
ß33
+33
+00
ß33
+33
+00
ß33
+33
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 63. 18-tone equal temperament
The Equally-Tempered Archetype 161
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 96. 18-tone equal temperament tuning matrix no. 1 (Program 23)
NOTE C DÝ D DÍ E¢ F² F¥ G³ A¸ A² B¸ B²
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +33 +00 +33 +00 ß33 +00 ß33 +00 ß33 +00 ß33
Table 97. 18-tone equal temperament tuning matrix no. 2 (Program 24)
NOTE – – CÚ – – DË – – F£ – – G£ – – AË – – B£
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß33 +00 ß33 +00 +33 +00 +33 +00 +33 +00 +33
Nineteen-Tone Equal Temperament Nineteen-tone equal temperament is generated by the factor 19 2 , and presents nineteen equal tones
63.158 cents in size, which represent the 19-et just diatonic semitone (approximately 531/512).237 It is a
temperament advocated by American musicologist, organist, and conductor Joseph Yasser (1893-
1981), and theorized in his book of 1932 entitled Theory of Evolving Tonality. A. R. McClure offers the
following discussion:
“Yasser’s speculations on the evolution of music led him to conclude that the diatonic scale of seven tones
plus five accidentals would, in time, give place to a supra-diatonic scale of twelve tones plus seven
accidentals. Then, on certain premises of his own and from assumptions of physical theorists, he
developed, by most ingenious logic, a fanciful scheme with fresh notation and terminology, designed to
serve the needs of composers in the future – the detailed plan of a hypothetical new order.”238
The temperament was theoretically perceived by Joseph Yasser as the historical expansion of a diatonic
system into a chromatic one (with the addition of five auxiliary tones), and a chromatic one into a ‘supra-
diatonic scale’ (with the further addition of seven auxiliary tones). The nineteen-tone scale is
accommodated by Yasser with a notation system incorporating a ten-line staff and two additional
symbols for accidentals. “On the theoretical side there are many points of interest, for the nineteen-tone
237 Daniélou, Tableau Comparatif des Intervalles Musicaux 18.
238 A. R. McClure, “Studies in Keyboard Temperaments,” The Galpin Society Journal 1 (Mar., 1948): 32-34
162 The Equally-Tempered Archetype
system allowed one to actually distinguish augmented intervals and their inversions, which on the normal
pianoforte have no separate existence,” points out McClure.239 Joseph Yasser envisaged systems of
temperament beyond the ‘supra-diatonic scale’, with the theorization of the ‘ultra-diatonics’ and systems
with tonal structures such as 19+12=31, and 31+19=50. The complete evolutionary chain is
expressed as the following combinations of scale degrees: 2+3=5, 5+2=7, 7+5=12, 12+7=19,
19+12=31, and 31+19=50.240
With regards to the aesthetic qualities of nineteen-tone equal temperament, Blackwood makes the
following observations: “Triads are smooth, but the scale sounds slightly out of tune because the leading
tone seems low with respects to the tonic. Diatonic behaviour is virtually identical to that of 12-note
tuning, but chromatic behaviour is very different. For example, a perfect fourth is divisible into two equal
parts, while an augmented sixth and a diminished seventh sound identical.”241 The pitch allocated as the
leading-tone that Blackwood refers to is B² at 1073.684 cents ( 1719 ]2[ , 19-et grave or small major
seventh, or approximately 119/64),242 which represents the just diatonic major seventh (15/8) with a
falsity of ß14.585 cents. Blackwood also makes note that “nineteen-note equal tuning contains diatonic
scales in which a major second spans three chromatic degrees, while a minor second spans two.”243
The nineteen-tone division of the octave (along with five, seven, twelve, thirty-one, forty-one, fifty-
three, three-hundred and six, and three-hundred and forty-seven) are represented in the list of nine
seminal propositions by theorists with regards to the cyclic division of the octave, or the Pythagorean
cycle of fifths. Although, the fact that the nineteenth 3/2 produces a tone 137.145 cents (3ñù/2óð, or
cyclic great limma) above the eleventh 2/1 in the nineteen cycle, reveals somewhat of an inferiority when
compared to the twelve-tone cycle, where the twelfth 3/2 produces a tone 23.460 cents (3ñò/2ñù, or
Pythagorean comma) above the seventh 2/1.
Just minor thirds and major sixths (6/5s and 5/3s) are represented well in this temperament, with a
falsity of +0.148 and ß0.148 cents on each count, while just major thirds and minor sixths (5/4s and
8/5s), represented adequately, with a falsity +7.366 and ß7.366 cents on each count. This is a
significant improvement to the falsities of 15.641 cents for 6/5 and 5/3, and 13.686 cents for 5/4 and 8/5
expounded by twelve-tone equal temperament.244 Nineteen-tone equal temperament is the third
239 Read, 20th-Century Microtonal Notation 98.
240 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 431-32.
241 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
242 Daniélou, Tableau Comparatif des Intervalles Musicaux 37.
243 Easley Blackwood, “Modes and Chord Progressions in Equal Tunings,” Perspectives of New Music 29.2
(Summer, 1991): 168. 244 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 400-32.
The Equally-Tempered Archetype 163
temperament to contain what Blackwood terms as recognizable diatonic scales and “perfect fifths within
the range of recognisability.”245
The nineteen-tone equally-tempered major scale generates consistent major triads, which produce
the just major third (5/4) and just perfect fifth (3/2) with a falsity of +7.366 and ß7.218 cents, as in C
major (C, E, and G), F major (F, A, and C), and G major (G, B², and D²). Equally consistent minor triads are
available, which produce the just minor third (6/5) and just perfect fifth with a falsity of +0.148 and
ß7.218 cents, as in D minor (D, F, and A), E minor (E, G, and B²), and A minor (A, C, and E); while the B
diminished triad (B², D², and F), produces the just minor third (6/5) and just tritone (45/32) with a falsity
of +0.148 and +41.355 cents. “In sum, all diatonic progressions of triads and seventh chords have the
same behaviour and produce the same musical effect in twelve-note and nineteen-note tuning,”
comments Blackwood, “save for slight differences only, the most noticeable being the peculiar tuning of
the nineteen-note major scale.” In striking contrast, “many nineteen-note chromatic progressions will
bring about alien melodic intervals.”246
Ex. 64. The nineteen-tone equally-tempered major scale
The work outlines a nineteen-tone equally-tempered minor triad in first inversion, which includes the
pitches C, EÝ, and G, and presents a minor third 315.789 cents in size ( 519 ]2[ , 19-et just minor third, or
approximately 6/5) and a perfect fifth 694.737 cents in size ( 1119 ]2[ , 19-et just perfect fifth, or
approximately 115/77).247 The comparison with the just minor third (6/5) and perfect fifth (3/2) reveal a
falsity of +0.148 and ß7.218 cents on each count. G4 (390.806Hz), or the 19-et just perfect fifth
( 1119 ]2[ ), presents 3.266 beats between the third harmonic of C4 and the second harmonic of G4
(781.611Hz), and 6.531 beats between the sixth harmonic of C4 and the fourth harmonic of G4 (1563.222Hz); while EÝ4 (313.978Hz), or the 19-et just minor third ( 519 ]2[ ), 0.134 beats between the
sixth harmonic of C4 and the fifth harmonic of EÝ4 (1569.888Hz).
245 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
246 Blackwood, “Modes and Chord Progressions in Equal Tunings,” Perspectives of New Music 169-72.
247 Daniélou, Tableau Comparatif des Intervalles Musicaux 82-125.
164 The Equally-Tempered Archetype
Ex. 65. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 97
Table 98. The beating characteristics of the nineteen-tone equally-tempered minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
EÝ4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
313.978
– – – –
– – – –
627.955
– – – –
941.933
– – – –
– – – –
1255.910
– – – –
1569.888
– – – –
1883.865
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.134
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
390.806
– – – –
– – – –
781.611
– – – –
– – – –
1172.417
– – – –
– – – –
1563.222
– – – –
– – – –
1954.028
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
3.266
– – – –
– – – –
– – – –
– – – –
– – – –
6.531
– – – –
– – – –
– – – –
– – – –
Nineteen-tone equal temperament provides two options for the approximation of the minor second,
being D¸ or DÝ, or the 19-et just diatonic semitone ( 19 2 ) and 19-et great limma 219 ]2[ , approximately
128/119, or 126.316 cents); two for the major second, being D² or DË, or the 19-et just minor tone
( 319 ]2[ , approximately 512/459, or 189.474 cents) and 19-et five quarter-tones ( 419 ]2[ , approximately
81/70, or 252.632 cents); two for the major third, being E or EË, or the 19-et just major third ( 619 ]2[ ,
approximately 61/49, or 378.947 cents) and 19-et nine quarter-tones ( 719 ]2[ , approximately 661/512,
or 442.105 cents); two for the tritone, being FÚ or GÝ, or the 19-et grave or small augmented fourth
( 919 ]2[ , approximately 711/512, or 568.421 cents) and 19-et acute or large diminished fifth ( 1019 ]2[ ,
The Equally-Tempered Archetype 165
approximately 1024/711, or 631.579 cents); two for the minor sixth, being A¸ or Aí, or the 19-et
Pythagorean minor sixth ( 1219 ]2[ , approximately 793/512, or 757.895 cents) and 19-et seventeen
quarter-tones ( 1319 ]2[ , approximately 98/61, or 821.053 cents); two for the major sixth, being A or AË,
or the 19-et just major sixth ( 1419 ]2[ , approximately 5/3, or 884.211 cents) and 19-et nineteen quarter-
tones ( 1519 ]2[ , approximately 140/81, or 947.368 cents); as well as two options for the major seventh,
being B² or BË, or the 19-et grave or small major seventh ( 1719 ]2[ ) and 19-et twenty-three quarter-tones
( 1819 ]2[ , approximately 27/14, or 1136.842 cents).248
Simultaneous sonorities include the dyads A and C, or the 19-et just minor third ( 519 ]2[ ); DÚ and
G, EÚ and A, Aí and D¸, or 19-et nine quarter-tones ( 619 ]2[ ); C and F, D² and G, EÝ and Aí, E and A, F
and BÝ, G and C, or the 19-et just perfect fourth ( 819 ]2[ , approximately 154/115, or 505.263 cents); Aí
and D², BÝ and E, AÚ and EÝ, C and FÚ, DÝ and G, or the 19-et grave or small augmented fourth ( 919 ]2[ );
A¸ and D², A and EÝ, C and GÝ, or the 19-et acute or large diminished fifth ( 1019 ]2[ ); C and G, or the 19-
et just perfect fifth ( 1119 ]2[ ); and C and Aí, or 19-et seventeen quarter-tones ( 1319 ]2[ ). Bar 94 and 95
highlights intonational reinterpretations of simultaneous sonorities based on the two options for the major
second (D² and G, and DÚ and G); two for the major third (E and A, and EÚ and A); two for the minor
sixth (A¸ and D², and Aí and D²); two for the major sixth (A and EÝ, and AÚ and EÝ); and the two
options for the tritone (C and FÚ, and C and GÝ).249
Ex. 66. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bars 94-95
The following table depicts the tonal resources of nineteen-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
248 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-139.
249 The juxtaposition of three just major thirds (5/4), equal to 1158.941 cents is a figure that diminishes the octave
(2/1) by 41.059 cents, and the interval termed as the ‘great diesis’ with a frequency ratio of 128/125. For a further
discussion, see Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 453.
166 The Equally-Tempered Archetype
Table 99. The nineteen-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÐÑ
C
D¸
DÝ
D²
DË / Eì
EÝ
E
EË / Fí
F
FÚ
GÝ
G
A¸
GÍ / Aí
A
AË / Bì
BÝ
B²
BË / Cí
C
unison
19-et just diatonic semitone, or major half-tone
19-et great limma, or large half-tone
19-et just minor tone
19-et five quarter-tones
19-et just minor third
19-et just major third
19-et nine quarter-tones
19-et just and Pythagorean perfect fourth
19-et grave or small augmented fourth
19-et acute or large diminished fifth
19-et just and Pythagorean perfect fifth
19-et Pythagorean minor sixth
19-et seventeen quarter-tones
19-et just major sixth
19-et nineteen quarter-tones
19-et acute or large minor seventh
19-et grave or small major seventh
19-et twenty-three quarter-tones
octave
1.000000
1.037155
1.075691
1.115658
1.157110
1.200103
1.244693
1.290939
1.338904
1.388651
1.440247
1.493759
1.549260
1.606822
1.666524
1.728444
1.792664
1.859271
1.928352
2.000000
261.626
271.346
281.428
291.885
302.730
313.978
325.643
337.743
350.292
363.307
376.805
390.806
405.326
420.386
436.005
452.205
469.007
486.433
504.506
523.251
0.000
63.158
126.316
189.474
252.632
315.789
378.947
442.105
505.263
568.421
631.579
694.737
757.895
821.053
884.211
947.368
1010.526
1073.684
1136.842
1200.000
+00
ß37
+26
ß11
+53
+16
ß21
+42
+05
ß32
+32
ß05
ß42
ß79
ß16
+47
+11
ß26
+37
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 67. 19-tone equal temperament
The Equally-Tempered Archetype 167
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 100. 19-tone equal temperament tuning matrix no. 1 (Program 25)
NOTE C DÝ D² EÝ E F F! G Aí A BÝ B²
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +26 ß11 +16 ß21 +05 ß32 ß05 +21 ß16 +11 ß26
Table 101. 19-tone equal temperament tuning matrix no. 2 (Program 26)
NOTE – – D¸ – – DË EË – – GÝ – – A¸ AË – – BË
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß37 +00 ß47 +42 +00 +32 +00 ß42 +47 +00 +37
Twenty-Tone Equal Temperament Twenty-tone equal temperament ( 20 2 ) presents a subdivision of the 5-et supermajor second ( 5 2 ) into
four equal tones 60.000 cents in size, which represent the 20-et Greek enharmonic quarter-tone
(approximately 265/256).250 The temperament may alternatively be expressed as the juxtaposition of two
sets of ten-tone equally-tempered intervals, four sets of five-tone equally-tempered intervals, five sets of
four-tone equally-tempered intervals, or ten sets of two-tone equally-tempered intervals interlocked a
20-et Greek enharmonic quarter-tone ( 20 2 ) apart. Blackwood makes the following statement with
regards to the temperament: “Triads are very bad, and the most consonant harmony this tuning offers is a
minor triad with an added major sixth, along with its inversions.”251 The minor triad with an added major
sixth that Blackwood refers to includes the pitches C, E¸, G¢, and A¢, and is made up of the equal major
third ( 3 2 ), the 5-et acute or large fifth ( 35 ]2[ ), and the equal major sixth ( 34 ]2[ ). All these pitches are
also available in four-tone, eight-tone, twelve-tone, sixteen-tone, and twenty-four-tone equal
temperaments, but for G¢ ( 35 ]2[ , or 5-et acute or large fifth), which is only available in five-tone, ten-
tone, fifteen-tone, and twenty-tone equal temperaments.
250 Daniélou, Tableau Comparatif des Intervalles Musicaux 16.
251 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
168 The Equally-Tempered Archetype
Twenty-tone equal temperament provides two options for the approximation of the minor second,
being Dì or D¸, or the 20-et Greek enharmonic quarter-tone ( 20 2 ) and 10-et just diatonic semitone
( 10 2 ); two for the major second, being D² or D£, or the 20-et just minor tone ( 320 ]2[ , approximately
81/73, or 180.000 cents) and 5-et supermajor second ( 5 2 ); two for the major third, being E² or E£, or
the 10-et grave or small major third ( 310 ]2[ ) and 20-et acute or large major third ( 720 ]2[ , approximately
25088/19683, or 420.000 cents); two for the fifth, being G³ or G¢, or the 20-et subfifth ( 1120 ]2[ ,
approximately 41/28, or 660.000 cents) and 5-et acute or large fifth ( 35 ]2[ ); two for the major sixth,
being A³ or A¢, or the 10-et neutral sixth ( 710 ]2[ , or approximately 13/8) and equal major sixth ( 34 ]2[ );
two for the augmented sixth/minor seventh, being AÚ or BÝ, or the 5-et augmented sixth ( 45 ]2[ ) and 20-
et acute or large minor seventh ( 1720 ]2[ , approximately 146/81, or 1020.000 cents); as well as two
options for the major seventh, being B or BË, or the 10-et just diatonic major seventh ( 910 ]2[ ,
approximately 28/15, or 1080.000 cents) and 20-et twenty-three quarter-tones ( 1920 ]2[ , approximately
989/512, or 1140.000 cents).252 Simultaneous sonorities include the dyads C and E¸, or the equal minor
third ( 4 2 ); E¸ and G³, or the 10-et grave or small major third ( 310 ]2[ ); E¸ and G¢, and C and E£, or the
20-et acute or large major third ( 720 ]2[ ); as well as C and A³, or the 10-et neutral sixth ( 710 ]2[ ). Bar
100 illustrates the utilization of intonational reinterpretations of simultaneous sonorities based on the two
options for the fifth, G³ and G¢, or the 20-et subfifth ( 1120 ]2[ ) and 5-et acute or large fifth ( 35 ]2[ ). In
this case, the juxtaposition with E¸, or the equal minor third ( 4 2 ) generates the 20-et acute or large
major third ( 720 ]2[ ) and 10-et grave or small major third ( 310 ]2[ ), which in each case present the just
major third (5/4) with a falsity of +33.686 and ß26.314 cents.
Ex. 68. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 100
The following table depicts the tonal resources of twenty-tone equal temperament, indicating degree,
notation, interval, ratio, frequency, cents, and tuning.
252 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-133.
The Equally-Tempered Archetype 169
Table 102. The twenty-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÐÑ
C
CË / Dì
D¸
D²
D£
E¸
E²
E£
F²
F£
F¥
G³
G¢
GÚ
A³
A¢
AÚ
BÝ
B
BË / Cí
C
unison
20-et Greek enharmonic or septimal quarter-tone
10-et just diatonic semitone, or major half-tone
20-et just minor tone
5-et supermajor second
equal or Pythagorean minor third, or trihemitone
10-et grave or small major third
20-et acute or large major third
5-et grave or small fourth
20-et superfourth
equal or acute or large tritone, or augmented fourth
20-et subfifth
5-et acute or large fifth
20-et augmented fifth
10-et neutral sixth
equal or Pythagorean major sixth
5-et augmented sixth
20-et acute or large minor seventh
10-et just diatonic major seventh
20-et twenty-three quarter-tones
octave
1.000000
1.035265
1.071773
1.109569
1.148698
1.189207
1.231144
1.274561
1.319508
1.366040
1.414214
1.464086
1.515717
1.569168
1.624505
1.681793
1.741101
1.802501
1.866066
1.931873
2.000000
261.626
270.852
280.403
290.292
300.529
311.127
322.099
333.458
345.217
357.391
369.994
383.042
396.550
410.535
425.012
440.000
455.517
471.580
488.211
505.427
523.251
0.000
60.000
120.000
180.000
240.000
300.000
360.000
420.000
480.000
540.000
600.000
660.000
720.000
780.000
840.000
900.000
960.000
1020.000
1080.000
1140.000
1200.000
+00
ß40
+20
ß20
+40
+00
ß40
+20
ß20
+40
+00
ß40
+20
ß20
ß60
+00
ß40
+20
ß20
+40
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 69. 20-tone equal temperament
170 The Equally-Tempered Archetype
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 103. 20-tone equal temperament tuning matrix no. 1 (Program 27)
NOTE C D¸ D² E¸ E² F² F¥ G¢ G! A£ BÝ B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +20 ß20 +00 ß40 ß20 +00 +20 ß20 +00 +20 ß20
Table 104. 20-tone equal temperament tuning matrix no. 2 (Program 28)
NOTE – – Dì D£ – – E£ F£ – – G³ A³ – – A! BË
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß40 +40 +00 +20 +40 +00 ß40 +40 +00 ß40 +40
Twenty-One-Tone Equal Temperament Twenty-one-tone equal temperament ( 212 ), with its twofold tripartite division of the octave, presents a
subdivision of the 7-et grave or small tone ( 7 2 ) into three equal tones 57.143 cents in size, which
represent the 21-et Greek enharmonic quarter-tone (approximately 1323/1280).253 The temperament
may alternatively be expressed as the juxtaposition of three sets of seven-tone equally-tempered intervals
interlocked an equal 21-et Greek enharmonic quarter-tone ( 212 ) apart. According to Blackwood,
“Major and minor triads and keys are relatively consonant here. Scales sound somewhat out of tune,
however, due to the impossibility of dividing a major third into two equal parts.”254 The twenty-one-tone
equally-tempered major scale generates inconsistent major triads, which produce the just major third
(5/4) and just perfect fifth (3/2) with a falsity of +13.686 cents and ß16.241 cents, as in C major (C, E¢,
and G²); ß43.457 and ß16.241 cents, as in F major (F¢, A², and C); and +13.686 and +40.902 cents,
as in G major (G², B, and D¢). Equally inconsistent minor triads are available, deviating from the just minor
third (6/5) and just perfect fifth (3/2) by ß29.927 and ß73.384 cents, as in D minor (D, F¢, and A²);
ß29.927 and ß16.241 cents, as in E minor (E¢, G², and B); and +27.216 and +40.902 cents, as in A
253 Daniélou, Tableau Comparatif des Intervalles Musicaux 16.
254 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
The Equally-Tempered Archetype 171
minor (A², C, and E¢); while the B diminished triad (B, D¢, and F¢), deviating from the just minor third (6/5)
and just tritone (45/32) by +27.216 and +38.348 cents. The D minor triad produces a just perfect fifth
(3/2) with a falsity of ß73.384 cents, which is an extremely dissonant interval 628.571 cents in size
( 1121 ]2[ , 21-et acute or large diminished fifth, or approximately 23/16), and therefore certainly no form of
recognizable perfect fifth.255
Ex. 70. The twenty-one-tone equally-tempered major scale
Twenty-one-tone equal temperament provides two options for the approximation of the minor second,
being Dì or D¸, or the 21-et Greek enharmonic quarter-tone ( 212 ) and 21-et just diatonic semitone
( 221 ]2[ , approximately 47/44, or 114.286 cents); two for the major second, being D³ or D¢, or the 7-et
grave or small tone ( 7 2 ) and 21-et acute or large tone ( 421 ]2[ , approximately 81/71, or 228.571
cents); two for the major third, being E³ or E¢, or the 7-et neutral third ( 27 ]2[ ) and equal major third
( 3 2 ); two for the fourth, being F³ or F¢, or the 21-et subfourth ( 821 ]2[ , approximately 125/96, or
457.143 cents) and 7-et acute or large fourth ( 37 ]2[ ); two for the tritone, being FÚ or GÝ, or the 21-et
grave or small augmented fourth ( 1021 ]2[ , approximately 32/23, or 571.429 cents) and 21-et acute or
large diminished fifth ( 1121 ]2[ ); two for the fifth, being G² or G£, or the 7-et grave or small fifth ( 47 ]2[ )
and 21-et superfifth ( 1321 ]2[ , approximately 43/28, or 742.857 cents); two for the major sixth, being A²
or A¢, or the 7-et grave or small major sixth ( 57 ]2[ ) and 21-et Pythagorean major sixth ( 1621 ]2[ ,
approximately 27783/16384, or 914.286 cents); two for the minor seventh, being B¹ or Bí, or the 21-et
grave or small minor seventh ( 1721 ]2[ , approximately 142/81, or 971.429 cents) and 7-et twenty-one
quarter-tones ( 67 ]2[ ); as well as two options for the major seventh, being B or BË, or the 21-et just
diatonic major seventh ( 1921 ]2[ , approximately 88/47, or 1085.714 cents) and 21-et twenty-three
quarter-tones ( 2021 ]2[ , approximately 2560/1323, or 1142.857 cents).256 Bar 107 highlights a
descending melodic passage, which incorporates all the primary, secondary, and auxiliary pitches
available within the pélog scheme – the series C, (BË), B, Bí, (B¹), A¸, (G£), G², (GÝ), FÚ, F¢, (F³), E¸, D¸,
(Dì), and C.
255 Daniélou, Tableau Comparatif des Intervalles Musicaux 141.
256 Daniélou, Tableau Comparatif des Intervalles Musicaux 17-140.
172 The Equally-Tempered Archetype
Ex. 71. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 107
Simultaneous sonorities utilized in this temperament are limited to C and G², or the 7-et grave or small fifth
( 47 ]2[ ), and F¢ and Bí, and F³ and B¹, or the 7-et acute or large fourth ( 37 ]2[ ). The following table
depicts the tonal resources of twenty-one-tone equal temperament, indicating degree, notation, interval,
ratio, frequency, cents, and tuning.
Table 105. The twenty-one-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ ÑÓ ÑÔ ÑÕ ÑÖ Ñ× ÑØ ÑÙ ÒÐ ÒÑ
ÐÑ
C CË / Dì
D¸ D³ D¢ EÝ E³ E¢ F³ F¢ FÚ GÝ G² G£ A¸ A² A¢ B¹
AÍ / Bí B
BË / Cí
C
unison
21-et Greek enharmonic or septimal quarter-tone
21-et just diatonic semitone, or major half-tone
7-et grave or small tone
21-et acute or large tone
21-et just minor third
7-et neutral third
equal or Pythagorean major third, or ditone
21-et subfourth
7-et acute or large fourth
21-et grave or small augmented fourth
21-et acute or large diminished fifth
7-et grave or small fifth
21-et superfifth
equal or Pythagorean minor sixth
7-et grave or small major sixth
21-et Pythagorean major sixth
21-et grave or small minor seventh
7-et twenty-one quarter-tones
21-et just diatonic major seventh
21-et twenty-three quarter-tones
octave
1.000000
1.033558
1.068242
1.104090
1.141140
1.179434
1.219014
1.259921
1.302201
1.345900
1.391066
1.437747
1.485994
1.535861
1.587401
1.640671
1.695728
1.752633
1.811447
1.872235
1.935064
2.000000
261.626
270.405
279.479
288.858
298.551
308.570
318.925
329.628
340.689
352.122
363.938
376.151
388.774
401.820
415.305
429.241
443.646
458.534
473.921
489.825
506.262
523.251
0.000
57.143
114.286
171.429
228.571
285.714
342.857
400.000
457.143
514.286
571.429
628.571
685.714
742.857
800.000
857.143
914.286
971.429
1028.571
1085.714
1142.857
1200.000
+00
ß43
+14
ß29
+29
ß14
+43
+00
ß43
+14
ß29
+29
ß14
+43
+00
ß43
+14
ß29
ß71
ß14
+43
+00
The Equally-Tempered Archetype 173
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 72. 21-tone equal temperament
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 106. 21-tone equal temperament tuning matrix no. 1 (Program 29)
NOTE C D¸ D¢ E¸ E¢ F¢ F! G² A¸ A² Bí B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +14 +29 ß14 +00 +14 ß29 ß14 +00 ß43 +29 ß14
Table 107. 21-tone equal temperament tuning matrix no. 2 (Program 30)
NOTE – – Dì D³ E³ – – F³ GÝ G£ – – A¢ B¹ BË
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß43 ß29 +43 +00 ß43 +29 +43 +00 +14 ß29 +43
Twenty-Two-Tone Equal Temperament Twenty-two-tone equal temperament ( 22 2 ) presents a subdivision of the 11-et just diatonic semitone
( 112 ) into two equal tones 54.545 cents in size, which represent the 22-et Greek enharmonic quarter-
tone (approximately 4096/3969).257 The temperament may alternatively be expressed as the
juxtaposition of two sets of eleven-tone equally-tempered intervals, or eleven sets of two-tone equally-
257 Daniélou, Tableau Comparatif des Intervalles Musicaux 14.
174 The Equally-Tempered Archetype
tempered intervals interlocked a 22-et Greek enharmonic quarter-tone ( 22 2 ) apart. Blackwood makes
the following observations: “This tuning contains triads that are very smooth – in some respects, even a
bit smoother than those of 12-note tuning. The smoothness of the triads is offset, however, by the out-
of-tune scale which sharply restricts the tuning’s diatonic vocabulary.”258 Twenty-two-tone equal
temperament is the fourth temperament to contain what Blackwood terms as recognizable diatonic
scales and “perfect fifths within the range of recognisability.”259 The twenty-two-tone equally-tempered
major scale generates consistent major triads, which produce the just major third (5/4) and just perfect
fifth (3/2) with a falsity of ß4.496 cents and +7.136 cents, as in C major (C, E, and G), F major (F, A², and
C), and G major (G, B, and D¢). A very dissonant minor triad is available, which presents the just minor
third (6/5) and perfect fifth (3/2) with a falsity of by ß42.914 and ß47.410 cents, as in D minor (D, F, and A²); while another two, display more consistency, as well as consonance, with a deviation of
+11.631 and +7.136 cents, as in E minor (E, G, and B) and A minor (A², C, and E). The B diminished
triad (B, D¢, and F) presents the just minor third (6/5) and just tritone (45/32) with a falsity of +11.631
and ß9.776 cents. The out-of-tune scale Blackwood is referring to is made up of tones and semitones
deviating from just major tones (9/8s), just minor tones (10/9s), and just diatonic semitones (16/15s) by
+14.272, ß18.767, and ß2.640 cents. Twelve-tone equal temperament yields ß3.910, +17.596, and
ß11.731 cents on the same basis.
Ex. 73. The twenty-two-tone equally-tempered major scale
The work outlines the triad C, DÚ, and G, which when enharmonically reinterpreted as C, E¹, and G may be stated as being a twenty-two-tone equally-tempered minor triad with the minor third and perfect fifth
272.727 and 709.091 cents in size respectively (the interval of the augmented second converted into a
subminor third). The comparison with the just minor third (6/5) and just perfect fifth (3/2) reveal a falsity
of ß42.914 and +7.136 cents on each count. G4 (394.059Hz), or the 22-et just perfect fifth ( 1322 ]2[ ,
or approximately 122/81), presents 3.242 beats between the third harmonic of C4 and the second
harmonic of G4 (788.119Hz), and 6.484 beats between the sixth harmonic of C4 and the fourth
harmonic of G4 (1576.237Hz); while E¹4 (306.264Hz), or 22-et subminor third ( 522 ]2[ , or
approximately 2560/2187), 38.433 beats between the sixth harmonic of C4 and the fifth harmonic of
E¹4 (1531.320Hz).260
258 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
259 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
260 Daniélou, Tableau Comparatif des Intervalles Musicaux 72-123.
The Equally-Tempered Archetype 175
Ex. 74. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 109
Table 108. The beating characteristics of the nineteen-tone equally-tempered minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E¹4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
306.264
– – – –
– – – –
612.528
– – – –
918.792
– – – –
– – – –
1225.056
– – – –
1531.320
– – – –
1837.585
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
38.433
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
394.059
– – – –
– – – –
788.119
– – – –
– – – –
1182.178
– – – –
– – – –
1576.237
– – – –
– – – –
1970.296
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
3.242
– – – –
– – – –
– – – –
– – – –
– – – –
6.484
– – – –
– – – –
– – – –
– – – –
Twenty-two-tone equal temperament provides two options for the approximation of the major second,
being D³ or D¢, or the 22-et grave or small tone ( 322 ]2[ , approximately 256/233, or 163.636 cents) and
11-et acute or large tone ( 211 ]2[ ); two for the augmented second, being DÚ or DÍ, or the 22-et
augmented second ( 522 ]2[ ) and 11-et seven quarter-tones ( 311 ]2[ ); two for the major third, being E or
EË, or the 22-et just major third ( 722 ]2[ , approximately 96/77, or 381.818 cents) and 11-et nine quarter-
tones ( 411 ]2[ ); two for the fourth, being F or F£, or the 22-et just perfect fourth ( 922 ]2[ , approximately
81/61, or 490.909 cents) and 11-et superfourth ( 511 ]2[ , approximately 37/27, or 545.455 cents); two
for the fifth, being G³ or G, or the 11-et subfifth ( 611 ]2[ ) and 22-et just perfect fifth ( 1322 ]2[ ); two for
the minor sixth, being Aì or AÝ, or 11-et fifteen quarter-tones ( 711 ]2[ , approximately 199/128, or
763.636 cents) and the 22-et just minor sixth ( 1522 ]2[ , approximately 77/48, or 818.182 cents); two for
176 The Equally-Tempered Archetype
the major sixth, being A² or A£, or the 11-et grave or small major sixth ( 811 ]2[ , approximately 48/29, or
872.727 cents) and 22-et acute or large major sixth ( 1722 ]2[ , approximately 2187/1280, or 927.273
cents); two for the minor seventh, being B¹ or Bí, or the 11-et grave or small minor seventh ( 911 ]2[ ,
approximately 432/245, or 981.818 cents) and 22-et twenty-one quarter-tones ( 1922 ]2[ , approximately
233/128, or 1036.364 cents); as well as two options for the major seventh, being B or BË, or the 11-et
just diatonic major seventh ( 1011 ]2[ , approximately 77/41, or 1090.909 cents) and 22-et twenty-three
quarter-tones 2122 ]2[ , approximately 3969/2048, or 1145.455 cents).261 A microtonal cluster in bar 108
made up of the pitches D¸, Eí, F', F¥, G, AÝ, Bí, and B' highlights four significant twenty-two-tone
equally-tempered intervals, measuring 54.545, 109.091, 163.636, and 218.182 cents.
Ex. 75. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 108
The only simultaneous sonority utilized in this temperament is Bí and F, or the 11-et subfifth ( 611 ]2[ ).
The following table depicts the tonal resources of twenty-two-tone equal temperament, indicating
degree, notation, interval, ratio, frequency, cents, and tuning.
261 Daniélou, Tableau Comparatif des Intervalles Musicaux 31-134.
The Equally-Tempered Archetype 177
Table 109. The twenty-two-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÐÑ
C
CË / Dì
D¸
D³
D¢
DÚ
DÍ / Eí
E
EË / Fí
F
F£
F¥
G³
G
GË / Aì
AÝ
A²
A£
B¹
AÍ / Bí
B
BË / Cí
C
unison
22-et Greek enharmonic or septimal quarter-tone
11-et just diatonic semitone, or major half-tone
22-et grave or small tone
11-et acute or large tone
22-et augmented second
11-et seven quarter-tones
22-et just major third
11-et nine quarter-tones
22-et just and Pythagorean perfect fourth
11-et superfourth
equal or acute or large tritone, or augmented fourth
11-et subfifth
22-et just and Pythagorean perfect fifth
11-et fifteen quarter-tones
22-et just minor sixth
11-et grave or small major sixth
22-et acute or large major sixth
11-et grave or small minor seventh
22-et twenty-one quarter-tones
11-et just diatonic major seventh
22-et twenty-three quarter-tones
octave
1.000000
1.032008
1.065041
1.099131
1.134313
1.170620
1.208089
1.246758
1.286665
1.327849
1.370351
1.414214
1.459480
1.506196
1.554406
1.604160
1.655507
1.708496
1.763183
1.819619
1.877862
1.937969
2.000000
261.626
270.000
278.642
287.561
296.765
306.264
316.067
326.184
336.624
347.399
358.519
369.994
381.837
394.059
406.672
419.689
433.123
446.986
461.294
476.059
491.297
507.022
523.251
0.000
54.545
109.091
163.636
218.182
272.727
327.273
381.818
436.364
490.909
545.455
600.000
654.545
709.091
763.636
818.182
872.727
927.273
981.818
1036.364
1090.909
1145.455
1200.000
+00
+55
+09
ß36
+18
ß27
+27
ß18
+36
ß09
+45
+00
ß45
+09
ß36
+18
ß27
+27
ß18
ß64
ß09
+45
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 76. 22-tone equal temperament
178 The Equally-Tempered Archetype
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 110. 22-tone equal temperament tuning matrix no. 1 (Program 31)
NOTE C D¸ D¢ DÍ E F F¥ G AÝ A² Bí B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +09 +18 +27 ß18 ß09 +00 +09 +18 ß27 +36 ß09
Table 111. 22-tone equal temperament tuning matrix no. 2 (Program 32)
NOTE – – CË D³ D! EË F£ – – G³ Aì A£ B¹ BË
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß45 ß36 ß27 +36 +45 +00 ß45 ß36 +27 ß18 +45
Twenty-Three-Tone Equal Temperament Twenty-three-tone equal temperament is generated by the 23 2 , and presents twenty-three equal tones
52.174 cents in size, which represent the 23-et Greek enharmonic quarter-tone (approximately 34/33),
and approximate the equal quarter-tone ( 24 2 ) with a falsity of +2.174 cents.262 According to
Blackwood, twenty-three-tone equal temperament presents the following characteristics: “A particular
challenge, 23-note tuning contains no diatonic configurations and no chromatic structures in common
with any of the other tunings of this study. However, it does contain an intriguing arrangement of the two
distinct pentatonic modes of Java and Bali, known as sléndro and pélog – modes that cannot be
realistically approximated in 12-note tuning.”263 The twenty-three-tone equally-tempered major scale
generates inconsistent major triads, which produce the just major third (5/4) and just perfect fifth (3/2)
262 Daniélou, Tableau Comparatif des Intervalles Musicaux 14.
263 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
The Equally-Tempered Archetype 179
with a falsity of ß21.096 cents and ß23.694 cents, as in C major (C, E², and G²) and F major (F¢, A, and
C); and with +31.078 cents and +28.480 cents, as in G major (G, B, and D). Somewhat more
consonant minor triads are available, which produce the just minor third (6/5) and just perfect fifth (3/2)
with a falsity of ß2.598 and ß23.694 cents, as in D minor (D, F¢, and A) and A minor (A, C, and E²); and
with ß2.598 and +28.480 cents, as in E minor (E, G², and B); while the B diminished triad (B, D, and F¢)
produces the just minor third (6/5) and just tritone (45/32) with a falsity of ß2.598 and +35.863 cents.
Ex. 77. The twenty-three-tone equally-tempered major scale
The sléndro scale is represented in twenty-three-tone equal temperament with the pitches C, D, F¢, G²,
and B¸, and equal to 0.000, 208.696, 521.739, 678.261, and 991.304 cents; while the pélog scale, with
the pitches C, D¸, EÝ, F¢, G², A¸, and B, and equal to 0.000, 104.348, 313.043, 521.739, 678.261,
782.609, and 1095.652 cents. A comparison with the tuning of the gamelan gedhé analyzed by Perlman
(presenting sléndro measurements equal to 0.000, 238.000, 475.000, 727.000, and 950.000 cents; and
pélog measurements equal to 0.000, 116.000, 281.000, 550.000, 669.000, 769.000, and 861.000
cents) reveals a falsity of between 9.261 and 32.043 cents on the first count (not accounting for the
+134.652 cent falsity of B), and a falsity between 29.304 and 48.739 cents on the second count.264 A
further comparison of the two scales with five-tone and seven-tone equal temperaments produces an
even greater level of error.
The work outlines a twenty-three-tone equally-tempered minor triad, which includes the pitches C,
EÝ, and G², and produces a minor third 313.043 cents in size ( 623 ]2[ , 23-et just minor third, or
approximately 1024/855) and a perfect fifth 678.261 cents in size ( 1323 ]2[ , 23-et grave or small fifth, or
approximately 262144/177147).265 The comparison with the just minor third (6/5) and perfect fifth (3/2)
reveal a falsity of ß2.598 and ß23.694 cents on each count. G²4 (387.104Hz), or the 23-et grave or
small fifth ( 1323 ]2[ ), presents 10.669 beats between the third harmonic of C4 and the second harmonic
of G²4 (774.208Hz), and 21.338 beats between the sixth harmonic of C4 and the fourth harmonic of G²4
(1548.416Hz); while EÝ4 (313.480Hz), or the 23-et just minor third ( 623 ]2[ ), 2.354 beats between the
sixth harmonic of C4 and the fifth harmonic of EÝ4 (1567.400Hz).
264 Perlman, Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory 41.
265 Daniélou, Tableau Comparatif des Intervalles Musicaux 82-129.
180 The Equally-Tempered Archetype
Ex. 78. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 112
Table 112. The beating characteristics of the nineteen-tone equally-tempered minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
EÝ4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G²4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
313.480
– – – –
– – – –
626.960
– – – –
940.440
– – – –
– – – –
1253.920
– – – –
1567.400
– – – –
1880.880
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
2.354
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
387.104
– – – –
– – – –
774.208
– – – –
– – – –
1161.312
– – – –
– – – –
1548.416
– – – –
– – – –
1935.520
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
10.669
– – – –
– – – –
– – – –
– – – –
– – – –
21.338
– – – –
– – – –
– – – –
– – – –
Twenty-three-tone equal temperament provides two options for the approximation of the major second,
being D³ or D, or the 23-et grave or small tone ( 323 ]2[ , approximately 81/74, or 156.522 cents) and 23-
et just major tone ( 423 ]2[ , approximately 44/39, or 208.696 cents); two for the augmented
second/minor third, being DË or EÝ, or 23-et five quarter-tones ( 523 ]2[ , approximately 93/80, or 260.870
cents) and the 23-et just minor third ( 623 ]2[ ); two for the major third, being E² or E¢, or the 23-et grave
or small major third ( 723 ]2[ , approximately 2401/1944, or 365.217 cents) and 23-et Pythagorean major
third ( 823 ]2[ , approximately 14/11, or 417.391 cents); two for the fourth, being F² or F¢, or the 23-et
grave or small fourth ( 923 ]2[ , approximately 101/77, or 469.565 cents) and 23-et acute or large fourth
( 1023 ]2[ , approximately 173/128, or 521.739 cents); two for the tritone, being FÚ or GÝ, or the 23-et
grave or small augmented fourth ( 1123 ]2[ , approximately 1024/735, or 573.913 cents) and 23-et acute
The Equally-Tempered Archetype 181
or large diminished fifth ( 1223 ]2[ , approximately 735/512, or 626.087 cents); two for the fifth, being G²
or G¢, or the 23-et grave or small fifth ( 1323 ]2[ ) and 23-et acute or large fifth ( 1423 ]2[ , approximately
154/101, or 730.435 cents); two for the major sixth, being A³ or A, or the 23-et neutral sixth ( 1623 ]2[ ,
approximately 3888/2401, or 834.783 cents) and 23-et just major sixth ( 1723 ]2[ , approximately
855/512, or 886.957 cents); as well as two options for the major seventh, being B³ or B, or the 23-et
neutral seventh ( 2023 ]2[ , approximately 148/81, or 1043.478 cents) and 23-et just diatonic major
seventh ( 2123 ]2[ , approximately 145/77, or 1095.652 cents). The two approximations for the
augmented second/minor third and fifth is also highlighted in bar 112 with the twenty-three-tone equally-
tempered minor triad variant of C, DË, and G¢, which presents a triad with a just minor third (6/5) and just
perfect fifth (3/2) with a falsity of ß2.598 and ß23.694 cents.266
Simultaneous sonorities include the dyads F¢ and A¸, or 23-et five quarter-tones ( 523 ]2[ ); G¢ and
B³, and B¸ and D¸, or the 23-et just minor third ( 623 ]2[ ); GÝ and B¸, or the 23-et grave or small major
third ( 723 ]2[ ); A¸ and C, and B³ and DË, or the 23-et Pythagorean major third ( 823 ]2[ ); C and F², and DË
and G¢, or the 23-et grave or small fourth ( 923 ]2[ ); G² and C, and D¸ and GÝ, or the or 23-et acute or
large fourth ( 1023 ]2[ ). The following table depicts the tonal resources of twenty-three-tone equal
temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.
266 Daniélou, Tableau Comparatif des Intervalles Musicaux 29-141.
182 The Equally-Tempered Archetype
Table 113. The twenty-three-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÐÑ
C
CË / Dì
D¸
D³
D
DË / Eì
EÝ
E²
E¢
F²
F¢
FÚ
GÝ
G²
G¢
A¸
A³
A
AË / Bì
B¸
B³
B
BË / Cí
C
unison
23-et Greek enharmonic or septimal quarter-tone
23-et just diatonic semitone, or major half-tone
23-et grave or small tone
23-et just major tone
23-et five quarter-tones
23-et just minor third
23-et grave or small major third
23-et Pythagorean major third, or ditone
23-et grave or small fourth
23-et acute or large fourth
23-et grave or small augmented fourth
23-et acute or large diminished fifth
23-et grave or small fifth
23-et acute or large fifth
23-et Pythagorean minor sixth
23-et neutral sixth
23-et just major sixth
23-et nineteen quarter-tones
23-et Pythagorean minor seventh
23-et neutral seventh
23-et just diatonic major seventh
23-et twenty-three quarter-tones
octave
1.000000
1.030596
1.062127
1.094624
1.128114
1.162629
1.198201
1.234860
1.272642
1.311579
1.351707
1.393063
1.435685
1.479610
1.524880
1.571534
1.619616
1.669169
1.720239
1.772870
1.827112
1.883014
1.940626
2.000000
261.626
269.630
277.880
286.382
295.143
304.174
313.480
323.071
332.956
343.143
353.641
364.461
375.612
387.104
398.948
411.154
423.733
436.697
450.058
463.828
478.019
492.645
507.717
523.251
0.000
52.174
104.348
156.522
208.696
260.870
313.043
365.217
417.391
469.565
521.739
573.913
626.087
678.261
730.435
782.609
834.783
886.957
939.130
991.304
1043.478
1095.652
1147.826
1200.000
+00
+52
+04
ß43
+09
ß39
+13
ß35
+17
ß30
+22
ß26
+26
ß22
+30
ß17
ß65
ß13
+39
ß09
ß56
ß04
ß52
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
Ex. 79. 23-tone equal temperament
The Equally-Tempered Archetype 183
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 114. 23-tone equal temperament tuning matrix no. 1 (Program 33)
NOTE C D¸ D EÝ E² F¢ F! G² A¸ A B¸ B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +04 +09 +13 ß35 +22 ß26 ß22 ß17 ß13 ß09 ß04
Table 115. 23-tone equal temperament tuning matrix no. 2 (Program 34)
NOTE – – CË D³ DÍ E¢ F² GÝ G¢ A³ AË B³ Cí
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß48 ß43 ß39 +17 ß30 +26 +30 +35 +39 +44 +48
Twenty-Four-Tone Equal Temperament Twenty-four-tone equal temperament ( 24 2 ), with its division of the octave into twenty-four equally-
tempered quarter-tones 50.000 cents in size, marks the concluding tuning modulation of the work. The
temperament may alternatively be expressed as the juxtaposition of two sets of twelve-tone equally-
tempered intervals, three sets of eight-tone equally-tempered intervals, four sets of six-tone equally-
tempered intervals, six sets of four-tone equally-tempered intervals, eight sets of three-tone equally-
tempered intervals, or twelve sets of two-tone equally-tempered intervals interlocked an equal quarter-
tone ( 24 2 ) apart. Blackwood makes the following observations: “This familiar ‘quarter-tone’ tuning is
actually one of the most difficult to deal with in a practical situation. Without exception, the notes ‘in the
cracks’ make extreme discords with the other notes, and there are only a very few satisfactory harmonies
184 The Equally-Tempered Archetype
that combine the two.”267 Twenty-four-tone equal temperament is the fifth temperament to contain what
Blackwood terms as recognizable diatonic scales and “perfect fifths within the range of recognisability.”268
Twenty-four-tone equal temperament provides two options for the approximation of the minor
second, being Dì or D¹ , or the equal quarter-tone ( 24 2 ) and equal semitone ( 12 2 ); two for the major
second, being D or D£, or the equal major tone ( 6 2 ) and five equal quarter-tones ( 524 ]2[ ,
approximately 52/45, or 250.000 cents); two for the major third, being E³ or E, or seven equal quarter-
tones ( 724 ]2[ , approximately 60/49, or 350.000 cents) and the equal major third ( 3 2 ); two for the
fourth, being Fí or F, or nine equal quarter-tones ( 38 ]2[ ) and the equal perfect fourth ( 512 ]2[ ); two for
the tritone, being FË or F¥, or eleven equal quarter-tones ( 1124 ]2[ , approximately 703/512, or 550.000
cents) and the equal tritone ( 2 2 ); two for the fifth, being G or G£, or the equal perfect fifth ( 712 ]2[ )
and fifteen equal quarter-tones ( 58 ]2[ ); two for the minor sixth, being A² or A¢, or seventeen equal
quarter-tones ( 1724 ]2[ , approximately 49/30, or 850.00 cents) and the equal major sixth ( 34 ]2[ ); two
for the augmented sixth/minor seventh, being AÚ or B¸, or nineteen equal quarter-tones ( 1924 ]2[ ,
approximately 45/26, or 950.000 cents) and the equal minor seventh ( 56 ]2[ ); as well as two options for
the major seventh, being B³ or B¢, or twenty-one equal quarter-tones ( 78 ]2[ ) and the equal major
seventh ( 1112 ]2[ ).269 A five-figure rhythmically perpetuating descending melodic passage in bar 115
highlights all the primary, secondary, and auxiliary pitches available within the pélog scheme.
Ex. 80. Exposiciones for Sampled Microtonal Schoenhut Toy Piano, bar 115
Simultaneous sonorities in this temperament are limited to the dyads G and C, or the equal perfect fourth
( 512 ]2[ ), and its inversion of C and G, or the equal perfect fifth ( 712 ]2[ ). American composer Charles
Edward Ives’s (1874-1954) experiments with twenty-four-tone equal temperament conducted during
the 1925-62 period revealed that the division of the equal perfect fifth ( 712 ]2[ ) into seven equal quarter-
tones, or neutral thirds ( 724 ]2[ ) to produce a twenty-four-tone equally-tempered neutral triad, resulted
in a “chord that is heard as an ‘out of tune’ major or minor triad, not as an entity in itself,” but by the
addition of a fourth tone (“in a quarter-tone relationship to the root or fifth”) the simultaneous sonority
267 Blackwood, Microtonal Compositions by Easley Blackwood, n. pag.
268 Blackwood, The Structure of Recognizable Diatonic Tunings 197.
269 Daniélou, Tableau Comparatif des Intervalles Musicaux 66-134.
The Equally-Tempered Archetype 185
“seemed to establish an identity of its own.”270 The following table depicts the tonal resources of twenty-
four-tone equal temperament, indicating degree, notation, interval, ratio, frequency, cents, and tuning.
Table 116. The twenty-four-tone equally-tempered division of the octave Relative Pitch: A4=440Hz / C4 (middle C)=261.6255654Hz
DEGREE
NUMBER
NOTE INTERVAL RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS TUNING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÐÑ
C
CË / Dì
D¹
CÍ / Dí
D
D£
E¸
E³
E¢
EË / Fí
F
FË / Gì
F¥
FÍ / Gí
G
G£
A¸
A²
A¢
AÚ
B¸
B³
B¢
BË / Cí
C
unison
equal, Greek enharmonic or septimal quarter-tone
equal semitone, or Pythagorean limma
equal three-quarter-tone
equal or just major tone
five equal quarter-tones, or supermajor second
equal or Pythagorean minor third, or trihemitone
seven equal quarter-tones, or neutral third
equal or Pythagorean major third, or ditone
nine equal quarter-tones
equal, or just and Pythagorean perfect fourth
eleven equal quarter-tones
equal or acute or large tritone, or augmented fourth
thirteen equal quarter-tones
equal, or just and Pythagorean perfect fifth
fifteen equal quarter-tones, or superfifth
equal or Pythagorean minor sixth
seventeen equal quarter-tones
equal or Pythagorean major sixth
nineteen equal quarter-tones, or augmented sixth
equal or Pythagorean minor seventh
twenty-one equal quarter-tones, or neutral seventh
equal or Pythagorean major seventh
twenty-three equal quarter-tones
octave
1.000000
1.029302
1.059463
1.090508
1.122462
1.155353
1.189207
1.224054
1.259921
1.296840
1.334840
1.373954
1.414214
1.455653
1.498307
1.542211
1.587401
1.633915
1.681793
1.731073
1.781797
1.834008
1.887749
1.943064
2.000000
261.626
269.292
277.183
285.305
293.665
302.270
311.127
320.244
329.628
339.286
349.228
359.461
369.994
380.836
391.995
403.482
415.305
427.474
440.000
452.893
466.164
479.823
493.883
508.355
523.251
0.000
50.000
100.000
150.000
200.000
250.000
300.000
350.000
400.000
450.000
500.000
550.000
600.000
650.000
700.000
750.000
800.000
850.000
900.000
950.000
1000.000
1050.000
1100.000
1150.000
1200.000
+00
+50
+00
ß50
+00
+50
+00
ß50
+00
ß50
+00
ß50
+00
ß50
+00
+50
+00
ß50
+00
ß50
+00
ß50
+00
ß50
+00
The set of examples illustrate the capacity of the temperament to generate scalar pitch material, and to
represent sléndro and pélog scales.
270 Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 5.
186 The Equally-Tempered Archetype
Ex. 81. 24-tone equal temperament
The following two tuning matrixes contain the data required to represent the temperament chromatically
within the program memory of the Akai S3000XL.
Table 117. 24-tone equal temperament tuning matrix no. 1 (Program 35)
NOTE C D¹ D E¸ E¢ F F¥ G A¸ A¢ B¸ B¢
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00 +00
Table 118. 24-tone equal temperament tuning matrix no. 2 (Program 36)
NOTE CË Dí D£ E³ Fí FÍ Gí G£ A² A! B³ Cí
KEY C C! D D! E F F! G G! A A! B
CENTS +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50
The Equally-Tempered Archetype 187
Blackwood’s Dictum Blackwood offers the following summary of some the characteristics of equal tuning principles, and ‘the
equally-tempered archetype’:
“Mostly, I’ve explored the equal tunings from twelve to twenty-four notes per octave. The choice of which
tuning to use depends largely on the desired style. Certain tunings are more versatile or prettier than
others. Twelve, seventeen, nineteen, twenty-two, and twenty-four contain recognizable diatonic scales. If
random dissonance is what you want, then the worst of all tunings is twelve-note equal, because it
contains a greater concentration of relatively consonant intervals and harmonies in a smaller number of
notes. The most effective temperament for random dissonance is eleven notes. There aren’t two notes in
that tuning that make any kind of a consonance. Certain others tend toward modal arrangements that
coexist in twelve-note equal. For example, if the number of notes is divisible by four, you always have
families of octatonic scales. If the number of notes is divisible by six, there are always families of whole-
tone scales that can arise as altered chords, as they normally do in twelve-note tuning. If the number of
notes is divisible by three, there is a symmetric mode that alternates minor thirds with minor seconds,
creating a chromatic world all its own.”271
271 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on
Nonstandard Tunings,” Perspectives of New Music 177.
188 The Equally-Tempered Archetype
3. The Harmonic Consideration: La Homa Kanto for Harmonically Tuned Synthesizer Quartet
Just Intonation
“Extended just intonation is a tuning system based on the ‘pure’ intervals of the overtone series: intervals
generated therefore from whole number ratios. This is in contrast to temperaments, both equal and
unequal, which use compromised intervals, often based on irrational numbers,” explains John Fonville.
American composer and theorist Benjamin Burwell Johnston (1926-)272 personally elaborates on the
matter within a statement contained in the performance notes of the score to his String Quartet No. 9,
headed On the Performance Practice of Extended Just Intonation:
“Just intonation is simply the easiest way to tune musical intervals by ear. It results in greatly heightened
purity and clarity of sound for two reasons: it eliminates acoustic beats to the maximum possible, and
second, it exploits resonances by utilizing harmonically simple combinations of pitches. The term
extended refers to the use of higher overtones than the first six partials.”273
In order to further understand the basic concept of just intonation, one must refer to the natural acoustic
phenomenon known as the ‘harmonic series’. The overtones of a specific pitch are generally referred to
as the ‘harmonic series’, and the musical scale derived from this series is constructed around ‘pure’ or
‘just’ intervals. This system of just intonation is strikingly dissimilar to the twelve-tone equally-tempered
division of the octave, which is based on the division of the octave into twelve equal intervals; the
frequency ratio of each semitone therefore mathematically representing the twelfth root of two.274 Leta E.
Miller and Fredric Lieberman describe just intonation thus: “Pure intervals arise when the frequencies of
the individual tones reflect the precise mathematical proportions that occur in the series: 3/2 for the fifth,
4/3 for the fourth, etcetera.” Intervals manifested naturally within the harmonic series are particularly
favourable in just intonation, and certainly ones with “superparticular vibration ratios,” where the
272 “Johnston’s credentials are impressive. He studied with Harry Partch, Darius Milhaud, John Cage, and Burrill
Phillips, and although some of these teachers influenced Johnston’s early music, he has remained an individual and
followed his own creative path. Yet he has written in many different styles: jazz resulting from his Naval service,
neoclassicism, and serialism (which he has adapted to serve a functional purpose in just intonation). Much of his work has
intuitively predated important trends in contemporary music. He used combinatoriality in the late 1950’s, quotations in
the 1960’s, and returned to tonality in the early 1970’s. He even tried indeterminacy and electronic music but has been
less successful in these areas.” For a further discussion, see Heidi Von Gunden, The Music of Ben Johnston (Metuchen,
N.J.:The Scarecrow Press, 1986) vii.
273 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 106-07.
274 Olson, Music, Physics and Engineering 46-47.
“numerator exceeds the denominator by one” such as the just perfect fifth (3/2), just perfect fourth (4/3),
just major third (5/4) and just minor third (6/5).275
The Harmonic and Subharmonic Series
The concept of the vibrating string and its connection to music – via the sounding of an open string to
produce the fundamental (or the 1/1 ratio), with the string stopped at the midpoint producing the
octave (or the 2/1 ratio), and at the two-third point, the perfect fifth (or the 3/2 ratio) – is first presented
in the West by Pythagoras of Samos (570-504 B.C.), although it is not until Galileo Galilei (1564-1642) and
Marin Mersenne (1588-1648) that the connection between the actual time or period of the cycle or
vibration, and string length, tension and density is made. John Wallis (1616-1703) and Joseph Sauveur
(1653-1716) are acknowledged for contributing further to the study with their discovery that
proportional vibrations (in the ratios of 1/1, 1/2, 1/3, 1/4, and so on) are also simultaneously produced –
this fact elucidating the notion of the vibrating string as an amalgam of the fundamental and octave
proposed by Aristotle (382-322 B.C.), and consequently stimulating scientific study that in time reveals
more and more upper partials. “Thus the note produced in the ear by a vibrating string was shown to be
composed of a series of pure tones (partials), sounding simultaneously, and corresponding to
component or partial vibrations of the main vibration whose periods (or string lengths) formed an
harmonic series,” explains Lloyd and Boyle.276 The following table illustrates the harmonic characteristics of
the first eight partials of the harmonic series.277
Table 119. The beating characteristics of the first eight partials of the harmonic series
PARTIAL
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
FREQUENCY
(HERTZ)
CENTS
1
2
3
4
5
6
7
8
C4 C5 G5 C6 E6 G6 B;6 C7
fundamental
octave
twelfth
fifteenth, or double octave
tierce (octave tenth)
octave twelfth
harmonic seventh
triple octave
1/1
2/1
6/2
4/1
20/4
12/2
28/4
8/1
1.000000
2.000000
3.000000
4.000000
5.000000
6.000000
7.000000
8.000000
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
0.000
1200.000
1901.955
2400.000
2786.314
3101.955
3368.826
3600.000
275 Leta E. Miller, and Fredric Lieberman, Lou Harrison: Composing a World (New York: Oxford U. Press, 1998) 107.
276 Lloyd, and Boyle, Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation 2-4.
277 R. H. M. Bosanquet, An Elementary Treatise on Musical Intervals and Temperament, ed. Rudolf Rasch (Utrecht,
The Netherlands: Diapason Press, 1987) 16.
190 The Harmonic Consideration
When just intonation practitioners speak of eliminating “acoustic beats to the maximum possible” they are
of course referring to the beats that occur when unmatched pure tones sounding simultaneously move in
and out-of-phase, which generate shifts in amplitude, and hence difference tones that are manifested as
beats per second.278 The chart for the beating characteristics of the mistuned and properly tuned unison
illustrates the fact that the partials of a properly tuned unison correspond precisely with those generated
by the fundamental, and result in an amalgam “absolutely smooth and free from any disturbance.” In
striking contrast, the consequence of a mistuned unison (1.000Hz sharp) is a continuous series of
mistuned pairs of partials.279
Table 120. The beating characteristics of the mistuned and properly tuned unison
FUNDAMENTAL MISTUNED UNISON PROPERLY TUNED UNISON
C4
(PARTIAL)
FREQUENCY
(HERTZ)
C4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
C4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
2
3
4
5
6
7
8
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
1
2
3
4
5
6
7
8
262.626
525.251
787.877
1050.502
1313.128
1575.753
1838.379
1050.502
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
1
2
3
4
5
6
7
8
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
The case for the octave is identical but for the fact that every second partial of a properly tuned octave
corresponds precisely with those generated by the fundamental, while a mistuned octave (1.000Hz
sharp) exponentially beats at rate of an additional 1.000Hz every second partial.280
278 Roderick D. Gordon, The World of Musical Sound (Dubuque, IA: Kendall/Hunt, 1979) 41.
279 “Beats, which have now to be considered, are essentially distinguished from combinational tones as follows: In
combinational tones the composition of vibrations in the elastic vibrating bodies which are either within or without the
ear, undergoes certain disturbances, although the ear resolves the motion which is finally conducted to it, into a series of
simple tones, according to the usual law. In beats, on the contrary, the objective motions of the elastic bodies follow the
simple law; but the composition of the sensations is disturbed. As long as several simple tones of a sufficiently different
pitch enter the ear together, the sensation due to each remains undisturbed in the ear, probably because entirely different
bundles of nerve fibres are affected. But tones of the same, or of nearly the same pitch, which therefore affect the same
nerve fibres, do not produce a sensation which is the sum of the two they would have separately excited, but new and
peculiar phenomena arises which we term ‘interference’, when caused by two perfectly equal simple tones, and ‘beats’
when due to nearly equal simple tones.” For a further discussion, see Helmholtz, On the Sensations of Tone: As a
Physiological Basis for the Theory of Music 159-60.
280 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 20.
The Harmonic Consideration 191
Table 121. The beating characteristics of the mistuned and properly tuned octave
FUNDAMENTAL MISTUNED OCTAVE PROPERLY TUNED OCTAVE
C4
(PARTIAL)
FREQUENCY
(HERTZ)
C5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
C5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
2
3
4
5
6
7
8
261.626
523.251
784.877
1046.502
1308.128
1569.753
1831.379
2093.005
– – – –
1
– – – –
2
– – – –
3
– – – –
4
– – – –
524.251
– – – –
1048.502
– – – –
1572.753
– – – –
2097.005
– – – –
1.000
– – – –
2.000
– – – –
3.000
– – – –
4.000
– – –
1
– – – –
2
– – – –
3
– – – –
4
– – – –
523.251
– – – –
1046.502
– – – –
1569.753
– – – –
2093.005
– – – –
0.000
– – – –
0.000
– – – –
0.000
– – – –
0.000
According to Helmholtz, ‘dissonance’ is explained as a beating condition between the corresponding
partials of a simultaneous sonority, and “there are certain determinate ratios between pitch numbers, for
which this rule suffers an exception, and either no beats at all are formed, or at least only such as have so
little intensity that they produce no unpleasant disturbance of the united sound. These exceptional cases
are called consonances.” James Tenney explains: “Helmholtz equates the dissonance of a simultaneous
aggregate with ‘roughness’ of the sensation caused by beats between adjacent partials (and to a lesser
extent, between ‘combinational tones’) in the combined spectrum of the tones forming the aggregate.”281
Helmholtz categorizes consonances as “the most perfect consonances,” or “absolute,” which include
the octave (2/1), twelfth (6/2), and double octave (4/1). Next come “perfect consonances” such as the
fifth (3/2) and fourth (4/3), which are followed by “medial consonances” such as major sixth (5/3) and
major third (5/4); with the minor third (6/5) and minor sixth (8/5) categorized as “imperfect
consonances.” The “essence of dissonance” is ultimately summarized by Helmholtz as follows:
“It is apparent to the simplest natural observation that the essence of dissonance consists merely in very
rapid beats. The nerves of hearing feel these rapid beats as ‘rough’ and unpleasant, because every
intermittent excitement of any nervous apparatus affects us more powerfully than one that lasts unaltered.
The individual pulses of tone in a dissonant combination give us certainly the same impression of separate
pulses as slow beats, although we are unable to recognize them separately and count them; hence they
form a ‘tangled’ mass of tone, which cannot be analyzed into its constituents. The cause of the
unpleasantness of dissonance we attribute to this ‘roughness’ and ‘entanglement’. The meaning of this
distinction may be thus briefly stated: ‘Consonance is a continuous, dissonance an intermittent sensation of
tone.’ Two consonant tones flow quietly side by side in an undisturbed stream; dissonant tones cut one
another up into separate pulses of tone. This description of the distinction at which we have arrived
281 James Tenney, A History of ‘Consonance’ and ‘Dissonance’ (New York: Excelsior Music Publishing Co., 1988)
87-88.
192 The Harmonic Consideration
agrees precisely with Euclid’s old definition, ‘Consonance is the blending of a higher with a lower tone.
Dissonance is incapacity to mix, when two tones cannot blend, but appear rough to the ear.’”282
Tenney offers the following observations with regards to the general acceptance of Helmholtz’s beat
theory by music theorists: “The fact that the consonance or dissonance predicted by the beat theory for
a given dyad would vary with the absolute frequencies of its tones, rather than simply the intervals
between them, has been pointed out by many other writers – and generally used as an argument against
the validity of Helmholtz’s theory.”283 It is interesting to note that Bosanquet makes a distinction “between
‘beating dissonances’ and ‘unsatisfied combinations’,” classifying the harmonic seventh (7/4), along with
the just perfect fourth (4/3) dyads in the latter group;284 while Johnston, “between different kinds of
dissonance: the dissonance of complex ratios and the dissonance of higher prime numbers,” utilizing the
numerical similarity of the grave or small just chromatic semitone (25/24) and tridecimal third tone (26/25,
or 67.900 cents) as examples of the relationship between the dissonant curve and prime limit.285
An important factor in just intonation rationale is the further notion of the complement or mirror
image of the harmonics series, which is referred to as the ‘subharmonic series’, and represents the
foundation of Partch’s theory of ‘otonalities’ (pitches derived from the ascending series) and ‘utonalities’
(pitches derived from the descending series). “Unlike the harmonic series, the subharmonic series is not
represented in the partials of any known sounding bodies,” notes Doty. “Theorists in earlier centuries
anxiously sought sounds in nature with subharmonic partials, but none were ever discovered.”286 Henry
Cowell presents the following discussion on the musical significance of research conducted on the
subharmonic series:
“A very interesting approach to the theoretical explanation of minor is a consideration of the theory of
undertones. Until recently undertones were a theory only. Their existence was contested by scientists on
the ground that a string or vibrating body could not vibrate at a length greater than its complete length,
which gives the fundamental tone. Hence, it was contended, no deeper tones in such a series would be
possible of formation on the string. Now, however, Professor Nicolas A. Garbusov, of the Moscow State
Institute for Musicology, has built an instrument on which at least the first nine undertones are easily heard
without the aid of resonators. The principle is not that the original sounding body produces the
undertones, but that it is difficult to avoid them in resonation.”287
282 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 194-226.
283 Tenney, A History of ‘Consonance’ and ‘Dissonance’ 90. 284 R. H. M. Bosanquet, “Temperament; Or, the Division of the Octave (Part II),” Proceedings of the Musical
Association, 2nd Sess. (1874-75): 127.
285 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on
Nonstandard Tunings,” Perspectives of New Music 202.
286 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 28-29.
287 Henry Cowell, New Musical Resources (Cambridge: Cambridge U. Press, 1996) 21-22.
The Harmonic Consideration 193
Ex. 82. The harmonics and subharmonics of the fundamental C – first partial, through the 16th partial
The following comparative table of intonation depicts the twelve basic intervals of just intonation,
Pythagorean intonation, meantone temperament, and equal temperament, indicating interval, ratio, and
cents for each system of tuning.
Table 122. Comparative table of intonation
COMPARATIVE
TABLE
JUST
INTONATION
PYTHAGOREAN
INTONATION
MEANTONE
TEMPERAMENT
EQUAL
TEMPERAMENT
INTERVAL RATIO CENTS RATIO CENTS RATIO CENTS RATIO CENTS
unison
diatonic semitone
major tone
minor third
major third
perfect fourth
tritone
perfect fifth
minor sixth
major sixth
minor seventh
major seventh
octave
1/1
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1
0.000
111.731
203.910
315.641
386.314
498.045
590.224
701.955
813.686
884.359
1017.596
1088.269
1200.000
1/1
2187/2048
9/8
32/27
81/64
4/3
729/512
3/2
128/81
27/16
16/9
243/128
2/1
0.000
113.685
203.910
294.135
407.820
498.045
611.730
701.955
792.180
905.865
996.090
1109.775
1200.000
1.000000
1.044907
1.118034
1.196279
1.250000
1.337481
1.397542
1.495349
1.600000
1.671851
1.788854
1.869186
2.000000
0.000
76.049
193.157
310.265
386.314
503.422
579.471
696.578
813.686
889.735
1006.843
1082.892
1200.000
1.000000
1.059463
1.122462
1.189207
1.259921
1.334840
1.414214
1.498307
1.587401
1.681793
1.781797
1.887749
2.000000
0.000
100.000
200.000
300.000
400.000
500.000
600.000
700.000
800.000
900.000
1000.000
1100.000
1200.000
194 The Harmonic Consideration
Tabl
e 12
3. T
he h
arm
onic
ser
ies
– fir
st p
artia
l, th
roug
h th
e 12
8th
par
tial
Rela
tive
Pitc
h: A
4=44
0Hz
/ C4 (
mid
dle
C)=
261.
6255
654H
z DE
GRE
E
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
C C.
C,
CP
CX(
D+ D D!
EW
E©
EO E E(
F+(
uniso
n (1
st h
arm
onic
)
trid
ecim
al c
omm
a (6
5th
harm
onic
)
und
ecim
al c
omm
a (3
3rd
and
66t
h ha
rmon
ic)
67th
har
mon
ic
sep
tend
ecim
al c
hrom
atic
sem
itone
(17t
h, 3
4th
and
68t
h ha
rmon
ic)
trivi
gesim
al c
hrom
atic
sem
itone
(69
th h
arm
onic
)
sep
timal
neu
tral s
econ
d (3
5th
and
70th
har
mon
ic)
71st
har
mon
ic
just
maj
or to
ne (
9th,
18t
h, 3
6th
and
72nd
har
mon
ic)
73rd
har
mon
ic
37t
h an
d 74
th h
arm
onic
augm
ente
d se
cond
(75t
h ha
rmon
ic)
nona
dec
imal
sub
min
or, o
r ove
rtone
min
or th
ird (1
9th,
38t
h an
d 76
th h
arm
onic
)
und
ecim
al n
eutra
l thi
rd (
77th
har
mon
ic)
trid
ecim
al g
rave
or s
mal
l neu
tral t
hird
(39
th a
nd 7
8th
harm
onic
)
79th
har
mon
ic
just
maj
or th
ird (
5th,
10t
h, 2
0th,
40t
h an
d 80
th h
arm
onic
)
Pyth
agor
ean
maj
or th
ird, o
r dito
ne (8
1st h
arm
onic
)
41st
and
82n
d ha
rmon
ic
83rd
har
mon
ic
sep
timal
sub
four
th (
21st
, 42n
d an
d 84
th h
arm
onic
)
1/1
65/6
4
33/3
2 (6
6/64
)
67/6
4
17/1
6 (3
4/32
, 68/
64)
69/6
4
35/3
2 (7
0/64
)
71/6
4
9/8
(18/
16, 3
6/32
, 72/
64)
73/6
4
37/3
2 (7
4/64
)
75/6
4
19/1
6 (3
8/32
, 76/
64)
77/6
4
39/3
2 (7
8/64
)
79/6
4
5/4
(10/
8, 2
0/16
, 40/
32, 8
0/64
)
81/6
4
41/3
2 (8
2/64
)
83/6
4
21/1
6 (4
2/32
, 84/
64)
1.00
0000
1.01
5625
1.03
1250
1.04
6875
1.06
2500
1.07
8125
1.09
3750
1.10
9375
1.12
5000
1.14
0625
1.15
6250
1.17
1875
1.18
7500
1.20
3125
1.21
8750
1.23
4375
1.25
0000
1.26
5625
1.28
1250
1.29
6875
1.31
2500
261.
626
265.
713
269.
801
273.
889
277.
977
282.
065
286.
153
290.
241
294.
329
298.
417
302.
505
306.
592
310.
680
314.
768
318.
856
322.
944
327.
032
331.
120
335.
208
339.
296
343.
384
0.00
0
26.8
41
53.2
73
79.0
70
104.
955
130.
229
155.
140
179.
697
203.
910
227.
789
251.
344
274.
582
297.
513
320.
144
342.
483
364.
537
386.
314
407.
820
429.
062
450.
047
470.
781
+00
+27
ß47
ß21
+05
+30
ß45
ß20
+04
+28
ß49
ß25
ß02
+20
+42
ß35
ß14
+08
+29
ß50
ß29
The Harmonic Consideration 195
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
Ò×
ÒØ
ÒÙ
ÓÐ
ÓÑ
ÓÒ
ÓÓ
ÓÔ
ÓÕ
ÓÖ
Ó×
ÓØ
ÓÙ
ÔÐ
ÔÑ
ÔÒ
ÔÓ
ÔÔ
EP
F6(
F, F!
(
GÞ
FX(
F`(
G2
G
AŒ(
G,
GÚ GP
AO
A+(
sep
tend
ecim
al s
uper
augm
ente
d th
ird (
85th
har
mon
ic)
43rd
and
86t
h ha
rmon
ic
nona
vige
simal
sup
erfo
urth
(87
th h
arm
onic
)
und
ecim
al s
uper
four
th (
11th
, 22n
d, 4
4th
and
88t
h ha
rmon
ic)
89th
har
mon
ic
just
trito
ne, o
r aug
men
ted
four
th (
45th
and
90t
h ha
rmon
ic)
trid
ecim
al d
imin
ished
fifth
(91
st h
arm
onic
)
trivi
gesim
al s
uper
augm
ente
d fo
urth
(23r
d, 4
6th
and
92n
d h
arm
onic
)
untri
gesim
al s
uper
augm
ente
d fo
urth
(93
rd h
arm
onic
)
47th
and
94t
h ha
rmon
ic
nona
dec
imal
sub
fifth
(95
th h
arm
onic
)
just
and
Pyt
hago
rean
per
fect
fifth
(3r
d, 6
th, 1
2th,
24t
h, 4
8th
and
96t
h ha
rmon
ic)
97th
har
mon
ic
sep
timal
dim
inish
ed s
ixth
(49
th a
nd 9
8th
harm
onic
)
und
ecim
al s
uper
fifth
(99t
h ha
rmon
ic)
augm
ente
d fi
fth (
25th
, 50t
h an
d 1
00th
har
mon
ic)
101s
t har
mon
ic
sept
ende
cim
al s
uper
augm
ente
d fif
th (5
1st a
nd 1
02nd
har
mon
ic)
103r
d ha
rmon
ic
trid
ecim
al g
rave
or s
mal
l neu
tral,
or o
verto
ne s
ixth
(13
th, 2
6th,
52n
d a
nd 1
04th
har
mon
ic)
sept
imal
neu
tral s
ixth
(10
5th
harm
onic
)
53rd
and
106
th h
arm
onic
107t
h ha
rmon
ic
85/6
4
43/3
2 (8
6/64
)
87/6
4
11/8
(22/
16, 4
4/32
, 88/
64)
89/6
4
45/3
2 (9
0/64
)
91/6
4
23/1
6 (4
6/32
, 92/
64)
93/6
4
47/3
2 (9
4/64
)
95/6
4
3/2
(6/4
, 12/
8, 2
4/16
, 48/
32, 9
6/64
)
97/6
4
49/3
2 (9
8/64
)
99/6
4
25/1
6 (5
0/32
, 100
/64)
101/
64
51/3
2 (1
02/6
4)
103/
64
13/8
(26/
16, 5
2/32
, 104
/64)
105/
64
53/3
2 (1
06/6
4)
107/
64
1.32
8125
1.34
3750
1.35
9375
1.37
5000
1.39
0625
1.40
6250
1.42
1875
1.43
7500
1.45
3125
1.46
8750
1.48
4375
1.50
0000
1.51
5625
1.53
1250
1.54
6875
1.56
2500
1.57
8125
1.59
3750
1.60
9375
1.62
5000
1.64
0625
1.65
6250
1.67
1875
347.
471
351.
559
355.
647
359.
735
363.
823
367.
911
371.
999
376.
087
380.
175
384.
263
388.
350
392.
438
396.
526
400.
614
404.
702
408.
790
412.
878
416.
966
421.
054
425.
142
429.
229
433.
317
437.
405
491.
269
511.
518
531.
532
551.
318
570.
880
590.
224
609.
354
628.
274
646.
991
665.
507
683.
827
701.
955
719.
895
737.
652
755.
228
772.
627
789.
854
806.
910
823.
801
840.
528
857.
095
873.
505
889.
760
ß09
+12
+32
ß49
ß29
ß10
+09
+28
+47
ß34
ß16
+02
+20
+38
ß45
ß27
ß10
+07
+24
+41
ß43
ß26
ß10
196 The Harmonic Consideration
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÔÕ
ÔÖ
Ô×
ÔØ
ÔÙ
ÕÐ
ÕÑ
ÕÒ
ÕÓ
ÕÔ
ÕÕ
ÕÖ
Õ×
ÕØ
ÕÙ
ÖÐ
ÖÑ
ÖÒ
ÖÓ
ÖÔ ÐÑ
A( A, B;
BW
AX(
B_
BO Bß
B BÏ)
B8
B!
C+(
C
Pyth
agor
ean
maj
or s
ixth
(27t
h, 5
4th
and
108t
h ha
rmon
ic)
109t
h ha
rmon
ic
und
ecim
al s
uper
maj
or s
ixth
(55t
h an
d 11
0th
harm
onic
)
111t
h ha
rmon
ic
sep
timal
sub
min
or s
even
th (
7th,
14t
h, 2
8th,
56t
h an
d 11
2th
harm
onic
)
113t
h ha
rmon
ic
nona
dec
imal
sub
min
or s
even
th (
57th
and
114
th h
arm
onic
)
trivi
gesim
al s
uper
augm
ente
d six
th (
115t
h ha
rmon
ic)
nona
vige
simal
gra
ve o
r sm
all n
eutra
l sev
enth
(29
th, 5
8th
and
116t
h ha
rmon
ic)
trid
ecim
al n
eutra
l sev
enth
(117
th h
arm
onic
)
59th
and
118
th h
arm
onic
sept
ende
cim
al n
eutra
l sev
enth
(11
9th
harm
onic
)
just
dia
toni
c m
ajor
sev
enth
(15
th, 3
0th,
60t
h an
d 12
0th
harm
onic
)
grav
e or
sm
all n
eutra
l sev
enth
(12
1st h
arm
onic
)
61st
and
122
nd h
arm
onic
123r
d ha
rmon
ic
untri
gesim
al s
uper
maj
or s
even
th (
31st
, 62n
d an
d 12
4th
harm
onic
)
augm
ente
d s
even
th (1
25th
har
mon
ic)
sep
timal
sub
dim
inish
ed o
ctav
e (6
3rd
and
126
th h
arm
onic
)
127t
h ha
rmon
ic
octa
ve (
2nd
har
mon
ic)
(4th
, 8th
, 16t
h, 3
2nd,
64t
h an
d 12
8th
harm
onic
)
27/1
6 (5
4/32
, 108
/64)
109/
64
55/3
2 (1
10/6
4)
111/
64
7/4
(14/
8, 2
8/16
, 56/
32, 1
12/6
4)
113/
64
57/3
2 (1
14/6
4)
115/
64
29/1
6 (5
8/32
, 116
/64)
117/
64
59/3
2 (1
18/6
4)
119/
64
15/8
(30/
16, 6
0/32
, 120
/64)
121/
64
61/3
2 (1
22/6
4)
123/
64
31/1
6 (6
2/32
, 124
/64)
125/
64
63/3
2 (1
26/6
4)
127/
64
2/1
(4/2
, 8/4
, 16/
8, 3
2/16
, 64/
32, 1
28/6
4)
1.68
7500
1.70
3125
1.71
8750
1.73
4375
1.75
0000
1.76
5625
1.78
1250
1.79
6875
1.81
2500
1.82
8125
1.84
3750
1.85
9375
1.87
5000
1.89
0625
1.90
6250
1.92
1875
1.93
7500
1.95
3125
1.96
8750
1.98
4375
2.00
0000
441.
493
445.
581
449.
669
453.
757
457.
845
461.
933
466.
021
470.
108
474.
196
478.
284
482.
372
486.
460
490.
548
494.
636
498.
724
502.
812
506.
900
510.
987
515.
075
519.
163
523.
251
905.
865
921.
821
937.
632
953.
299
968.
826
984.
215
999.
468
1014
.588
1029
.577
1044
.438
1059
.172
1073
.781
1088
.269
1102
.636
1116
.885
1131
.017
1145
.036
1158
.941
1172
.736
1186
.422
1200
.000
+06
+22
+38
ß47
ß31
ß16
ß01
+15
+30
+44
ß41
ß26
ß12
+03
+17
+31
+45
ß41
ß27
ß14
+00
The Harmonic Consideration 197
It is important to note that odd-numbered partials in the harmonic and subharmonic series represent
unique entities, while even-numbered partials, merely their respective octave duplications, and therefore
in harmonic analysis the former needs only be considered. The term designated to these unique entities
of the harmonic and subharmonic series is ‘identities’.288
The Monochord
A monochord is a simple instrument that consists of a metal string extended over two bridges, and is a
useful tool for illustrating the nature of ratios (a bamboo pipe, or air column deliver similar results). If for
example, striking the open string produced 100 cycles when set in vibration (cycles referring to the
number of vibrations per second which a tone makes), striking half of that length (the string stopped via a
third bridge placed in the centre) would produce 200 cycles, or a 200 to 100 ratio (2/1); sounding the
perfect octave of the original pitch. The logic then follows that striking a third of the full length would
produce 300 cycles, or a 300 to 200 ratio (3/2) of the half length, and hence now sounding the perfect
fifth.289 Cecil Adkins provides the following historical perspective:
“The monochord in its early form, and in the form utilized throughout the Middle Ages was a table or plank
(AC) upon which were erected two fixed bridges (EB and FD). The string was stretched across the
bridges (EF) and securely fastened at the ends (AC). A movable bridge (K) was then placed underneath
the string, dividing it into two sections (EK and KF). The marks indicating the placement of the movable
bridge were then inscribed on the table underneath the string, between the two end bridges (B and D).
The resonating box, generally considered an integral part of the instrument, is not mentioned in the treatises
of the Middle Ages, but is depicted in miniatures after the twelfth century. It was probably a late medieval
addition directed at increasing the portability of the instrument as much as enhancing its tone.”
Fig. 6. Medieval figure of the monochord
The traditional function of the monochord is the facilitation of aural representations of intervals or scales,
established via mathematical calculation, and generally articulated via the principles of proportions, string
lengths, or cents. The term ‘monochord division’ in this context refers to what Adkins prefers to call the
“manual division,” and therefore implicates the actual physical application of the mathematical formula
288 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 30.
289 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 79-81.
198 The Harmonic Consideration
required to produce a specific interval. The Pythagorean principle of “monochord division by
proportions” takes into account the “arithmetic mean and the harmonic mean” in a system of four
principal intervals directly derived from the “smallest whole integers” that is capable of expressing the
intervallic “relationships of these two means.” In this case, the principle is represented by the numbers 6,
8, 9, and 12, and by the resulting ratios of 12/6 (the diapason, or octave), 9/6 and 12/8 (the diapente, or
fifth), 8/6 and 12/9 (the diatessaron, or fourth), and 9/8 (the tone, or major second). In different terms,
these four intervals may be further theorized as the “multiplex and superparticular proportions” of 2/1
(dupla), 3/2 (sesquialtera), 4/3 (sesquitertia), and 9/8 (sesquioctava).290 The employment of string lengths
and cents as a measuring device in the Pythagorean chromatic context represent 2/1 (octave) by the
figure 314928, or 1200.000 cents; 3/2 (just perfect fifth) by 419904, or 701.955 cents; 4/3 (just perfect
fourth) by 472392, or 498.045 cents; and 9/8 (just major tone) by 559872, or 203.910 cents. The
interval of the unison is represented by the string length of 629856, or 0.000 cents.291
Table 124. The Pythagorean monochord
DEGREE
NUMBER
NOTE LENGTHS RATIO
(DECIMAL)
CENTS FALSITY
(CENTS)
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ
ÐÑ
C4
C!4
D4
D!4
E4
F4
F!4
G4
G!4
A4
A!4
B4
C5
629856
589824
559872
531441
497664
472392
442368
419904
393216
373248
354294
321776
314928
1.000000
1.067871
1.125000
1.185185
1.265625
1.333333
1.423828
1.500000
1.601807
1.687500
1.777778
1.898438
2.000000
0.000
113.685
203.910
294.135
407.820
498.045
611.730
701.955
815.640
905.865
996.090
1109.775
1200.000
+0.000
+1.954
+0.000
ß21.506
+21.506
+0.000
+21.506
+0.000
+1.954
+21.506
ß21.506
+21.506
+0.000
290 Cecil Adkins, “The Technique of the Monochord,” Acta Musicologica 39 (Jan.-Jun., 1967): 34-37.
291 Barbour, Tuning and Temperament: A Historical Survey 90.
The Harmonic Consideration 199
Combinational Tones
The discovery of combinational tones in 1745 may be attributed to German organist Georg Andreas
Sorge (1703-1778), and was followed by the later classification of Italian violinist Giuseppe Tartini (1692-
1770). According to Helmholtz, “These tones are heard whenever two musical tones of different
pitches are sounded together, loudly and continuously.” There are two classes of combinational tones,
with the first being differential tones (whereby the resultant tone equals f2 minus f1), and the second,
summation tones (whereby the resultant tone equals f2 plus f1), which represent Helmholtz’s own
contribution to the theory. Primary and upper partial tones are both capable of producing the two
classes of combinational tones. Differential tones are more prominent when the interval between the
two generating tones is inferior to an octave, as this condition produces a resultant tone below the
primary tones. The theory may be further expanded to differentiate between first and second-order
differential tones, with the latter being phenomena of an infinite nature, and the result of the secondary
differential tones produced by the combination of the primary differential tones and the actual generating
tones. Summation tones are less prominent than combinational tones; and in view of the fact that the
resultant frequency is equal to the sum of the two primary tones, the pitch of these tones will always
exceed their generators. It should be further noted that upper partial tones are also capable of
producing combinational tones.292 The following example illustrates the differential tones produced by
the octave (2/1), just perfect fifth (3/2), just perfect fourth (4/3), just major third (5/4), just minor sixth
(8/5), just minor third (6/5), and just major sixth (5/3).
Ex. 83. Differential tones produced by 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, and 5/3
Summation tones for the same generating tones on the other hand present the following intervals: just
perfect fifth (3/2), just major third (5/4), septimal subminor third (7/6), just major tone (9/8), tridecimal
subdiminished fourth (13/10), undecimal acute or large neutral second (11/10), and just perfect fourth
(4/3).
292 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 152-56.
200 The Harmonic Consideration
Ex. 84. Summation tones produced by 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, and 5/3
Periodicity Pitch
“When two or more pure tones separated by intervals greater than the critical band are sounded
simultaneously, the auditory mechanism is sensitive to the period or frequency of the resulting composite
waveform. This gives rise to a phenomenon variously known as the periodicity pitch, virtual pitch,
subjective pitch, residue tone, or the missing fundamental,” explains Doty. The simultaneous sounding of
two dissimilar pitched musical tones may be mathematically explained as f2= nm Ïf1, with m and n
representative of the relevant integers (numerator and denominator) that denote the frequency ratio of
the interval. The formula for calculating the periodicity pitch of a dyad is �0= n1 �1, with n, the
denominator, utilized as the dividing factor, and applied to the fundamental, or lower tone. The process
may be illustrated via the calculations of the just perfect fifth (3/2) and just perfect fourth (4/3) in relation
to the unison (1/1), which presents 21 Ï261.626Hz (or C4)=130.8136Hz (or C3); and 3
1 Ï261.626Hz
(or C4)=87.209Hz (or F2). Doty alludes to the facts that the “smaller the value of n, the shorter the
period and the higher the frequency of the resulting pattern,” and that “musical consonance is associated
with high periodicity pitch.” Dyads generated via simple integer-ratio intervals with integers inferior to
eight or nine produce “unambiguous periodicity pitch,” while superior integers generally produce an
“ambiguous” approximation of a simpler ratio.293
Ex. 85. Periodicity pitches produced by 2/1, 3/2, 4/3, 5/3, 5/4, 7/4, 6/5, 7/5, 8/5, and 7/6
293 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 17-18.
The Harmonic Consideration 201
Sethares has the following to say about the “periodicity theory of pitch perception,” and its theoretical
significance in relation to JND, or Just Noticeable Difference:
“An alternative hypothesis (to JND, or Just Noticeable Difference), called ‘periodicity’ theory of pitch
perception suggests that information is extracted directly from the time behaviour of the sound. For
instance, the time interval over which a signal repeats may be used to determine its frequency. In fact,
there is now (and has been for the past hundred years or so) considerable controversy between
advocates of the place and periodicity theories, and it is probably safe to say that there is not enough
evidence to decide between them. Indeed, J. R. Pierce (Periodicity and Pitch Perception, 1889-93
[1991]) suggests that both mechanisms may operate in tandem, and a growing body of recent
neurophysiological research (such as P. Cariani and his co-workers [Temporal Coding of Periodicity Pitch in
the Auditory System: An Overview [1999], and A Temporal Model for Pitch Multiplicity and Tonal
Consonance [2004]) reinforce this.”294
Prime Numbers, Primary Intervals, and Prime Limits
Another aspect of just intonation principles is the theory of prime limits, which is based on the concept
of prime numbers, or integers with only factors of one and themselves. This is an infinite series, but
musical significance is accorded to primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. The primary intervals
of these eleven primes are 2/1, 3/2, 5/4, 7/4, 11/8, 13/8, 17/16, 19/16, 23/16, 29/16, and 31/16
(produced by the formula: p/2n=p [prime], with “2n being the greatest power of 2 less than p”), while
its subharmonic complements may be represented by 2/1, 4/3, 8/5, 8/7, 16/11, 16/13, 32/17, 32/19,
32/32, 32/29, and 32/31 (produced by the formula: 2n/p=p [prime], with “2n being the smallest power
of 2 less than p”). An example of this concept is ‘five limit just intonation’, which should be taken to
mean an intonation system based only on the primes 2, 3, and 5, with all intervals directly derived from
2/1, 3/2, and 5/4 (the octave, just perfect fifth, and just major third), or 2/1, 4/3, and 8/5 (octave, just
perfect fourth, and just minor sixth) from a subharmonic perspective.295
The Just Diatonic Scale
The construction of a just diatonic scale begins with the fundamental or unison (the frequency ratio 1/1),
and the establishment of a just major triad in the ratio of 1/1:5/4:3/2.296 The harmonic relationship of this
root position chord, which has the 1, 5, and 3 identities of the harmonic series, is also expressible as
4:5:6, or in the key of C major as the pitches C, E, and G. In first inversion, the triad is represented by the
294 Sethares, Tuning, Timbre, Spectrum, Scale 44.
295 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 28-30. 296 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters,” 108.
202 The Harmonic Consideration
ratio 5:6:8, and in second, by 3:4:5. This triad is beatless, made up of the first, fifth, and third partials of
the harmonic series, and contains the intervals of the unison (1/1), just major third (5/4), and just perfect
fifth (3/2). Analysis of this simultaneous sonority further reveals a 6/5 ratio (or just minor third) between
the third and the fifth. G4 (392.438Hz), or the just perfect fifth (3/2), presents 0.000 beats between the
third harmonic of C4 (1/1) and the second harmonic of G4 (784.877Hz), and 0.000 beats between the
sixth harmonic of C4 and the fourth harmonic of G4 (1569.753Hz); while E4 (327.032Hz), or the just
major third (5/4) then consistently also presents 0.000 beats between the fifth harmonic of C4 and the
fourth harmonic of E4 (1308.128Hz).
Table 125. The beating characteristics of the just major triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
327.032
– – – –
– – – –
654.064
– – – –
981.096
– – – –
– – – –
1308.128
– – – –
1635.160
– – – –
1962.192
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
392.438
– – – –
– – – –
784.877
– – – –
– – – –
1177.315
– – – –
1569.753
– – – –
– – – –
1962.192
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The next stage in the generation of a just diatonic scale requires simple mathematics, or the multiplication
of ratios, to deal with the addition of ratios. If one were to take the C major triad as an example, which
has a 6/5 ratio between 5/4 and 3/2, one can ascertain that adding 6/5 to 5/4 will produce the equation:
6/5 plus 5/4=(6Ï5):(5Ï4)=30/20=3/2. In this particular case, 3/2 represents the reduction of the
unnecessary larger 30/20 ratio. Subtraction of ratios is alternatively resolved via the multiplication of ratios
with the subtraction in question inverted. In other words, the subtraction of 6/5 from 3/2 is resolved
thus: 3/2 minus 6/5=(3Ï5):(2Ï6)=15/12=5/4. In cases where addition results in a figure that exceeds
the octave (2/1), or where the “numerator is greater than twice the denominator,” subtraction of an
appropriate number of 2/1s should be undertaken. For example, the equation of 3/2 plus
The Harmonic Consideration 203
3/2=(3Ï3):(2Ï2)=9/4 should be followed by 9/4 minus 2/1=(9Ï1):(4Ï2)=9/8. On the other hand,
in cases where subtraction results in a figure that exceeds the unison (1/1), or where the result is a “ratio
with a denominator greater than its numerator”, addition of an appropriate number of 2/1s should be
undertaken. For example, 9/8 minus 3/2=(9Ï2):(8Ï3)=18/24=3/4 should be followed by 3/4 plus
2/1=(3Ï2):(4Ï1)=6/4=3/2.297 The final stage in the establishment of the just diatonic scale involves the
further generation of just major triads on the fifth degree, 3/2 (or G) and its inversion, 4/3 (or F), which
produce the intervals 3/2, 15/8, and 9/8 (or the pitches G, B, and D), and 4/3, 5/3, and 1/1 (or the
pitches F, A, and C). The equations involved in obtaining 4:5:6 harmonic relationships (or appropriate
5/4s and 3/2s) for 1/1, 3/2, and 4/3 are:
i. C (unison) generated by the ratio 1/1
ii. E (just major third) by the equation 1/1 plus 5/4=(1Ï5):(1Ï4)=5/4
iii. G (just and Pythagorean perfect fifth) by 1/1 plus 3/2=(1Ï3):(1Ï2)=3/2
iv. G (just and Pythagorean perfect fifth) by the ratio 3/2
v. B (just diatonic major seventh) by the equation 3/2 plus 5/4=(3Ï5):(2Ï4)=15/8
vi D (just major tone) by 3/2 plus 3/2=(3Ï3):(2Ï2)=9/4; and 9/4 minus 2/1=(9Ï1):(4Ï2)=9/8
vii. F (just and Pythagorean perfect fourth) by the ratio 4/3
viii. A (just major sixth) by the equation 4/3 plus 5/4=(4Ï5):(3Ï4)=20/12=5/3
ix. C (unison) by 4/3 plus 3/2=(4Ï3):(3Ï2)=12/6=2/1; and 2/1 minus 2/1=(2Ï1):(1Ï2)=2/2=1/1
Ex. 86. Just major triads on 1/1, 3/2, and 4/3
The resulting scale consists of two dissimilar tetrachords separated by a just major tone, or the ratio 9/8,
which in relation to the fundamental, or unison (1/1), presents the intervals of the just major tone (9/8),
just major third (5/4), just perfect fourth (4/3), just perfect fifth (3/2), just major sixth (5/3), just diatonic
major seventh (15/8), and octave (2/1).
297 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 22-26.
204 The Harmonic Consideration
Ex. 87. The harmonic characteristics of the just diatonic scale
The Just Chromatic Scale The next stage involves the construction of a “secondary group of triads” with 4:5:6 harmonic
relationships to D, E, A, and B, or 9/8, 5/4, 5/3, and 15/8, which are produced via the following
equations:
i. D (just major tone) generated by the ratio 9/8
ii. F!( (just tritone, or augmented fourth) by the equation 9/8 plus 5/4=(9Ï5):(8Ï4)=45/32
iii. A( (Pythagorean major sixth) by 9/8 plus 3/2=(9Ï3):(8Ï2)=27/16
iv. E (just major third) by the ratio 5/4
v. G! (augmented fifth) by the equation 5/4 plus 5/4=(5Ï5):(4Ï4)=25/16
vi. B (just diatonic major seventh) by 5/4 plus 3/2=(5Ï3):(4Ï2)=15/8
vii. A (just major sixth) by the ratio 5/3
viii. C! (grave or small just chromatic semitone) by the equation 5/3 plus
5/4=(5Ï5):(3Ï4)=25/12; and 25/12 minus 2/1=(25Ï1):(12Ï2)=25:24
ix. E (just major third) by 5/3 plus 3/2=(5Ï3):(3Ï2)=15/6; and 15/6 minus
2/1=(15Ï1):(6Ï2)=15/12=5/4
x. B (just diatonic major seventh) by the ratio 15/8
xi. D! (augmented second) by 15/8 plus 5/4=(15Ï5):(8Ï4)=75/32; and 75/32 minus
2/1=(75Ï1):(32Ï2)=75/64
xii. F!( (just tritone, or augmented fourth) by 15/8 plus 3/2=(15Ï3):(8Ï2)=45/16; and 45/16
minus 2/1=(45Ï1):(16Ï2)=45/32
The Harmonic Consideration 205
Ex. 88. Just major triads on 9/8, 5/4, 5/3, and 15/8
Of particular note is the fact that the A major triad now presents the grave or small just chromatic
semitone (25/24), or what may be assumed to be the ‘sharp’ (!) ratio, and hence the ratio “applied to a
diatonic note to produce the sharpened form of that note.” Following from that, in order to generate the
correct ratios for the 4:5:6 triads built on D and B will require the raising of the 25/24 ratio by an
additional interval of a syntonic comma (81/80). Theorists generally refer to this interval (the sum of 25/24
and 81/80) as the large limma (135/128, or 92.179 cents), which may be mathematically expressed as:
25/24 plus 81/80=(25Ï81):(24Ï80)=2025/1920=135/128. 135/128, and its reciprocal, 128/135,
may also be theoretically referred to as the sharp, or greater limma, and flat, or hypolimma respectively.298
The generation of flats then utilizes Partch’s otonal and utonal theoretical basis to incorporate
pitches derived from the complement or mirror image of the harmonics series. The retrograde of a C
major triad – the initiating factor – is therefore utilized to produce an F minor triad, or what Fonville
describes as an inversion explained as a “subharmonic relationship or subharmonic series descending
from 1/1,” or in different terms; the intervallic complements of 1/1, 5/4, and 3/2, equal to 1/1, 8/5, and
4/3. The equations involved in obtaining 5:6:10 relationships (or appropriate 8/5s and 4/3s) for 1/1, 3/2,
and 4/3 are:
i. C (unison) generated by the ratio 1/1
ii. A" (just minor sixth) by the equation 1/1 minus 5/4=(1Ï4):(1Ï5)=4/5; and 4/5 plus
2/1=(4Ï2):(5Ï1)=8/5
iii. F (just and Pythagorean perfect fourth) by 1/1 minus 3/2=(1Ï2):(1Ï3)=2/3; and 2/3 plus
2/1=(2Ï2):(3Ï1)=4/3
iv. G (just and Pythagorean perfect fifth) by the ratio 3/2
v. E" (just minor third) by the equation 3/2 minus 5/4=(3Ï4):(2Ï5)=12/10=6/5
vi C (unison) by 3/2 minus 3/2=(3Ï2):(2Ï3)=6/6=1/1
vii. F (just and Pythagorean perfect fourth) by the ratio 4/3
298 Alexander J. Ellis, “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments with Fixed
Tones,” Proceedings of the Royal Society of London 13 (1863-64): 95.
206 The Harmonic Consideration
viii. D") (just diatonic semitone, or major half-tone) by the equation 4/3 minus
5/4=(4Ï4):(3Ï5)=16/15
ix. B") (Pythagorean minor seventh) by 4/3 minus 3/2=(4Ï2):(3Ï3)=8/9; and 8/9 plus
2/1=(8Ï2):(9Ï1)=16/9
Ex. 89. Just minor triads on 1/1, 3/2, and 4/3
The chromatic sonorities in the F and G minor triads now present the subtraction of a grave or small just
chromatic semitone (25/24) from a diatonic note, and hence, the introduction of the flat (") symbol. The
symbol could for example be applied to D, or 9/8 (or any other diatonic interval for that matter) to
produce the flattened form of that note or ratio simply by the subtraction of 25/24 from 9/8, or 9/8
minus 25/24=(9Ï24):(8Ï25)=216/200=27/25. Theorists generally refer to this interval as the great
limma, acute or large half-tone, which measures 133.238 cents. The correct ratios for the 5:6:10 triad
built on F require the lowering of the 9/8 and 9/5 ratios by an additional syntonic comma (81/80). For
example, D"), or the just diatonic semitone (16/15) is the result of the subtraction of 25/24 and 81/80
(equal to 135/128) from 9/8, or 9/8 minus 135/128=(9Ï128):(8Ï135)=1152/1080=16/15. The
otonal process produces the just minor scale (aeolian mode, natural minor, or descending melodic minor
scale). The scale consists of two dissimilar tetrachords separated by a just major tone, or the ratio 9/8,
which in relation to the fundamental, or unison (1/1) presents the intervals of just major tone (9/8), just
minor third (6/5), just perfect fourth (4/3), just perfect fifth (3/2), just minor sixth (8/5), acute or large
minor seventh (9/5), and octave (2/1).
Ex. 90. The just minor scale
The Harmonic Consideration 207
The process is then continued to include 5:6:10 relationships (or appropriate 8/5s and 4/3s) for 9/8, 5/4,
5/3, and 15/8, which are produced via the following equations:
i. D (just major tone) generated by the ratio 9/8
ii. B" (acute or large minor seventh) by the equation 9/8 minus 5/4=(9Ï4):(8Ï5)=36/40=9/5
iii. G (just and Pythagorean perfect fifth) by 9/8 minus 3/2=(9Ï2):(8Ï3)=18/24=3/4; and 3/4
plus 2/1=(3Ï2):(4Ï1)=6/4=3/2
iv. E (just major third) by the ratio 5/4
v. C (unison) by the equation 5/4 minus 5/4=(5Ï4):(4Ï5)=20/20=1/1
vi. A (just major sixth) by 5/4 minus 3/2=(5Ï2):(4Ï3)=10/12; and 10/12 plus
2/1=(10Ï2):(12Ï1)=20/12=5/3
vii. A (just major sixth) by the ratio 5/3
viii. F (just and Pythagorean perfect fourth) by the equation 5/3 minus 5/4=(5Ï4):(3Ï5)=20/15=4/3
ix. D) (just minor tone) by 5/3 minus 3/2=(5Ï2):(3Ï3)=10/9
x. B (just diatonic major seventh) by the ratio 15/8
xi. G (just and Pythagorean perfect fifth) by 15/8 minus 5/4=(15Ï4):(8Ï5)=60/40=3/2
xii. E (just major third) by 15/8 minus 3/2=(15Ï2):(8Ï3)=30/24=5/4
Ex. 91. Just minor triads on 9/8, 5/4, 5/3, and 15/8
Repeating the process of triadic construction to include all major and minor triads, as well as dominant
triads to all the basic just scales degrees (1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, and 15/8)
results in a twenty-five-note just enharmonic scale.299
299 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 108-11.
208 The Harmonic Consideration
Ex. 92. The just enharmonic scale
Ben Johnston’s Fifty-Three-Tone Just Intonation Scale Johnston’s elaboration of the twenty-five-note just enharmonic scale – premised on a system of infinite
intervallic regeneration – firstly produces a twelve-note just scale, via the combination of the just diatonic
and just minor scales, and the further subdivision of the two just major tones of the structure; the process
resulting in the addition of D" and G" between C and D, and F and G respectively. The scale is then
developed into a nineteen-note just enharmonic scale, via the establishment of an enharmonic equivalent
of the previous twelve-note model, and therefore inclusive of C!, D!, F!, G!, and A!, as well as E! and F",
and B! and C"; and finally, expanded into a 2, 3, 5-limit just intonation fifty-three-tone enharmonic scale
that confers each of the nineteen pitches with a function as root, fifth, major third, and minor third – a
process limited to the overlapping of the “twelve chromatic regions of the octave.”
Each whole-tone, minor whole-tone, and diatonic semitone in Johnston’s just intonation fifty-three-
tone enharmonic scale features the identical pattern of adjacent intervals, which include either the
syntonic comma (81/80), diaskhisma (2048/2025),300 or grave or small diesis (3125/3072, or 29.614
cents). The grave or small diesis may be characterized as the difference between the grave or small just
chromatic semitone (25/24) and the great diesis (128/125).301
300 “The diaskhisma is a small interval composed of two major thirds down and four perfect fifths down
= 2411 532 ßß , also expressed as the 2,3,5-monzo [11ß4,ß2>. Its ratio is 2048/2025 [=0.20 semitones
=~19.55256881 or ~195/9 cents]. The term was used by Alexander Ellis in his English translation of Helmholtz, On the
Sensations of Tone (1875). It had been referred to earlier by Rameau (Traité de l’harmonie, 1722) as the ‘diminished
comma’. The standard epimoric approximation to the diaskhisma is the ratio 89:88 (=~19.56217479 cents), whose
2,3,5,11,89-monzo is ß3, 0, 0, ß1, 1>. Tuning treatises before c.1970 sometimes defined the diaskhisma as the 89:88
ratio without emphasis on the fact of its being an approximation, particularly in the German literature of c.1850-1950. A
good example is Helmholtz in On the Sensations of Tone.” For a further discussion, see Monzo, “Encyclopedia of
Microtonal Music Theory,” n. pag.
301 Ben Johnston, “Scalar Order as a Compositional Resource,” Perspectives of New Music 2.2 (Summer, 1964): 69-
73.
The Harmonic Consideration 209
Tabl
e 12
6. B
en J
ohns
ton’
s fif
ty-th
ree-
tone
just
into
natio
n sc
ale
Rela
tive
Pitc
h: A
4=44
0Hz
/ C4 (
mid
dle
C)=
261.
6255
654H
z
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(F
RACT
ION
) RA
TIO
(D
ECIM
AL)
FR
EQUE
NCY
(H
ERTZ
) CE
NTS
TU
NIN
G
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
C C(
D$)
C!
C!(
D"
)
D"
C#(
D)
D D(
E$
D!
E")
E"
D#
E)
E E(
F"
E!
F)
F F(
G
$)
F!
uniso
n sy
nton
ic c
omm
a d
imin
ished
sec
ond
, or g
reat
die
sis
grav
e or
sm
all j
ust c
hrom
atic
sem
itone
, or m
inor
hal
f-ton
e Py
thag
orea
n ac
ute
or la
rge
limm
a ju
st d
iato
nic
sem
itone
, or m
ajor
hal
f-ton
e gr
eat l
imm
a, a
cute
or l
arge
hal
f-ton
e ac
ute
or la
rge
dou
ble
augm
ente
d o
ctav
e ju
st m
inor
tone
ju
st m
ajor
tone
(9t
h ha
rmon
ic)
acut
e or
larg
e to
ne
dim
inish
ed th
ird
augm
ente
d s
econ
d (7
5th
harm
onic
) Py
thag
orea
n m
inor
third
, or t
rihem
itone
ju
st m
inor
third
d
oubl
e au
gmen
ted
seco
nd
grav
e or
sm
all m
ajor
third
ju
st m
ajor
third
(5t
h ha
rmon
ic)
Pyth
agor
ean
maj
or th
ird, o
r dito
ne (8
1st h
arm
onic
) d
imin
ished
four
th
augm
ente
d th
ird
grav
e or
sm
all f
ourth
ju
st a
nd P
ytha
gore
an p
erfe
ct fo
urth
ac
ute
or la
rge
four
th
grav
e or
sm
all d
oubl
e d
imin
ished
fifth
gr
ave
or s
mal
l aug
men
ted
four
th
1/1
81/8
0 12
8/12
5 25
/24
135/
128
16/1
5 27
/25
1125
/102
4 10
/9
9/8
729/
640
144/
125
75/6
4 32
/27
6/5
625/
512
100/
81
5/4
81/6
4 32
/25
125/
96
320/
243
4/3
27/2
0 51
2/37
5 25
/18
1.00
0000
1.
0125
00
1.02
4000
1.
0416
67
1.05
4688
1.
0666
67
1.08
0000
1.
0986
33
1.11
1111
1.
1250
00
1.13
9063
1.
1520
00
1.17
1875
1.
1851
85
1.20
0000
1.
2207
03
1.23
4568
1.
2500
00
1.26
5625
1.
2800
00
1.30
2083
1.
3168
72
1.33
3333
1.
3500
00
1.36
5333
1.
3888
89
261.
626
264.
896
267.
905
272.
527
275.
933
279.
067
282.
556
287.
430
290.
695
294.
329
298.
008
301.
393
306.
592
310.
075
313.
951
319.
367
322.
995
327.
032
331.
120
334.
881
340.
658
344.
527
348.
834
353.
195
357.
206
363.
369
0.00
0 21
.506
41
.059
70
.672
92
.179
11
1.73
1 13
3.23
8 16
2.85
1 18
2.40
4 20
3.91
0 22
5.41
6 24
4.96
9 27
4.58
2 29
4.13
5 31
5.64
1 34
5.25
5 36
4.80
7 38
6.31
4 40
7.82
0 42
7.37
3 45
6.98
6 47
6.53
9 49
8.04
5 51
9.55
1 53
9.10
4 56
8.71
7
+00
+
22
+41
ß
29
ß08
+
12
+33
ß
37
ß18
+
04
+25
+
45
ß25
ß
06
+16
+
45
ß35
ß
14
+08
+
27
ß43
ß
23
ß02
+
20
+39
ß
31
210 The Harmonic Consideration
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(F
RACT
ION
) RA
TIO
(D
ECIM
AL)
FR
EQUE
NCY
(H
ERTZ
) CE
NTS
TU
NIN
G
Ò×
ÒØ
ÒÙ
ÓÐ
ÓÑ
ÓÒ
ÓÓ
ÓÔ
ÓÕ
ÓÖ
Ó×
ÓØ
ÓÙ
ÔÐ
ÔÑ
ÔÒ
ÔÓ
ÔÔ
ÔÕ
ÔÖ
Ô×
ÔØ
ÔÙ
ÕÐ
ÕÑ
ÕÒ
ÕÓ ÐÑ
F!(
G")
G"
F#
(
G)
G G(
A$
G!
A")
A"
G#
A)
A A(
B$
)
A!
A!(
B"
)
B"
A#(
B)
B B(
C"
B!
C)
C
just
trito
ne, o
r aug
men
ted
four
th (
45th
har
mon
ic)
dim
inish
ed fi
fth
acut
e or
larg
e d
imin
ished
fifth
ac
ute
or la
rge
dou
ble
augm
ente
d fo
urth
gr
ave
or s
mal
l fift
h ju
st a
nd P
ytha
gore
an p
erfe
ct fi
fth (
3rd
har
mon
ic)
acut
e or
larg
e fif
th
dim
inish
ed s
ixth
au
gmen
ted
fifth
(25
th h
arm
onic
) Py
thag
orea
n m
inor
six
th
just
min
or s
ixth
d
oubl
e au
gmen
ted
fifth
gr
ave
or s
mal
l maj
or s
ixth
ju
st m
ajor
six
th
Pyth
agor
ean
maj
or s
ixth
(27t
h ha
rmon
ic)
dim
inish
ed s
even
th
augm
ente
d s
ixth
ac
ute
or la
rge
augm
ente
d s
ixth
Py
thag
orea
n m
inor
sev
enth
ac
ute
or la
rge
min
or s
even
th
acut
e or
larg
e d
oubl
e au
gmen
ted
six
th
grav
e or
sm
all m
ajor
sev
enth
ju
st d
iato
nic
maj
or s
even
th (
15th
har
mon
ic)
Pyth
agor
ean
maj
or s
even
th
dim
inish
ed o
ctav
e m
eant
one
augm
ente
d s
even
th (
125t
h ha
rmon
ic)
grav
e or
sm
all o
ctav
e
octa
ve
45/3
2 64
/45
36/2
5 37
5/25
6 40
/27
3/2
243/
160
192/
125
25/1
6 12
8/81
8/
5 62
5/38
4 40
0/24
3 5/
3 27
/16
128/
75
125/
72
225/
128
16/9
9/
5 18
75/1
024
50/2
7 15
/8
243/
128
48/2
5 12
5/64
16
0/81
2/1
1.40
6250
1.
4222
22
1.44
0000
1.
4648
44
1.48
1481
1.
5000
00
1.51
8750
1.
5360
00
1.56
2500
1.
5802
47
1.60
0000
1.
6276
04
1.64
6091
1.
6666
67
1.68
7500
1.
7066
67
1.73
6111
1.
7578
13
1.77
7778
1.
8000
00
1.83
1055
1.
8518
52
1.87
5000
1.
8984
38
1.92
0000
1.
9531
25
1.97
5309
2.00
0000
367.
911
372.
090
376.
741
383.
241
387.
593
392.
438
397.
344
401.
857
408.
790
413.
433
418.
601
425.
823
430.
659
436.
043
441.
493
446.
508
454.
211
459.
889
465.
112
470.
926
479.
051
484.
492
490.
548
496.
680
502.
321
510.
987
516.
791
52
3.25
1
590.
224
609.
776
631.
283
660.
896
680.
449
701.
955
723.
014
743.
014
772.
627
792.
180
813.
686
843.
300
862.
852
884.
359
905.
865
925.
418
955.
031
976.
537
996.
090
1017
.596
10
47.2
10
1066
.762
10
88.2
69
1109
.775
11
29.3
28
1158
.941
11
78.4
94
12
00.0
00
ß10
+
10
+31
ß
39
ß20
+
02
+23
+
43
ß27
ß
08
+14
+
43
ß37
ß
16
+06
+
25
ß45
ß
23
ß04
+
18
+47
ß
33
ß12
+
10
+29
ß
41
ß22
+00
The Harmonic Consideration 211
Harry Partch’s Forty-Three-Tone Just Intonation Scale Partch’s forty-three-tone just intonation scale, with its inclusion of eleven-limit intervals, represents a
harmonic expansion of Johnston’s 2, 3, 5-limit just intonation fifty-three-tone enharmonic scale. The
aesthetic rationale behind discontinuing beyond the seven-limit is illustrated by Partch within these
colourful remarks:
“The reasons why Monophony proceeds to the limit of 11 are basic and quite specific, as will be seen, but
the reason for resting at the limit of 11 is a purely personal and arbitrary one. When a hungry man has a
large table of aromatic and unusual viands spread before him he is unlikely to go tramping along the
seashore and in the woods for still another exotic fare. And however sceptical he is of the many warnings
regarding the unwholesomeness of his fare – like the ‘poison’ of the ‘love-apple’ tomato of a comparatively
few generations ago – he has no desire to provoke further alarums.”
According to Partch, the harmonic argument behind the abovementioned intonation system is based on
the premise that “the expansion of identities 1-3-5 through 7-9-11 provides a new and highly intriguing
triad, and immediately makes possible a wide variety in quality.” The identities 7-9-11 represent the
intervals 7/4, 9/8, and 11/8, and the ratio 7:9:11, which form an unusual variety of major triad consisting
of the septimal supermajor third (9/7, or 435.084 cents) and the undecimal augmented fifth (11/7, or
782.492 cents). Twenty unique triads, fifteen unique tetrads, as well as six unique pentads are now
made possible via the combination of the six identities of each individual otonality and utonality, and this
is excluding the further augmentation of harmonic resources effectuated via the inversion and extension
of simultaneous sonorities.
Ex. 93. Eleven-limit simultaneous sonorities i. The twenty eleven-limit triads
ii. The fifteen eleven-limit tetrads
212 The Harmonic Consideration
iii. The six eleven-limit pentads
With regards to the historical argument, Partch gives reference to Alexandrian astronomer, mathematician,
and geographer of the second century, Claudius Ptolemy (c.87-150).302
“In Ptolemy’s scales there is enough evidence to warrant the conclusion that his procedure was generally
governed by the principle of appropriating the smallest-number ratios permissible to the purpose of the
scale in question. In this light it is quite natural that he should have used all the ratios of the 11-limit as a
body.”
Partch’s forty-three-tone scale firstly adopts pairs of complementary five-limit intervals made available by
the first five partials of the harmonic series – the octave (2/1), unison (1/1), just perfect fifth (3/2), just
perfect fourth (4/3), just major third (5/4), and just minor third (6/5) – to then incorporate seven-limit and
eleven-limit intervallic ratios to construct a scale of twenty-nine degrees. To resolve some of the scalar
discontinuities of this unequal scale, the system is then expanded to include secondary ratios, which
result in a scale consisting of forty-three-tones. An example of one of these secondary ratios is 33/32
(undecimal comma, or 33rd harmonic), which is derived by calculating the 3/2 of 11/8 via the equation:
11/8 plus 3/2=(11Ï3):(8Ï2)=33/16; followed by 33/16 minus 2/1=(33Ï1):(16Ï2)=33/32. The final
set of intervals include four types of tritones: the undecimal superfourth (11/8), septimal subdiminished
fifth (7/5), and their complements; septimal tritone, or superaugmented fourth (10/7), and undecimal
subfifth (16/11); six types of thirds: the septimal subminor third (7/6), Pythagorean minor third (32/27),
just minor third (6/5), undecimal acute or large neutral third (11/9), just major third (5/4), septimal
supermajor third (9/7), and their complements; the septimal subminor sixth (14/9), just minor sixth (8/5),
undecimal grave or small neutral sixth (18/11), just major sixth (5/3), Pythagorean major sixth (27/16), and
septimal supermajor sixth (12/7); seven types of seconds: the septimal diatonic semitone (21/20), just
diatonic semitone (16/15), undecimal grave or small neutral second (12/11), undecimal acute or large
neutral second (11/10), just minor tone (10/9), just major tone (9/8), septimal supermajor second (8/7),
and their complements; the septimal subminor seventh (7/4), Pythagorean minor seventh (16/9), acute or
large minor seventh (9/5), undecimal grave or small neutral seventh (20/11), undecimal acute or large
neutral seventh (11/6), just diatonic major seventh (15/8), and septimal supermajor seventh (40/21).303
302 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 123-35.
303 Bob Gilmore, “On Harry Partch’s Seventeen Lyrics by Li Po,” Perspectives of New Music 30.2 (Summer, 1992):
26-27.
The Harmonic Consideration 213
Tabl
e 12
7. H
arry
Par
tch’
s fo
rty-th
ree-
tone
just
into
natio
n sc
ale
Rela
tive
Pitc
h: A
4=44
0Hz
/ C4 (
mid
dle
C)=
261.
6255
654H
z
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
C C (
C,
D ;
D")
D-
D E)
D )
D D *)
E ;
E ")
E "
E E)
E F K(
E *
F +(
F F (
F ,
uniso
n
synt
onic
com
ma
und
ecim
al c
omm
a (3
3rd
har
mon
ic)
sep
timal
chr
omat
ic s
emito
ne
just
dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
und
ecim
al g
rave
or s
mal
l neu
tral s
econ
d
und
ecim
al a
cute
or l
arge
neu
tral s
econ
d
just
min
or to
ne
just
maj
or to
ne (
9th
harm
onic
)
sep
timal
sup
erm
ajor
sec
ond
sep
timal
sub
min
or th
ird
Pyth
agor
ean
min
or th
ird, o
r trih
emito
ne
just
min
or th
ird
und
ecim
al a
cute
or l
arge
neu
tral t
hird
just
maj
or th
ird (
5th
harm
onic
)
und
ecim
al d
imin
ished
four
th
sep
timal
sup
erm
ajor
third
sep
timal
sub
four
th (
21st
har
mon
ic)
just
and
Pyt
hago
rean
per
fect
four
th
acut
e or
larg
e fo
urth
und
ecim
al s
uper
four
th (
11th
har
mon
ic)
1/1
81/8
0
33/3
2
21/2
0
16/1
5
12/1
1
11/1
0
10/9
9/8
8/7
7/6
32/2
7
6/5
11/9
5/4
14/1
1
9/7
21/1
6
4/3
27/2
0
11/8
1.00
0000
1.01
2500
1.03
1250
1.05
0000
1.06
6667
1.09
0909
1.10
0000
1.11
1111
1.12
5000
1.14
2857
1.16
6667
1.18
5185
1.20
0000
1.22
2222
1.25
0000
1.27
2727
1.28
5714
1.31
2500
1.33
3333
1.35
0000
1.37
5000
261.
626
264.
896
269.
801
274.
707
279.
067
285.
410
287.
788
290.
695
294.
329
299.
001
305.
230u
310.
075
313.
951
319.
765
327.
032
332.
978
336.
376
343.
384
348.
834
353.
195
359.
735
0.00
0
21.5
06
53.2
73
84.4
67
111.
731
150.
637
165.
004
182.
404
203.
910
231.
174
266.
871
294.
135
315.
641
347.
408
386.
314
417.
508
435.
084
470.
781
498.
045
519.
551
551.
318
+00
+22
ß47
ß16
+12
ß49
ß35
ß18
+04
+31
ß33
ß06
+16
+47
ß14
+18
+35
ß29
ß02
+20
ß49
214 The Harmonic Consideration
DEG
REE
NUM
BER
NO
TE
INTE
RVA
L RA
TIO
(FRA
CTIO
N)
RATI
O
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
TUN
ING
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
Ò×
ÒØ
ÒÙ
ÓÐ
ÓÑ
ÓÒ
ÓÓ
ÓÔ
ÓÕ
ÓÖ
Ó×
ÓØ
ÓÙ
ÔÐ
ÔÑ
ÔÒ
ÔÓ ÐÑ
G;
F :
G-
G)
G
G*)
A;
GJ)
A"
A-(
A
A(
A*
B ;
B ")
B "
B -
B E)
B B *)
C -
C ) C
sep
timal
sub
dim
inish
ed fi
fth
sep
timal
trito
ne, o
r sup
erau
gmen
ted
four
th
und
ecim
al s
ubfif
th
grav
e or
sm
all f
ifth
just
and
Pyt
hago
rean
per
fect
fifth
(3r
d h
arm
onic
)
sep
timal
sup
erfif
th
sep
timal
sub
min
or s
ixth
und
ecim
al a
ugm
ente
d fi
fth
just
min
or s
ixth
und
ecim
al g
rave
or s
mal
l neu
tral s
ixth
just
maj
or s
ixth
Pyth
agor
ean
maj
or s
ixth
(27t
h ha
rmon
ic)
sep
timal
sup
erm
ajor
six
th
sep
timal
sub
min
or s
even
th (
7th
harm
onic
)
Pyth
agor
ean
min
or s
even
th
acut
e or
larg
e m
inor
sev
enth
und
ecim
al g
rave
or s
mal
l neu
tral s
even
th
und
ecim
al a
cute
or l
arge
neu
tral s
even
th
just
dia
toni
c m
ajor
sev
enth
(15
th h
arm
onic
)
sep
timal
sup
erm
ajor
sev
enth
und
ecim
al s
ubd
imin
ished
oct
ave
grav
e or
sm
all o
ctav
e
octa
ve
7/5
10/7
16/1
1
40/2
7
3/2
32/2
1
14/9
11/7
8/5
18/1
1
5/3
27/1
6
12/7
7/4
16/9
9/5
20/1
1
11/6
15/8
40/2
1
64/3
3
160/
81
2/1
1.40
0000
1.42
8571
1.45
4545
1.48
1481
1.50
0000
1.52
3810
1.55
5556
1.57
1429
1.60
0000
1.63
6364
1.66
6667
1.68
7500
1.71
4286
1.75
0000
1.77
7778
1.80
0000
1.81
8182
1.83
3333
1.87
5000
1.90
4762
1.93
9394
1.97
5309
2.00
0000
366.
276
373.
751
380.
546
387.
593
392.
438
398.
668
406.
973
411.
126
418.
601
428.
115
436.
043
441.
493
448.
501
457.
845
465.
112
470.
926
475.
683
479.
647
490.
548
498.
334
507.
395
516.
791
523.
251
582.
512
617.
488
648.
682
680.
449
701.
955
729.
219
764.
916
782.
492
813.
686
852.
592
884.
359
905.
865
933.
129
968.
826
996.
090
1017
.596
1034
.996
1049
.363
1088
.269
1115
.533
1146
.727
1178
.494
1200
.000
ß17
+17
+49
ß20
+02
+29
ß35
ß18
+14
ß47
ß16
+06
+33
ß31
ß04
+18
+35
+49
ß12
+16
+47
ß22
+00
The Harmonic Consideration 215
Adriaan Daniël Fokker’s Thirty-One-Tone Equally-Tempered Division of the Octave According to Wolf, Dutch physicist, music theorist, and composer Adriaan Daniël Fokker (1987-1972)304
“appears to be the first theorist to represent a 7-limit tuning system graphically with three implied axes or
dimensions of tonal space, the horizontal axis is assigned to fifths (3/2s), the vertical axis to major thirds
(5/4s), and an oblique axis, implying a third dimension, to tones generated by the 7/4 relationship.”
Fokker, inspired by the musical writings of Dutch mathematician, astronomer, and physicist Christian
Huygens (1629-95), became an ardent advocate of the tonal resources of seven-limit just intonation, and
their “practical realization” within the domain of thirty-one-tone equal temperament.305 Huygens, via the
recognition of the significance of the interval of the great diesis (128/125), or one-fifth of a whole-tone
( 589 ), which is calculated via the amount the octave (2/1) exceeds three consecutive just major thirds
(2/1÷125/64[5/4Ï5/4Ï5/4]=128/125, or 1200ß1158.941=41.059 cents), developed a theoretical
basis for the “octave partitioned into thirty-one steps.” Fokker’s extensive research into thirty-one-tone
equal temperament appears in his 1966 publication of New Music with 31 Notes.306
Notably, the thirty-one-tone equally-tempered division of the octave ( 312 ) produces the just
major third (5/4) and septimal subminor seventh (7/4), with a falsity of +0.783 and ß1.084 cents on
each count, which is a “faithful rendering” of the former, while a “nearly exact reproduction” of the latter.
The untrigesimal, or ‘tricesimoprimal’ equal temperament is therefore capable of approximating the
septimal dominant seventh tetrad (4:5:6:7) with the intervallic measurements of 387.097, 696.774, and
967.742 cents, which are extremely close to the ‘true’ seven-limit equivalents (386.314, 701.955, and
968.826 cents).307 “Thirty-one equal has an honourable history,” notes Rapoport, and “partly because it
has excellent approximations to the intervals represented by harmonics five and seven, and is the closed
regular expansion of quarter-comma meantone temperament.”308
304 “Fokker was born to Dutch parents in Indonesia. He received his doctorate in physics at Leiden in 1913,
pursued advanced work with Rutherford and Einstein (with whom he seems to have collaborated in research and
publication), and became a distinguished theoretical physicist and professor, with important publications on relativity
theory, atomic physics, acoustics, electricity, and magnetism.” For a further discussion, see Douglas Leedy, “Selected
Musical Compositions (1948-1972),” Notes 46.1 (Sep., 1989): 224. 305 Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 8. 306 “New Music with 31 Notes appeared originally in German in 1966. It is a distillation of the ideas which have
punctuated his theoretical writings and make a fine introduction to the works of this seminal figure. The book is in two
parts, the first narrating the history of Fokker’s involvement in 31-tone temperament and the early fruits of this interest. The
longer second part is theoretical and deals with the aspects of temperament which have especially preoccupied Fokker.”
For a further discussion, see Joel Mandelbaum, “Toward the Expansion of Our Concepts of Intonation,” Perspectives of
New Music 13.1 (Autumn-Winter, 1974): 220.
307 A. D. Fokker, “Equal Temperament and the Thirty-One-Keyed Organ,” The Scientific Monthly 81.4 (Oct., 1955):
162-63.
308 Paul Rapoport, “Towards the Infinite Expansion of Tonal Resources,” Tempo 144 (Mar., 1983): 9.
216 The Harmonic Consideration
Tabl
e 12
8. A
dria
an D
anië
l Fok
ker’s
thirt
y-on
e-to
ne e
qua
lly-te
mp
ered
div
ision
of t
he o
ctav
e Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
iddl
e C)
=26
1.62
5565
4Hz
DE
GRE
E N
UMBE
R N
OTE
INTE
RVA
LFA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY(H
ERTZ
) CE
NTS
TUN
ING
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
Ò×
ÒØ
ÒÙ
ÓÐ
ÓÑ ÐÑ
CCà
/ D$
C !
/ Dé
C è
/ D"
C #
/ Dá
D
D à /
E$
D ! /
Eé
D è /
E"
D # /
Eá
E E à
/ F"
E !
/ Fá
F
F à /
G$
F !
/ G
é F è
/ G
" F #
/ G
á G
G
à / A
$
G!
/ Aé
Gè
/ A"
G#
/ Aá
A
A
à / B
$
A!
/ Bé
Aè
/ B"
A#
/ Bá
B
B à /
C"
B ! /
Cá
C
uniso
n31
-et s
uper
octa
ve, o
r dim
inish
ed s
econ
d
31-e
t aug
men
ted
oct
ave,
or s
ubm
inor
sec
ond
31-e
t sup
erau
gmen
ted
oct
ave,
or m
inor
sec
ond
31
-et d
oubl
e au
gmen
ted
oct
ave,
or n
eutra
l sec
ond
31
-et m
ajor
tone
31
-et s
uper
maj
or s
econ
d, o
r dim
inish
ed th
ird
31-e
t aug
men
ted
sec
ond
, or s
ubm
inor
third
31
-et s
uper
augm
ente
d s
econ
d, o
r min
or th
ird
31-e
t dou
ble
augm
ente
d s
econ
d, o
r neu
tral t
hird
31
-et m
ajor
third
31
-et s
uper
maj
or th
ird, o
r dim
inish
ed fo
urth
31
-et a
ugm
ente
d th
ird, o
r sub
four
th
31-e
t per
fect
four
th
31-e
t sup
erfo
urth
, or d
imin
ished
fifth
31
-et a
ugm
ente
d fo
urth
, or s
ubd
imin
ished
fifth
31
-et s
uper
augm
ente
d fo
urth
, or d
imin
ished
fifth
31
-et d
oubl
e au
gmen
ted
four
th, o
r sub
fifth
31
-et p
erfe
ct fi
fth
31-e
t sup
erfif
th, o
r dim
inish
ed s
ixth
31
-et a
ugm
ente
d fi
fth, o
r sub
min
or s
ixth
31
-et s
uper
augm
ente
d fi
fth, o
r min
or s
ixth
31
-et d
oubl
e au
gmen
ted
fifth
, or n
eutra
l six
th
31-e
t maj
or s
ixth
31
-et s
uper
maj
or s
ixth
, or d
imin
ished
sev
enth
31
-et a
ugm
ente
d s
ixth
, or s
ubm
inor
sev
enth
31
-et s
uper
augm
ente
d s
ixth
, or m
inor
sev
enth
31
-et d
oubl
e au
gmen
ted
six
th, o
r neu
tral s
even
th
31-e
t maj
or s
even
th
31-e
t sup
erm
ajor
sev
enth
, or d
imin
ished
oct
ave
31-e
t aug
men
ted
sev
enth
, sub
octa
ve
oc
tave
1/1
312
231
)2
(
331
)2
(
431
)2
(
531
)2
(
631
)2
(
731
)2
(
831
)2
(
931
)2
(
1031
)2
(
1131
)2
(
1231
)2
(
1331
)2
(
1431
)2
(
1531
)2
(
1631
)2
(
1731
)2
(
1831
)2
(
1931
)2
(
2031
)2
(
2131
)2
(
2231
)2
(
2331
)2
(
2431
)2
(
2531
)2
(
2631
)2
(
2731
)2
(
2831
)2
(
2931
)2
(
3031
)2
(
2/
1
1.00
0000
1.02
2611
1.
0457
34
1.06
9380
1.
0935
60
1.11
8287
1.
1435
73
1.16
9431
1.
1958
73
1.22
2914
1.
2505
66
1.27
8843
1.
3077
59
1.33
7329
1.
3675
68
1.39
8491
1.
4301
13
1.46
2450
1.
4955
18
1.52
9334
1.
5639
14
1.59
9276
1.
6354
38
1.67
2418
1.
7102
34
1.74
8905
1.
7884
50
1.82
8889
1.
8702
43
1.91
2532
1.
9557
77
2.
0000
00
261.
626
267.
541
273.
591
279.
777
286.
103
292.
572
299.
188
305.
953
312.
871
319.
945
327.
180
334.
578
342.
143
349.
880
357.
791
365.
881
374.
154
382.
614
391.
266
400.
113
409.
160
418.
412
427.
872
437.
547
447.
441
457.
558
467.
904
478.
484
489.
303
500.
367
511.
681
52
3.25
1
0.00
038
.710
77
.419
11
6.12
9 15
4.83
9 19
3.54
8 23
2.25
8 27
0.96
8 30
9.67
7 34
8.38
7 38
7.09
7 42
5.80
6 46
4.51
6 50
3.22
6 54
1.93
5 58
0.64
5 61
9.35
5 65
8.06
5 69
6.77
4 73
5.48
4 77
4.19
4 81
2.90
3 85
1.61
3 89
0.32
3 92
9.03
2 96
7.74
2 10
06.4
52
1045
.161
10
83.8
71
1122
.581
11
61.2
90
12
00.0
00
+00
+
39
ß23
+
16
ß45
ß
06
+32
ß
29
+10
ß
52
ß13
+
26
ß35
+
03
+42
ß
19
+19
ß
42
ß03
+
35
ß26
+
13
ß48
ß
10
+29
ß
32
+06
ß
55
ß16
+
23
ß39
+00
The Harmonic Consideration 217
The notation of fifth tones, or the equal untrigesimal equal diesis ( 312 , or 38.710 cents) of thirty-one
equal temperament, was devised by Fokker, together with Dutch composer Henk Badings (1907-87),309
and is represented with the following nine symbols: semi-sharp (à), sharp (!), sesqui-sharp (è), double
sharp (#), semi-flat (á), flat ("), sesqui-flat (é), double flat ($), and natural ('). The selection of symbols
also represents the recommendations made by the International Musicological Society, following their
1967 meeting in Ljubljana, Slovenia, where it was agreed upon that the “best microtonal symbology”
should incorporate the “five standard accidental signs,” along with the semi-sharp and sesqui-sharp
symbols, as well as the semi-flat and sesqui-flat symbols introduced by Italian composer and violinist
Guiseppe Tartini (1692-1770) in 1754.310
La Homa Kanto for Harmonically Tuned Synthesizer Quartet La Homa Kanto, or ‘The Human Song’ in Esperanto,311 is a dedication to the late American composer Lou
Harrison (1917-2003),312 and its pitch material has been derived directly from Harrison’s five-tone scales,
presented in Lou Harrison’s Music Primer: Various Items About Music to 1970. Included are the first five
in the series, with the first (the diatonic or major pentatonic scale) acknowledged by Harrison as the
“prime pentatonic,” and “practically the Human Song.” According to Harrison, “These first five are the
most widespread, the core, the principal modes of Human Music. They also constitute the bone-work,
the firmest compositional basis for seven-tone music.”313
Ex. 94. Major Pentatonic Scale “The Human Song”
309 Richard Orton, “The 31-Note Organ,” The Musical Times 107.1478 (Apr., 1966): 342.
310 Read, 20th-Century Microtonal Notation 19-20. 311 Esperanto is “an artificial language invented as a means of international communication.” The origin of the name
derived from “Dr. Esperanto, a pen name of the inventor.” For a further discussion, see Catherine Soanes, ed., Oxford
Dictionary of Current English, 3rd ed. (Lodon: Oxford U. Press, 2001) 304. 312 “Lou Harrison, who celebrated his eightieth birthday in 1997, has often been cited as one of America’s most
original and influential composers. In addition to his prolific musical output, Harrison is also a skilled painter, calligrapher,
essayist, critic, poet, and instrument-builder. During his long and varied career, he has explored dance, Asian music,
tuning systems, and universal languages, and has actively championed political causes ranging from pacifism to gay rights.”
For a further discussion, see Miller, and Lieberman, Lou Harrison: Composing a World n.pag.
313 Lou Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 (New York: C. F. Peters, 1971) 27-
29.
218 The Harmonic Consideration
In his music primer, Harrison also states that the “fullest musical bounty of any tradition is the mode,” and
expresses the following sentiments about five-tone modes:
“Time’s endless flowering of tone and tune here streams across to each; embodied matrix song; hearts gift
of ancestry and dreams. Behind the singer’s song stands quietly (or dancing) the strange remembered
body of the mode. Or is it fashioned piecewise (that body) from the rhymes and purlings of the tune?
Sure, scales are not. These sit, emotive, on systematic right. Perhaps modes meld from melodies
themselves.”314
Modes one, two, and five may be considered to be anhemitonic pentatonic forms (scales incorporating
half steps, or semitones), while three and four, hemitonic pentatonic forms (scales not incorporating half
steps, or semitones).315 Harrison’s five pentatonic scales include the diatonic or major pentatonic, minor
pentatonic (or fifth mode major pentatonic), Japanese hirajoshi, fifth mode Indonesian pélog, and
Indonesian sléndro (or second mode major pentatonic). The consideration of their complements
presents the third mode major pentatonic, fourth mode major pentatonic, third mode Indonesian pélog,
fourth mode Japanese hirajoshi, and Indonesian sléndro (or second mode major pentatonic).316
Ex. 95. Lou Harrison’s five pentatonic scales i. Lou Harrison pentatonic #2
ii. Lou Harrison pentatonic #2
314 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 27.
315 J. H. Kwabena Nketia, The Music of Africa (New York: W. W. Norton & Co., 1974) 118. 316 Persichetti, Twentieth-Century Harmony: Creative Aspects and Practice 50-51.
The Harmonic Consideration 219
ii. Lou Harrison pentatonic #3
iv. Lou Harrison pentatonic #4
v. Lou Harrison pentatonic #5
220 The Harmonic Consideration
The Harpsichord Sample The ‘1967 William Dowd French Double Harpsichord’ samples (recorded in a recital room space by
sound designer and programmer Peter Grech at the School of Music, Victorian College of the Arts,
Melbourne, Australia, utilizing an Audio-Technica AT4050/CM5 condenser microphone [with switchable
cardioid, omnidirectional, or figure-8 operation, and a frequency response from 20Hz-20kHz] and an
Akai S3000XL Midi Stereo Digital Sampler) include two sets of twenty (five-octave span) samples
(encompassing the upper and lower manuals of the harpsichord), as well as one corresponding set of
keyboard release clicks. All these sounds multi-sampled on an Akai S3000XL Midi Stereo Digital Sampler
– tuned firstly to standard A=440Hz twelve-tone equal temperament, and then modified within nineteen
patches collectively adhering to the tuning matrixes required to represent the ten unique systems of just
intonation utilized in the composition. In view of the superior aesthetic qualities of the upper manual,
and for the sake of harmonic clarity, the lower manual and keyboard release clicks are omitted from the
final 63-key ‘1967 William Dowd French Double Harpsichord’ sample. The technical requirements for the
realization of the live performance of the work is an Akai S3000XL Midi Stereo Digital Sampler, together
with four 61-key Midi Keyboards.
Fig. 7. Alex Pertout, 1967 William Dowd French Double Harpsichord.317
317 1967 William Dowd French Double Harpsichord, personal photograph of Alex Pertout, 3 Feb. 2007.
The Harmonic Consideration 221
Ben Johnston’s System of Notation
The system of notation utilized in the score is based on Johnston’s ‘extended just intonation’. It contains
twenty-three unique symbols, which may be combined in any manner – five identical to conventional
sharp (Ú), flat (Û), double sharp (#), double flat ($), and natural (Ö) accidentals (raising, lowering or
neutralizing a tone by 25/24 and 625/576, or 70.672 and 141.345 cents). Additional symbols include
(() and ()), raising or lowering a tone by 81/80 (one syntonic comma), or 21.506 cents, as well as the
following utonal and otonal sets of symbols for each partial chroma up to the thirty-first harmonic: (+) and
(*), lowering or raising a tone by 36/35 (one septimal comma), or 48.770 cents; (,) and (-), raising or
lowering a tone by 33/32 (one undecimal comma), or 53.273 cents; (.) and (/), raising or lowering a
tone by 65/64 (one tridecimal comma), or 26.841 cents; (0) and (1), raising or lowering a tone by
51/50 (one septendecimal comma), or 34.283 cents; (2) and (3), lowering or raising a tone by 96/95
(one nonadecimal comma), or 18.128 cents; (4) and (5), raising or lowering a tone by 46/45 (one
trivigesimal comma), or 38.051 cents; (6) and (7), raising or lowering a tone by 145/144 (one
nonavigesimal comma), or 11.981 cents; and finally, (8) and (9), raising or lowering a tone by 31/30
(one untrigesimal comma), or 56.767 cents.318 Johnston “infinitely expandable” system of notation for
extended just intonation categorizes commas above the seventh harmonic under the nomenclature of
‘chromas’. “These accidentals are used in combination for the more complex ratios,” notes Bob Gilmore,
with reference to Johnston’s not uncommon use of “three such symbols applied to one note.”319
Table 129. Ben Johnston’s notational symbols for just intonation commas and chromas through thirty-one
RAISE LOWER INTERVAL RATIO
(FRACTION)
CENTS AMOUNT BY
WHICH…
EXCEEDS
Ú ( * , . 0 3 4 6 8
Û ) + - / 1 2 5 7 9
chromatic semitone
syntonic comma
septimal comma
undecimal comma
tridecimal comma
septendecimal comma
nonadecimal comma
trivigesimal comma
nonavigesimal comma
untrigesimal comma
25/24
81/80
36/35
33/32
65/64
51/50
96/95
46/45
145/144
31/30
70.672
21.506
48.770
53.273
26.841
34.283
18.128
38.051
11.981
56.767
5/4
9/8
9/5
11/8
13/8
17/16
6/5
23/16
29/16
31/16
6/5
10/9
7/4
4/3
8/5
25/24
19/16
45/32
9/5
15/8
318 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 109-15.
319 Bob Gilmore, “Changing the Metaphor: Ratio Models of Musical Pitch in the Work of Harry Partch, Ben Johnston,
and James Tenney,” Perspectives of New Music 33.1/2 (Winter-Summer, 1995): 480.
222 The Harmonic Consideration
Compositional Strategy
Harrison offers the following commentary about the microtonal compositional process, and the utilization
of just intonation principles:
“After only a brief study of intervals it becomes clear that there are two ways of composing with them: 1)
arranging them into a fixed mode, or gamut, and then composing within that structure. This is Strict Style,
and is the vastly predominant world method. However, another way is possible – 2) to freely assemble, or
compose with whatever intervals one feels that he needs as he goes along. This is Free Style, and I used
this method first in my Simfony in Free Style.”320
The compositional strategy for the work involves the creation of two complimentary sets of melodic
material, consisting of what is essentially a four-bar melodic sentence based on Harrison’s first pentatonic
(or major pentatonic), with an additional four-bar variant based on its complement (or third mode major
pentatonic). In some aspects, this second sentence could be considered an almost comparable
inversion, although it does not represent an ‘authentic’ inversion in the serial sense of the word, but rather
an inversion of contour utilizing the inverted pitch material of the original major pentatonic. The third bar
of both versions represents the only literal inversion of intervals. A further observation reveals the
respective major and minor tonalities of the two four-bar melodic sentences.
Ex. 96. Two four-bar melodic sentences i. Four-bar melodic sentence
ii. Complimentary sentence
The work also features ten distinct tuning modulations: three-limit, five-limit, seven-limit, eleven-limit,
thirteen-limit, seventeen-limit, nineteen-limit, twenty-three-limit, twenty-nine-limit, and thirty-one-limit just
intonation systems, based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth,
twenty-third, twenty-ninth, and thirty-first partials of the harmonic series – each system adding its own
microtonal nuances to the recurring melodic material, which is further transformed via the introduction of
alternative scalar material, as well as via harmonic development pertinent to each individual just intonation
320 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 6.
The Harmonic Consideration 223
system. The structural framework of the composition is directly related to the primary motive of the four-
bar melodic sentence. In five-limit terms, the primary motive denotes the following intervals: 5/4, 3/2, and
9/8, and 5/4, and 3/2; and is expressed in the score as section one (rehearsal letters A and B –
incorporating Lou Harrison Pentatonic #1, and both three-limit and five-limit just intonation), which is
designated a 5/4 metrical structure; section 2 (rehearsal letters C and D – incorporating Lou Harrison
Pentatonic #2, and both seven-limit and eleven-limit just intonation), a 3/2 metrical structure; section 3
(rehearsal letters E and F – incorporating Lou Harrison Pentatonic #3, and both thirteen-limit and
seventeen-limit just intonation), a 9/8 metrical structure; section 4 (rehearsal letters G and H –
incorporating Lou Harrison Pentatonic #4, and both nineteen-limit and twenty-three-limit just intonation),
a 5/4 metrical structure; and section 5 (rehearsal letters I and J – incorporating Lou Harrison Pentatonic
#5, and both twenty-nine-limit and thirty-one-limit just intonation), a 12/8 metrical structure (equal to
3/2). A sequential series of metrical structures is therefore utilized in the work to reflect the primary
motive and its frequency ratios of 5/4, 3/2, 9/8, 5/4, and 3/2.
Composing With Melodicles
According to Harrison, composing with melodicles, or neumes “in some form is the oldest known
method of musical composition, probably deriving from Mesopotamia and Egypt. One makes a mosaic,
so to speak.” Acknowledging Henry Cowell as the source for this technique, Harrison proposes that
beginning with a selection of melodicles, the composer may: combine melodicles – essentially motivic
material – to form phrases; diatonically transpose inside, or chromatically transpose outside of a specific
mode; invert, retrograde, or retrograde-invert melodicles; while maintaining pitch integrity, alter rhythmic
design; as well as individually or collectively displace octaves.321 The abovementioned is no doubt
representative of a series of elementary suggestions later subjected to considerable development in the
compositional process. The technique is adopted in La Homa Kanto; theorized utilizing the first bar of
the work’s principal melodic sentence (a two-note motive), and remodelled within the following three
categories of motivic manipulation: melodic transformation of motive, rhythmic transformation of motive,
and harmonic transformation of motive.
Ex. 97. Original, complement, retrograde, complement retrograde of motive
321 Harrison, Lou Harrison’s Music Primer: Various Items About Music to 1970 1.
224 The Harmonic Consideration
The following series of examples represent a selection of the first category of motivic manipulation, or the
melodic transformation of motive.
Ex. 98. Melodic transformation of motive i. Motivic diatonic expansion (start)
ii. Motivic diatonic expansion (middle)
iii. Motivic diatonic expansion (end)
iv. Motivic diatonic expansion (upbeat)
v. Motivic chromatic expansion (start)
vi. Motivic chromatic expansion (middle)
vii. Motivic chromatic expansion (end)
The Harmonic Consideration 225
viii. Motivic chromatic expansion (upbeat)
ix. Motivic exclusion
x. Intervallic inclusion
xi. Motivic diatonic ornamentation
xii. Motivic chromatic ornamentation
xiii. Intervallic augmentation
xiv. Intervallic diminution
xv. Intonational reinterpretation
226 The Harmonic Consideration
xvi. Intervallic saturation
xvii. Intervallic redirection
xviii. Intervallic reordering
xix. Diatonic transposition
xx. Chromatic transposition
xxi. Contour transposition
xxii. Conjunct octave displacement
xxiii. Disjunct octave displacement
The Harmonic Consideration 227
The following series of examples represent a selection of the second category of motivic manipulation, or
the rhythmic transformation of motive.
Ex. 99. Rhythmic transformation of motive i. Pitch recapitulation
ii. Motivic recapitulation
iii. Motivic ostinato
iv. Rhythmic recapitulation
v. Rhythmic augmentation
vi. Rhythmic diminution
vii. Irregular rhythmic augmentation
228 The Harmonic Consideration
viii. Irregular rhythmic diminution
ix. Conjunct rhythmic displacement
x. Disjunct rhythmic displacement
xi. Rhythmic serialism
xii. Polyrhythmic representation
xiii. Time signature change
The following series of examples represent a selection of the third category of motivic manipulation, or
the harmonic transformation of motive.
Ex. 100. Harmonic transformation of motive i. Diatonic harmonic extension
The Harmonic Consideration 229
ii. Chromatic harmonic extension
iii. Two-part canon
iv. Two-part rhythmically displaced canon
v. Two-part inverted canon
vi. Two-part complementary canon
vii. Three-part canon
viii. Double canon
230 The Harmonic Consideration
ix. Augmentation canon
x. Diminution canon
Three-Limit Just Intonation
The primary interval of three-limit just intonation is the third harmonic, or just perfect fifth, which is
represented by the frequency ratio 3/2, and in relation to C (1/1), notated simply as G. The complement,
4/3, or just perfect fourth, is notated simply as F. The notational symbols introduced in three-limit just
intonation ([(] and [)], raising or lowering a tone by 81/80 [one syntonic comma], or 21.506 cents) are
in reference to the amount a series of four fifths minus a major third exceeds the unison
(3/2Ï3/2Ï3/2Ï3/2÷5/4=81/80).322
The essential three-limit pitch material is arrived at via a one-dimensional process, and the
generation of an ascending series of fifths from the fundamental C (1/1) to produce the six ‘otonal’
pitches: G, D, A(, E(, B(, and F!(( (3/2, 9/8, 27/16, 81/64, 243/128, and 729/512); while a descending
series to produce the other five ‘utonal’ pitches: F, B"), E"), A"), D")), and G")) (4/3, 16/9, 32/27,
128/81, 256/243, and 1024/729). The pitch material is then sequentially arranged as:
C D")) D E") E( F F!(( G")) G A") A( B") B
11
243256
89
2732
6481
34
512729
729
102423
81
128
1627
9
16
2140
The series of intervals presents the Pythagorean limma (256/243), just major tone (9/8), Pythagorean minor
third (32/27), Pythagorean major third (81/64), just perfect fourth (4/3), Pythagorean tritone, or
augmented fourth (729/512), Pythagorean diminished fifth (1024/729), just perfect fifth (3/2),
Pythagorean minor sixth (128/81), Pythagorean major sixth (27/16), Pythagorean minor seventh (16/9),
and Pythagorean major seventh (243/128). The following table depicts the essential tonal resources of
three-limit just intonation, indicating degree, notation, interval, ratio, and cents.
322 Helmholtz, On the Sensations of Tone: As a Physiological Basis for the Theory of Music 452-53.
The Harmonic Consideration 231
Table 130. Essential three-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
D"))
D
E")
E(
F
F!((
G
A")
A(
B")
B(
C
unison
Pythagorean limma
just major tone (9th harmonic)
Pythagorean minor third, or trihemitone
Pythagorean major third, or ditone (81st harmonic)
just and Pythagorean perfect fourth
Pythagorean tritone, or augmented fourth
just and Pythagorean perfect fifth (3rd harmonic)
Pythagorean minor sixth
Pythagorean major sixth (27th harmonic)
Pythagorean minor seventh
Pythagorean major seventh
octave
1/1
256/243
9/8
32/27
81/64
4/3
729/512
3/2
128/81
27/16
16/9
243/128
2/1
1.000000
1.053498
1.125000
1.185185
1.265625
1.333333
1.423828
1.500000
1.580247
1.687500
1.777778
1.898438
2.000000
0.000
90.225
203.910
294.135
407.820
498.045
611.730
701.955
792.180
905.865
996.090
1109.775
1200.000
The adaptation of Harrison’s pentatonic #1 to three-limit intonation presents a collection of pitches that
include: C, D, E(, G, and A( (1/1, 9/8, 81/64, 3/2, and 27/16); as well as the complements: C, B"), A"),
F, and E") (2/1, 16/9, 128/81, 4/3, and 32/27). It must be noted that auxiliary pitches will be added to
the original pentatonic forms in order to beget a heptatonic scalar source.
Ex. 101. Lou Harrison pentatonic #1 (three-limit intonation)
232 The Harmonic Consideration
The following tuning matrix contains the data required to represent the system of intonation chromatically
within the program memory of the Akai S3000XL.
Table 131. 3-limit tuning matrix no. 1 ‘key of C’ (Program 01)
NOTE C D")) D E") E( F – – G A") A( B") B(
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß10 +04 ß06 +08 ß02 +00 +02 ß08 +06 ß04 +10
The first four bars of the composition serve to introduce the four-bar melodic sentence in three-limit
intonation, and what follows is essentially a two-part inverted canon highlighting the complement or
inversion of the principal melodic material. The end of the second system presents a D"("5/omit 3)
simultaneous sonority within a 7-1 suspension in the bass (the pitches D")), D")), and G, or the ratio
243:512:729), while the following harmony delineates D"("5)/F (the pitches F, D")) and G, or the ratio
324:512:729). In the latter example, D")) and G represents the interval of a Pythagorean tritone
(729/512), and F and D")), a Pythagorean minor sixth (128/81, or 792.180 cents). The third system now
introduces the four-bar variant based on the complement pentatonic scale. A hint of the C fundamental
highlights two characteristic intervals of three-limit just intonation: B"), or Pythagorean minor seventh
(16/9, or 996.090 cents), and A"), or Pythagorean minor sixth (128/81). Another two-part inverted
canon in the fourth system produces the succession: E") and B"), A") and D, and A") and E(, which
present the just perfect fifth (3/2), Pythagorean diminished fifth (1024/729, or 588.270 cents), and
Pythagorean diminished fourth (8192/6561, or 384.360 cents).
Another interesting simultaneous sonority presented in the fourth system is one proposed by
pitches E") and G, which pronounce the Pythagorean major third, or ditone (81/64, or 407.820 cents).
The nomenclature (ditone) is in direct reference to the resulting interval derived from the summation
process of two just major tones, or 9/8×9/8=81/64. It is important to note that whilst the three-limit
intonation system renders consonant just perfect fourths and fifths, as well as the just major tone (9/8), its
one-dimensional process of juxtaposed ascending fifths (series of 3/2s) and descending fourths (series
of 4/3s) presents thirds and sixths as complex and dissonant intervals. Kyle Gann offers the following
discussion with regards to the 81/64 interval:
“Before the advent of meantone tuning, the French academy at Notre Dame (13th and 14th centuries)
followed a medieval tradition since Boethius (4th century) in decreeing that only a series of perfect fifths
could make up a scale; their ratio was 3/2, and 3, after all, was the perfect number, connoting the Trinity
among other things. Thus the Pythagorean scale is a just intonation scale on a series of perfect fifths, all the
ratio numbers powers of either 3 or 2. This was an appropriate scale for a music in which perfect fifths
The Harmonic Consideration 233
and fourths were the overwhelmingly dominant simultaneous sonority, and in which the pitches CÚ, FÚ, and
GÚ hardly appeared if at all. Though used, the thirds were theoretical dissonances, and therefore avoided
at final cadences.”323
The root position three-limit major triad is represented by the ratio 64:81:96, identities 1-81-3, intervals
1/1, 81/64, and 3/2, and the pitches C, E(, and G; and presents the just major third (5/4) and just perfect
fifth (3/2) with a falsity of +21.506 and +0.000 cents on each count, which is a deviation of a syntonic
comma (81/80) for the major third. G4, or the just perfect fifth (3/2), presents 0.000 beats between the
third harmonic of C4 (1/1) and the second harmonic of G4, and 0.000 beats between the sixth harmonic
of C4 and the fourth harmonic of G4; while E(4 (331.120Hz), or the Pythagorean major third (81/64),
presents 16.352 beats between the fifth harmonic of C4 and the fourth harmonic of E(4 (1324.479Hz).
Table 132. The beating characteristics of the three-limit major triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
E(4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
1308.128
1569.753
– – – –
1831.379
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
331.120
– – – –
– – – –
662.240
– – – –
993.360
– – – –
– – – –
1324.479
– – – –
1655.599
– – – –
1896.719
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.352
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
392.438
– – – –
– – – –
784.877
– – – –
– – – –
1177.315
– – – –
1569.753
– – – –
– – – –
1962.192
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
Doty makes the following observation with regards to the intervallic properties of Pythagorean intonation
and its application during the European Middle Ages:
“Although three-limit scales are poorly suited to chordal harmony, they work well melodically and are
eminently suitable for polyphony based on fourths and fifths, such as the motets and organa of the
323 Kyle Gann, “An Introduction to Historical Tunings,” Kyle Gann’s Home Page, 1997, 15 December 2005,
<http://www.kylegann.com/histune.html>.
234 The Harmonic Consideration
European Ars Antiqua period (c. 1110-1300 C.E.). Pythagorean scales were the sole theoretical basis for
intonation throughout the European Middle Ages. The compositional practices of this period, which treat
thirds and sixths as dissonances, are understandable in light of the properties of Pythagorean tuning.”324
The root position three-limit minor triad is represented by the ratio 54:64:81, identities 27-1-81, intervals
27/16, 1/1, and 81/64, and the pitches A(, C, and E(; and presents the just minor third (6/5) and just
perfect fifth (3/2) with a falsity of ß21.506 and +0.000 cents on each count, which is a deviation of a
syntonic comma (81/80) for the major third. E(5 (662.240Hz), or the Pythagorean major third (81/64),
presents 0.000 beats between the third harmonic of A(4 (1324.479Hz), or the Pythagorean major sixth
(27/16), and the second harmonic of E(5 (1324.479Hz), and 0.000 beats between the sixth harmonic
of A(4 (2648.959Hz) and the fourth harmonic of D5 (2648.959Hz); while C5 (523.251Hz), or the octave
(2/1), 32.703 beats between the sixth harmonic of A(4 and the fifth harmonic of C5 (2616.256Hz).
Table 133. The beating characteristics of the three-limit minor triad
A(4
(PARTIAL)
FREQUENCY
(HERTZ)
C5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
E(5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
441.493
– – – –
– – – –
882.986
– – – –
1324.479
– – – –
1765.973
– – – –
– – – –
2207.466
2648.959
3090.452
– – – –
– – – –
3531.945
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
523.251
– – – –
– – – –
1046.502
– – – –
1569.753
– – – –
– – – –
2093.005
– – – –
2616.256
– – – –
3139.507
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
32.703
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
662.240
– – – –
– – – –
1324.479
– – – –
– – – –
1986.719
– – – –
– – – –
2648.959
– – – –
– – – –
3311.199
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The concluding sonorities of the fourth system include a G("5)/B triad resolving to a C tonal centre within
two successive 2-3 and 7-1 suspensions in the bass. This harmony presents the pitches B(, G, and
324 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 37-38.
The Harmonic Consideration 235
D")), and within the ratio 59049:93312:131072, characterizes the Pythagorean minor sixth (128/81),
and Pythagorean diminished fifth (1024/729).
Five-Limit Just Intonation
The primary interval of five-limit just intonation is the fifth harmonic, or just major third, which is
represented by the frequency ratio 5/4, and in relation to C (1/1), notated simply as E. The complement,
8/5, or just minor sixth, is notated simply as A". The essential five-limit pitch material is arrived at via a now
two-dimensional process, which requires a two-dimensional 2, 3, 5-limit pitch lattice. Pitch lattices allow
for the graphic representation of pitch relationships, with one dimension in this case represented by
horizontal columns defining just major thirds (the sequential addition or subtraction of 5/4s to the right or
left of any particular pitch), while the other dimension; vertical rows defining just perfect fifths (the
sequential addition or subtraction of 3/2s to the top or bottom of any particular pitch). In different terms,
north and south vertical links (0.00° and 180.00°) within the lattice produce 3/2 and 4/3 relationships;
east and west horizontal links (90.00° and 270.00°), 5/4 and 8/5; northeast and southwest diagonal links
(45.00° and 225.00°), 15/8 and 16/15; while northwest and southeast diagonal links (315.00° and
135.00°), 6/5 and 5/3.325
A five-limit scale is constructed via the establishment of just major triads (4:5:6) on the fundamental
C (1/1), G (3/2), and D (9/8), which generate the six otonal pitches: D, E, F, G, A, and B (9/8, 5/4, 4/3,
3/2, 5/3, and 15/8); as well as the establishment of complement minor triads (5:6:10), which generate
the six utonal pitches: B"), A", G, F, E", and D") (16/9, 8/5, 3/2, 4/3, 6/5, and 16/15).
Fig. 8. Five-limit otonal pitch generation
Fig. 9. Five-limit utonal pitch generation
325 Von Gunden, The Music of Ben Johnston 60.
G (3/2)
C (1/1) E (5/4)
D (9/8)
G (3/2) B (15/8) F (4/3) A (5/3)
C (1/1)
C (1/1) A" (8/5)
F (4/3)
F (3/2) D") (16/15)
B") (16/9)
E" (6/5)
C (1/1)
G (3/2)
236 The Harmonic Consideration
F!(, or the just tritone, or augmented fourth (45/32) is adopted to represent the interval of an augmented
fourth, along with the complement, G"), or the diminished fifth (64/45, or 609.776). The pitch material is
then sequentially arranged as:
C D") D E" E F F!( G") G A" A B") B
11
1516
89
56
45
34
3245
4564
23
58
35
9
16
815
The series of intervals presents the just diatonic semitone (16/15), just major tone (9/8), just minor third
(6/5), just major third (5/4), just perfect fourth (4/3), just tritone, or augmented fourth (45/32), diminished
fifth (64/45), just perfect fifth (3/2), just minor sixth (8/5), just major sixth (5/3), Pythagorean minor seventh
(16/9), and just diatonic major seventh (15/8). The prime number five is a prerequisite for consonant
thirds and sixths, and together with their respective complements, 5/4, 6/5, 5/3, and 8/5 represent the
“principle consonances of the five-limit.” The dissonant Pythagorean major third (81/64) is now replaced
by the consonant just major third with a superparticular vibrating ratio of 5/4. “This interval is the serene
consonance we expect a major third to be,” notes Doty.326 The following table depicts the essential
tonal resources of five-limit just intonation, indicating degree, notation, interval, ratio, and cents.
Table 134. Essential five-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ
ÐÑ
C
D")
D E"
E
F
F!(
G
A"
A
B")
B
C
unison
just diatonic semitone, or major half-tone
just major tone (9th harmonic)
just minor third
just major third (5th harmonic)
just and Pythagorean perfect fourth
just tritone, or augmented fourth (45th harmonic)
just and Pythagorean perfect fifth (3rd harmonic)
just minor sixth
just major sixth
Pythagorean minor seventh
just diatonic major seventh (15th harmonic)
octave
1/1
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
16/9
15/8
2/1
1.000000
1.066667
1.125000
1.200000
1.250000
1.333333
1.406250
1.500000
1.600000
1.666667
1.777778
1.875000
2.000000
0.000
111.731
203.910
315.641
386.314
498.045
590.224
701.955
813.686
884.359
996.090
1088.269
1200.000
326 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 38-39.
The Harmonic Consideration 237
Fig. 1
0. 2
, 3, 5
-lim
it p
itch
latti
ce (
nota
tion,
ratio
and
cen
t val
ues)
238 The Harmonic Consideration
Five-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #1 within the
pitches C, D, E, G, and A (1/1, 9/8, 5/4, 3/2, and 5/3); and the complements: C, B"), A", F, and E" (2/1,
16/9, 8/5, 4/3, and 6/5).
Ex. 102. Lou Harrison pentatonic #1 (five-limit intonation)
The following two tuning matrixes contain the data required to represent the system of intonation
chromatically within the program memory of the Akai S3000XL.
Table 135. 5-limit tuning matrix no. 1 ‘key of C’ (Program 02)
NOTE C D") D E" E F – – G A" A B") B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +12 +04 +16 ß14 ß02 +00 +02 +14 ß16 ß04 ß12
Table 136. 5-limit tuning matrix no. 2 ‘key of C’ (Program 03)
NOTE C D") D E" E F – – G A" A B" B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +12 +04 +16 ß14 ß02 +00 +02 +14 ß16 +18 ß12
As well as expressing a diatonic harmonic extension of the four-bar melodic sentence, the first
simultaneous sonority of section B represents the 1, 3, and 5 identities of the harmonic series that form
the consonant or ‘pure’ major triad of just intonation (4:5:6). “The five-limit major triad conforms in all
aspects to our definition of a consonant chord in just intonation,” notes Doty. “It is free from disturbing
interference beats (assuming a harmonic timbre), and is accompanied by first-order difference tones that
The Harmonic Consideration 239
reinforce the identity of the series from which the chord derives.” The first-order difference tones that
Doty refers to include C2 and C3 for the root position five-limit major triad (4:5:6), which in combination
with the fundamental tones essentially form the pitch series C2, C3, C4, E4, and G4, or the first, second, fourth, fifth, and sixth partials of the harmonic series. The inversions of the triad on the other hand include
C2, , C3, and G3 for the first inversion five-limit major triad (5:6:8); and C3 and C4 for the second inversion
five-limit major triad (3:4:5). This chord is obtained via the two-dimensional five-limit process that now
incorporates not only 3/2 vertical derivatives of the C fundamental, but also horizontal 5/4 relationships
that produce the just major third (5/4), as well as the just perfect fifth (3/2) of three-limit just intonation;
not to mention the complements – the 8/5 relationships that produce the just major sixth (5/3), as well as
the just perfect fourth (4/3) of three-limit just intonation. “In root position, the fifth harmonic of the root
(1 identity) coincides with the fourth harmonic of the major third (5 identity), and the sixth harmonic of
the major third coincides with the fifth harmonic of the perfect fifth (3 identity),” explains Doty. The
following example illustrates the consonant nature of the five-limit major triad. Differential tones are
represented in the bass clef by filled noteheads, while periodicity pitch, by triangular noteheads.
Ex. 103. Differential tones and periodicity pitches produced by the five-limit major triad
The second most important chord of five-limit intonation is the five-limit minor triad (10:12:15), which
according to Doty is more consonant than the three-limit Pythagorean minor triad, yet nevertheless less
consonant than the five-limit major triad due to the fact of possessing identities 3, 5, and 15, but no 1
identity. Analysis of this simultaneous sonority further reveals a 6/5 ratio (or just minor third) between the
fifth and the seventh. B4 (490.548Hz), or the just diatonic major seventh (15/8), presents 0.000 beats
between the third harmonic of E4 (981.096Hz), or the just major third (5/4), and the second harmonic of
B4 (981.096Hz), and 0.000 beats between the sixth harmonic of E4 (1962.192Hz) and the fourth
harmonic of B4 (1962.192Hz); while G4, or the just perfect fifth (3/2), then consistently also presents
0.000 beats between the sixth harmonic of E4 and the fifth harmonic of G4 (1962.192Hz).327
327 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 46-47.
240 The Harmonic Consideration
Table 137. The beating characteristics of the just minor triad
E4
(PARTIAL)
FREQUENCY
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
B4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
327.032
– – – –
– – – –
654.064
– – – –
981.096
– – – –
1308.128
– – – –
– – – –
1635.160
1962.192
2289.224
– – – –
– – – –
2616.256
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
392.438
– – – –
– – – –
784.877
– – – –
1177.315
– – – –
– – – –
1569.753
– – – –
1962.192
– – – –
2354.630
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
490.548
– – – –
– – – –
981.096
– – – –
– – – –
1471.644
– – – –
– – – –
1962.192
– – – –
– – – –
2452.740
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The beating characteristics of the just minor triad display equal consonance in comparison to the just
major triad, but upon analysis of the inherent differential tones and periodicity pitch it becomes evident
that in the case of the three inversions (10:12:15, 12:15:20, and 15:20:24), one or more tones alien to
the fundamental triad are introduced. These tones include A" and B", or 8/5 and 9/5, which have no
relevance to the conventional musical application of the minor triad. Although, according to Partch’s
theories, these two triads may be stated as being equally consonant, for they form the theoretical basis of
‘otonalities’ and ‘utonalities’ – the major and minor tonalities of the harmonics and subharmonic series.
Partch makes the following observations:
“In utonality (‘minor’) the conception is somewhat different (to otonality [‘major’]), since the series of
identities descends in pitch from its unity, though the practical results are exactly the same; the unity is here
the ‘fifth of the chord’. The long controversy as to the correct location of the ‘root’ of the ‘minor’ triad is
rhetoric, so far as creative music goes, since the composer needs no greater authority than his fancy to put
the ‘root’ wherever he wants to put it.” 328
328 Partch, Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments 110-12.
The Harmonic Consideration 241
In opposition to the subharmonic argument for the minor triad, Doty states that, “the Partchian view,
however, ignores the implications of difference tones and periodicity pitches. These phenomena will
always reinforce a harmonic interpretation of a chord.”329
Ex. 104. Differential tones and periodicity pitches produced by the five-limit minor triad
Other simultaneous sonorities of interest in five-limit just intonation include the root position major seventh
tetrad (8:10:12:15) with identities 1, 5, 3, and 15; minor seventh tetrad (10:12:15:18) with identities 5, 3,
15, and 9, but no 1 identity; major ninth pentad (8:10:12:15:18) with identities 1, 5, 3, 15, and 9; and
minor ninth pentad (20:24:30:36:45) with identities 5, 3, 15, 9, and 45, and no 1 identity.
Ex. 105. Five-limit chords
The second system features a two-part inverted canon, but with intervallic augmentation and saturation
applied to the complement or inversion of the principal melodic material. The result, is the melodic line’s
transformation from the sequence A" and F, into A", G, and E. Also, the fact that the transformed motive
now occupies the upper position of the harmony (soprano and alto) means that the final sonority in the
system is now D"("5) – the pitches D"), G, and F, or the ratio 32:45:80 (as opposed to the three-limit first
inversion D"("5)/F sonority). In the third system, a minor alteration in the tuning scheme is required, due to
the resulting dissonant Cmin7(omit 5) sonority – the pitches C, E", and B"), or the ratio 135:162:200.
Analysis of the harmonic properties of the simultaneous sonority reveals an appropriate just minor third
(6/5) between the bottom two sonorities, yet a grave or small fifth (40/27, or 680.449 cents) between
the upper two sonorities. In order to adhere to the correct intervallic properties of the five-limit minor
seventh tetrad (10:12:15:18, with identities 5-3-15-9), it is then simply a matter of retuning the minor
seventh from a B"), or Pythagorean minor seventh (16/9), to a B", or acute or large minor seventh (9/5),
329 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 47.
242 The Harmonic Consideration
which essentially forms a 3/2 relationship between the upper two sonorities, to produce a Cmin7(omit 5)
sonority with the ratio 10:12:15. Interesting sonorities presented in the fourth system – which also
features a two-part inverted canon, with the complement or inversion of the principal melodic material
occupying the upper position of the harmony – include the final sonorities at the end of the system (a
double inverted canon) that facilitate a modulation to G major via an E" augmented triad – the pitches E",
G, and B, or the ratio 16:20:25, moving to a second inversion G major triad – the pitches D, G, and B, or
the ratio 3:4:5.
Seven-Limit Just Intonation
The primary interval of seven-limit just intonation is the seventh harmonic, or septimal subminor seventh,
which is represented by the frequency ratio 7/4 (968.826 cents), and notated as B;. The complement,
8/7 (231.174 cents), or septimal supermajor second, is notated as D*). According to Gayle Young, “this
interval is closer to a major second than to a minor third,” yet Partch reports that it was accepted as a
“consonance by Mersenne in the seventh century.”330 The notational symbols for the partial chroma ([+]
and [*], lowering or raising a tone by 36/35 [one septimal comma], or 48.770 cents) are in reference to
“the amount the acute or large minor seventh (9/5) exceeds the seventh harmonic” (7/4Ï36/35=9/5).331
7/4 is 31.174 cents flat from the equal minor seventh ( 56 ]2[ ), while 8/7, 31.174 sharp from the equal
major second ( 6 2 ). Doty states the following about the 7/4 ratio: “7/4 is a powerful consonance,
whereas tempered, three-limit, and five-limit minor sevenths are all quite dissonant. Thus the harmonic
use of 7/4 demands a departure from common practice. 7/4 is a necessary constituent of consonant
dominant seventh chords and diminished triads.”332
Seven-limit intonation now requires a three-dimensional 2, 3, 5, 7-limit pitch lattice, capable of
displaying relationships not just within seven-limit intonation, but also front and back interrelationships
between the four specified limits. According to Heidi Von Gunden:
“Johnston began designing scales using three, four, and even five prime numbers. As the number of
generating ratios increases, certain complications result. One is the inability to show the network of
relationships on a two-dimensional plane, such as a piece of paper. A system using four generating ratios
needs to be represented with a three-dimensional design. Notice that this (2, 3, 5, 7-limit pitch) lattice
maintains the 3/2 ratios on the vertical axes, the 5/4 ratios on the horizontal axes, and the 7/4 ratios are the
third dimensions seen as axes behind (for the seventh above) and in front (for the seventh below) of the
3/2 axes. This lattice produces natural dominant seventh chords if one reads vertically one block north (for
330 Gayle Young, “The Pitch Organization of Harmonium for James Tenney,” Perspectives of New Music 26.2
(Summer, 1988): 205. 331 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 113.
332 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 51.
The Harmonic Consideration 243
the 3/2 ratio), one block horizontally east (for the 5/4 ratio), and one block to the rear of the 1/1 ratio (for
the 7/4 ratio).”333
The process adopted for the creation of seven-limit intonation pitch material involves the establishment of
just major triads (4:5:6) from the seventh harmonics of the fundamental C (1/1), G (3/2), and D (9/8),
which generate the six otonal pitches: D;, E;, F+(, G;, A;, and B; (21/20, 7/6, 21/16, 7/5, 14/9, and
7/4); as well as the establishment of complement minor triads (5:6:10), which generate the six utonal
pitches: B*), A*, G*), F:, E*, and D*) (40/21, 12/7, 32/21, 10/7, 9/7, and 8/7). It must be noted that
F+( (21/16) and G*) (32/21) will purely serve a theoretical purpose, as the work adopts the three-limit
perfect fourth (4/3) and fifth (3/2) in actual scale formation.
Fig. 11. Seven-limit otonal pitch generation
Fig. 12. Seven-limit utonal pitch generation
The pitch material is then sequentially arranged as:
C D; D*) E; E* F F: G; G A; A* B; B*)
11
2021
78
67
79
34
7
10
57
23
9
14
712
47
2140
F+( G*)
1621
2132
The series of intervals presents the septimal diatonic semitone (21/20), septimal supermajor second (8/7),
septimal subminor third (7/6), septimal supermajor third (9/7), just perfect fourth (4/3), septimal tritone, or
superaugmented fourth (10/7), septimal subdiminished fifth (7/5), septimal subminor sixth (14/9),
septimal supermajor sixth (12/7), septimal subminor seventh (7/4), and septimal supermajor seventh
333 Von Gunden, The Music of Ben Johnston 128-29.
A* (12/7)
D*) (8/7) F: (10/7)
D*) (8/7)
G*) (32/21) B*) (40/21) A* (12/7) C: (15/14)
E* (9/7)
B; (7/4) G; (7/5)
E; (7/6)
F+( (21/16) D; (21/20)
B; (7/4)
C; (28/15)
A; (14/9)
E; (7/6)
244 The Harmonic Consideration
(40/21); as well as the non-essential tones septimal subfourth (21/16) and septimal superfifth (32/21).
The following table depicts the essential tonal resources of seven-limit just intonation, indicating degree,
notation, interval, ratio, and cents.
Table 138. Essential seven-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
D;
D*)
E;
E*
F+(
F:
G*)
A;
A*
B;
B*)
C
unison
septimal diatonic semitone
septimal supermajor second
septimal subminor third
septimal supermajor third
septimal subfourth (21st harmonic)
septimal tritone, or superaugmented fourth
septimal superfifth
septimal subminor sixth
septimal supermajor sixth
septimal subminor seventh (7th harmonic)
septimal supermajor seventh
octave
1/1
21/20
8/7
7/6
9/7
21/16
10/7
32/21
14/9
12/7
7/4
40/21
2/1
1.000000
1.050000
1.142857
1.166667
1.285714
1.312500
1.428571
1.523810
1.555556
1.714286
1.750000
1.904762
2.000000
0.000
84.467
231.174
266.871
435.084
470.781
617.488
729.219
764.916
933.129
968.826
1115.533
1200.000
The adaptation of Harrison’s pentatonic #2 to seven-limit intonation presents a collection of pitches that
include: C, E;, F, G, and B; (1/1, 7/6, 4/3, 3/2, and 7/4); as well as the complements: C, A*, G, F, and
D*) (2/1, 12/7, 3/2, 4/3, and 8/7).
Ex. 106. Lou Harrison pentatonic #2 (seven-limit intonation)
The Harmonic Consideration 245
Fig. 1
3. 2
, 3, 5
, 7-li
mit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
246 The Harmonic Consideration
The modulation to G (presented in the tuning matrixes) effectively results in the following transposition of
the pitch series:
G A;( A* B; B* C C: D; D E; E* F+( F:
23
4063
7
12
47
1427
11
1415
2021
89
67
79
1621
7
10
C7( D*)
3263
78
The following two tuning matrixes contain the data required to represent the system of intonation
chromatically within the program memory of the Akai S3000XL.
Table 139. 7-limit tuning matrix no. 1 ‘key of G’ (Program 04)
NOTE C C: D E; E* F+( F: G A;( A* B; B*
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +19 +04 ß33 +35 ß29 +17 +02 ß14 +33 ß31 +37
Table 140. 7-limit tuning matrix no. 2 ‘key of G’ (Program 05)
NOTE C C: D E; E* F+( F: G A;( A* B; B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 +19 +04 ß33 +35 ß29 +17 +02 ß14 +33 ß31 ß12
The first simultaneous sonorities of section C represent the 1, 5, 3, and 7 identities that form the septimal
dominant seventh tetrad (4:5:6:7). “The most important chord in the seven-limit is unquestionably the
dominant seventh chord,” notes Doty. “This is the most consonant possible chord consisting of four
unique identities. The difference tones of the just dominant seventh chord strongly reinforce the 1 and 3
identities of the series from which the chord derives, adding to its stability and clarity.”
The Harmonic Consideration 247
Ex. 107. Differential tones produced by the seven-limit septimal dominant seventh tetrad
The septimal dominant seventh tetrad is represented within the work in the forms of G7(omit 3), or the
ratio 4:6:7; and G7/D, or the ratio 3:4:5:6:7. Norden offers the following discussion with regards to the
disparity between theory and performance practice in the Western classical music tradition, and its
misconception of the seven-limit septimal dominant seventh tetrad:
“We hear just intonation many times in fine performances of choral, chamber, and orchestral music, but we
have not recognized it theoretically. Theory has differed widely from practice. The dominant seventh
chord with the harmonic seventh (ratios 4:5:6:7), frequently sounded in performance uninfluenced by
fixed-pitch instruments, is completely ignored in theory books. This is a beautiful chord, easily tuned. The
chord we theorize about is the diatonic dominant-seventh chord (ratios 36:45:54:64), which in equal
temperament is distorted from (386, 316, and 294 cents) to (400, 300, and 300 cents).”334
Another interesting seven-limit chord is the septimal dominant ninth pentad (with the ratio 4:5:6:7:9, and
identities 1-5-3-7-9), which adds the major ninth (9/4) to the tetrad. Doty makes the following
observations:
“Depending on the register in which the major ninth is sounded, beating may occur between some of the
lower harmonics of a pair of tones in this relation, adding roughness to the interval. When added to the
dominant seventh chord, the major ninth spawns another relatively dissonant interval, the acute minor
seventh, 9/5. Nevertheless, the overall impression created by the just dominant ninth chord, at least in root
position, is one of stability.”
It is important to note that the septimal dominant seventh and ninth chords additional present a series of
subsets, which include the septimal diminished triad (5:6:7, with identities 5-3-7), septimal half-
diminished seventh tetrad (5:6:7:9, with identities 5-3-7-9), “incomplete dominant seventh” triad (4:6:7,
with identities 1-3-7), and “added-second” tetrad (8:9:10:12, with identities 1-9-5-3). The septimal
diminished triad is the second most consonant chord in seven-limit just intonation (preceded by the
334 Norden, “A New Theory of Untempered Music: A Few Important Features with Special Reference to ‘A Capella’
Music,” The Musical Quarterly 232.
248 The Harmonic Consideration
seven-limit septimal major triad), and due to there being no 5 identity, and hence no tritone, the sonority
has little in common with the traditional function of a dominant.
Ex. 108. Differential tones produced by the seven-limit septimal dominant ninth pentad
The first system of section C also features the septimal major and minor triads (14:18:21 and 6:7:9, with
identities 7-9-21 and 3-7-9). According to Doty, the subminor septimal minor triad is “quite distinct from
the five-limit minor triad (10:12:15), and is, in the opinion of some listeners, more consonant.” It is
interesting to note that the differential tones produced by the root position septimal minor triad imply “a
fundamental a 3/2 below the root”, as opposed to the five-limit, “5/4 below the root.”335
Ex. 109. Differential tones produced by the seven-limit septimal major triad
The root position seven-limit major triad is represented by the ratio 14:18:21, identities 7-9-21, intervals
7/4, 9/8, and 21/16, and the pitches B;, D, and F+(; and presents the just major third (5/4) and just
perfect fifth (3/2) with a falsity of +48.770 and +0.000 cents on each count, which is a deviation of a
septimal comma (36/35) for the major third. F+(5 (686.767Hz), or the septimal subfourth (21/16, or
470.781 cents), presents 0.000 beats between the third harmonic of B;4 (1373.534Hz), or the septimal
subminor seventh (7/4), and the second harmonic of F+(5 (1373.534Hz), and 0.000 beats between the
sixth harmonic of B;4 (2747.068Hz) and the fourth harmonic of F+(5 (2747.068Hz); while D5
(588.658Hz), or the just major tone (9/8), 65.406 beats between the fifth harmonic of B;4 (2289.224Hz) and the fourth harmonic of D5 (2354.630Hz).
335 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 56-7.
The Harmonic Consideration 249
Table 141. The beating characteristics of the seven-limit major triad
B;4
(PARTIAL)
FREQUENCY
(HERTZ)
D5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
F+(5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
457.845
– – – –
– – – –
915.689
– – – –
1373.534
– – – –
1831.379
– – – –
2289.224
2747.068
– – – –
3204.913
– – – –
3662.758
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
5
– – – –
– – – –
6
– – – –
– – – –
588.658
– – – –
– – – –
1177.315
– – – –
1765.973
– – – –
– – – –
2354.630
– – – –
2943.288
– – – –
3531.945
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
65.406
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
686.767
– – – –
– – – –
1373.534
– – – –
– – – –
2060.301
– – – –
2747.068
– – – –
– – – –
3433.836
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
Ex. 110. Differential tones produced by the seven-limit septimal minor triad
The root position seven-limit minor triad is represented by the ratio 6:7:9, identities 3-7-9, intervals 3/2,
7/4, and 9/8, and the pitches G, B;, and D; and presents the just minor third (6/5) and just perfect fifth
(3/2) with a falsity of ß48.770 and +0.000 cents on each count, which is a deviation of a septimal
comma (36/35) for the minor third. D5, or the just major tone (9/8), presents 0.000 beats between the
third harmonic of G4 (1177.315Hz), or the just perfect fifth (3/2), and the second harmonic of D5
(1177.315Hz), and 0.000 beats between the sixth harmonic of G4 (2354.630Hz) and the fourth
harmonic of D5; while B;4 (457.845Hz), or the septimal subminor seventh (7/4), 65.406 beats between
the sixth harmonic of G4 and the fifth harmonic of B;4, and 0.000 beats between the seventh harmonic
of G4 (2747.068Hz) and the sixth harmonic of B;4.
250 The Harmonic Consideration
Table 142. The beating characteristics of the seven-limit minor triad
G4
(PARTIAL)
FREQUENCY
(HERTZ)
B;4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
D5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
8
392.438
– – – –
– – – –
784.877
– – – –
1177.315
– – – –
1569.753
– – – –
– – – –
1962.192
2354.630
2747.068
– – – –
3139.507
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
6
– – – –
– – – –
– – – –
457.845
– – – –
– – – –
915.690
– – – –
1373.534
– – – –
– – – –
1831.379
– – – –
2289.224
2747.068
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
65.406
0.000
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
– – – –
– – – –
588.658
– – – –
– – – –
1177.315
– – – –
– – – –
1765.973
– – – –
– – – –
2354.630
– – – –
2943.288
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
The principal melodic material is developed further via utonal and otonal intonational reinterpretations
(from quintal to septimal major and minor chromatic harmonic extensions), as well as via pitch and
rhythmic recapitulation. The beginning of the second system features motivic chromatic extension
applied to both the start and end of the two-part inverted canon.
The sonorities at the end of the second system (bar 40) imply a Gmin9 within the pitches G, F+(,
G, A*, and D (28:49:56:64:84). Although a minor third is omitted in literal terms, it does make a
presence two beats earlier in the same bar. In the third system, presented are Gmin7(omit 5) within the
pitches G, B;, and F+( (12:14:21); and Edim/G within the pitches G, B;, and E* (192:224:329); with the
entries of the complimentary sentence transformed via intervallic diminution and motivic diatonic
ornamentation on each count. In the fourth and final system, compositional techniques introduced
include intervallic inclusion and rhythmic diminution. The concluding sonorities of the fourth system (bar
48) include a Dmin7(add 11)/F chord resolving to a G tonal centre – featuring a G5 chord (2:3:4:8) –
within two successive 2-3 and 7-1 suspensions, and the pitches F+(, G, D, A*, and C
(147:168:252:384:448).
The Harmonic Consideration 251
Eleven-Limit Just Intonation The primary interval of eleven-limit just intonation is the eleventh harmonic, or undecimal superfourth,
which is represented by the frequency ratio 11/8 (551.318 cents), and notated as F,. The complement,
16/11 (648.682 cents), or undecimal subfifth, is notated as G-. “The frequency of a sound lying exactly
midway between the equal tempered F that lies just below and the equal tempered FÚ that lies just
above our 11th partial can be found by multiplying (523.251) by the 11th power of the 24th root of 2
(523.251Ï 1124 ]2[ =718.923Hz),” notes Wilford W. Berard.336 The accuracy of this numerical
approximation may be stated as being within two cents of the ‘true’ frequency of 11/8. The notational
symbols for the partial chroma ([,] and [-], raising or lowering a tone by 33/32 [one undecimal comma],
or 53.273 cents) are in reference to the “amount the eleventh harmonic exceeds the just perfect fourth”
(4/3Ï33/32=11/8). 11/8 is 51.318 cents sharp from the equal perfect fifth ( 712 ]2[ ), while 16/11,
48.682 flat from the equal perfect fifth ( 712 ]2[ ).337
The process adopted for the creation of eleven-limit intonation pitch material involves the
establishment of just minor triads (5:6:10) from the eleventh harmonics of the fundamental C (1/1), G
(3/2), and F (4/3), which generate the six otonal pitches: DE), EE), F,, GE), AE, and BE) (11/10, 11/9,
11/8, 22/15, 33/20, and 11/6); as well as the establishment of complement major triads (4:5:6), which
generate the six utonal pitches: B-, A-(, G-, FD(,E-, and D- (20/11, 18/11, 16/11, 15/11, 40/33, and
12/11).
Fig. 14. Eleven-limit otonal pitch generation
Fig. 15. Eleven-limit utonal pitch generation
336 Wilford W. Berard, “The Eleventh and Thirteenth Partials,” Journal of Music Theory 5.1 (Spring, 1961): 96-7. 337 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 114.
D- (12/11)
G- (16/11) B- (20/11)
G- (16/11)
C- (64/33) E- (40/33) D- (12/11) FD( (15/11)
A-( (18/11)
F, (11/8) DE) (11/10)
BE) (11/6)
C, (33/32) AE (33/20)
F, (11/8)
GE) (22/15)
EE) (11/9)
BE) (11/6)
252 The Harmonic Consideration
The pitch material is then sequentially arranged as:
C DE) D- EE) E- F FD( GE) G AE A-( BE) B-
11
1011
1112
911
3340
34
1115
1522
23
2033
1118
611
1120
F, G-
811
1116
The series of intervals presents the undecimal acute or large neutral second (11/10), undecimal grave or
small neutral second (12/11), undecimal acute or large neutral third (11/9), undecimal grave or small
neutral third (40/33), just perfect fourth (4/3), undecimal tritone, or augmented fourth (15/11), undecimal
subdiminished fifth (22/15), just perfect fifth (3/2), undecimal acute or large neutral sixth (33/20),
undecimal grave or small neutral sixth (18/11), undecimal acute or large neutral seventh (11/6), undecimal
grave or small neutral seventh (20/11); as well as the non-essential tones undecimal superfourth (11/8)
and undecimal subfifth (16/11). The following table depicts the essential tonal resources of eleven-limit
just intonation, indicating degree, notation, interval, ratio, and cents.
Table 143. Essential eleven-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
DE)
D-
EE)
E-
F,
FD(
G-
AE
A-(
BE)
B-
C
unison
undecimal acute or large neutral second
undecimal grave or small neutral second
undecimal acute or large neutral third
undecimal grave or small neutral third
undecimal superfourth (11th harmonic)
undecimal tritone, or augmented fourth
undecimal subfifth
undecimal acute or large neutral sixth
undecimal grave or small neutral sixth
undecimal acute or large neutral seventh
undecimal grave or small neutral seventh
octave
1/1
11/10
12/11
11/9
40/33
11/8
15/11
16/11
33/20
18/11
11/6
20/11
2/1
1.000000
1.100000
1.090909
1.222222
1.212121
1.375000
1.363636
1.454545
1.650000
1.636364
1.833333
1.818181
2.000000
0.000
165.004
150.637
347.408
333.041
551.318
536.951
648.682
866.959
852.592
1049.363
1034.996
1200.000
The Harmonic Consideration 253
Fig. 1
6. 2
, 3, 5
, 11-
limit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
254 The Harmonic Consideration
Eleven-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #2 within the
pitches C, EE), F, G, and BE) (1/1, 11/9, 4/3, 3/2, and 11/6); and the complements: C, A-(, G, F, and D-
(2/1, 18/11, 3/2, 4/3, and 12/11).
Ex. 111. Lou Harrison pentatonic #2 (eleven-limit intonation)
The modulation to G (presented in the tuning matrixes) effectively results in the following transposition of
the pitch series:
G AE A-) BE) B- C CD( DE) D EE E-( F, FD(
23
2033
1118
611
1120
11
4445
1011
89
8099
2227
811
1115
C, D-
3233
1112
The following four tuning matrixes contain the data required to represent the system of intonation
chromatically within the program memory of the Akai S3000XL.
Table 144. 11-limit tuning matrix no. 1 ‘key of G’ (Program 06)
NOTE C C, D – – E-( FD( F, G – – A-( BE) – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß47 +04 +00 ß45 +37 ß49 +02 +00 ß47 +49 +00
The Harmonic Consideration 255
Table 145. 11-limit tuning matrix no. 2 ‘key of G’ (Program 07)
NOTE C C, D – – E-( F+( F, G – – A( BE) B
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß47 +04 +00 ß45 ß29 ß49 +02 +00 +06 +49 ß12
Table 146. 11-limit tuning matrix no. 3 ‘key of G’ (Program 08)
NOTE CD( – – D – – EE FD( F, G – – A-( B- – –
KEY C C! D D! E F F! G G! A A! B
CENTS +39 +00 +04 +00 ß31 +37 ß49 +02 +00 ß47 +35 +00
Table 147. 11-limit tuning matrix no. 4 ‘key of G’ (Program 09)
NOTE C C, D – – E-( FD( F, G – – AE BE) – –
KEY C C! D D! E F F! G G! A A! B
CENTS +00 ß47 +04 +00 ß45 +37 ß49 +02 +00 ß33 +49 +00
The first simultaneous sonorities of section D represent the 1, 5, 3, 7, 9, and 11 identities that form the
undecimal dominant eleventh hexad (4:5:6:7:9:11). “In root position and in a reasonably high register,
the addition of the 11 identity to the dominant-ninth chord doesn’t seriously injure its stability or
consonance, although it does add an indescribable, piquant quality to the sonority,” notes Doty.338
Ex. 112. Differential tones produced by the eleven-limit undecimal dominant eleventh hexad
338 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 62.
256 The Harmonic Consideration
The undecimal dominant eleventh hexad is represented within the work in the forms of G7(add 11/omit 3),
or the ratio 4:6:7:11; and G11/D, or the ratio 3:4:5:6:7:8:11. Also introduced in the first system of
section D are undecimal major and minor triads. The root position eleven-limit major triad is represented
by the ratio 66:80:99, identities 33-5-99, intervals 33/32, 5/4, and 99/64, and the pitches C,, E, and G,;
and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity of ß53.273 and +0.000
cents on each count, which is a deviation of an undecimal comma (33/32) for the major third. G,4
(404.702Hz), or the undecimal superfifth (99/64, or 755.228 cents), presents 0.000 beats between the
third harmonic of C,4 (809.404Hz), or the undecimal comma (33/32), and the second harmonic of G,4
(809.404Hz), and 0.000 beats between the sixth harmonic of C,4 (1618.808Hz) and the fourth
harmonic of G,4 (1618.808Hz); while E4, or the just major third (5/4), 40.879 beats between the fifth
harmonic of C,4 (1349.007Hz) and the fourth harmonic of E4 (1308.128Hz).
Table 148. The beating characteristics of the eleven-limit major triad
C,4
(PARTIAL)
FREQUENCY
(HERTZ)
E4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G,4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
269.801
– – – –
– – – –
539.603
– – – –
809.404
– – – –
1079.205
– – – –
1349.007
1618.808
– – – –
1888.610
– – – –
2158.411
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
327.032
– – – –
– – – –
654.064
– – – –
981.096
– – – –
– – – –
1308.128
– – – –
1635.160
– – – –
1962.192
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
40.879
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
404.702
– – – –
– – – –
809.404
– – – –
– – – –
1214.106
– – – –
1618.808
– – – –
– – – –
2023.510
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The root position eleven-limit minor triad is represented by the ratio 18:22:27, identities 9-11-27, intervals
9/8, 11/8, and 27/16, and the pitches D, F,, and A(; and presents the just minor third (6/5) and just
perfect fifth (3/2) with a falsity of +53.273 and +0.000 cents on each count, which is a deviation of an
undecimal comma (33/32) for the minor third. A(4 (441.493Hz), or the Pythagorean major sixth (27/16),
presents 0.000 beats between the third harmonic of D4 (882.986Hz), or the just major tone (9/8), and
The Harmonic Consideration 257
the second harmonic of A(4 (882.986Hz), and 0.000 beats between the sixth harmonic of D4
(1765.973Hz) and the fourth harmonic of A(4 (1765.973Hz); while F,4 (359.735Hz), or the undecimal
superfourth (11/8), 32.703 beats between the sixth harmonic of D4 and the fifth harmonic of F,4
(1798.676Hz).
Table 149. The beating characteristics of the eleven-limit minor triad
D4
(PARTIAL)
FREQUENCY
(HERTZ)
F,4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
A(4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
294.329
– – – –
– – – –
588.658
– – – –
882.986
– – – –
1177.315
– – – –
– – – –
1471.644
1765.973
2060.301
– – – –
– – – –
2354.630
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
359.735
– – – –
– – – –
719.470
– – – –
1079.205
– – – –
– – – –
1438.941
– – – –
1798.676
– – – –
2158.411
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
32.703
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
441.493
– – – –
– – – –
882.986
– – – –
– – – –
1324.479
– – – –
– – – –
1765.973
– – – –
– – – –
2207.466
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The sonorities at the end of the second system (bar 56) imply a Gm9 within the pitches G, D, BE), A-(,
and F, (198:297:484:864:1089); and in the third system, Gmin7(add11/omit 5) within the pitches G, BE),
F,, and C, (36:44:66:99); and Edim/G within the pitches G, BE), and E-( (180:220:297). The
concluding sonorities of the fourth system (bar 65) facilitate a modulation to D major via a B" augmented
triad – the pitches BE), D, and FD(, or the ratio 110:135:162, moving to a second inversion D major triad
– the pitches A-(, D, and FD(, or the ratio 40:55:66.
258 The Harmonic Consideration
Thirteen-Limit Just Intonation
The primary interval of thirteen-limit just intonation is the thirteenth harmonic, or tridecimal grave or small
neutral, or overtone sixth, which is represented by the frequency ratio 13/8 (840.528 cents), and
notated as AO. The complement, 16/13 (359.472 cents), or tridecimal acute or large neutral third, is
notated as E/. The notational symbols for the partial chroma ([.] and [/], raising or lowering a tone by
65/64 [one tridecimal comma], or 26.841 cents) is in reference to the “amount the thirteenth harmonic
exceeds the just minor sixth” (8/5Ï65/43=13/8).339 13/8 is 40.528 cents sharp from the equal minor
sixth ( 23 ]2[ ), while 16/13, 40.528 flat from the equal major third ( 3 2 ).
The process adopted for the creation of thirteen-limit intonation pitch material involves the
establishment of just minor triads (5:6:10) from the thirteen harmonics of the fundamental C (1/1), G
(3/2), and F (4/3), which generate the four otonal pitches: DO), EO, GO), and AO (13/12, 39/32, 13/9,
and 13/8); as well as the establishment of complement major triads (4:5:6), which generate the four
utonal pitches: B/, A/, FN(, and E/ (24/13, 64/39, 18/13, and 16/13). In this particular case, the
exercise omits necessary major second, perfect fourth, perfect fifth, and minor seventh pitch material.
Fig. 17. Thirteen-limit otonal pitch generation
Fig. 18. Thirteen-limit utonal pitch generation
The necessary additional pitch material is obtained via a secondary process that involves the calculation
of 5/4 relationships with DO) (13/12) and GO) (13/9), which generate the otonal pitches F. (65/48)
and BO) (65/36), and utonal complements G/ (96/65) and D/ (72/65). The pitch material is then
sequentially arranged as:
339 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.
B/ (24/13)
E/ (16/13) GN (20/13)
E/ (16/13)
A/ (64/39) CN (40/39) B/ (24/13) DN (15/13)
FN( (18/13)
AO (13/8) FO (13/10)
DO) (13/12)
EO (39/32) CO (39/20)
AO (13/8)
Bg) (26/15)
GO) (13/9)
DO) (13/12)
The Harmonic Consideration 259
C DO) D/ EO E/ F FN( GO) G AO A/ BO) B/
11
1213
6572
3239
1316
34
1318
913
23
8
13
3964
3665
1324
F. G/
4865
6596
The series of intervals presents the tridecimal grave or small neutral second (13/12), tridecimal acute or
large neutral second (72/65), tridecimal grave or small neutral third (39/32), tridecimal acute or large
neutral third (16/13), just perfect fourth (4/3), tridecimal tritone, or augmented fourth (18/13), tridecimal
subdiminished fifth (13/9), just perfect fifth (3/2), tridecimal grave or small neutral, or overtone sixth
(13/8), tridecimal acute or large neutral sixth (64/39), tridecimal grave or small neutral seventh (65/36),
tridecimal acute or large neutral seventh (24/13); as well as the non-essential tones tridecimal superfourth
(65/48) and tridecimal subfifth (96/65). The following table depicts the essential tonal resources of
thirteen-limit just intonation, indicating degree, notation, interval, ratio, and cents.
Table 150. Essential thirteen-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
DO)
D/
EO
E/
F.
FN(
G/
AO
A/
BO)
B/
C
unison
tridecimal grave or small neutral second
tridecimal acute or large neutral second
tridecimal grave or small neutral third (39th harmonic)
tridecimal acute or large neutral third
tridecimal superfourth
tridecimal tritone, or augmented fourth
tridecimal subfifth
tridecimal grave or small neutral, or overtone sixth (13th harmonic)
tridecimal acute or large neutral sixth
tridecimal grave or small neutral seventh
tridecimal acute or large neutral seventh
octave
1/1
13/12
72/65
39/32
16/13
65/48
18/13
96/65
13/8
64/39
65/36
24/13
2/1
1.000000
1.083333
1.107692
1.218750
1.230769
1.354167
1.384615
1.476923
1.625000
1.641026
1.805556
1.846154
2.000000
0.000
138.573
177.069
342.483
359.472
524.886
563.382
675.114
840.528
857.517
1022.931
1061.427
1200.000
260 The Harmonic Consideration
Fig. 1
9. 2
, 3, 5
, 13-
limit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
The Harmonic Consideration 261
The adaptation of Harrison’s pentatonic #3 to thirteen-limit intonation presents a collection of pitches
that include: C, D/, EO, G, and AO (1/1, 72/65, 39/32, 3/2, and 13/8); as well as the complements: C,
BO), A/, F, and E/ (2/1, 65/36, 64/39, 4/3, and 16/13).
Ex. 113. Lou Harrison pentatonic #3 (thirteen-limit intonation)
The modulation to D (presented in the tuning matrixes) effectively results in the following transposition of
the pitch series:
D EO E/( F.( FN( G GN( AO A( BO B/ C. CN(
89
3239
6581
256351
1318
23
5281
8
13
1627
64
117
1324
6465
2627
G. A/(
128195
65
108
The following two tuning matrixes contain the data required to represent the system of intonation
chromatically within the program memory of the Akai S3000XL.
Table 151. 13-limit tuning matrix no. 1 ‘key of D’ (Program 10)
NOTE C. CN( D EO E/( F.( FN( G GN( A( BO B/
KEY C C! D D! E F F! G G! A A! B
CENTS +27 ß35 +04 +42 ß19 +46 ß37 +02 ß33 +06 +44 ß39
262 The Harmonic Consideration
Table 152. 13-limit tuning matrix no. 2 ‘key of D’ (Program 11)
NOTE C. CN( D EO E/( F.( FN( G AO A( BO B/
KEY C C! D D! E F F! G G! A A! B
CENTS +27 ß35 +04 +42 ß19 +46 ß37 +02 +41 +06 +44 ß39
Introduced in the first system of section E (bars 65-66), are tridecimal major and minor triads. The root
position thirteen-limit major triad is represented by the ratio 26:32:39, identities 13-1-39, intervals 13/8,
1/1, and 39/32, and the pitches AO, C, and EO; and presents the just major third (5/4) and just perfect
fifth (3/2) with a falsity of +26.841 and +0.000 cents on each count, which is a deviation of a tridecimal
comma (65/64) for the major third. EO5 (637.712Hz), or the tridecimal grave or small neutral third
(39/32, or 342.483 cents), presents 0.000 beats between the third harmonic of AO4 (1275.425Hz), or
the tridecimal grave or small neutral sixth (13/8), and the second harmonic of EO5 (1275.425Hz), and
0.000 beats between the sixth harmonic of AO4 (2550.849Hz) and the fourth harmonic of EO5
(2550.849Hz); while C5, or the octave (2/1), 32.703 beats between the fifth harmonic of AO4
(2125.708Hz) and the fourth harmonic of C5 (2093.005Hz).
Table 153. The beating characteristics of the thirteen-limit major triad
AO 4
(PARTIAL)
FREQUENCY
(HERTZ)
C5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
EO5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
425.142
– – – –
– – – –
850.283
– – – –
1275.425
– – – –
1700.566
– – – –
2125.708
2550.849
– – – –
2975.991
– – – –
3401.132
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
523.251
– – – –
– – – –
1046.502
– – – –
1569.753
– – – –
– – – –
2093.005
– – – –
2616.256
– – – –
3139.507
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
32.703
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
637.712
– – – –
– – – –
1275.425
– – – –
– – – –
1913.137
– – – –
2550.849
– – – –
– – – –
3188.562
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The Harmonic Consideration 263
The root position thirteen-limit minor triad is represented by the ratio 32:39:48, identities 1-39-3, intervals
1/1, 39/32, and 3/2, and the pitches C, EO, and G; and presents the just minor third (6/5) and just
perfect fifth (3/2) with a falsity of ß26.841 and +0.000 cents on each count, which is a deviation of a
tridecimal comma (65/64) for the minor third. G4, or the just perfect fifth (3/2), presents 0.000 beats
between the third harmonic of C4 (784.877Hz), or the unison (1/1), and the second harmonic of G4,
and 0.000 beats between the sixth harmonic of C4 (1569.753Hz) and the fourth harmonic of G4; while
EO4 (318.856Hz), or the tridecimal grave or small neutral third (39/32), 24.527 beats between the sixth
harmonic of C4 and the fifth harmonic of EO4 (1594.281Hz).
Table 154. The beating characteristics of the thirteen-limit minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
EO4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
– – – –
318.856
– – – –
– – – –
637.712
– – – –
956.568
– – – –
– – – –
1275.425
– – – –
1594.281
– – – –
1913.137
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
24.527
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
392.438
– – – –
– – – –
784.877
– – – –
– – – –
1177.315
– – – –
– – – –
1569.753
– – – –
– – – –
1962.192
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The sonorities at the beginning of the second system (bars 69-70) present intonational reinterpretations
of an implied thirteen-limit tridecimal dominant thirteenth heptad via the simultaneous sonorities
D9(!11)/F!, featuring the pitches D, FN(, A(, C., E/(, and GN(
(30420:37440:45630:54925:67392:84240); and Dmin9(!11)/F, featuring the pitches D, F.(, A(, C.,
E/(, and GN( (56160:68445:84240:101400:124416:155520). In striking contrast, the ‘harmonic’ root
position thirteen-limit tridecimal dominant thirteenth heptad has a ratio of 4:5:6:7:9:11:13, and features
264 The Harmonic Consideration
the just major third (5/4), just perfect fifth (3/2), septimal subminor seventh (7/4), undecimal superfourth
(11/8), and tridecimal grave or small neutral sixth (13/8).
Ex. 114. Differential tones produced by the thirteen-limit tridecimal dominant thirteenth heptad
The simultaneous sonorities in the fourth system (bars 77-78) include a B"7/D tetrad, with the pitches D,
BO, F.(, and AO (144:234:351:416).
Numerous compositional techniques are simultaneously utilized in the development of the four-bar
melodic sentence and complimentary sentence in thirteen-limit just intonation. Notary amongst these
includes the time signature change (from quadruple, or double duple time [4/4] to compound triple time
[9/8]) applied to section E (bar 65), which significantly alters the rhythmic proportions of the motivic
material. Intervallic redirection and utonal and otonal harmonic extension (bars 65 and 66 essentially
outlining a major/minor tonality shift) is also applied to the two-note motive. A motivic ostinato featuring
chromatic expansion and rhythmic recapitulation is additionally introduced, which generates a sense of
moto perpetuo, or perpetual motion.340
340 “Perpetuum mobile [Lat., perpetual motion; It. moto perpetuo]. A composition in which rhythmic motion, often
in a single-note-value at rapid tempo, is continuous from beginning to end. Among composers who have used the term
as title for such a piece are Paganini (op. 11), Weber (Piano Sonata op. 24, last movement), Mendelssohn (op. 119), and
Johann Strauss, Jr. (op. 257). The technique is also encountered in some Chopin etudes.” For a further discussion, see
Randel, ed., The New Harvard Dictionary of Music 628.
The Harmonic Consideration 265
Seventeen-Limit Just Intonation
The primary interval of seventeen-limit just intonation is the seventeenth harmonic, or septendecimal
chromatic semitone, which is represented by the frequency ratio 17/16 (104.955 cents), and notated as
CP. The complement, 32/17 (1095.045 cents), or septendecimal superdiminished octave, is notated as
CQ. The notational symbols for the partial chroma ([0] and [1], raising or lowering a tone by 51/50 [one
septendecimal comma], or 34.283 cents) is in reference to the “amount the seventeenth harmonic
exceeds the grave or small just chromatic semitone” (25/24Ï51/50=17/16). 17/16 is 4.955 cents sharp
from the equal semitone ( 12 2 ), while 32/17, 4.955 flat from the equal diminished octave ( 1112 ]2[ ).341
The process adopted for the creation of seventeen-limit intonation pitch material involves the
establishment of just minor triads (5:6:10) from the seventeenth harmonics of the fundamental C (1/1), G
(3/2), and F (4/3), which generate the five otonal pitches: D0), E0, FP, A0, and B0) (17/15, 51/40,
17/12, 17/10, and 17/9); as well as the establishment of complement major triads (4:5:6), which
generate the five utonal pitches: BQ, AQ, GQ, EQ, and DQ (30/17, 80/51, 24/17, 20/17, and 18/17). In
this particular case, the exercise omits necessary perfect fourth and perfect fifth pitch material.
Fig. 20. Seventeen-limit otonal pitch generation
Fig. 21. Seventeen-limit utonal pitch generation
The necessary additional pitch material is obtained via a secondary process that involves the calculation
of an 8/5 relationship with B0) (17/9), which generates the otonal pitch G0) (68/45), and utonal
complement F1( (45/34). A minor adjustment is also made with regards to E0 (51/40), and due to the
fact this pitch does not represent the simplest ratio available within the seventeen-limit intonation scheme.
E0) (34/27) and AQ( (27/17) are adopted as a consequence. The pitch material is then sequentially
arranged as:
341 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.
GQ (24/17)
CQ (32/17) EQ (20/17)
CQ (32/17)
FQ (64/51) AQ (80/51) GQ (24/17) BQ (30/17)
DQ (18/17)
CP (17/16) A0 (17/10)
FP (17/12)
GP (51/32) E0 (51/40)
CP (17/16)
D0) (17/15)
B0) (17/9)
FP (17/12)
266 The Harmonic Consideration
C DQ D0) EQ E0) F FP GQ G AQ( A0 BQ B0)
11
1718
1517
1720
2734
34
1217
1724
23
1727
1017
1730
9
17
F1( G0)
3445
4568
The series of intervals presents the septendecimal diatonic semitone (18/17), septendecimal supermajor
second (17/15), septendecimal subminor third (20/17), septendecimal supermajor third (34/27), just
perfect fourth (4/3), septendecimal tritone, or superaugmented fourth (17/12), just perfect fifth (3/2),
septendecimal subdiminished fifth (24/17), septendecimal subminor sixth (27/17), septendecimal
supermajor sixth (17/10), septendecimal subminor seventh (30/17), and septendecimal supermajor
seventh (17/9); as well as the non-essential tones septendecimal subfourth (45/34) and septendecimal
superfifth (68/45). The following table depicts the essential tonal resources of seventeen-limit just
intonation, indicating degree, notation, interval, ratio, and cents.
Table 155. Essential seventeen-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
DQ
D0)
EQ
E0)
F1(
FP
G0)
AQ(
A0
BQ
B0)
C
unison
septendecimal diatonic semitone
septendecimal supermajor second
septendecimal subminor third
septendecimal supermajor third
septendecimal subfourth
septendecimal tritone, or superaugmented fourth
septendecimal superfifth
septendecimal subminor sixth
septendecimal supermajor sixth
septendecimal subminor seventh
septendecimal supermajor seventh
octave
1/1
18/17
17/15
20/17
34/27
45/34
17/12
68/45
27/17
17/10
30/17
17/9
2/1
1.000000
1.058824
1.133333
1.176471
1.259259
1.323529
1.416667
1.511111
1.588235
1.700000
1.764706
1.888889
2.000000
0.000
98.955
216.687
281.358
399.090
485.268
603.000
714.732
800.910
918.642
983.313
1101.045
1200.000
Seventeen-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #3 within the
pitches C, D0), EQ, G, and AQ( (1/1, 17/15, 20/17, 3/2, and 27/17); and the complements: C, BQ, A0,
F, and E0) (2/1, 30/17, 17/10, 4/3, and 34/27).
The Harmonic Consideration 267
Fig. 2
2. 2
, 3, 5
, 17-
limit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
268 The Harmonic Consideration
Ex. 115. Lou Harrison pentatonic #3 (seventeen-limit intonation)
The modulation to D (presented in the tuning matrixes) effectively results in the following transposition of
the pitch series:
D EQ( E0 F1( FP G GP AQ( A( BQ( B0 C1( CP
89
6881
4051
3445
1217
23
3251
1727
1627
136243
80153
68
135
1617
G1( A0
272405
1017
The following two tuning matrixes contain the data required to represent the system of intonation
chromatically within the program memory of the Akai S3000XL.
Table 156. 17-limit tuning matrix no. 1 ‘key of D’ (Program 12)
NOTE C1( CP D EQ( E0 F1( FP G GP A( BQ( B0
KEY C C! D D! E F F! G G! A A! B
CENTS ß13 +05 +04 +03 +21 ß15 +03 +02 +07 +06 +05 +23
Table 157. 17-limit tuning matrix no. 2 ‘key of D’ (Program 13)
NOTE C1( DQ D EQ( E0 F1( FP G AQ( A( BQ( B0
KEY C C! D D! E F F! G G! A A! B
CENTS ß13 ß01 +04 +03 +21 ß15 +03 +02 +01 +06 +05 +23
The Harmonic Consideration 269
Introduced in the first system of section F (bars 81-82), are septendecimal major and minor triads. The
root position seventeen-limit major triad is represented by the ratio 54:68:81, identities 27-17-81,
intervals 27/16, 17/16, and 81/64, and the pitches A(, CP, and E(; and presents the just major third
(5/4) and just perfect fifth (3/2) with a falsity of +34.283 and +0.000 cents on each count, which is a
deviation of a septendecimal comma (51/50) for the major third. E(5, or the Pythagorean major third
(81/64), presents 0.000 beats between the third harmonic of A(4, or the Pythagorean major sixth
(27/16), and the second harmonic of E(5, and 0.000 beats between the sixth harmonic of A(4 and the
fourth harmonic of E(5 (2648.959Hz); while CP5 (555.954Hz), or the septendecimal chromatic
semitone (17/16), 16.352 beats between the fifth harmonic of A(4 (2207.466Hz) and the fourth
harmonic of CP5 (2223.817Hz).
Table 158. The beating characteristics of the seventeen-limit major triad
A(4
(PARTIAL)
FREQUENCY
(HERTZ)
CP5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
E(5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
– – – –
6
7
– – – –
8
441.493
– – – –
– – – –
882.986
– – – –
1324.479
– – – –
1765.973
– – – –
2207.466
– – – –
2648.959
3090.452
– – – –
3531.945
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
5
– – – –
– – – –
6
– – – –
– – – –
555.954
– – – –
– – – –
1111.909
– – – –
1667.863
– – – –
– – – –
2223.817
2616.256
– – – –
– – – –
3335.726
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.352
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
– – – –
– – – –
662.240
– – – –
– – – –
1324.479
– – – –
– – – –
1986.719
– – – –
– – – –
2648.959
– – – –
3311.199
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
The root position seventeen-limit minor triad is represented by the ratio 34:40:51, identities 17-5-51,
intervals 17/16, 5/4, and 51/32, and the pitches CP, E, and GP; and presents the just minor third (6/5)
and just perfect fifth (3/2) with a falsity of ß34.283 and +0.000 cents on each count, which is a
deviation of a septendecimal comma (51/50) for the minor third. GP4 (416.966Hz), or the
septendecimal superaugmented fifth (51/32, or 806.910 cents), presents 0.000 beats between the third
harmonic of CP4 (833.931Hz), or the septendecimal chromatic semitone (17/16), and the second
270 The Harmonic Consideration
harmonic of GP4 (833.931Hz), and 0.000 beats between the sixth harmonic of CP4 (1667.863Hz) and
the fourth harmonic of GP4 (1667.863Hz); while E4 (327.032Hz), or the just major third (5/4), 32.703
beats between the sixth harmonic of CP4 and the fifth harmonic of E4 (1635.160Hz).
Table 159. The beating characteristics of the seventeen-limit minor triad
CP4
(PARTIAL)
FREQUENCY
(HERTZ)
E4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
GP4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
277.977
– – – –
– – – –
555.954
– – – –
833.931
– – – –
1111.909
– – – –
– – – –
1389.886
1667.863
1945.840
– – – –
– – – –
2223.817
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
327.032
– – – –
– – – –
654.064
– – – –
981.096
– – – –
– – – –
1308.128
– – – –
1635.160
– – – –
1962.192
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
32.703
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
416.966
– – – –
– – – –
833.931
– – – –
– – – –
1250.897
– – – –
– – – –
1667.863
– – – –
– – – –
2084.829
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The sonorities at the beginning of the second system (bars 85-86) present an intonational reinterpretation
of the simultaneous sonorities presented in the previous E section, which include hexads D9(!11)/F!,
featuring the pitches D, FP, A(, C1(, E0, and GP (4590:5780:6885:8100:10404:13005); and
Dmin9(!11)/F, featuring the pitches D, F1(, A(,C1(, E0, and GP (4590:5400:6885:8100:10404:13005).
The motivic ostinato introduced in section E (thirteen-limit just intonation) featuring chromatic expansion
and rhythmic recapitulation is now further developed in section F (seventeen-limit just intonation), with
chromatic ornamentation, as well as transformation via intonational reinterpretation. The simultaneous
sonorities in the fourth system (bars 93-94) include a B"7/D tetrad with the pitches D, BQ(, F1(, and AQ(
(170:270:405:486). The concluding sonorities of the fourth system (bar 96) facilitate a modulation to A
major (in section G) via an E"(!11) hexad – the pitches EQ(, G, BQ(,DQ, F1( and FD(, or the ratio
54:68:81:96:120:153.
The Harmonic Consideration 271
Nineteen-Limit Just Intonation
The primary interval of nineteen-limit just intonation is the nineteenth harmonic, or nonadecimal subminor,
or overtone minor third, which is represented by the frequency ratio 19/16 (297.513 cents), and
notated as EW. The complement, 32/19 (902.487 cents), or nonadecimal supermajor sixth, is notated as
A3. The notational symbols for the partial chroma ([2] and [3], lowering or raising a tone by 96/95 [one
nonadecimal comma], or 18.128 cents) is in reference to the “amount the just minor third (6/5) exceeds
the nineteenth harmonic” (19/16Ï96/95=6/5).342 19/16 is 2.487 cents flat from the equal minor third
( 4 2 ), while 32/19, 2.487 sharp from the equal major sixth ( 34 ]2[ ).
The process adopted for the creation of nineteen-limit intonation pitch material involves the
establishment of just minor triads (5:6:10) from the nineteenth harmonics of C (1/1), G (3/2), and F (4/3),
which generate the five otonal pitches: DW), EW, GW, AW, and BW (19/18, 19/16, 57/40, 19/12, and
57/32); as well as the establishment of the complement major triads (4:5:6), which generate five utonal
pitches: B3, A3, FV, E3, and D3) (36/19, 32/19, 80/57, 24/19, and 64/57). In this particular case, the
exercise omits necessary perfect fourth and perfect fifth pitch material.
Fig. 23. Nineteen-limit otonal pitch generation
Fig. 24. Nineteen-limit utonal pitch generation
The necessary additional pitch material is obtained via a secondary process that involves the calculation
of a 5/4 relationship with DW) (19/18), which generates the otonal pitch F2 (95/72) and utonal
complement G3 (144/95). A minor adjustment is also made with regards to GW (57/40), and due to the
fact this pitch does not represent the simplest ratio available within the nineteen-limit intonation scheme.
GW) (38/27) and FV( (27/19) are adopted as a consequence. The pitch material is then sequentially
arranged as:
342 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.
E3 (24/19)
A3 (32/19) CV (20/19)
A3 (32/19)
D3) (64/57) FV (80/57) E3 (24/19) FV (30/19)
B3 (36/19)
EW (19/16) CW (19/10)
AW (19/12)
BW (57/32) GW (57/40)
EW (19/16)
FW (19/15)
DW) (19/18)
AW (19/12)
272 The Harmonic Consideration
C DW) D3) EW E3 F FV( GW) G AW A3 BW B3
11
1819
5764
1619
1924
34
1927
2738
23
1219
1932
3257
1936
F2 G3
7295
95
144
The series of intervals presents the nonadecimal diatonic semitone (19/18), nonadecimal supermajor
second (64/57), nonadecimal subminor, or overtone minor third (19/16), nonadecimal supermajor third
(24/19), just perfect fourth (4/3), nonadecimal tritone, or superaugmented fourth (27/19), just major fifth
(3/2), nonadecimal subdiminished fifth (38/27), nonadecimal subminor sixth (19/12), nonadecimal
supermajor sixth (32/19), nonadecimal subminor seventh (57/32), nonadecimal supermajor seventh
(36/19); as well as the non-essential tones nonadecimal subfourth (95/72) and nonadecimal superfifth
(144/95). The following table depicts the essential tonal resources of nineteen-limit just intonation,
indicating degree, notation, interval, ratio, and cents.
Table 160. Essential nineteen-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
DW)
D3)
EW
E3
F2
FV(
G3
AW
A3
BW
B3
C
unison
nonadecimal diatonic semitone
nonadecimal supermajor second
nonadecimal subminor, or overtone minor third (19th harmonic)
nonadecimal supermajor third
nonadecimal subfourth
nonadecimal tritone, or superaugmented fourth
nonadecimal superfifth
nonadecimal subminor sixth
nonadecimal supermajor sixth
nonadecimal subminor seventh (57th harmonic)
nonadecimal supermajor seventh
octave
1/1
19/18
64/57
19/16
24/19
95/72
27/19
144/95
19/12
32/19
57/32
36/19
2/1
1.000000
1.055556
1.122807
1.187500
1.263158
1.319444
1.421053
1.515789
1.583333
1.684211
1.781250
1.894737
2.000000
0.000
93.603
200.532
297.513
404.442
479.917
608.352
720.083
795.558
902.487
999.468
1106.397
1200.000
The Harmonic Consideration 273
Fig. 2
5. 2
, 3, 5
, 19-
limit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
274 The Harmonic Consideration
The adaptation of Harrison’s pentatonic #4 to nineteen-limit intonation presents a collection of pitches
that include: C, E3, F, G, and B3 (1/1, 24/19, 4/3, 3/2, and 36/19); as well as the complements: C, AW,
G, F, and DW) (2/1, 19/12, 3/2, 4/3, and 19/18).
Ex. 116. Lou Harrison pentatonic #4 (nineteen-limit intonation)
The modulation to A( (presented in the tuning matrix) effectively results in the following transposition of
the pitch series:
A( BW B3 C2( CV( D DV( EW E( F2( FV( G2( GV(
1627
3257
1936
512513
7681
89
608729
1619
6481
128171
1927
10241539
152243
D2 E3(
256285
190243
The following tuning matrix contains the data required to represent the system of intonation chromatically
within the program memory of the Akai S3000XL.
Table 161. 19-limit tuning matrix no. 1 ‘key of A’ (Program 14)
NOTE C2( CV( D DV( E( F2( FV( G2( GV( A( BW B3
KEY C C! D D! E F F! G G! A A! B
CENTS +13 +10 +04 +14 +08 +01 +08 +05 +12 +06 ß01 +06
The root position nineteen-limit major triad is represented by the ratio 38:48:57, identities 19-3-57,
intervals 19/16, 3/2, and 57/32, and the pitches EW, G, and BW; and presents the just major third (5/4)
and just perfect fifth (3/2) with a falsity of ß18.128 and +0.000 cents on each count, which is a
The Harmonic Consideration 275
deviation of a nonadecimal comma (96/95) for the major third. BW4 (466.021Hz), or the nonadecimal
subminor seventh (57/32, or 999.468 cents), presents 0.000 beats between the third harmonic of EW4
(932.041Hz), or the nonadecimal subminor third (19/16), and the second harmonic of BW4
(932.041Hz), and 0.000 beats between the sixth harmonic of EW4 (1864.082Hz) and the fourth
harmonic of BW4 (1864.082Hz); while G4, or the just perfect fifth (3/2), 16.352 beats between the fifth
harmonic of EW4 (1553.402Hz) and the fourth harmonic of G4.
Table 162. The beating characteristics of the nineteen-limit major triad
EW4
(PARTIAL)
FREQUENCY
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
BW4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
310.680
– – – –
– – – –
621.361
– – – –
932.041
– – – –
1242.721
– – – –
1553.402
1864.082
– – – –
2174.763
– – – –
2485.443
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
392.438
– – – –
– – – –
784.877
– – – –
1177.315
– – – –
– – – –
1569.753
– – – –
1962.192
– – – –
2354.630
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.352
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
466.021
– – – –
– – – –
932.041
– – – –
– – – –
1398.062
– – – –
1864.082
– – – –
– – – –
2330.103
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The root position nineteen-limit minor triad is represented by the ratio 16:19:24, identities 1-19-3,
intervals 1/1, 19/16, and 3/2, and the pitches C, EW, and G; and presents the just minor third (6/5) and
just perfect fifth (3/2) with a falsity of +18.128 and +0.000 cents on each count, which is a deviation of
a nonadecimal comma (96/95) for the minor third. The nonadecimal subminor third (19/16) has been
proposed by a number of theorists as a possible substitute for the just minor third (6/5) in the five-limit
minor triad (10:12:15), which yields identities 3, 5, and 15, but no 1 identity. The alternative septimal
subminor third (7/6) in the seven-limit minor triad (6:7:9) presents identities 3, 7, and 9, and therefore
also lacks 1. In striking contrast, the harmonic structure of the nineteen-limit minor triad (16:19:24) allows
for the representation of the 1 identity, which coincides with the root the chord.
276 The Harmonic Consideration
Doty offers the following discussion with regards to the musical significance of the nonadecimal subminor
third:
“However, 19/16 is definitely not a consonance, no more so than is the tempered minor third, and
therefore does not yield a consonant triad. In any case, it appears that a group of tones with relative
frequencies higher than 8 or 9 is likely to produce an ambiguous sensation of periodicity pitch, so that
nothing of practical value is achieved making the root of the chord the fundamental of the harmonic
series.”343
G4, or the just perfect fifth (3/2), presents 0.000 beats between the third harmonic of C4, or the unison
(1/1), and the second harmonic of G4, and 0.000 beats between the sixth harmonic of C4 and the fourth
harmonic of G4; while EW4 (310.680Hz), or the nonadecimal subminor third (19/16), 16.352 beats
between the sixth harmonic of C4 and the fifth harmonic of EW4.
Table 163. The beating characteristics of the nineteen-limit minor triad
C4
(PARTIAL)
FREQUENCY
(HERTZ)
EW4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
G4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
261.626
– – – –
– – – –
523.251
– – – –
784.877
– – – –
1046.502
– – – –
– – – –
1308.128
1569.753
1831.379
– – – –
– – – –
2093.005
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
310.680
– – – –
– – – –
621.361
– – – –
932.041
– – – –
– – – –
1242.721
– – – –
1553.402
– – – –
1864.082
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.352
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
392.438
– – – –
– – – –
784.877
– – – –
– – – –
1177.315
– – – –
– – – –
1569.753
– – – –
– – – –
1962.192
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The end of the second system in section G presents a B"("5/omit 3) simultaneous sonority within a 7-1
suspension in the bass – the pitches BW, BW, and E(, or the ratio 19:38:54, with the following harmony
343 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 64-65.
The Harmonic Consideration 277
delineating the triad B"("5)/D – the pitches D, BW, and E(, or the ratio 12:19:27. The concluding
sonorities of the fourth system include a E("5)/G! triad (171:216:304) resolving to an A tonal centre within
a chord inversion and a 7-1 suspension in the bass.
Twenty-Three-Limit Just Intonation
The primary interval of twenty-three-limit just intonation is the twenty-third harmonic, or trivigesimal tritone,
or superaugmented fourth, which is represented by the frequency ratio 23/16 (628.274 cents), and
notated as FX(. The complement, 32/23 (571.726 cents), or trivigesimal subdiminished fifth, is notated
as GY). The notational symbols for the partial chroma ([4] and [5], raising or lowering a tone by 46/45
[one trivigesimal comma], or 38.051 cents) is in reference to the “amount the twenty-third harmonic
exceeds the just tritone” (45/32Ï46/45=23/16). 23/16 is 28.274 cents sharp from the equal
augmented fourth ( 2 2 ), while 32/23, 28.274 flat from the equal diminished fifth ( 2 2 ).344
The process adopted for the creation of twenty-three-limit intonation pitch material involves the
establishment of just minor triads (5:6:10) from the twenty-third harmonics of the fundamental C (1/1), G
(3/2), and F (4/3), which generate the six otonal pitches: D4, E4, FX(, G4, A4(, and B4 (23/20, 23/18,
23/16, 23/15, 69/40, and 23/12); as well as the establishment of complement major triads (4:5:6), which
generate the six utonal pitches: BY), AY, GY), F5, EY), and DY) (40/23, 36/23, 32/23, 30/23, 80/69,
and 24/23).
Fig. 26. Twenty-three-limit otonal pitch generation
Fig. 27. Twenty-three-limit utonal pitch generation
344 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.
DY) (24/23)
GY) (32/23) BY) (40/23)
GY) (32/23)
CY) (128/69) EY) (80/69) DY) (24/23) F5 (30/23)
AY (36/23)
FX( (23/16) D4 (23/20)
B4 (23/12)
CX( (69/64)A4( (69/40)
FX( (23/16)
G4 (23/15)
E4 (23/18)
B4 (23/12)
278 The Harmonic Consideration
A minor adjustment is also made with regards to A4( (69/40), and due to the fact this pitch does not
represent the simplest ratio available within the twenty-three-limit intonation scheme. A4 (46/27) and EY
(27/23) is adopted as a consequence. The pitch material is then sequentially arranged as:
C DY) D4 EY E4 F FX( GY) G AY A4 BY) B4
11
2324
2023
2327
1823
34
1623
2332
23
2336
2746
2340
1223
F5 G4
2330
1523
The series of intervals presents the trivigesimal diatonic semitone (24/23), trivigesimal supermajor second
(23/20), trivigesimal subminor third (27/23), trivigesimal supermajor third (23/18), just perfect fourth (4/3),
trivigesimal tritone, or superaugmented fourth (23/16), just major fifth (3/2), trivigesimal subdiminished fifth
(32/23), trivigesimal subminor sixth (36/23), trivigesimal supermajor sixth (46/27), trivigesimal subminor
seventh (40/23), trivigesimal supermajor seventh (23/12); as well as the non-essential tones trivigesimal
subfourth (30/23) and trivigesimal subminor sixth (36/23). The following table depicts the essential tonal
resources of twenty-three-limit just intonation, indicating degree, notation, interval, ratio, and cents.
Table 164. Essential twenty-three-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÐÑ
C
DY)
D4
EY
E4
F5
FX(
G4
AY
A4
BY)
B4
C
unison
trivigesimal diatonic semitone
trivigesimal supermajor second
trivigesimal subminor third
trivigesimal supermajor third
trivigesimal subfourth
trivigesimal tritone, or superaugmented fourth (23rd harmonic)
trivigesimal superfifth
trivigesimal subminor sixth
trivigesimal supermajor sixth
trivigesimal subminor seventh
trivigesimal supermajor seventh
octave
1/1
24/23
23/20
27/23
23/18
30/23
23/16
23/15
36/23
46/27
40/23
23/12
2/1
1.000000
1.043478
1.150000
1.173913
1.277778
1.304348
1.437500
1.533333
1.565217
1.703704
1.739130
1.916667
2.000000
0.000
73.681
241.961
277.591
424.364
459.994
628.274
740.006
775.636
922.409
958.039
1126.319
1200.000
The Harmonic Consideration 279
Fig. 2
8. 2
, 3, 5
, 23-
limit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
280 The Harmonic Consideration
Twenty-three-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #4 within
the pitches C, E4, F, G, and B4 (1/1, 23/18, 4/3, 3/2, and 23/12); and the complements: C, BQ, G, F, and
DY) (2/1, 30/17, 3/2, 4/3, and 24/23).
Ex. 117. Lou Harrison pentatonic #4 (twenty-three-limit intonation)
The modulation to A( (presented in the tuning matrixes) effectively results in the following transposition
of the pitch series:
A( BY B4( C5( CX( D DX( EY E( F5( FX( G5 GX(
1627
4681
320621
368729
6469
89
512621
2327
6481
184243
1623
92
135
128207
D5 E4(
368405
160207
The following two tuning matrixes contain the data required to represent the system of intonation
chromatically within the program memory of the Akai S3000XL.
Table 165. 23-limit tuning matrix no. 1 ‘key of A’ (Program 15)
NOTE C5( CX( D DX( E( F5( FX( G5 GX( A( BY B4(
KEY C C! D D! E F F! G G! A A! B
CENTS ß17 +30 +04 +34 +08 ß18 +28 ß36 +32 +06 ß20 +48
The Harmonic Consideration 281
Table 166. 23-limit tuning matrix no. 2 ‘key of A’ (Program 16)
NOTE C5( CX( D DX( E( F5( FX( G5( GX( A( BY B4(
KEY C C! D D! E F F! G G! A A! B
CENTS ß17 +30 +04 +34 +08 ß18 +28 ß15 +32 +06 ß20 +48
The first system of section H features the trivigesimal major triad. The root position twenty-three-limit
major triad is represented by the ratio 18:23:27, identities 9-23-27, intervals 9/8, 23/16, and 27/16, and
the pitches D, FX(, and A(; and presents the just major third (5/4) and just perfect fifth (3/2) with a falsity
of +38.051 and +0.000 cents on each count, which is a deviation of a trivigesimal comma (46/45) for
the major third. A(4, or the Pythagorean major sixth (27/16), presents 0.000 beats between the third
harmonic of D4, or the just major tone (9/8), and the second harmonic of A(4, and 0.000 beats between
the sixth harmonic of D4 and the fourth harmonic of A(4; while FX(4 (376.087Hz), or the trivigesimal
tritone, or superaugmented fourth (23/16), 32.703 beats between the fifth harmonic of D4 (1471.644Hz)
and the fourth harmonic of FX(4 (1504.347Hz).
Table 167. The beating characteristics of the twenty-three-limit major triad
D4
(PARTIAL)
FREQUENCY
(HERTZ)
FX(4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
A(4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
294.329
– – – –
– – – –
588.658
– – – –
882.986
– – – –
1177.315
– – – –
1471.644
1765.973
– – – –
2060.301
– – – –
2354.630
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
376.087
– – – –
– – – –
752.174
– – – –
1128.260
– – – –
– – – –
1504.347
– – – –
1880.434
– – – –
2256.521
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
32.703
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
441.493
– – – –
– – – –
882.986
– – – –
– – – –
1324.479
– – – –
1765.973
– – – –
– – – –
2207.466
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
282 The Harmonic Consideration
The root position twenty-three-limit minor triad is represented by the ratio 46:54:69, identities 23-27-69,
intervals 23/16, 27/16, and 69/64, and the pitches FX(, A(, and CX(; and presents the just minor third
(6/5) and just perfect fifth (3/2) with a falsity of ß38.051 and +0.000 cents on each count, which is a
deviation of a trivigesimal comma (46/45) for the minor third. CX(5 (564.130Hz), or the trivigesimal
chromatic semitone (69/64, or 130.229 cents), presents 0.000 beats between the third harmonic of
FX(4 (1128.260Hz), or the trivigesimal superaugmented fourth (23/16), and the second harmonic of
CX(5 (1128.260Hz) and 0.000 beats between the sixth harmonic of FX(4 (2256.521Hz) and the fourth
harmonic of CX(5 (2256.521Hz); while A(4, or the Pythagorean major sixth (27/16), 49.059 beats
between the sixth harmonic of FX(4 and the fifth harmonic of A(4.
Table 168. The beating characteristics of the twenty-three-limit minor triad
FX(4
(PARTIAL)
FREQUENCY
(HERTZ)
A(4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
CX(5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
– – – –
8
376.087
– – – –
– – – –
752.174
– – – –
1128.260
– – – –
1504.347
– – – –
– – – –
1880.434
2256.521
2632.607
– – – –
– – – –
3008.694
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
– – – –
441.493
– – – –
– – – –
882.986
– – – –
1324.479
– – – –
– – – –
1765.973
– – – –
2207.466
– – – –
2648.959
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
49.059
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
564.130
– – – –
– – – –
1128.260
– – – –
– – – –
1692.390
– – – –
– – – –
2256.521
– – – –
– – – –
2820.651
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The final chord in the second system is the triad B"("5), featuring the pitches BY, E(, and D, or the ratio
144:207:368. The third system requires a minor alteration in the tuning scheme, due to the resulting
dissonant Amin7(omit 5) simultaneous sonority – the pitches A(, C5(, and G5, or the ratio 23:27:40.
Analysis of the harmonic properties of the simultaneous sonority reveals the appropriate trivigesimal
subminor third (27/23) between the bottom two sonorities, yet a grave or small fifth (40/27) between
the upper two sonorities. In order to adhere to the more aesthetically pleasing intervallic properties of
the twenty-three-limit minor seventh tetrad it is then simply a matter of retuning G5 (135/92) up a
The Harmonic Consideration 283
syntonic comma (81/80) to G5( (2187/1472), which essentially forms a 3/2 relationship between the
upper two sonorities, and generates a Amin7(omit 5) simultaneous sonority with the ratio 46:54:81.
Interesting sonorities presented in the fourth system include the final sonorities at the end of the system,
which facilitate a modulation to E major via an Emaj7/D! tetrad – the pitches DX(,E(, GX(, and B4(, or
the ratio 345:360:460:552, moving to a second inversion E major triad – the pitches B4(,E(, and GX(,
or the ratio 207:270:345.
Twenty-Nine-Limit Just Intonation
The primary interval of twenty-nine-limit just intonation is the twenty-ninth harmonic, or grave or small
nonavigesimal neutral seventh, which is represented by the frequency ratio 29/16 (1029.577 cents), and
notated as B_. The complement, 32/29 (170.423 cents), or nonavigesimal acute or large neutral second,
is notated as D7). The notational symbols for the partial chroma ([6] and [7], raising or lowering a tone
by 145/144 [one nonavigesimal comma], or 11.981 cents) is in reference to the “amount the twenty-
ninth harmonic exceeds the acute or large minor seventh” (9/5Ï145/144=29/16). 29/16 is 29.577
cents sharp from the equal minor seventh ( 56 ]2[ ), while 32/29, 29.577 flat from the equal major second
( 12 2 ).345
The process adopted for the creation of twenty-nine-limit intonation pitch material involves the
establishment of just minor triads (5:6:10) from the twenty-ninth harmonics of the fundamental C (1/1), G
(3/2), and F (4/3), which generate the six otonal pitches: D_, E_, F6(, G_, A_, and B_ (87/80, 29/24,
87/64, 29/20, 29/18, and 29/16); as well as the establishment of complement major triads (4:5:6), which
generate the six utonal pitches: B7), A7, G7), F^, E7, and D7) (160/87, 48/28, 128/87, 40/29, 36/29,
and 32/29).
Fig. 29. Twenty-nine-limit otonal pitch generation
Fig. 30. Twenty-nine-limit utonal pitch generation
345 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.
A7 (48/29)
D7) (32/29) F^ (40/29)
D7) (32/29)
G7) (128/87) B7) (160/87) A7 (48/29) C^ (30/29)
E7 (36/29)
B_ (29/16) G_ (29/20)
E_ (29/24)
F6( (87/64) D_ (87/80)
B_ (29/16)
C_ (29/15)
A_ (29/18)
E_ (29/24)
284 The Harmonic Consideration
A minor adjustment is also made with regards to D_ (87/80), and due to the fact this pitch does not
represent the simplest ratio available within the twenty-nine-limit intonation scheme. D_) (29/27) and
B7 (54/29) is adopted as a consequence. The pitch material is sequentially arranged as:
C D_) D7) E_ E7 F F^ G_ G A_ A7 B_ B7
11
2729
2932
2429
2936
34
2940
2029
23
1829
2948
1629
2954
F6( G7)
6487
87
128
The series of intervals presents the nonavigesimal grave or small neutral second (29/27), nonavigesimal
acute or large neutral second (32/29), nonavigesimal grave or small neutral third (29/24), nonavigesimal
acute or large neutral third (36/29), just perfect fourth (4/3), nonavigesimal tritone, or augmented fourth
(40/29), just perfect fifth (3/2), nonavigesimal subdiminished fifth (29/20), nonavigesimal grave or small
neutral sixth (29/18), nonavigesimal acute or large neutral sixth (48/29), nonavigesimal grave or small
neutral seventh (29/16), nonavigesimal acute or large neutral seventh (54/29); as well as the non-essential
tones nonavigesimal superfourth (87/64) and nonavigesimal subfifth (128/87). The following table
depicts the essential tonal resources of twenty-nine-limit just intonation, indicating degree, notation,
interval, ratio, and cents.
Table 169. Essential twenty-nine-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ
ÐÑ
C
D_)
D7)
E_
E7
F6(
F^
G7)
A_
A7
B_
B7
C
unison
nonavigesimal grave or small neutral second
nonavigesimal acute or large neutral second
nonavigesimal grave or small neutral third
nonavigesimal acute or large neutral third
nonavigesimal superfourth (87th harmonic)
nonavigesimal tritone, or augmented fourth
nonavigesimal subfifth
nonavigesimal grave or small neutral sixth
nonavigesimal acute or large neutral sixth
nonavigesimal grave or small neutral seventh (29th harmonic)
nonavigesimal acute or large neutral seventh
octave
1/1
29/27
32/29
29/24
36/29
87/64
40/29
128/87
29/18
48/29
29/16
54/29
2/1
1.000000
1.074074
1.103448
1.208333
1.241379
1.359375
1.379310
1.471264
1.611111
1.655172
1.812500
1.862069
2.000000
0.000
123.712
170.423
327.622
374.333
531.532
556.737
668.468
825.667
872.378
1029.577
1076.288
1200.000
The Harmonic Consideration 285
Fig. 3
1. 2
, 3, 5
, 29-
limit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
286 The Harmonic Consideration
The adaptation of Harrison’s pentatonic #5 to twenty-nine-limit intonation presents a collection of
pitches that include: C, D7), F, G, and B_ (1/1, 32/29, 4/3, 3/2, and 29/16); as well as the complements:
C, B_, G, F, and D7) (2/1, 29/16, 3/2, 4/3, and 32/29).
Ex. 118. Lou Harrison pentatonic #5 (twenty-nine-limit intonation)
The modulation to E( (presented in the tuning matrixes) effectively results in the following transposition of
the pitch series:
E( F6( F^( G6( G^( A( A^( B_( B( C6( C^( D6( D^(
6481
6487
5881
512783
464729
1627
232405
12802349
128243
256261
232243
20482349
18562187
A6(( B7
40967047
2954
The following tuning matrix contains the data required to represent the system of intonation chromatically
within the program memory of the Akai S3000XL.
Table 170. 29-limit tuning matrix no. 1 ‘key of E’ (Program 17)
NOTE C6( C^( D6( D^( E( F6( F^( G6( G^( A( A^( B(
KEY C C! D D! E F F! G G! A A! B
CENTS +33 ß20 +37 ß16 +08 +32 ß22 +35 ß18 +06 ß35 +10
The first system of section I (bars 128-29) features the nonavigesimal minor and major triads. The root
position twenty-nine-limit major triad is represented by the ratio 58:72:87, identities 29-9-87, intervals
29/16, 9/8, and 87/64, and the pitches B_, D, and F6(; and presents the just major third (5/4) and just
The Harmonic Consideration 287
perfect fifth (3/2) with a falsity of +11.981 and +0.000 cents on each count, which is a deviation of a
nonavigesimal comma (145/144) for the major third. F6(5 (711.295Hz), or the nonavigesimal
superfourth (87/64, or 531.532 cents), presents 0.000 beats between the third harmonic of B_4
(1422.589Hz), or the nonavigesimal grave or small neutral seventh (29/16), and the second harmonic of
F6(5 (1422.589Hz), and 0.000 beats between the sixth harmonic of B_4 (2845.178Hz) and the fourth
harmonic of F6(5 (2845.178Hz); while D5, or the just major tone (9/8), 16.352 beats between the fifth
harmonic of B_4 (2370.982Hz) and the fourth harmonic of D5.
Table 171. The beating characteristics of the twenty-nine-limit major triad
B_4
(PARTIAL)
FREQUENCY
(HERTZ)
D5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
F6(5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
474.196
– – – –
– – – –
948.393
– – – –
1422.589
– – – –
1896.785
– – – –
2370.982
2845.178
– – – –
3319.374
– – – –
3793.571
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
588.658
– – – –
– – – –
1177.315
– – – –
1765.973
– – – –
– – – –
2354.630
– – – –
2943.288
– – – –
3531.945
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.352
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
711.295
– – – –
– – – –
1422.589
– – – –
– – – –
2133.884
– – – –
2845.178
– – – –
– – – –
3556.473
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The root position twenty-nine-limit minor triad is represented by the ratio 24:29:36, identities 3-29-9,
intervals 3/2, 29/16, and 9/8, and the pitches G, B_, and D; and presents the just minor third (6/5) and
just perfect fifth (3/2) with a falsity of ß11.981 and +0.000 cents on each count, which is a deviation of
a nonavigesimal comma (145/144) for the minor third. D5, or the just major tone (9/8), presents 0.000
beats between the third harmonic of G4, or the just perfect fifth (3/2), and the second harmonic of D5,
and 0.000 beats between the sixth harmonic of G4 and the fourth harmonic of D5; while B_4
(474.196Hz), or the nonavigesimal grave or small neutral seventh (29/16), 16.352 beats between the
sixth harmonic of G4 and the fifth harmonic of B_4.
288 The Harmonic Consideration
Table 172. The beating characteristics of the twenty-nine-limit minor triad
G4
(PARTIAL)
FREQUENCY
(HERTZ)
B_4
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
D5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
7
– – – –
8
392.438
– – – –
– – – –
784.877
– – – –
1177.315
– – – –
1569.753
– – – –
– – – –
1962.192
2354.630
2747.068
– – – –
3139.507
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
474.196
– – – –
– – – –
948.393
– – – –
1422.589
– – – –
– – – –
1896.785
– – – –
2370.982
– – – –
2845.178
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
16.352
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
– – – –
– – – –
588.658
– – – –
– – – –
1177.315
– – – –
– – – –
1765.973
– – – –
– – – –
2354.630
– – – –
2943.288
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
The sonorities at the beginning of the second system (bars 133-34) present hexads Emin9(!11)/G,
featuring the pitches E(, G6(, B(, D6(, F^(, and A^( (1392:1682:2088:2523:3072:3840); and
E9(!11)/G!, featuring the pitches E(, G^(, B(, D6(, F^(, and A^( (464:576:696:841:1024:1280). The
simultaneous sonorities in the fourth and fifth systems (bars 141-42) include an Emin9/D pentad with the
pitches E(, B(, D6(, F^(, and G6( (1392:2088:2523:3072:3364).
Time signature change (from quadruple, or double duple time [4/4] to compound quadruple time
[12/8]) is now applied to section I (bar 129); significantly altering not only the rhythmic proportions of the
motivic material yet again, but also expanding the motivic ostinato introduced in section E (thirteen-limit
just intonation. The technique of otonal and utonal harmonic extension (bars 129 and 130 essentially
now outline a minor/major tonality shift) is additionally applied to the two-note motive. Thirty-One-Limit Just Intonation The primary interval of thirty-one-limit just intonation is the thirty-first harmonic, or untrigesimal supermajor
seventh, which is represented by the frequency ratio 31/16 (1145.036 cents), and notated as B8. The
complement, 32/31 (54.964 cents), or untrigesimal diatonic semitone, or Greek enharmonic quarter-tone
is notated as Da). The notational symbols for the partial chroma ([8] and [9], raising or lowering a tone
The Harmonic Consideration 289
by 31/30 [one untrigesimal comma], or 56.767 cents) is in reference to the “amount the thirty-first
harmonic exceeds the just diatonic major seventh” (15/8Ï31/30=31/16). 31/16 is 45.036 cents sharp
from the equal major seventh ( 1112 ]2[ , or approximately 967/512,346 while 32/31, 45.036 cents flat from
the equal minor second ( 12 2 ).347
The process adopted for the creation of thirty-one-limit intonation pitch material involves the
establishment of just minor triads (5:6:10) from the thirty-first harmonics of the fundamental C (1/1), G
(3/2), and F (4/3), which generate the six otonal pitches: D8, E8, F`(, G8, A8, and B8 (93/80, 31/24,
93/64, 31/20, 31/18, and 31/16); as well as the establishment of complement major triads (4:5:6), which
generate the six utonal pitches: Ba), Aa, Ga), F9, Ea, and Da) (160/93, 48/31, 128/93, 40/31, 36/31,
and 32/31).
Fig. 32. Thirty-one-limit otonal pitch generation
Fig. 33. Thirty-one-limit utonal pitch generation
A minor adjustment is also made with regards to D8 (93/80), and due to the fact this pitch does not
represent the simplest ratio available within the thirty-one-limit intonation scheme. D8) (31/27) and Ba)
(54/31) are adopted as a consequence. The pitch material is then sequentially arranged as:
C Da) D8) Ea E8 F F`( Ga) G Aa A8 Ba B8
11
3132
2731
3136
2431
34
6493
93
128
23
3148
1831
3154
1631
F9 G8
3140
2031
346 Daniélou, Tableau Comparatif des Intervalles Musicaux 29.
347 Fonville, “Ben Johnston’s Extended Just Intonation: A Guide for Interpreters” 115.
Aa (48/31)
Da) (32/31) F9 (40/31)
Da) (32/31)
Ga) (128/93) Ba) (160/93) Aa (48/31) C9 (60/31)
Ea (36/31)
B8 (31/16) G8 (31/20)
E8 (31/24)
F`( (93/64) D8 (93/80)
B8 (31/16)
C8 (31/30)
A8 (31/18)
E8 (31/24)
290 The Harmonic Consideration
The series of intervals presents the nonavigesimal grave or small neutral second (29/27), nonavigesimal
acute or large neutral second (32/29), nonavigesimal grave or small neutral third (29/24), nonavigesimal
acute or large neutral third (36/29), just perfect fourth (4/3), nonavigesimal tritone, or augmented fourth
(40/29), just perfect fifth (3/2), nonavigesimal subdiminished fifth (29/20), nonavigesimal grave or small
neutral sixth (29/18), nonavigesimal acute or large neutral sixth (48/29), nonavigesimal grave or small
neutral seventh (29/16), nonavigesimal acute or large neutral seventh (54/29); as well as the non-essential
tones nonavigesimal superfourth (87/64) and nonavigesimal subfifth (128/87). The following table
depicts the essential tonal resources of thirty-one-limit just intonation, indicating degree, notation, interval,
ratio, and cents.
Table 173. Essential thirty-one-limit pitch material
DEGREE
NUMBER
NOTE INTERVAL RATIO
(FRACTION)
RATIO
(DECIMAL)
CENTS
ÐÑ ÐÒ ÐÓ ÐÔ ÐÕ ÐÖ Ð× ÐØ ÐÙ ÑÐ ÑÑ ÑÒ
ÐÑ
C
Da)
D8)
Ea
E8
F9
F`(
G8
Aa
A8
Ba
B8
C
unison
untrigesimal diatonic semitone, or Greek enharmonic quarter-tone
untrigesimal supermajor second
untrigesimal subminor third
untrigesimal supermajor third
untrigesimal subfourth
untrigesimal tritone, or superaugmented fourth (93rd harmonic)
untrigesimal superfifth
untrigesimal subminor sixth
untrigesimal supermajor sixth
untrigesimal subminor seventh
untrigesimal supermajor seventh (31st harmonic)
octave
1/1
32/31
31/27
36/31
31/24
40/31
93/64
31/20
48/31
31/18
54/31
31/16
2/1
1.000000
1.032258
1.148148
1.161290
1.291667
1.290323
1.453125
1.550000
1.548387
1.722222
1.741935
1.937500
2.00000
0.000
54.964
239.171
258.874
443.081
441.278
646.991
758.722
756.919
941.126
960.829
1145.036
1200.000
The thirty-one limit is the source of one of La Monte Young’s tunings for the ‘dream chord’ – a sonority
utilized in a number of his ‘dream’ compositions, such as The Four Dreams of China, and The Second
Dream of the High Tension Line Step-Down Transformer. The essence of the ‘dream chord’ is
represented by the ratio 6:8:9, and identities 3-1-9, which is then extended via the inclusion of an
additional tone generated by the division of the 9/8 interval. Three varieties of the ‘dream chord’ include
the tetrads 24:32:35:36 (3-1-35-9), 42:56:62:63 (21-7-31-63), and 12:16:17:18 (3-1-17-9). The
second example features the untrigesimal ratio of 31/28 (equal to 176.210 cents).348
348 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 64.
The Harmonic Consideration 291
Fig. 3
4. 2
, 3, 5
, 31-
limit
pitc
h la
ttice
(no
tatio
n, ra
tio a
nd c
ent v
alue
s)
292 The Harmonic Consideration
Thirty-one-limit intonation presents an intonational reinterpretation of Harrison’s pentatonic #4 within the
pitches C, D8), F, G, and Ba (1/1, 31/27, 4/3, 3/2, and 54/31); and the complements: C, Ba, A0, F, and
D8) (2/1, 54/31, 3/2, 4/3, and 31/27).
Ex. 119. Lou Harrison pentatonic #5 (thirty-one-limit intonation)
The modulation to E( (presented in the tuning matrixes) effectively results in the following transposition of
the pitch series:
E( F9( F`( G9( G`( A( A`(( Ba B( C9( C`( D9( D`(
6481
6281
6493
496729
512837
1627
40967533
3154
128243
124243
256279
19842187
20482511
A9( B8(
248405
12802511
The following two tuning matrixes contain the data required to represent the system of intonation
chromatically within the program memory of the Akai S3000XL.
Table 174. 31-limit tuning matrix no. 1 ‘key of E’ (Program 18)
NOTE C9( C`( D9( D`( E( F9( F`( G9( G`( A( – – B(
KEY C C! D D! E F F! G G! A A! B
CENTS ß35 +49 ß31 +53 +08 ß37 +47 ß33 +51 +06 +00 +10
The Harmonic Consideration 293
Table 175. 31-limit tuning matrix no. 2 ‘key of E’ (Program 19)
NOTE C9( C`( D9( D`( E( F9( F`( G9( G`( A( – – A`((
KEY C C! D D! E F F! G G! A A! B
CENTS ß35 +49 ß31 +53 +08 ß37 +47 ß33 +51 +06 +00 ß45
The first system of section J (bars 145-46) features the untrigesimal minor and major triads. The root
position thirty-one-limit major triad is represented by the ratio 24:31:36, identities 3-31-9, intervals 3/2,
31/16, and 9/8, and the pitches G, B8, and D; and presents the just major third (5/4) and just perfect fifth
(3/2) with a falsity of +56.767 and +0.000 cents on each count, which is a deviation of a untrigesimal
comma (31/30) for the major third. D5, or the just major tone (9/8), presents 0.000 beats between the
third harmonic of G4, or the just perfect fifth (3/2), and the second harmonic of D5, and 0.000 beats
between the sixth harmonic of G4 and the fourth harmonic of D5; while B84, or the untrigesimal
supermajor seventh (31/16), 65.406 beats between the fifth harmonic of G4 and the fourth harmonic of
B84 (2027.598Hz).
Table 176. The beating characteristics of the thirty-one-limit major triad
G4
(PARTIAL)
FREQUENCY
(HERTZ)
B84
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
D5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
5
6
– – – –
7
– – – –
8
392.438
– – – –
– – – –
784.877
– – – –
1177.315
– – – –
1569.753
– – – –
1962.192
2354.630
– – – –
2747.068
– – – –
3139.507
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
– – – –
6
– – – –
– – – –
506.900
– – – –
– – – –
1013.799
– – – –
1520.699
– – – –
– – – –
2027.598
– – – –
2534.498
– – – –
3531.945
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
65.406
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
588.658
– – – –
– – – –
1177.315
– – – –
– – – –
1765.973
– – – –
2354.630
– – – –
– – – –
2943.288
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
294 The Harmonic Consideration
The root position thirty-one-limit minor triad is represented by the ratio 62:72:93, identities 31-9-93,
intervals 31/16, 9/8, and 93/64, and the pitches B8, D, and F`(; and presents the just minor third (6/5)
and just perfect fifth (3/2) with a falsity of ß56.767 and +0.000 cents on each count, which is a
deviation of a untrigesimal comma (31/30) for the minor third. F`(5 (760.349Hz), or the untrigesimal
tritone, or superaugmented fourth (93/64), presents 0.000 beats between the third harmonic of B84
(1520.699Hz), or the untrigesimal supermajor seventh (31/16), and the second harmonic of F`(5
(1520.699Hz), and 0.000 beats between the sixth harmonic of B84 (3041.397Hz) and the fourth
harmonic of F`(5 (3041.397Hz); while D5, or the just major tone (9/8), 98.110 beats between the sixth
harmonic of B84 and the fifth harmonic of D5.
Table 177. The beating characteristics of the thirty-one-limit minor triad
B84
(PARTIAL)
FREQUENCY
(HERTZ)
D5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
F`(5
(PARTIAL)
FREQUENCY
(HERTZ)
BEATS
(HERTZ)
1
– – – –
– – – –
2
– – – –
3
– – – –
4
– – – –
– – – –
5
6
– – – –
7
– – – –
8
506.900
– – – –
– – – –
1013.799
– – – –
1520.699
– – – –
2027.598
– – – –
– – – –
2534.498
3041.397
– – – –
3548.297
– – – –
4055.196
– – – –
1
– – – –
– – – –
2
– – – –
3
– – – –
– – – –
4
– – – –
5
6
– – – –
– – – –
– – – –
– – – –
588.658
– – – –
– – – –
1177.315
– – – –
1765.973
– – – –
– – – –
2354.630
– – – –
2943.288
3531.945
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
98.110
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
1
– – – –
– – – –
2
– – – –
– – – –
3
– – – –
– – – –
4
– – – –
– – – –
5
– – – –
– – – –
– – – –
760.349
– – – –
– – – –
1520.699
– – – –
– – – –
2281.048
– – – –
– – – –
3041.397
– – – –
– – – –
3801.746
– – – –
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
– – – –
0.000
– – – –
– – – –
– – – –
– – – –
The sonorities in the second system (bars 149-50) present hexads Emin9(!11)/G, featuring the pitches E(,
G9(, B(, D9(, F`(, and A`(( (26784:31104:40176:46656:61504:77841); and E9(!11)/G!, featuring
the pitches E(, G`(, B(, D9(, F`(, and A`(( (26784:34596:40176:46656:61504:77841). The
simultaneous sonorities in the beginning of the fifth system (bars 157-58) include an Emin9/B hexad with
the pitches B(, D9(, E(, B(, F`(, and G9( (2511:2916:3348:5022:7688:7776). The work is
concluded with an Emin7 tetrad, with the pitches E(, B(, D9(, and G9( (62:93:108:144, with identities
31-93-27-9), which essentially outline an untrigesimal minor seventh simultaneous sonority harmonically
The Harmonic Consideration 295
constructed from the tonic (1/1), untrigesimal subminor third (36/31), just perfect fifth (3/2), and
untrigesimal subminor seventh (54/31).
Johnston’s Dictum
Johnston offers the following summary of some the characteristics of extended just intonation principles,
and ‘the harmonic consideration’:
“Definitely the affect is unique with each tuning. Each overtone is a unique rasa. The third partial, which
generates perfect fifths and fourths, contributes stability and strength. The fifth partial, which is the third and
sixth, contributes warmth of emotion; ordinary human warmth. The seventh partial creates sensuality, for
example in vernacular music like the blues. The eleventh partial introduces ambiguity, because the intervals
of 12/11 and 11/10, which are the overtone intervals surrounding the eleventh partial, are neutral seconds,
squarely in between major and minor. The 11/9 is a neutral third. The 11/8 is in between a perfect fourth
and augmented fourth. The thirteenth partial has a melancholy, dark quality. Nearly every time I’ve used it, it
has something to do with death, which would square with the meaning of thirteen in numerology. The
seventeenth and nineteenth don’t really bring anything new, because seventeen is almost exactly a
tempered half-step and nineteen is close to a tempered minor third.”349
349 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on
Nonstandard Tunings,” Perspectives of New Music 198.
296 The Harmonic Consideration
Conclusion
‘Manual’ of Microtonal Composition The thesis, via its presentation of an articulated exposition of three ‘original’ and unique microtonal
composition models individually exploring the expanded tonal resources of Pythagorean intonation,
equal temperament, and just intonation was not only able to demonstrate the uniqueness, as well as
musical potential of such systems, but also mark some directions for further exploration. The works –
Àzàdeh for santñr and tape, Exposiciones for sampled microtonal Schoenhut toy piano, and La Homa
Kanto for harmonically tuned synthesizer quartet – together with the classification of 724 intervals in the
octave presented at the close (the culmination of research directly related to the compositions), stand as
a testament to the limited potential of the vast resources of the microtonal paradigm; merely three grains
of sand in a gargantuan ocean. The study will nevertheless have the capacity to serve as one possible
‘manual’ of microtonal composition, and hopefully inspire the composer of tomorrow to adopt the
notion, if not some of the principles. A Vast Universe of Subtle Intervallic Relationships In conclusion, it may be stated that the virtues of the microtonal paradigm are best summed up by some
of it proponents. What follows is a collection of observations that characterize the general appeal of
microtonal composition, at the same time disclosing the rationale of some composers for the total
abandonment of twelve-tone equally-tempered composition. In a Perspectives of New Music article
entitled Six American Composers on Nonstandard Tunings, Douglas Keislar asks a selection of American
composers actively involved in the application of “microtonal scales or other non-standard tunings” to
describe what they find most interesting about composing with nonstandard tunings. The replies of
Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and William Schottstaedt
follow:
Blackwood: “The aspect that intrigues me most is finding conventional harmonic progressions, or at least
coherent progressions found by extension of their analogues in the more familiar tunings.”
Eaton: “Microtones permit a greater variety of harmonic and melodic motion, which in opera helps
delineate and define character. My interest in microtones came from three directions. First, I wrote some
of the very first pieces that involved woodwind multiphonics in the early 1960s, and I was intrigued by the
‘out-of-tuneness’ of the multiphonics Secondly, I was interested in cluster music. After a while, though, it
seemed like a lot of sound and fury signifying absolutely nothing. But by changing the tuning between or
within clusters, I could again generate harmonic and melodic motion and have events of some significance
occur. Finally, during this period of my life I was making a living as a jazz musician. With jazz I could get
involved immediately with microtonal intervals.”
Harrison: “The reason for my interest (in nonstandard tunings) is very simple. Real intervals – the ones with
whole number ratios – grab you; they’re beautiful; they draw you into the music; whereas fake intervals like
those of equal temperament don’t do much.”
Johnston: “I love extending my vocabulary and trying to imagine unfamiliar sounds. You can generally
imagine a melodic line, but it’s very difficult to imagine what combination of strange intervals will sound
like.”
Mandelbaum: “I find that extending the consonance to the seventh partial provides a fascinating means of
enrichment. It retains traditional consonance and dissonance, unlike atonal music, which has to abolish the
old in order to arrive at the new.”
Schottstaedt: “I like the unusual sounds, the intense dissonances in particular, such as the squeezed minor
seconds and stretched fourth in Dinosaur Music. I’ve never been much interested in getting cleaner
consonances; beats don’t offend me.”350
Doty offers an interesting discussion on the ‘golden age’ of Western music, and the direct benefits that
may be attributed to the adoption of equal temperament in the common practice era, which unlike the
previous meantone system of tuning, facilitated unrestricted modulation, and hence the development of
complex chromatic harmony. Although in spite of its benevolent influences on harmonic music, twelve-
tone equal temperament is also acknowledged for ultimately leading to its “demise as a vital
compositional style.” Doty offers the following conclusions:
“Twelve-tone equal temperament is a limited and closed system. Once you have modulated around the
so-called circle of fifths, through its twelve major and twelve minor keys, and once you have stacked up
every combination of tones that can reasonably be considered a chord, there is nowhere left to go in
search of new resources. This is essentially where Western composers found themselves at the beginning
of the twentieth century. Everything that could be done with the equally-tempered scale and the
principles of tonal harmony had been tried, and the system was breaking down. This situation led many
composers to the erroneous conclusion that consonance, tonality, and even pitch had been exhausted as
organizing principles. What was really exhausted were merely the very limited resources of the tempered
scale. By substituting twelve equally spaced tones for a vast universe of subtle intervallic relationships, the
composers and theorists of the eighteenth and nineteenth centuries effectively painted Western music into
a corner from which it has not, as yet, extradited itself.”351
350 Keislar, Blackwood, Eaton, Harrison, Johnston, Mandelbaum, and Schottstaedt, “Six American Composers on
Nonstandard Tunings,” Perspectives of New Music 184-85.
351 Doty, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation 5.
298 Conclusion
In the final report presented at the end of his residency at the University of Wisconsin in Madison
between May 1944 and May 1947 (the setting for the completion of Genesis of a Music – published in
1949 by University of Wisconsin Press), Partch paints an even bleaker picture of the twelve-tone equally-
tempered continuum when he speaks of “Western world’s current three-hundred years of twelve-tone
paralysis.” Ronald V. Wiecki makes the following statement about the legacy of Partch:
“Partch’s work may remain a curiosity, a rare and esoteric treasure, even if the present glimmer of interest in
microtonality grows into a significant aspect of musical practice. But his aesthetic theories, both in their
origins and their expression, provide an interesting look into the sense of failure present in American music
in the early 1920s, a period in which Partch’s attitudes must have taken shape. That Partch found it
necessary to react in such an extreme manner to this sense of crisis is eloquent testimony to its
pervasiveness. How conscious Partch was of this situation needs to be examined further, but it can hardly
be disputed that Partch’s music is passionate, that it contains truths distilled from the American experience,
and that it successfully addresses the essential problem of all American composers of the time – that of
establishing a uniquely American musical practice, one not slavishly or unthinkingly based on an imported
model. Partch’s response to this problem simply attempted to penetrate much more deeply into the
historical roots of the problem, and his solution will remain to inspire others.”352
352 Ronald V. Wiecki, “Relieving ‘12-Tone Paralysis’: Harry Partch in Madison, Wisconsin, 1944-1947” American Music
9.1 (Spring, 1991): 43-60.
Conclusion 299
Bibliography Books
Backus, John. The Acoustical Foundations of Music. 2nd ed. New York: Norton, 1977.
Barbour, J. Murray. Tuning and Temperament: A Historical Survey. New York: Dover Publications, 2004.
Benade, Arthur H. Fundamentals of Musical Acoustics. 2nd ed. New York: Dover Publications, 1990.
— . Horn, Strings, and Harmony. New York: Dover Publications, 1992.
Bineš, Taqi. The Short History of Persian Music. Tehran: Àrvin Publication, 1995.
Blackwood, Easley. The Structure of Recognizable Diatonic Tunings. Princeton, NJ: Princeton U. Press,
1985.
Bosanquet, R. H. M. An Elementary Treatise on Musical Intervals and Temperament. Ed. Rudolf Rasch.
Utrecht, The Netherlands: Diapason Press, 1987.
Brinner, Benjamin. Knowing Music, Making Music: Javanese Gamelan and the Theory of Musical
Competence and Interaction. Chicago: U. of Chicago Press, 1995.
Chalmers, John H. Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales.
Hanover, NH: Frog Peak Music, 1993.
Collier, Graham. “Compositional Devices.” Vol. 1. Boston, MA: Berklee Press, 1975.
Cope, David. Techniques of the Contemporary Composer. New York: Schirmer Books, 1997.
Cowell, Henry. New Musical Resources. Cambridge: Cambridge U. Press, 1996.
Daniélou, Alain. Introduction to the Study of Musical Scales. London: India Society, 1943.
— . Tableau Comparatif des Intervalles Musicaux. Pondichéry, India: Institut Français d'Indologie, 1958.
— . Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness. Rochester,
VT: Inner Traditions, 1995.
Davies, Hugh. “Toy Piano.” The New Grove Dictionary of Music and Musicians. Ed. Stanley Sadie and
John Tyrrell. 2nd ed. Vol. 12. London: Macmillan Reference, 2001. 615.
Deva, B. Chaitanya. Indian Music. New Delhi: Indian Council for Cultural Relations, 1974.
Doty, David D. The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation.
3rd ed. San Francisco: Other Music, 2002.
During, Jean, Scheherazade Q Hassan, and Alastair Dick. “Santñr.” The New Grove Dictionary of Musical
Instruments. Ed. Stanley Sadie. Vol. 3. London: Macmillan Reference, 1984. 291-92.
Farhat, Hormoz. The Dastgàh Concept in Persian Music. New York: Cambridge U. Press, 1990.
— . “Iran: Classical Traditions.” The New Grove Dictionary of Music and Musicians. Ed. Stanley Sadie and
John Tyrrell. 2nd ed. Vol. 12. London: Macmillan Reference, 2001. 530-37.
— . “The Music of Islam.” Ancient and Oriental Music. Ed. Egon Wellesz. London: Oxford U. Press,
1957. 421-77.
Farmer, Henry George. A History of Arabian Music (to the XIIIth Century). London: Luzac & Co., 1967.
— . Historical Facts for the Arabian Musical Influence. London: William Reeves, 1930.
Gilmore, Bob. Harry Partch: A Biography. New Haven, CT: Yale U. Press, 1998.
Gordon, Roderick D. The World of Musical Sound. Dubuque, IA: Kendall/Hunt, 1979.
Harrison, Lou. Lou Harrison’s Music Primer: Various Items About Music to 1970. New York: C. F. Peters,
1971.
Helmholtz, Hermann L. F. On the Sensations of Tone: As a Physiological Basis for the Theory of Music.
2nd ed. New York: Dover Publications, 1954.
Henderson, Isobel. “Ancient Greek Music.” Ancient and Oriental Music. Ed. Egon Wellesz. London:
Oxford U. Press, 1957. 336-403.
Isacoff, Stuart. Temperament: How Music Became a Battleground for the Great Minds of Western
Civilization. New York: Vintage, 2003.
Johnston, Ian. Measured Tones: The Interplay of Physics and Music. Bristol; Philadelphia: Adam Hilger,
1989.
Katzner, Kenneth. The Languages of the World. London: Routledge, 1986.
Kostka, Stefan. Materials and Techniques of Twentieth-Century Music. 2nd ed. Upper Saddle River, NJ:
Prentice-Hall, 1999.
Kunst, Jaap. Music in Java: Its History, its Theory and its Technique. Ed. E. L. Heins. 3rd ed. Vol. 1. The
Hague: Martinus Nijhoff, 1973.
Lindley, Mark. Mathematical Models of Musical Scales: A New Approach. Bonn: Verlag für Systematische
Musikwissenschaft, 1993.
Lloyd, Llewelyn Southworth, and Hugh Boyle. Intervals, Scales and Temperaments: An Introduction to the
Study of Musical Intonation. London: McDonald and Jane’s, 1978.
Lloyd-Watts, Valery, Carole L. Bigler, and Willard A. Palmer. Ornamentation: A Question and Answer
Manual. Van Nuys, CA: Alfred Publishing Co., 1995.
Macpherson, Stewart, and Anthony Payne. Rudiments of Music. London: Steiner & Bell, 1994.
Malm, William P. Music Cultures of the Pacific, the Near East, and Asia. Englewood Cliffs, NJ: Prentice-Hall,
1967.
McCombie, Ian. The Piano Handbook. London: David & Charles, 1980.
Miller, Leta E., and Fredric Lieberman. Lou Harrison: Composing a World. New York: Oxford U. Press,
1998.
Moin, Moäamad. An Intermediate Iranian Dictionary. 6 vols. Tehran: Amirkabir Publication, 1974.
Marñfi, Mñsà. Les Systems de la Musique Traditionelle de L'Iran (Radif). Tehran: Iranian Music Association
Publisher, 1995.
302 Bibliography
Nakanishi, Akira. Writing Systems of the World: Alphabets, Syllabaries, Pictograms. Rutland, VT: Charles E.
Tuttle Co., 1980.
Nketia, J. H. Kwabena. The Music of Africa. New York: W. W. Norton & Co., 1974.
Olson, Harry F. Music, Physics and Engineering. 2nd ed. New York: Dover Publications, 1967.
Pacholczyk, Josef M. “Secular Classical Music in the Arabic Near East.” Musics of Many Cultures. Berkeley,
CA: U. of California Press, 1980. 253-68.
Partch, Harry. Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments. 2nd ed.
New York: Da Capo, 1974.
Perlman, Marc. Unplayed Melodies: Javanese Gamelan and the Genesis of Music Theory. Berkeley: U. of
California Press, 2004.
Persichetti, Vincent. Twentieth-Century Harmony: Creative Aspects and Practice. New York: W. W.
Norton & Co., 1961.
Picken, Laurence. “The Music of Far Eastern Asia: 2. Other Countries.” Ancient and Oriental Music. Ed.
Egon Wellesz. London: Oxford U. Press, 1957. 135-94.
Piston, Walter. Counterpoint. 4th ed. London: Victor Gollancz, 1964.
Rameau, Jean-Philippe. Treatise on Harmony. Trans. Philip Gossett. New York: Dover Publications, 1971.
Randel, Don Michael, ed. The New Harvard Dictionary of Music. Cambridge, MA: Belknap Press of
Harvard U Press, 1986.
Read, Gardner. 20th-Century Microtonal Notation. Westport, CT: Greenwood Press, 1990.
Romanowski, Patricia, Holly George-Warren, and Jon Pareles, ed. The New Rolling Stone Encyclopedia of
Rock & Roll. New York: Fireside, 1995.
Sadie, Stanley, ed. The Grove Concise Dictionary of Music. London: Macmillan, 1994.
— . The New Grove Dictionary of Musical Instruments. 3 vols. London: Macmillan, 1984.
Salvador-Daniel, Francesco, and Henry George Farmer. The Music and Musical Instruments of the Arab:
With Introduction on How to Appreciate Arab Music. Portland, ME: Longwood Press, 1976.
Seashore, Carl E. Psychology of Music. New York: Dover Publications, 1967.
Sethares, William A. Tuning, Timbre, Spectrum, Scale. 2nd ed. London: Springer-Verlag, 2005.
Slonimsky, Nicolas. Music of South America. New York: Da Capo Press, 1972.
Soanes, Catherine, ed. Oxford Dictionary of Current English. 3rd ed. London: Oxford U. Press, 2001.
Steiglitz, Ken. A DSP Primer: with Applications to Digital Audio and Computer Music. Menlo Park, CA:
Addison-Wesley Pub., 1995.
Talai, Dariush. “A New Approach to the Theory of Persian Art Music: The Radåf and the Modal System.”
Ed. Virginia Danielson, Scott Marcus and Dwight Reynolds. The Garland Encyclopedia of World
Music: The Middle East. Vol. 6. New York: Routledge, 2002. 865-74.
Bibliography 303
— . Traditional Persian Art Music: The Radif of Mirza Abdollah. Trans. Manoochehr Sadeghi. Coasta
Mesa, CA: Mazda, 2000.
Tenney, James. A History of ‘Consonance’ and ‘Dissonance’. New York: Excelsior Music Publishing Co.,
1988.
Tenzer, Michael. Balinese Music. Singapore: Periplus Editions, 1991.
Thomson, Virgil. Virgil Thomson. New York: Da Capo Press, 1966.
Touma, Habib Hassan. The Music of the Arabs. Trans. Laurie Schwartz. Portland, OR: Amadeus Press,
1957.
Zonis, Ella. “Classical Iranian Music.” Musics of Many Cultures. Berkeley, CA: U. of California Press, 1980.
269-83.
— . Classical Persian Music: An Introduction. Cambridge, MA: Harvard U. Press, 1973.
Von Gunden, Heidi. The Music of Ben Johnston. Metuchen, N.J.:The Scarecrow Press, 1986.
Wright, Owen. “Arab Music: Art Music.” The New Grove Dictionary of Music and Musicians. Ed. Stanley
Sadie and John Tyrrell. 2nd ed. Vol. 12. London: Macmillan Reference, 2001. 797-833.
— . The Modal System of Arab and Persian Music A.D. 1250-1300. London Oriental Series. Vol. 28.
Oxford: Oxford U. Press, 1978. Discography Blackwood, Easley. Microtonal Compositions by Easley Blackwood. Perf. Easley Blackwood (polyfusion
synthesizer), and Jeffrey Kust (guitar). Rec. 16 Sep. 1990. Liner notes by Easley Blackwood.
Cedille, 1994. CDR 90000 018. Dissertations
Ayers, Lydia. “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their
Compositional Applications.” DMA diss. U. of Illinois, Urbana-Champaign, 1994. PA 9512292.
Piraglu, Qmars. “Faràmarz Pàyvar and His Place in Iranian Music.” Diss. Melbourne U., Austral., 2002.
— . “The Persian Music Tradition (In Iran) under Qajar and Pahlavi Dynasties.” Diss. Göteborg U., Swed.,
1998. Electronic Publications Antares Audio Technologies. 2006. Antares Audio Technologies. 15 Dec. 2006.
<http://www.antarestech.com/>.
304 Bibliography
Barrett, Kevin. “Understanding Polyrhythms.” Funkster’s Groove Theory. 23 Mar. 2004. 18 Jan. 2006.
<http://www.funkdrums.com/polyrhythm.html>.
“BBC Weather Centre: World Weather.” BBC Home Page. 6 Apr. 2006. British Broadcasting Corporation.
6 Apr. 2006. <http://212.58.224.86/weather/>
Braun, Martin. “The Gamelan Pélog Scale of Central Java as an Example of a Non-Harmonic Musical Scale.”
Neuroscience of Music. 2 Aug. 2002: S-671 95 Klässbol, Sweden, 10 Apr. 2006.
<http://web.telia.com/~u57011259/pelog_main.htm>.
dB Audioware: Professional Audio Software. 2006. dB Audioware Limited. 15 Dec. 2006.
<http://www.db-audioware.com/>.
Dhomont, Francis. “Acousmatic Update.” Contact! 8.2 Spring, 1995. CEC – Communauté
Électroacoustique Canadienne / Canadian Electroacoustic Community. 27 Jan. 2006.
<http://cec.concordia.ca/contact/contact82Dhom.html>
DSound. 2005. DSound. 15 Dec. 2006. <http://www.dsound1.com/>.
“Encyclopedia of Microtonal Music Theory.” Microtonal, Just Intonation Electronic Music Software. 2005.
Tonalsoft. 17 Nov. 2006. <http://www.tonalsoft.com/>.
Extensible Toy Piano Project, The. Ed. David Claman and Matt Malsky. 1 Jan. 2005. Clark U., Worcester,
MA. 21 Aug. 2005. <http://www.clarku.edu/xtp/xtp.html>.
Gann, Kyle. “Anatomy of an Octave.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005.
<http://www.kylegann.com/Octave.html>.
— . “An Introduction to Historical Tunings.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005.
<http://www.kylegann.com/histune.html>.
— . “Just Intonation Explained.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005.
<http://www.kylegann.com/tuning.html>.
— . “My Idiosyncratic Reasons for Using Just Intonation.” Kyle Gann’s Home Page. 1997. 15 Dec. 2005.
<http://www.kylegann.com/JIreasons.html>.
Keenan, David C. “A Note on the Naming of Musical Intervals.” David Keenan’s Home Page. 3 Nov.
2001. 22 Nov. 2006. <http://users.bigpond.net.au/d.keenan/Music/IntervalNaming.htm>.
Lewis, Martha Beth. “Tuning Your Piano: Why Pianos Go Out of Tune.” Martha Beth Lewis’ Home Page.
1999. 8 Apr. 2006. <http://www.serve.com/marbeth/tune_piano.html>.
Op de Coul, Manuel. “Huygens-Fokker: List of intervals.” Huygens-Fokker Foundation: Centre for
Microtonal Music. 2006. Huygens-Fokker Foundation. 22 Nov. 2006.
<http://www.xs4all.nl/~huygensf/doc/intervals.html>.
Peterson, James B. “Names of Bases.” The Math Forum: Ask Dr. Math. 15 Apr. 2002. Drexel U.,
Philadelphia, PA. 22 Nov. 2006. <http://mathforum.org/library/drmath/view/60405.html>.
Bibliography 305
PSP Audioware: Audio Processors and Effects Plug-ins. 2006. PSP Audioware. 15 Dec. 2006.
<http://www.pspaudioware.com/>.
Schoenhut Toy Piano Company: Over 130 Years. 2005. Schoenhut Toy Piano Company. 21 Aug. 2005.
<http://www.toypiano.com/6625.htm>.
Sony Media Software: Home for Vegas, Sound Forge and Acid. 2006. Sony Corporation of America.
15 Dec. 2006. <http://www.sonymediasoftware.com/>.
Spin Audio Software. 2006. Spin Audio Software. 15 Dec. 2006. <http://www.spinaudio.com/>.
Stearns, Dan. “Some Thoughts on an Alternative Definition of Equal Temperament.” Kronosonic. 2006.
The International Society for Creative Guitar and String Music. 1 Aug. 2006.
<http://kronoson.station185.com/kronosonic/forum/index>.
Tatlow, Ruth. “Golden Number (Golden Section).” Grove Music Online. Ed. L. Macy. 2004. 16 Dec.
2004. <http://www.grovemusic.com.ezproxy.lib.unimelb.edu.au>.
Vai, Steve. “Little Black Dots: Tempo Metal.” The Official Steve Vai Website. 1983. 18 Jan. 2006.
<http://www.vai.com/LittleBlackDots/tempomental.html>.
Waves: The World’s Leading Developer of Audio Signal Processing Software. 2006. Waves
Incorporated. 15 Dec. 2006. <http://www.waves.com/>. Multimedia Encyclopaedia Britannica. “Humidity.” Encyclopaedia Britannica 2001. Deluxe ed. CD-ROM. Chicago:
Encyclopaedia Britannica, 2001.
— . “Sound.” Encyclopaedia Britannica 2001. Deluxe ed. CD-ROM. Chicago: Encyclopaedia Britannica,
2001.
— . “Statistics.” Encyclopaedia Britannica 2001. Deluxe ed. CD-ROM. Chicago: Encyclopaedia
Britannica, 2001. Periodicals Adkins, Cecil. “The Technique of the Monochord.” Acta Musicologica 39 (Jan.-Jun., 1967): 34-43.
Barbour, J. Murray. “Irregular Systems of Temperament.” Journal of the American Musicological Society
1.3 (Autumn, 1948): 20-26.
Benjamin, Gerald R. “Julian Carrillo and ‘Sonido Trece’ (Dedicated to the Memory of Nabor Carrillo).”
Anuario 3 (1967): 33-68.
Berard, Wilford W. “The Eleventh and Thirteenth Partials.” Journal of Music Theory 5.1 (Spring, 1961): 95-
107.
306 Bibliography
Blackwood, Easley. “Modes and Chord Progressions in Equal Tunings.” Perspectives of New Music 29.2
(Summer, 1991): 166-200.
Bobbitt, Richard. “The Physical Basis of Intervallic Quality and its Application to the Problem of
Dissonance.” Journal of Music Theory 3.2 (Nov., 1959): 173-207.
Bosanquet, R. H. M. “Temperament; Or, the Division of the Octave (Part I).” Proceedings of the Musical
Association. 1st Sess. (1874-75): 4-17.
— . “Temperament; Or, the Division of the Octave (Part II).” Proceedings of the Musical Association.
2nd Sess. (1874-75): 112-58.
— . “The Theory of the Division of the Octave, and the Practical Treatment of the Musical Systems Thus
Obtained.” Proceedings of the Royal Society of London 23 (1874-1875): 390-408.
Brown, J. P., and William B. Rose. “Humidity and Moisture in Historic Buildings: The Origins of Buildings and
Object Conservation.” APT Bulletin 27.3 (1996): 12-23.
Desantos, Sandra. “Acousmatic Morphology: An Interview with François Bayle.” Computer Music Journal
21.3 (Fall, 1997): 11-19.
De Klerk, Dirk. “Equal Temperament.” Acta Musicologica 51.1 (Jan.-Jun., 1979): 140-50.
De Vitry, Philippe. “Philippe de Vitry's ‘Ars Nova’: A Translation.” Journal of Music Theory 5.2 (Winter,
1961): 204-23.
Dunne, Edward, and Mark McConnell. “Pianos and Continued Fractions.” Mathematics Magazine 72.2
(Apr., 1999): 104-15.
Ellis, Alexander J. “On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments
wit Fixed Tones.” Proceedings of the Royal Society of London 13 (1863-64): 93-108.
Ellis, Alexander J., and Alfred J. Hipkins. “Tonometrical Observations on Some Existing Non-Harmonic
Musical Scales.” Proceedings of the Royal Society of London 37 (1884): 368-85.
Fokker, A. D. “Equal Temperament and the Thirty-One-Keyed Organ.” The Scientific Monthly 81.4 (Oct.,
1955): 161-66.
Fonville, John. ”Ben Johnston’s Extended Just Intonation: A Guide for Interpreters.” Perspectives of New
Music 29.2 (Summer, 1991): 106-37.
Fuller, Ramon. “A Study of Microtonal Equal Temperaments.” Journal of Music Theory 35.1/2 (Spring-
Autumn, 1991): 211-37.
Gilmore, Bob. “Changing the Metaphor: Ratio Models of Musical Pitch in the Work of Harry Partch, Ben
Johnston, and James Tenney.” Perspectives of New Music 33.1/2 (Winter-Summer, 1995): 458-
503.
— . “On Harry Partch's Seventeen Lyrics by Li Po.” Perspectives of New Music 30.2 (Summer, 1992): 22-
58.
Bibliography 307
Jairazbhoy, N. A., and A. W. Stone. “Intonation in Present-Day North Indian Classical Music.” Bulletin of
the School of Oriental and African Studies, University of London 26.1 (1963): 119-32.
Johnston, Ben. ”Scalar Order as a Compositional Resource.” Perspectives of New Music 2.2 (Summer,
1964): 56-76.
Keislar, Douglas. ”Introduction.” Perspectives of New Music 29.1 (Winter, 1991): 173-75.
Keislar, Douglas, Easley Blackwood, John Eaton, Lou Harrison, Ben Johnston, Joel Mandelbaum, and
William Schottstaedt. “Six American Composers on Nonstandard Tunings.” Perspectives of New
Music 29.1 (Winter, 1991): 176-211.
Kelley, Truman L. “How Many Figures are Significant.” Science 60.1562 (Dec. 5, 1924): 524.
Kuttner, Fritz A. “A Musicological Interpretation of the Twelve Lüs in China's Traditional Tone System.”
Ethnomusicology 9.1 (Jan., 1965): 22-38.
— . “Prince Chu Tsai-Yu's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament
Theory.” Ethnomusicology 19.2 (May, 1975): 163-206.
Leedy, Douglas. “A Venerable Temperament Rediscovered.” Perspectives of New Music 29.2 (Summer,
1991): 201-11.
— . “Selected Musical Compositions (1948-1972).” Notes 46.1 (Sep., 1989): 224-26.
Mandelbaum, Joel. “Toward the Expansion of Our Concepts of Intonation.” Perspectives of New Music
13.1 (Autumn-Winter, 1974): 216-26.
Marcus, Scott. “The Interface Between Theory and Practice: Intonation in Arab Music.” Asian Music 24.2
(Spring-Summer, 1993): 39-58.
McClure, A. R. “Studies in Keyboard Temperaments.” The Galpin Society Journal 1 (Mar., 1948): 28-40.
Miller, Terry E., and Sam-ang Sam. “The Classical Musics of Cambodia and Thailand: A Study of
Distinctions.” Ethnomusicology 39.2 (Spring-Summer, 1995): 229-43.
Nooshin, Laudan. “The Song of the Nightingale: Processes of Improvisation in Dastgàh Segàh (Iranian
Classical Music).” British Journal of Ethnomusicology 7 (1998): 69-116.
Norden, N. Lindsay. “A New Theory of Untempered Music: A Few Important Features with Special
Reference to ‘A Capella’ Music.” The Musical Quarterly 22.2 (Apr., 1936): 217-33.
Orton, Richard. “The 31-Note Organ.” The Musical Times 107.1478 (Apr., 1966): 342-43.
Pearson, E. S., and Joan Haines. “The Use of Range in Place of Standard Deviation in Small Samples.”
Supplement in the Journal of the Royal Statistical Society 2.1 (1935): 83-98.
Pertout, Andrián. “Siamak Noory: Pardis.” Interview with Siamak Noory. Australian Musician 30 (Winter,
Jun. 2002): 35.
— . “Siamak Noory: Pardis.” Interview with Siamak Noory. Mixdown 104 (Dec. 2002): 38.
— . “Siamak Noory: The Santurist – Part 1.” Interview with Siamak Noory. Mixdown 97 (May 2002): 25.
— . “Siamak Noory: The Santurist – Part 2.” Interview with Siamak Noory. Mixdown 98 (Jun. 2002): 25.
308 Bibliography
— . “Siamak Noory: The Santurist – Part 3.” Interview with Siamak Noory. Mixdown 101 (Sep. 2002):
26.
— . “Siamak Noory: The Santurist – Part 4.” Interview with Siamak Noory. Mixdown 103 (Nov. 2002):
32.
Rapoport, Paul. “The Notation of Equal Temperaments.” Xenharmonikôn: An Informal Journal of
Experimental Music 16 (Autumn, 1995): 61-84.
— . “The Structural Relationships of Fifths and Thirds in Equal Temperaments.” Journal of Music Theory
37.2 (Autumn, 1993): 351-89.
— . “Towards the Infinite Expansion of Tonal Resources.” Tempo 144 (Mar., 1983): 7-11.
Reider, Joseph. “Jewish and Arabic Music.” The Jewish Quarterly Review 7.4 (Apr., 1917): 635-44.
Sacks, Oliver, G. Schlaug, L. Jäncke, Y. Huang, and H. Steinmetz. “Musical Ability.” Science 268.5211
(May 5, 1995): 621-22.
Stanford, Charles Villiers. “On Some Recent Tendencies in Composition.” Proceedings of the Musical
Association, 47th Sess. (1920): 39-53.
Swinburne, James. “The Ideal Scale: Its AEtiology, Lysis and SequelAE.” Proceedings of the Musical
Association, 63rd sess. (1936-1937): 39-64.
Tischler, Hans. “Musica Ficta in the Thirteenth Century.” Music & Letters 54.1 (Jan., 1973): 38-56.
Touma, Habib Hassan. “The Maqam Phenomenon: An Improvisation Technique in the Music of the
Middle East.” Ethnomusicology 15.1 (Jan., 1971): 38-48.
Wiecki, Ronald V. “Relieving ‘12-Tone Paralysis’: Harry Partch in Madison, Wisconsin, 1944-1947.”
American Music 9.1 (Spring, 1991): 43-66.
Winnington-Ingram, R. P. “Aristoxenus and the Intervals of Greek Music.” The Classical Quarterly 26.3/4
(Jul.-Oct., 1932): 195-208.
Wolf, Daniel James. “Alternative Tunings, Alternative Tonalities.” Contemporary Music Review 22.1-2
(2003): 3-14.
Young, Gayle. “The Pitch Organization of Harmonium for James Tenney.” Perspectives of New Music
26.2 (Summer, 1988): 204-12.
Zonis, Ella. “Contemporary Art Music in Persia.” The Music Quarterly 51.4 (Oct., 1965): 636-48. Photographs 1967 William Dowd French Double Harpsichord. Personal photograph of Alex Pertout. 3 Feb. 2007.
Qmars Piraglu. Personal photograph of Andrián Pertout. 22 Oct. 2006.
Bibliography 309
Malsky, Matt. “Schoenhut Model 6625: 25-Key Toy Piano.” Feb. 2005. The Extensible Toy Piano Project.
Ed. David Claman and Matt Malsky. 1 Mar. 2005. Clark U., Worcester, MA. 21 Aug. 2005.
<http://www.clarku.edu/xtp/photo.html>
310 Bibliography
Appendix A
Comparative Table of Musical Intervals
Tabl
e 17
8. C
omp
arat
ive
tabl
e of
mus
ical
inte
rval
s Re
lativ
e Pi
tch:
A4=
440H
z / C
4 (m
idd
le C
)=26
1.62
5565
4Hz
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÐÑ
ÐÒ
ÐÓ
ÐÔ
ÐÕ
ÐÖ
Ð×
ÐØ
ÐÙ
ÑÐ
ÑÑ
ÑÒ
ÑÓ
ÑÔ
ÑÕ
ÑÖ
Ñ×
ÑØ
ÑÙ
ÒÐ
ÒÑ
uniso
n (1
st h
arm
onic
)
equa
l hun
dre
dth
-sem
itone
one-
elev
enth
syn
toni
c co
mm
a, o
r skh
isma
one-
sixth
syn
toni
c co
mm
a
cycl
ic o
ctav
e (A
) LII
I
one-
fifth
syn
toni
c co
mm
a
two-
nint
h sy
nton
ic c
omm
a
one-
qua
rter s
ynto
nic
com
ma
two-
seve
nth
synt
onic
com
ma
one-
third
syn
toni
c co
mm
a
one-
half
synt
onic
com
ma
nona
vige
simal
com
ma
equa
l six
teen
th-to
ne
thre
e-q
uarte
r syn
toni
c co
mm
a
equa
l tw
elfth
-tone
nona
dec
imal
com
ma
subd
imin
ished
sec
ond
, or d
iask
hism
a
synt
onic
com
ma
53-e
t syn
toni
c co
mm
a
Pyth
agor
ean
com
ma
(A)
XII
equa
l eig
hth-
tone
1/1
1200
2, o
r ap
pro
xim
atel
y 17
31/1
730
118081
, or 3
2805
/327
68
68081
3õó/2
øô 5
8081
4.5
8081
48081
3.5
8081
38081
28081
145/
144
962
1.33
3333
8081
722
96/9
5
2048
/202
5
81/8
0 53
2
3ñò/2
ñù, o
r 531
441/
5242
88
482
1.00
0000
1.00
0578
1.00
1130
1.00
2073
1.00
2090
1.00
2488
1.00
2764
1.00
3110
1.00
3556
1.00
4149
1.00
6231
1.00
6944
1.00
7246
1.00
9360
1.00
9674
1.01
0526
1.01
1358
1.01
2500
1.01
3164
1.01
3643
1.01
4545
261.
626
261.
777
261.
921
262.
168
262.
172
262.
276
262.
349
262.
439
262.
556
262.
711
263.
256
263.
442
263.
521
264.
074
264.
156
264.
380
264.
597
264.
896
265.
070
265.
195
265.
431
0.00
0
1.00
0
1.95
5
3.58
4
3.61
5
4.30
1
4.77
9
5.37
7
6.14
5
7.16
9
10.7
53
11.9
81
12.5
00
16.1
30
16.6
67
18.1
28
19.5
53
21.5
06
22.6
42
23.4
60
25.0
00
312 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÒÒ
ÒÓ
ÒÔ
ÒÕ
ÒÖ
Ò×
ÒØ
ÒÙ
ÓÐ
ÓÑ
ÓÒ
ÓÓ
ÓÔ
ÓÕ
ÓÖ
Ó×
ÓØ
ÓÙ
ÔÐ
ÔÑ
ÔÒ
ÔÓ
trid
ecim
al c
omm
a (6
5th
harm
onic
)
one
and
one
-qua
rter s
ynto
nic
com
ma
min
imal
die
sis
43-e
t dim
inish
ed s
econ
d
grav
e or
sm
all d
iesis
one
and
one
-hal
f syn
toni
c co
mm
a
equa
l six
th-to
ne
sep
tend
ecim
al c
omm
a
one
and
thre
e-q
uarte
r syn
toni
c co
mm
a
trivi
gesim
al c
omm
a
31-e
t sup
eroc
tave
, or d
imin
ished
sec
ond
und
ecim
al g
rave
or s
mal
l chr
omat
ic s
emito
ne
equa
l fift
h-to
ne
dim
inish
ed s
econ
d, o
r gre
at d
iesis
two
synt
onic
com
mas
, or M
athi
eu s
uper
die
sis
53-e
t gre
at d
iesis
grea
t die
sis (A
) XX
IV
two
and
one
-qua
rter s
ynto
nic
com
ma
sep
timal
com
ma
equa
l qua
rter-t
one
23-e
t Gre
ek e
nhar
mon
ic o
r sep
timal
qua
rter-t
one
und
ecim
al c
omm
a (3
3rd
har
mon
ic)
65/6
4
0.8
8081
2000
0/19
683
432
3125
/307
2
0.66
6667
8081
362
51/5
0
0.57
1429
8081
46/4
5
312
45/4
4
302
128/
125
0.5
8081, o
r 656
1/64
00
253
)2
(
3 òô/2
óø 0.
4444
448081
36/3
5
242
, or a
ppro
xim
atel
y 52
7/51
2 23
2
33/3
2
1.01
5625
1.01
5649
1.01
6105
1.01
6250
1.01
7253
1.01
8808
1.01
9441
1.02
0000
1.02
1977
1.02
2222
1.02
2611
1.02
2727
1.02
3374
1.02
4000
1.02
5156
1.02
6502
1.02
7473
1.02
8345
1.02
8571
1.02
9302
1.03
0596
1.03
1250
265.
713
265.
720
265.
839
265.
877
266.
139
266.
546
266.
712
266.
858
267.
375
267.
439
267.
541
267.
572
267.
741
267.
905
268.
207
268.
559
268.
813
269.
041
269.
101
269.
292
269.
630
269.
801
26.8
41
26.8
83
27.6
60
27.9
07
29.6
14
32.2
59
33.3
33
34.2
83
37.6
36
38.0
51
38.7
10
38.9
06
40.0
00
41.0
59
43.0
13
45.2
83
46.9
20
48.3
89
48.7
70
50.0
00
52.1
74
53.2
73
Appendices 313
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÔÔ
ÔÕ
ÔÖ
Ô×
ÔØ
ÔÙ
ÕÐ
ÕÑ
ÕÒ
ÕÓ
ÕÔ
ÕÕ
ÕÖ
Õ×
ÕØ
ÕÙ
ÖÐ
ÖÑ
ÖÒ
ÖÓ
ÖÔ
ÖÕ
two
and
one
-hal
f syn
toni
c co
mm
a
22-e
t Gre
ek e
nhar
mon
ic o
r sep
timal
qua
rter-t
one
untri
gesim
al d
iato
nic
sem
itone
, or G
reek
enh
arm
onic
qua
rter-t
one
43-e
t dou
ble
augm
ente
d s
even
th
Gre
ek e
nhar
mon
ic q
uarte
r-ton
e, o
r unt
riges
imal
com
ma
21-e
t Gre
ek e
nhar
mon
ic o
r sep
timal
qua
rter-t
one
two
and
thre
e-q
uarte
r syn
toni
c co
mm
a
20-e
t Gre
ek e
nhar
mon
ic o
r sep
timal
qua
rter-t
one
19-e
t jus
t dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
thre
e sy
nton
ic c
omm
as
trid
ecim
al g
rave
or s
mal
l chr
omat
ic s
emito
ne
18-e
t gra
ve o
r sm
all c
hrom
atic
sem
itone
, or e
qua
l thi
rd-to
ne
Pyth
agor
ean
dou
ble
dim
inish
ed th
ird
53-e
t gra
ve o
r sm
all c
hrom
atic
sem
itone
, or m
inor
hal
f-ton
e
thre
e an
d o
ne-q
uarte
r syn
toni
c co
mm
a
cycl
ic g
rave
or s
mal
l chr
omat
ic s
emito
ne, o
r min
or h
alf-t
one
(A)
XXXV
I
17-e
t gra
ve o
r sm
all c
hrom
atic
sem
itone
, or m
inor
hal
f-ton
e
grav
e or
sm
all j
ust c
hrom
atic
sem
itone
, or m
inor
hal
f-ton
e
trivi
gesim
al d
iato
nic
sem
itone
16-e
t gra
ve o
r sm
all c
hrom
atic
sem
itone
, or m
inor
hal
f-ton
e
thre
e an
d o
ne-h
alf s
ynto
nic
com
ma
mea
nton
e ch
rom
atic
sem
itone
, or m
inor
hal
f-ton
e (A
) V
II4
3 1ß
0.4
8081
222
32/3
1 243
)2
(
31/3
0
212
0.36
3636
8081
202
19
2
0.33
3333
8081, o
r 531
441/
5120
00
27/2
6
182
1342
1772
8/12
9140
163
353
)2
(
0.30
7692
8081
3 óö/2
õ÷ 17
2
25/2
4
24/2
3
162
0.28
5714
8081
2187
/204
8×0.
5714
298180
1.03
1544
1.03
2008
1.03
2258
1.03
2765
1.03
3333
1.03
3558
1.03
4752
1.03
5265
1.03
7155
1.03
7971
1.03
8462
1.03
9259
1.03
9318
1.04
0015
1.04
1199
1.04
1491
1.04
1616
1.04
1667
1.04
3478
1.04
4274
1.04
4438
1.04
4907
269.
878
270.
000
270.
065
270.
198
270.
346
270.
405
270.
718
270.
852
271.
346
271.
560
271.
688
271.
897
271.
912
272.
094
272.
404
272.
481
272.
513
272.
527
273.
001
273.
209
273.
252
273.
374
53.7
66
54.5
45
54.9
64
55.8
14
56.7
67
57.1
43
59.1
42
60.0
00
63.1
58
64.5
19
65.3
37
66.6
67
66.7
65
67.9
25
69.8
95
70.3
80
70.5
88
70.6
72
73.6
81
75.0
00
75.2
72
76.0
49
314 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÖÖ
Ö×
ÖØ
ÖÙ
×Ð
×Ñ
×Ò
×Ó
×Ô
×Õ
×Ö
××
×Ø
×Ù
ØÐ
ØÑ
ØÒ
ØÓ
ØÔ
ØÕ
ØÖ
Ø×
31-e
t aug
men
ted
oct
ave,
or s
ubm
inor
sec
ond
67th
har
mon
ic
15-e
t gra
ve o
r sm
all c
hrom
atic
sem
itone
, or m
inor
hal
f-ton
e
nona
vige
simal
gra
ve o
r sm
all c
hrom
atic
sem
itone
thre
e an
d th
ree-
qua
rter s
ynto
nic
com
ma
43-e
t chr
omat
ic s
emito
ne, o
r min
or h
alf-t
one
sep
timal
dia
toni
c se
mito
ne
14-e
t Pyt
hago
rean
lim
ma
Pyth
agor
ean
limm
a, o
r dia
toni
c se
mito
ne (D
) V
53-e
t Pyt
hago
rean
lim
ma
acut
e or
larg
e Py
thag
orea
n lim
ma
13-e
t Pyt
hago
rean
lim
ma
nona
dec
imal
dia
toni
c se
mito
ne
cycl
ic P
ytha
gore
an li
mm
a (A
) XL
VIII
sep
tend
ecim
al d
iato
nic
sem
itone
equa
l sem
itone
23-e
t jus
t dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
sept
end
ecim
al c
hrom
atic
sem
itone
(17t
h ha
rmon
ic)
11-e
t jus
t dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
nona
deci
mal
chr
omat
ic s
emito
ne
43-e
t min
or s
econ
d
just
dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
231
)2
(
67/6
4 15
2
243/
232
0.26
6667
8081
343
)2
(
21/2
0
142
256/
243 4
53)
2(
135/
128
132
19/1
8
3 ôø/2
÷ö
18/1
7
122
, or a
ppro
xim
atel
y 10
24/9
67
223
)2
(
17/1
6
112
81/7
6 443
)2
(
16/1
5
1.04
5734
1.04
6875
1.04
7294
1.04
7414
1.04
7687
1.04
9547
1.05
0000
1.05
0757
1.05
3498
1.05
3705
1.05
4688
1.05
4766
1.05
5556
1.05
5700
1.05
8824
1.05
9463
1.06
2127
1.06
2500
1.06
5041
1.06
5789
1.06
6603
1.06
6667
273.
591
273.
889
273.
999
274.
030
274.
102
274.
588
274.
707
274.
905
275.
622
275.
676
275.
933
275.
954
276.
160
276.
198
277.
015
277.
183
277.
880
277.
977
278.
642
278.
838
279.
051
279.
067
77.4
19
79.0
70
80.0
00
80.1
98
80.6
49
83.7
21
84.4
67
85.7
14
90.2
25
90.5
66
92.1
79
92.3
08
93.6
03
93.8
40
98.9
55
100.
000
104.
348
104.
955
109.
091
110.
307
111.
628
111.
731
Appendices 315
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ØØ
ØÙ
ÙÐ
ÙÑ
ÙÒ
ÙÓ
ÙÔ
ÙÕ
ÙÖ
Ù×
ÙØ
ÙÙ
ÑÐÐ
ÑÐÑ
ÑÐÒ
ÑÐÓ
ÑÐÔ
ÑÐÕ
ÑÐÖ
ÑÐ×
ÑÐØ
ÑÐÙ
53-e
t jus
t dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
Pyth
agor
ean
apot
ome,
or c
hrom
atic
sem
itone
(A)
VII
21-e
t jus
t dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
31-e
t sup
erau
gmen
ted
oct
ave,
or m
inor
sec
ond
mea
nton
e m
inor
sec
ond
(D)
V41
1�
sep
timal
chr
omat
ic s
emito
ne
10-e
t jus
t dia
toni
c se
mito
ne, o
r maj
or h
alf-t
one
nona
vige
simal
gra
ve o
r sm
all n
eutra
l sec
ond
19-e
t gre
at li
mm
a, o
r lar
ge h
alf-t
one
trivi
gesim
al c
hrom
atic
sem
itone
(69
th h
arm
onic
)
grea
t lim
ma,
acu
te o
r lar
ge h
alf-t
one
9-et
gre
at li
mm
a, o
r lar
ge h
alf-t
one
53-e
t gre
at li
mm
a, a
cute
or l
arge
hal
f-ton
e
cycl
ic g
reat
lim
ma,
acu
te o
r lar
ge h
alf-t
one
(A)
XIX
trid
ecim
al g
rave
or s
mal
l neu
tral s
econ
d
43-e
t dou
ble
dim
inish
ed th
ird
17-e
t thr
ee-q
uarte
r-ton
e
thre
e-q
uarte
r-ton
e
untri
gesim
al c
hrom
atic
sem
itone
equa
l thr
ee-q
uarte
r-ton
e
und
ecim
al g
rave
or s
mal
l neu
tral s
econ
d
mea
nton
e d
oubl
e au
gmen
ted
oct
ave
(A)
XIV
213
ß
553
)2
(
3 ÷/2
ññ, o
r 218
7/20
48
221
)2
(
331
)2
(
256/
243×
0.8
8081
15/1
4 10
2
29/2
7 219
)2
(
69/6
4
27/2
5 9
2 6
53)
2(
3 ñù/2
óð
13/1
2 543
)2
(
217
)2
(
135/
124
279/
256
82
, or a
ppro
xim
atel
y 10
24/9
39
12/1
1
4782
969/
4194
304×
0.28
5714
8180
1.06
7577
1.06
7871
1.06
8242
1.06
9380
1.06
9984
1.07
1429
1.07
1773
1.07
4074
1.07
5691
1.07
8125
1.08
0000
1.08
0060
1.08
1630
1.08
2440
1.08
3333
1.08
3936
1.08
4964
1.08
8710
1.08
9844
1.09
0508
1.09
0909
1.09
1830
279.
305
279.
382
279.
479
279.
777
279.
935
280.
313
280.
403
281.
005
281.
428
282.
065
282.
556
282.
571
282.
897
283.
194
283.
428
283.
585
283.
854
284.
834
285.
131
285.
305
285.
410
285.
651
113.
208
113.
685
114.
286
116.
129
117.
108
119.
443
120.
000
123.
712
126.
316
130.
229
133.
238
133.
333
135.
849
137.
145
138.
573
139.
535
147.
143
141.
176
148.
946
150.
000
150.
637
152.
098
316 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÑÑÐ
ÑÑÑ
ÑÑÒ
ÑÑÓ
ÑÑÔ
ÑÑÕ
ÑÑÖ
ÑÑ×
ÑÑØ
ÑÑÙ
ÑÒÐ
ÑÒÑ
ÑÒÒ
ÑÒÓ
ÑÒÔ
ÑÒÕ
ÑÒÖ
ÑÒ×
ÑÒØ
ÑÒÙ
ÑÓÐ
ÑÓÑ
31-e
t dou
ble
augm
ente
d o
ctav
e, o
r neu
tral s
econ
d
sep
timal
neu
tral s
econ
d (3
5th
harm
onic
)
23-e
t gra
ve o
r sm
all t
one
53-e
t gra
ve o
r sm
all t
one
15-e
t gra
ve o
r sm
all t
one
cycl
ic g
rave
or s
mal
l ton
e (A
) XX
XI
grav
e or
sm
all t
one
acut
e or
larg
e d
oubl
e au
gmen
ted
oct
ave
22-e
t gra
ve o
r sm
all t
one
und
ecim
al a
cute
or l
arge
neu
tral s
econ
d
43-e
t dou
ble
augm
ente
d o
ctav
e
nona
vige
simal
acu
te o
r lar
ge n
eutra
l sec
ond
7-et
gra
ve o
r sm
all t
one
trid
ecim
al a
cute
or l
arge
neu
tral s
econ
d
71st
har
mon
ic
20-e
t jus
t min
or to
ne
Pyth
agor
ean
dim
inish
ed th
ird (D
) X
53-e
t jus
t min
or to
ne
just
min
or to
ne
cycl
ic m
inor
tone
(A)
XLIII
acut
e or
larg
e d
oubl
e su
per
augm
ente
d o
ctav
e
13-e
t jus
t min
or to
ne
431
)2
(
35/3
2 323
)2
(
753
)2
(
215
)2
(
3 óñ/2
ôù
800/
729
1125
/102
4 3
22)
2(
11/1
0 643
)2
(
32/2
9
72
72/6
5
71/6
4 320
)2
(
6553
6/59
049
853
)2
(
10/9
3 ôó/2
öø
1822
5/16
384
213
)2
(
1.09
3560
1.09
3750
1.09
4624
1.09
5869
1.09
6825
1.09
7208
1.09
7394
1.09
8633
1.09
9131
1.10
0000
1.10
1550
1.10
3448
1.10
4090
1.10
7692
1.10
9375
1.10
9569
1.10
9858
1.11
0295
1.11
1111
1.11
2178
1.11
2366
1.11
2531
286.
103
286.
153
286.
382
286.
707
286.
957
287.
058
287.
106
287.
430
287.
561
287.
788
288.
194
288.
690
288.
858
289.
801
290.
241
290.
292
290.
367
290.
482
290.
695
290.
974
291.
023
291.
067
154.
839
155.
140
156.
522
158.
491
160.
000
160.
605
160.
897
162.
851
163.
636
165.
004
167.
442
170.
423
171.
429
177.
069
179.
697
180.
000
180.
450
181.
132
182.
404
184.
065
184.
357
184.
616
Appendices 317
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÑÓÒ
ÑÓÓ
ÑÓÔ
ÑÓÕ
ÑÓÖ
ÑÓ×
ÑÓØ
ÑÓÙ
ÑÔÐ
ÑÔÑ
ÑÔÒ
ÑÔÓ
ÑÔÔ
ÑÔÕ
ÑÔÖ
ÑÔ×
ÑÔØ
ÑÔÙ
ÑÕÐ
ÑÕÑ
ÑÕÒ
ÑÕÓ
19-e
t jus
t min
or to
ne
third
-com
ma
mea
nton
e m
ajor
tone
(A)
II3
2ß
two-
seve
nth-
com
ma
mea
nton
e m
ajor
tone
(A)
II7
4ß
mea
nton
e m
ajor
tone
(A)
II21
ß
31-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
two-
nint
h-co
mm
a m
eant
one
maj
or to
ne (A
) II
94
ß
fifth
-com
ma
mea
nton
e m
ajor
tone
(A)
II5
2ß
43-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
sixth
-com
ma
mea
nton
e m
ajor
tone
(A)
II6
2ß
equa
l ton
e
nona
dec
imal
sup
erm
ajor
sec
ond
65-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
118-
et ju
st a
nd P
ytha
gore
an m
ajor
tone
53-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
just
and
Pyt
hago
rean
maj
or to
ne (A
) II (
9th
harm
onic
)
41-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
140-
et ju
st a
nd P
ytha
gore
an m
ajor
tone
99-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
87-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
55th
cyc
lic fi
fth (A
) LV
23-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
17-e
t jus
t and
Pyt
hago
rean
maj
or to
ne
319
)2
(
9/8×
1.5
8180
9/8×
1.75
8180
9/8×
28180
5
31)
2(
9/8×
2.25
8180
9/8×
2.5
8180
743
)2
(
9/8×
38180
6
2, o
r ap
pro
xim
atel
y 55
/49
64/5
7 1165
)2
(
2011
8)
2(
9
53)
2(
9/8
741
)2
(
2414
0)
2(
17
99)
2(
15
87)
2(
3 õõ/2
ø÷ 423
)2
(
317
)2
(
1.11
5658
1.11
5722
1.11
7042
1.11
8034
1.11
8287
1.11
8806
1.11
9424
1.11
9450
1.12
0351
1.12
2462
1.12
2807
1.12
4459
1.12
4662
1.12
4911
1.12
5000
1.12
5629
1.12
6173
1.12
6398
1.12
6942
1.12
7352
1.12
8114
1.13
0116
291.
885
291.
901
292.
247
292.
506
292.
572
292.
708
292.
870
292.
877
293.
113
293.
665
293.
755
294.
187
294.
240
294.
306
294.
329
294.
493
294.
636
294.
695
294.
837
294.
944
295.
143
295.
667
189.
474
189.
572
191.
621
193.
157
193.
548
194.
352
195.
307
195.
349
196.
741
200.
000
200.
532
203.
077
203.
390
203.
774
203.
910
204.
878
205.
714
206.
061
206.
897
207.
525
208.
696
211.
765
318 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÑÕÔ
ÑÕÕ
ÑÕÖ
ÑÕ×
ÑÕØ
ÑÕÙ
ÑÖÐ
ÑÖÑ
ÑÖÒ
ÑÖÓ
ÑÖÔ
ÑÖÕ
ÑÖÖ
ÑÖ×
ÑÖØ
ÑÖÙ
Ñ×Ð
Ñ×Ñ
Ñ×Ò
Ñ×Ó
Ñ×Ô
Ñ×Õ
sep
tend
ecim
al s
uper
maj
or s
econ
d
11-e
t acu
te o
r lar
ge to
ne
43-e
t dim
inish
ed th
ird
subd
imin
ished
third
, or a
cute
or l
arge
tone
16-e
t acu
te o
r lar
ge to
ne
acut
e or
larg
e to
ne
53-e
t acu
te o
r lar
ge to
ne
Pyth
agor
ean
dou
ble
augm
ente
d o
ctav
e, o
r cyc
lic a
cute
or l
arge
tone
(A)
XIV
73rd
har
mon
ic
21-e
t acu
te o
r lar
ge to
ne
sep
timal
sup
erm
ajor
sec
ond
31-e
t sup
erm
ajor
sec
ond
, or d
imin
ished
third
mea
nton
e d
imin
ished
third
(D)
X21
2�
untri
gesim
al s
uper
maj
or s
econ
d
5-et
sup
erm
ajor
sec
ond
trivi
gesim
al s
uper
maj
or s
econ
d
und
ecim
al g
rave
or s
mal
l aug
men
ted
sec
ond
sup
erm
ajor
sec
ond
dim
inish
ed th
ird
trid
ecim
al g
rave
or s
mal
l aug
men
ted
sec
ond
53-e
t sup
erm
ajor
sec
ond
five
equa
l qua
rter-t
ones
17/1
5 211
)2
(
843
)2
(
256/
225 3
16)
2(
729/
640
1053
)2
(
3 ñô/2
òò, o
r 478
2969
/419
4304
73/6
4 421
)2
(
8/7
631
)2
(
6553
6/59
049×
0.4
8081
31/2
7 5
2
23/2
0
405/
352
5904
9/51
300
144/
125
15/1
3 1153
)2
(
524
)2
(, o
r ap
pro
xim
atel
y 52
/45
1.13
3333
1.13
4313
1.13
7642
1.13
7778
1.13
8789
1.13
9063
1.13
9720
1.14
0349
1.14
0625
1.14
1140
1.14
2857
1.14
3573
1.14
4867
1.14
8148
1.14
8698
1.15
0000
1.15
0568
1.15
1053
1.15
2000
1.15
3846
1.15
4723
1.15
5353
296.
509
296.
765
297.
636
297.
672
297.
936
298.
008
298.
180
298.
344
298.
417
298.
551
299.
001
299.
188
299.
526
300.
385
300.
529
300.
869
301.
018
301.
145
301.
393
301.
876
302.
105
302.
270
216.
687
218.
182
223.
256
223.
463
225.
000
225.
416
226.
415
227.
370
227.
789
228.
571
231.
174
232.
258
234.
216
239.
171
240.
000
241.
961
242.
816
243.
545
244.
969
247.
741
249.
057
250.
000
Appendices 319
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
Ñ×Ö
Ñ××
Ñ×Ø
Ñ×Ù
ÑØÐ
ÑØÑ
ÑØÒ
ÑØÓ
ÑØÔ
ÑØÕ
ÑØÖ
ÑØ×
ÑØØ
ÑØÙ
ÑÙÐ
ÑÙÑ
ÑÙÒ
ÑÙÓ
ÑÙÔ
ÑÙÕ
ÑÙÖ
ÑÙ×
cycl
ic s
uper
maj
or s
econ
d (A
) XX
VI
43-e
t trip
le a
ugm
ente
d o
ctav
e
37th
har
mon
ic
19-e
t fiv
e q
uarte
r-ton
es
14-e
t fiv
e q
uarte
r-ton
es
untri
gesim
al s
ubm
inor
third
five
qua
rter-t
ones
23-e
t fiv
e q
uarte
r-ton
es
nona
vige
simal
gra
ve o
r sm
all a
ugm
ente
d s
econ
d
9-et
five
qua
rter-t
ones
sep
timal
sub
min
or th
ird
mea
nton
e au
gmen
ted
sec
ond
(A)
IX41
2ß
31-e
t aug
men
ted
sec
ond
, or s
ubm
inor
third
53-e
t aug
men
ted
seco
nd
22-e
t aug
men
ted
seco
nd
cycl
ic a
ugm
ente
d se
cond
(A)
XXXV
III
augm
ente
d se
cond
(75t
h ha
rmon
ic)
13-e
t aug
men
ted
seco
nd
trivi
gesim
al s
ubm
inor
third
43-e
t aug
men
ted
seco
nd
sep
tend
ecim
al s
ubm
inor
third
17-e
t aug
men
ted
seco
nd
3 òö/2
ôñ 943
)2
(
37/3
2 419
)2
(
314
)2
(
36/3
1
93/8
0 523
)2
(
135/
116 2
9)
2(
7/6
1968
3/16
384×
0.44
4444
8180
731
)2
(
1253
)2
(
522
)2
(
3 óø/2
öð
75/6
4 313
)2
(
27/2
3 1043
)2
(
20/1
7 417
)2
(
1.15
5907
1.15
6129
1.15
6250
1.15
7110
1.16
0129
1.16
1290
1.16
2500
1.16
2629
1.16
3793
1.16
6529
1.16
6667
1.16
8241
1.16
9431
1.16
9924
1.17
0620
1.17
1677
1.17
1875
1.17
3460
1.17
3913
1.17
4916
1.17
6471
1.17
7147
302.
415
302.
473
302.
505
302.
730
303.
520
303.
823
304.
140
304.
174
304.
478
305.
194
305.
230
305.
642
305.
953
306.
082
306.
264
306.
541
306.
592
307.
007
307.
126
307.
388
307.
795
307.
972
250.
830
251.
163
251.
344
252.
632
257.
143
258.
874
260.
677
260.
870
262.
602
266.
667
266.
871
269.
206
270.
968
271.
698
272.
727
274.
290
274.
582
276.
923
277.
591
279.
070
281.
358
282.
353
320 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÑÙØ
ÑÙÙ
ÒÐÐ
ÒÐÑ
ÒÐÒ
ÒÐÓ
ÒÐÔ
ÒÐÕ
ÒÐÖ
ÒÐ×
ÒÐØ
ÒÐÙ
ÒÑÐ
ÒÑÑ
ÒÑÒ
ÒÑÓ
ÒÑÔ
ÒÑÕ
ÒÑÖ
ÒÑ×
ÒÑØ
ÒÑÙ
21-e
t jus
t min
or th
ird
nona
dec
imal
sup
erau
gmen
ted
sec
ond
Pyth
agor
ean
min
or th
ird, o
r trih
emito
ne (D
) III
53-e
t Pyt
hago
rean
min
or th
ird, o
r trih
emito
ne
acut
e or
larg
e au
gmen
ted
sec
ond
nona
dec
imal
sub
min
or, o
r ove
rtone
min
or th
ird (1
9th
harm
onic
)
cycl
ic P
ytha
gore
an m
inor
third
, or t
rihem
itone
(A)
L
equa
l min
or th
ird
sixth
-com
ma
mea
nton
e m
inor
third
(D)
III6
3�
43-e
t jus
t min
or th
ird
fifth
-com
ma
mea
nton
e m
inor
third
(D)
III5
3�
two-
nint
h-co
mm
a m
eant
one
min
or th
ird (D
) III
96
�
sep
tend
ecim
al s
uper
augm
ente
d s
econ
d
31-e
t sup
erau
gmen
ted
sec
ond
, or j
ust m
inor
third
mea
nton
e m
inor
third
(D)
III4
33�
two-
seve
nth-
com
ma
mea
nton
e m
inor
third
(D)
III7
6�
trivi
gesim
al s
uper
augm
ente
d s
econ
d
23-e
t jus
t min
or th
ird
65-e
t jus
t min
or th
ird
99-e
t jus
t min
or th
ird
118-
et ju
st m
inor
third
third
-com
ma
mea
nton
e an
d ju
st m
inor
third
(D)
III1�
521
)2
(
45/3
8
32/2
7 1353
)2
(
1215
/102
4
19/1
6
3 õð/2
÷ù 4
2, o
r ap
pro
xim
atel
y 44
/37
32/2
7×2
8081
1143
)2
(
32/2
7×1.
6666
678081
32/2
7×1.
58081
153/
128 8
31)
2(
32/2
7×1.
3333
338081
32/2
7×1.
1666
678081
115/
96
623
)2
(
1765
)2
(
2699
)2
(
3111
8)
2(
32/2
7×81
/80,
or 6
/5
1.17
9434
1.18
4211
1.18
5185
1.18
5325
1.18
6523
1.18
7500
1.18
7663
1.18
9207
1.19
2570
1.19
4009
1.19
4052
1.19
5041
1.19
5313
1.19
5873
1.19
6279
1.19
7872
1.19
7917
1.19
8201
1.19
8756
1.19
9661
1.19
9732
1.20
0000
308.
570
309.
820
310.
075
310.
111
310.
425
310.
680
310.
723
311.
127
312.
007
312.
383
312.
395
312.
653
312.
724
312.
871
312.
977
313.
395
313.
406
313.
480
313.
625
313.
862
313.
880
313.
951
285.
714
292.
711
294.
135
294.
340
296.
089
297.
513
297.
750
300.
000
304.
888
306.
977
307.
039
308.
473
308.
865
309.
677
310.
265
312.
569
312.
633
313.
043
313.
846
315.
152
315.
254
315.
641
Appendices 321
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÒÒÐ
ÒÒÑ
ÒÒÒ
ÒÒÓ
ÒÒÔ
ÒÒÕ
ÒÒÖ
ÒÒ×
ÒÒØ
ÒÒÙ
ÒÓÐ
ÒÓÑ
ÒÓÒ
ÒÓÓ
ÒÓÔ
ÒÓÕ
ÒÓÖ
ÒÓ×
ÒÓØ
ÒÓÙ
ÒÔÐ
ÒÔÑ
19-e
t jus
t min
or th
ird
53-e
t jus
t min
or th
ird
140-
et ju
st m
inor
third
87-e
t jus
t min
or th
ird
Pyth
agor
ean
augm
ente
d s
econ
d, o
r cyc
lic m
inor
third
(A)
IX
15-e
t jus
t min
or th
ird
und
ecim
al n
eutra
l thi
rd (
77th
har
mon
ic)
41-e
t jus
t min
or th
ird
sep
timal
sup
erau
gmen
ted
sec
ond
11-e
t sev
en q
uarte
r-ton
es
nona
vige
simal
gra
ve o
r sm
all n
eutra
l thi
rd
seve
n q
uarte
r-ton
es
untri
gesim
al s
uper
augm
ente
d s
econ
d
und
ecim
al g
rave
or s
mal
l neu
tral t
hird
18-e
t sev
en q
uarte
r-ton
es
43-e
t dou
ble
dim
inish
ed fo
urth
grav
e or
sm
all n
eutra
l thi
rd
53-e
t neu
tral t
hird
cycl
ic n
eutra
l thi
rd (A
) XX
I
trid
ecim
al g
rave
or s
mal
l neu
tral t
hird
(39
th h
arm
onic
)
7-et
neu
tral t
hird
acut
e or
larg
e ne
utra
l thi
rd
519
)2
(
1453
)2
(
3714
0)
2(
23
87)
2(
3 ù/2
ñô, o
r 196
83/1
6384
4
15)
2(
77/6
4 1141
)2
(
135/
112 3
11)
2(
29/2
4
75/6
2
155/
128
40/3
3 518
)2
(
1243
)2
(
243/
200
1553
)2
(
3 òñ/2
óó
39/3
2 27
)2
(
8000
/656
1
1.20
0103
1.20
0929
1.20
1041
1.20
1110
1.20
1355
1.20
3025
1.20
3125
1.20
4382
1.20
5357
1.20
8089
1.20
8333
1.20
9677
1.21
0938
1.21
2121
1.21
2326
1.21
3412
1.21
5000
1.21
6738
1.21
7745
1.21
8750
1.21
9014
1.21
9326
313.
978
314.
194
314.
223
314.
241
314.
305
314.
742
314.
768
315.
097
315.
352
316.
067
316.
131
316.
483
316.
812
317.
122
317.
175
317.
460
317.
875
318.
330
318.
593
318.
856
318.
925
319.
007
315.
789
316.
981
317.
143
317.
241
317.
595
320.
000
320.
144
321.
951
323.
353
327.
273
327.
622
329.
547
331.
349
333.
041
333.
333
334.
884
337.
148
339.
623
341.
055
342.
483
342.
857
343.
301
322 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÒÔÒ
ÒÔÓ
ÒÔÔ
ÒÔÕ
ÒÔÖ
ÒÔ×
ÒÔØ
ÒÔÙ
ÒÕÐ
ÒÕÑ
ÒÕÒ
ÒÕÓ
ÒÕÔ
ÒÕÕ
ÒÕÖ
ÒÕ×
ÒÕØ
ÒÕÙ
ÒÖÐ
ÒÖÑ
ÒÖÒ
ÒÖÓ
dou
ble
augm
ente
d se
cond
und
ecim
al a
cute
or l
arge
neu
tral t
hird
31-e
t dou
ble
augm
ente
d s
econ
d, o
r neu
tral t
hird
seve
n eq
ual q
uarte
r-ton
es
17-e
t neu
tral t
hird
trid
ecim
al a
cute
or l
arge
neu
tral t
hird
10-e
t gra
ve o
r sm
all m
ajor
third
53-e
t gra
ve o
r sm
all m
ajor
third
43-e
t dou
ble
augm
ente
d s
econ
d
cycl
ic g
rave
or s
mal
l maj
or th
ird (A
) XX
XIII
79th
har
mon
ic
grav
e or
sm
all m
ajor
third
23-e
t gra
ve o
r sm
all m
ajor
third
13-e
t gra
ve o
r sm
all m
ajor
third
nona
vige
simal
acu
te o
r lar
ge n
eutra
l thi
rd
16-e
t gra
ve o
r sm
all m
ajor
third
19-e
t jus
t maj
or th
ird
third
-com
ma
mea
nton
e m
ajor
third
(A)
IV31
1ß
41-e
t jus
t maj
or th
ird
22-e
t jus
t maj
or th
ird
two-
seve
nth-
com
ma
mea
nton
e m
ajor
third
(A)
IV71
1ß
Pyth
agor
ean
dim
inish
ed fo
urth
(D)
VIII
625/
512
11/9
931
)2
(
724
)2
(, o
r ap
pro
xim
atel
y 60
/49
517
)2
(
16/1
3 310
)2
(
1653
)2
(
1343
)2
(
3 óó/2
õò
79/6
4
100/
81
723
)2
(
413
)2
(
36/2
9 516
)2
(
619
)2
(
81/6
4×0.
758180
13
41)
2(
7
22)
2(
81/6
4×0.
875
8180
8192
/656
1
1.22
0703
1.22
2222
1.22
2914
1.22
4054
1.22
6135
1.23
0769
1.23
1144
1.23
2756
1.23
3131
1.23
4359
1.23
4375
1.23
4568
1.23
4860
1.23
7726
1.24
1379
1.24
1858
1.24
4693
1.24
4835
1.24
5801
1.24
6758
1.24
7784
1.24
8590
319.
367
319.
765
319.
945
320.
244
320.
788
322.
001
322.
099
322.
520
322.
618
322.
940
322.
944
322.
995
323.
071
323.
821
324.
777
324.
902
325.
643
325.
681
325.
933
326.
184
326.
452
326.
663
345.
255
347.
408
348.
387
350.
000
352.
941
359.
472
360.
000
362.
264
362.
791
364.
515
364.
537
364.
807
365.
217
369.
231
374.
333
375.
000
378.
947
379.
145
380.
488
381.
818
383.
241
384.
360
Appendices 323
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÒÖÔ
ÒÖÕ
ÒÖÖ
ÒÖ×
ÒÖØ
ÒÖÙ
Ò×Ð
Ò×Ñ
Ò×Ò
Ò×Ó
Ò×Ô
Ò×Õ
Ò×Ö
Ò××
Ò×Ø
Ò×Ù
ÒØÐ
ÒØÑ
ÒØÒ
ÒØÓ
ÒØÔ
ÒØÕ
53-e
t jus
t maj
or th
ird
140-
et ju
st m
ajor
third
87-e
t jus
t maj
or th
ird
mea
nton
e an
d ju
st m
ajor
third
(A)
IV1ß
(5t
h ha
rmon
ic)
118-
et ju
st m
ajor
third
31-e
t jus
t maj
or th
ird
65-e
t jus
t maj
or th
ird
99-e
t jus
t maj
or th
ird
cycl
ic m
ajor
third
(A)
XLV
two-
nint
h-co
mm
a m
eant
one
maj
or th
ird (A
) IV
98
�
fifth
-com
ma
mea
nton
e m
ajor
third
(A)
IV5
4ß
43-e
t jus
t maj
or th
ird
sixth
-com
ma
mea
nton
e m
ajor
third
(A)
IV6
4ß
sep
tend
ecim
al s
uper
maj
or th
ird
equa
l maj
or th
ird
nona
dec
imal
sup
erm
ajor
third
grav
e or
sm
all d
imin
ished
four
th
53-e
t Pyt
hago
rean
maj
or th
ird, o
r dito
ne
Pyth
agor
ean
maj
or th
ird, o
r dito
ne (A
) IV
(81s
t har
mon
ic)
57th
cyc
lic fi
fth (A
) LV
II
23-e
t Pyt
hago
rean
maj
or th
ird, o
r dito
ne
und
ecim
al d
imin
ished
four
th
1753
)2
(
4514
0)
2(
28
87)
2(
81/6
4×80
/81,
or 5
/4
3811
8)
2(
10
31)
2(
21
65)
2(
32
99)
2(
3 ôõ/2
֖
81/6
4×1.
125
8180
81/6
4×1.
258180
14
43)
2(
81/6
4×1.
58180
34/2
7
32
, or a
pp
roxi
mat
ely
63/5
0
24/1
9
512/
405
1853
)2
(
81/6
4
3 õ÷/2
ùð 823
)2
(
14/1
1
1.24
8984
1.24
9567
1.24
9923
1.25
0000
1.25
0092
1.25
0566
1.25
0996
1.25
1131
1.25
1200
1.25
1727
1.25
3109
1.25
3169
1.25
5187
1.25
9259
1.25
9921
1.26
3158
1.26
4198
1.26
5426
1.26
5625
1.26
8271
1.27
2642
1.27
2727
326.
766
326.
919
327.
012
327.
032
327.
056
327.
180
327.
292
327.
328
327.
346
327.
484
327.
845
327.
861
328.
389
329.
454
329.
628
330.
474
330.
746
331.
068
331.
120
331.
812
332.
956
332.
978
384.
906
385.
714
386.
207
386.
314
386.
441
387.
097
387.
692
387.
879
387.
975
388.
703
390.
615
390.
698
393.
482
399.
090
400.
000
404.
442
405.
866
407.
547
407.
820
411.
435
417.
391
417.
508
324 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÒØÖ
ÒØ×
ÒØØ
ÒØÙ
ÒÙÐ
ÒÙÑ
ÒÙÒ
ÒÙÓ
ÒÙÔ
ÒÙÕ
ÒÙÖ
ÒÙ×
ÒÙØ
ÒÙÙ
ÓÐÐ
ÓÐÑ
ÓÐÒ
ÓÐÓ
ÓÐÔ
ÓÐÕ
ÓÐÖ
ÓÐ×
43-e
t dim
inish
ed fo
urth
20-e
t acu
te o
r lar
ge m
ajor
third
17-e
t acu
te o
r lar
ge m
ajor
third
trivi
gesim
al s
uper
maj
or th
ird
31-e
t sup
erm
ajor
third
, or d
imin
ished
four
th
mea
nton
e an
d d
imin
ished
four
th, o
r acu
te o
r lar
ge m
ajor
third
(D)
VIII
2�
14-e
t acu
te o
r lar
ge m
ajor
third
41st
har
mon
ic
53-e
t acu
te o
r lar
ge m
ajor
third
cycl
ic a
cute
or l
arge
maj
or th
ird (A
) XV
I
sep
timal
sup
erm
ajor
third
11-e
t nin
e q
uarte
r-ton
es
untri
gesim
al s
ubfo
urth
19-e
t nin
e q
uarte
r-ton
es
nine
qua
rter-t
ones
, or u
ntrig
esim
al s
uper
maj
or th
ird
43-e
t trip
le d
imin
ished
fifth
nine
eq
ual q
uarte
r-ton
es
83rd
har
mon
ic
53-e
t sub
four
th
cycl
ic s
ubfo
urth
(A)
XXV
III
augm
ente
d th
ird, o
r sub
four
th
21-e
t sub
four
th
1543
)2
(
720
)2
(
617
)2
(
23/1
8 1131
)2
(
8192
/656
1×0.
58081
, or 3
2/25
5
14)
2(
41/3
2 1953
)2
(
3 ñö/2
òõ
9/7
411
)2
(
40/3
1 719
)2
(
31/2
4 1643
)2
(
38
)2
(, o
r ap
pro
xim
atel
y 83
/64
83/6
4 2053
)2
(
3 òø/2
ôô
125/
96
821
)2
(
1.27
3534
1.27
4561
1.27
7162
1.27
7778
1.27
8843
1.28
0000
1.28
0887
1.28
1250
1.28
2084
1.28
2892
1.28
5714
1.28
6665
1.29
0323
1.29
0939
1.29
1667
1.29
4229
1.29
6840
1.29
6875
1.29
8961
1.30
0395
1.30
2083
1.30
2201
333.
189
333.
458
334.
138
334.
299
334.
578
334.
881
335.
113
335.
208
335.
426
335.
637
336.
376
336.
624
337.
581
337.
743
337.
933
338.
603
339.
286
339.
296
339.
841
340.
217
340.
658
340.
689
418.
605
420.
000
423.
529
424.
364
425.
806
427.
373
428.
571
429.
062
430.
189
431.
280
435.
084
436.
364
441.
278
442.
105
443.
081
446.
512
450.
000
450.
047
452.
830
454.
740
456.
986
457.
143
Appendices 325
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÓÐØ
ÓÐÙ
ÓÑÐ
ÓÑÑ
ÓÑÒ
ÓÑÓ
ÓÑÔ
ÓÑÕ
ÓÑÖ
ÓÑ×
ÓÑØ
ÓÑÙ
ÓÒÐ
ÓÒÑ
ÓÒÒ
ÓÒÓ
ÓÒÔ
ÓÒÕ
ÓÒÖ
ÓÒ×
ÓÒØ
ÓÒÙ
trivi
gesim
al s
ubfo
urth
13-e
t sub
four
th
mea
nton
e au
gmen
ted
third
(A) X
I4
32
ß
31-e
t aug
men
ted
third
, or s
ubfo
urth
18-e
t gra
ve o
r sm
all f
ourth
23-e
t gra
ve o
r sm
all f
ourth
sep
timal
sub
four
th (
21st
har
mon
ic)
43-e
t aug
men
ted
third
53-e
t gra
ve o
r sm
all f
ourth
grav
e or
sm
all f
ourth
cycl
ic g
rave
or s
mal
l fou
rth (A
) XL
acut
e or
larg
e au
gmen
ted
third
nona
dec
imal
sub
four
th
5-et
gra
ve o
r sm
all f
ourth
sep
tend
ecim
al s
ubfo
urth
22-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
sep
tend
ecim
al s
uper
augm
ente
d th
ird (
85th
har
mon
ic)
17-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
87-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
99-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
140-
et ju
st a
nd P
ytha
gore
an p
erfe
ct fo
urth
41-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
30/2
3 513
)2
(
1771
47/1
3107
2×0.
3636
368180
12
31)
2(
7
18)
2(
9
23)
2(
21/1
6 1743
)2
(
2153
)2
(
320/
243
3 ôð/2
öó
675/
512
95/7
2 25
)2
(
45/3
4 922
)2
(
85/6
4 717
)2
(
3687
)2
(
4199
)2
(
5814
0)
2(
17
41)
2(
1.30
4348
1.30
5512
1.30
6133
1.30
7759
1.30
9385
1.31
1579
1.31
2500
1.31
5261
1.31
6061
1.31
6872
1.31
8137
1.31
8359
1.31
9444
1.31
9508
1.32
3529
1.32
7849
1.32
8125
1.33
0312
1.33
2184
1.33
2505
1.33
2639
1.33
2961
341.
251
341.
555
341.
718
342.
143
342.
568
343.
143
343.
384
344.
106
344.
315
344.
527
344.
858
344.
917
345.
200
345.
217
346.
269
347.
399
347.
471
348.
044
348.
533
348.
617
348.
652
348.
737
459.
994
461.
538
462.
363
464.
516
466.
667
469.
565
470.
781
474.
419
475.
472
476.
539
478.
200
478.
492
479.
917
480.
000
485.
268
490.
909
491.
269
494.
118
496.
552
496.
970
497.
143
497.
561
326 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÓÓÐ
ÓÓÑ
ÓÓÒ
ÓÓÓ
ÓÓÔ
ÓÓÕ
ÓÓÖ
ÓÓ×
ÓÓØ
ÓÓÙ
ÓÔÐ
ÓÔÑ
ÓÔÒ
ÓÔÓ
ÓÔÔ
ÓÔÕ
ÓÔÖ
ÓÔ×
ÓÔØ
ÓÔÙ
ÓÕÐ
ÓÕÑ
just
and
Pyt
hago
rean
per
fect
four
th (D
) I
53-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
118-
et ju
st a
nd P
ytha
gore
an p
erfe
ct fo
urth
65-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
equa
l per
fect
four
th
sixth
-com
ma
mea
nton
e p
erfe
ct fo
urth
(D)
I61
�
cycl
ic p
erfe
ct fo
urth
(A)
LII
43-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
fifth
-com
ma
mea
nton
e p
erfe
ct fo
urth
(D)
I51
�
two-
nint
h-co
mm
a m
eant
one
per
fect
four
th (D
) I
92
�
31-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
mea
nton
e p
erfe
ct fo
urth
(D)
I 41
�
two-
seve
nth-
com
ma
mea
nton
e p
erfe
ct fo
urth
(D)
I7
2�
third
-com
ma
mea
nton
e p
erfe
ct fo
urth
(D)
I31
�
19-e
t jus
t and
Pyt
hago
rean
per
fect
four
th
43rd
har
mon
ic
7-et
acu
te o
r lar
ge fo
urth
subd
imin
ished
fifth
acut
e or
larg
e fo
urth
53-e
t acu
te o
r lar
ge fo
urth
Pyth
agor
ean
augm
ente
d th
ird, o
r cyc
lic a
cute
or l
arge
four
th (A
) XI
23-e
t acu
te o
r lar
ge fo
urth
4/3
2253
)2
(
4911
8)
2(
27
65)
2(
5
12)
2(
, or a
ppro
xim
atel
y 10
24/7
67
4/3×
68081
3 õò/2
øò 1843
)2
(
4/3×
58081
4/3×
4.5
8081
1331
)2
(
4/3×
48081
4/3×
3.5
8081
4/3×
38081
8
19)
2(
43/3
2 37
)2
(
8192
/607
5
27/2
0 2353
)2
(
3 ññ/2
ñ÷, o
r 177
147/
1310
72
1023
)2
(
1.33
3333
1.33
3386
1.33
3534
1.33
3654
1.33
4840
1.33
6097
1.33
6120
1.33
6634
1.33
6650
1.33
7019
1.33
7329
1.33
7481
1.33
8074
1.33
8866
1.33
8904
1.34
3750
1.34
5900
1.34
8477
1.35
0000
1.35
0939
1.35
1524
1.35
1707
348.
834
348.
848
348.
886
348.
918
349.
228
349.
557
349.
563
349.
698
349.
702
349.
798
349.
880
349.
919
350.
074
350.
282
350.
292
351.
559
352.
122
352.
796
353.
195
353.
440
353.
593
353.
641
498.
045
498.
113
498.
305
498.
462
500.
000
501.
629
501.
660
502.
326
502.
346
502.
824
503.
226
503.
422
504.
190
505.
214
505.
263
511.
518
514.
286
517.
598
519.
551
520.
755
521.
505
521.
739
Appendices 327
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÓÕÒ
ÓÕÓ
ÓÕÔ
ÓÕÕ
ÓÕÖ
ÓÕ×
ÓÕØ
ÓÕÙ
ÓÖÐ
ÓÖÑ
ÓÖÒ
ÓÖÓ
ÓÖÔ
ÓÖÕ
ÓÖÖ
ÓÖ×
ÓÖØ
ÓÖÙ
Ó×Ð
Ó×Ñ
Ó×Ò
Ó×Ó
trid
ecim
al s
uper
four
th
16-e
t acu
te o
r lar
ge fo
urth
43-e
t dou
ble
dim
inish
ed fi
fth
nona
vige
simal
sup
erfo
urth
(87
th h
arm
onic
)
9-et
sup
erfo
urth
und
ecim
al tr
itone
, or a
ugm
ente
d fo
urth
grav
e or
sm
all a
ugm
ente
d fo
urth
, or s
uper
four
th
20-e
t sup
erfo
urth
31-e
t sup
erfo
urth
, or d
imin
ished
fifth
53-e
t sup
erfo
urth
cycl
ic s
uper
four
th (A
) XX
III
11-e
t sup
erfo
urth
elev
en e
qua
l qua
rter-t
ones
und
ecim
al s
uper
four
th (
11th
har
mon
ic)
untri
gesim
al s
ubd
imin
ished
fifth
13-e
t ele
ven
qua
rter-t
ones
elev
en q
uarte
r-ton
es
nona
vige
simal
trito
ne, o
r aug
men
ted
four
th
43-e
t dou
ble
augm
ente
d th
ird
15-e
t ele
ven
qua
rter-t
ones
trid
ecim
al tr
itone
, or a
ugm
ente
d fo
urth
17-e
t gra
ve o
r sm
all a
ugm
ente
d fo
urth
65/4
8 716
)2
(
1943
)2
(
87/6
4 49
)2
(
15/1
1
512/
375 9
20)
2(
14
31)
2(
24
53)
2(
3 òó/2
óö 511
)2
(
1124
)2
(, o
r app
roxi
mat
ely
1024
/745
11/8
128/
93
613
)2
(
62/4
5
40/2
9 2043
)2
(
715
)2
(
18/1
3 817
)2
(
1.35
4167
1.35
4256
1.35
8355
1.35
9375
1.36
0790
1.36
3636
1.36
5333
1.36
6040
1.36
7568
1.36
8723
1.36
9964
1.37
0351
1.37
3954
1.37
5000
1.37
6344
1.37
7009
1.37
7778
1.37
9310
1.38
0429
1.38
1913
1.38
4615
1.38
5674
354.
285
354.
308
355.
380
355.
647
356.
017
356.
762
357.
206
357.
391
357.
791
358.
093
358.
417
358.
519
359.
461
359.
735
360.
087
360.
261
360.
462
360.
863
361.
155
361.
544
362.
251
362.
528
524.
886
525.
000
530.
233
531.
532
533.
333
536.
951
539.
104
540.
000
541.
935
543.
396
544.
965
545.
455
550.
000
551.
318
553.
009
553.
846
554.
812
556.
737
558.
140
560.
000
563.
382
564.
706
328 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
Ó×Ô
Ó×Õ
Ó×Ö
Ó××
Ó×Ø
Ó×Ù
ÓØÐ
ÓØÑ
ÓØÒ
ÓØÓ
ÓØÔ
ÓØÕ
ÓØÖ
ÓØ×
ÓØØ
ÓØÙ
ÓÙÐ
ÓÙÑ
ÓÙÒ
ÓÙÓ
ÓÙÔ
ÓÙÕ
53-e
t gra
ve o
r sm
all a
ugm
ente
d fo
urth
19-e
t gra
ve o
r sm
all a
ugm
ente
d fo
urth
cycl
ic g
rave
or s
mal
l aug
men
ted
four
th (A
) XX
XV
grav
e or
sm
all a
ugm
ente
d fo
urth
89th
har
mon
ic
21-e
t gra
ve o
r sm
all a
ugm
ente
d fo
urth
trivi
gesim
al s
ubd
imin
ished
fifth
23-e
t gra
ve o
r sm
all a
ugm
ente
d fo
urth
mea
nton
e tri
tone
, or a
ugm
ente
d fo
urth
(A)
VI
211ß
31-e
t aug
men
ted
four
th, o
r sub
dim
inish
ed fi
fth
sep
timal
sub
dim
inish
ed fi
fth
43-e
t jus
t trit
one,
or a
ugm
ente
d fo
urth
Pyth
agor
ean
dim
inish
ed fi
fth (D
) V
I
53-e
t jus
t trit
one,
or a
ugm
ente
d fo
urth
just
trito
ne, o
r aug
men
ted
four
th (
45th
har
mon
ic)
nona
dec
imal
sub
dim
inish
ed fi
fth
cycl
ic tr
itone
, or a
ugm
ente
d fo
urth
(A)
XLV
II
sep
tend
ecim
al s
ubd
imin
ished
fifth
equa
l trit
one,
or a
ugm
ente
d fo
urth
sep
tend
ecim
al tr
itone
, or s
uper
augm
ente
d fo
urth
nona
dec
imal
trito
ne, o
r sup
erau
gmen
ted
four
th
trid
ecim
al d
imin
ished
fifth
(91
st h
arm
onic
)
2553
)2
(
919
)2
(
3 óõ/2
õõ
25/1
8
89/6
4 1021
)2
(
32/2
3 1123
)2
(
729/
512×
0.66
6667
8180
1531
)2
(
7/5
2143
)2
(
1024
/729
26
53)
2(
45/3
2
38/2
7
3 ô÷/2
÷ô
24/1
7
22
, or a
ppro
xim
atel
y 18
1/12
8
17/1
2
27/1
9
91/6
4
1.38
6741
1.38
8651
1.38
8654
1.38
8889
1.39
0625
1.39
1066
1.39
1304
1.39
3063
1.39
7542
1.39
8491
1.40
0000
1.40
2861
1.40
4664
1.40
4996
1.40
6250
1.40
7407
1.40
7600
1.41
1765
1.41
4214
1.41
6667
1.42
1053
1.42
1875
362.
807
363.
307
363.
307
363.
369
363.
823
363.
938
364.
001
364.
461
365.
633
365.
881
366.
276
367.
024
367.
496
367.
583
367.
911
368.
214
368.
264
369.
354
369.
994
370.
636
371.
784
371.
999
566.
038
568.
421
568.
425
568.
717
570.
880
571.
429
571.
726
573.
913
579.
471
580.
645
582.
512
586.
047
588.
270
588.
679
590.
224
591.
648
591.
885
597.
000
600.
000
603.
000
608.
352
609.
354
Appendices 329
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÓÙÖ
ÓÙ×
ÓÙØ
ÓÙÙ
ÔÐÐ
ÔÐÑ
ÔÐÒ
ÔÐÓ
ÔÐÔ
ÔÐÕ
ÔÐÖ
ÔÐ×
ÔÐØ
ÔÐÙ
ÔÑÐ
ÔÑÑ
ÔÑÒ
ÔÑÓ
ÔÑÔ
ÔÑÕ
ÔÑÖ
ÔÑ×
dim
inish
ed fi
fth, o
r acu
te o
r lar
ge tr
itone
, or a
ugm
ente
d fo
urth
53-e
t acu
te o
r lar
ge tr
itone
, or a
ugm
ente
d fo
urth
Pyth
agor
ean
trito
ne, o
r aug
men
ted
four
th (A
) VI
43-e
t dim
inish
ed fi
fth
59th
cyc
lic fi
fth (A
) LIX
sep
timal
trito
ne, o
r sup
erau
gmen
ted
four
th
31-e
t sup
erau
gmen
ted
four
th, o
r dim
inish
ed fi
fth
mea
nton
e d
imin
ished
fifth
(D) V
I21
1�
23-e
t acu
te o
r lar
ge d
imin
ished
fifth
trivi
gesim
al s
uper
augm
ente
d fo
urth
(23r
d ha
rmon
ic)
21-e
t acu
te o
r lar
ge d
imin
ished
fifth
acut
e or
larg
e d
imin
ished
fifth
19-e
t acu
te o
r lar
ge d
imin
ished
fifth
53-e
t acu
te o
r lar
ge d
imin
ished
fifth
cycl
ic a
cute
or l
arge
dim
inish
ed fi
fth (A
) XV
III
17-e
t acu
te o
r lar
ge d
imin
ished
fifth
trid
ecim
al s
ubd
imin
ished
fifth
15-e
t thi
rteen
qua
rter-t
ones
43-e
t dou
ble
dim
inish
ed s
ixth
nona
vige
simal
sub
dim
inish
ed fi
fth
thirt
een
qua
rter-t
ones
13-e
t thi
rteen
qua
rter-t
ones
64/4
5 2753
)2
(
3 ö/2
ù, or
729
/512
22
43)
2(
3 õù/2
ùó
10/7
1631
)2
(
1024
/729
×0.
6666
678081
12
23)
2(
23/1
6 1121
)2
(
36/2
5 1019
)2
(
2853
)2
(
3 ñø/2
òø 917
)2
(
13/9
815
)2
(
2343
)2
(
29/2
0
90/6
2 713
)2
(
1.42
2222
1.42
3492
1.42
3828
1.42
5658
1.42
6804
1.42
8571
1.43
0113
1.43
1084
1.43
5685
1.43
7500
1.43
7747
1.44
0000
1.44
0247
1.44
2231
1.44
3254
1.44
3341
1.44
4444
1.44
7269
1.44
8825
1.45
0000
1.45
1613
1.45
2423
372.
090
372.
422
372.
510
372.
989
373.
289
373.
751
374.
154
374.
408
375.
612
376.
087
376.
151
376.
741
376.
805
377.
324
377.
592
377.
615
377.
904
378.
643
379.
050
379.
357
379.
779
379.
991
609.
776
611.
321
611.
730
613.
953
615.
345
617.
488
619.
355
620.
529
626.
087
628.
274
628.
571
631.
283
631.
579
633.
962
635.
190
635.
294
636.
618
640.
000
641.
860
643.
263
645.
188
646.
154
330 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÔÑØ
ÔÑÙ
ÔÒÐ
ÔÒÑ
ÔÒÒ
ÔÒÓ
ÔÒÔ
ÔÒÕ
ÔÒÖ
ÔÒ×
ÔÒØ
ÔÒÙ
ÔÓÐ
ÔÓÑ
ÔÓÒ
ÔÓÓ
ÔÓÔ
ÔÓÕ
ÔÓÖ
ÔÓ×
ÔÓØ
ÔÓÙ
untri
gesim
al s
uper
augm
ente
d fo
urth
(93
rd h
arm
onic
)
und
ecim
al s
ubfif
th
thirt
een
equa
l qua
rter-t
ones
11-e
t sub
fifth
mea
nton
e d
oubl
e au
gmen
ted
four
th (A
) XI
II41
3ß
53-e
t sub
fifth
31-e
t dou
ble
augm
ente
d fo
urth
, or s
ubfif
th
cycl
ic s
ubfif
th (A
) XX
X
20-e
t sub
fifth
acut
e or
larg
e d
oubl
e au
gmen
ted
four
th, o
r sub
fifth
und
ecim
al s
ubd
imin
ished
fifth
47th
har
mon
ic
9-et
sub
fifth
nona
vige
simal
sub
fifth
43-e
t dou
ble
augm
ente
d fo
urth
16-e
t gra
ve o
r sm
all f
ifth
trid
ecim
al s
ubfif
th
23-e
t gra
ve o
r sm
all f
ifth
Pyth
agor
ean
dim
inish
ed s
ixth
(D)
XI
53-e
t gra
ve o
r sm
all f
ifth
grav
e or
sm
all f
ifth
cycl
ic g
rave
or s
mal
l fift
h (A
) XL
II
93/6
4
16/1
1 13
24)
2(
, or a
ppro
xim
atel
y 74
5/51
2 6
11)
2(
1594
323/
1048
576×
0.30
7692
8180
2953
)2
(
1731
)2
(
3 óð/2
ô÷ 1120
)2
(
375/
256
22/1
5
47/3
2 59
)2
(
128/
87
2443
)2
(
916
)2
(
96/6
5 1323
)2
(
2621
44/1
7714
7 30
53)
2(
40/2
7
3 ôò/2
öö
1.45
3125
1.45
4545
1.45
5653
1.45
9480
1.46
0302
1.46
1216
1.46
2450
1.46
2944
1.46
4086
1.46
4844
1.46
6667
1.46
8750
1.46
9734
1.47
1264
1.47
2369
1.47
6826
1.47
6923
1.47
9610
1.47
9811
1.48
0452
1.48
1481
1.48
2904
380.
175
380.
546
380.
836
381.
837
382.
052
382.
292
382.
614
382.
744
383.
042
383.
241
383.
717
384.
263
384.
520
384.
920
385.
209
386.
375
386.
401
387.
104
387.
156
387.
324
387.
593
387.
966
646.
991
648.
682
650.
000
654.
545
655.
536
656.
604
658.
065
658.
650
660.
000
660.
896
663.
049
665.
507
666.
667
668.
468
669.
767
675.
000
675.
114
678.
261
678.
495
679.
245
680.
449
682.
110
Appendices 331
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÔÔÐ
ÔÔÑ
ÔÔÒ
ÔÔÓ
ÔÔÔ
ÔÔÕ
ÔÔÖ
ÔÔ×
ÔÔØ
ÔÔÙ
ÔÕÐ
ÔÕÑ
ÔÕÒ
ÔÕÓ
ÔÕÔ
ÔÕÕ
ÔÕÖ
ÔÕ×
ÔÕØ
ÔÕÙ
ÔÖÐ
ÔÖÑ
acut
e or
larg
e d
oubl
e su
per
augm
ente
d fo
urth
nona
dec
imal
sub
fifth
(95
th h
arm
onic
)
7-et
gra
ve o
r sm
all f
ifth
19-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
third
-com
ma
mea
nton
e p
erfe
ct fi
fth (A
) I31
ß
two-
seve
nth-
com
ma
mea
nton
e p
erfe
ct fi
fth (A
) I7
2ß
mea
nton
e p
erfe
ct fi
fth (A
) I41
ß
31-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
two-
nint
h-co
mm
a m
eant
one
per
fect
fifth
(A)
I9
2ß
fifth
-com
ma
mea
nton
e p
erfe
ct fi
fth (A
) I51
ß
43-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
sixth
-com
ma
mea
nton
e p
erfe
ct fi
fth (A
) I61
ß
equa
l per
fect
fifth
65-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
118-
et ju
st a
nd P
ytha
gore
an p
erfe
ct fi
fth
53-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
just
and
Pyt
hago
rean
per
fect
fifth
(A) I
(3rd
har
mon
ic)
41-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
140-
et ju
st a
nd P
ytha
gore
an p
erfe
ct fi
fth
99-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
87-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
54th
cyc
lic fi
fth (A
) LIV
6075
/409
6
95/6
4 47
)2
(
1119
)2
(
3/2×
38180
3/2×
3.5
8180
3/2×
48180
18
31)
2(
3/2×
4.5
8180
3/2×
58180
25
43)
2(
3/2×
68180
7
12)
2(
, or a
ppro
xim
atel
y 76
7/51
2 38
65)
2(
69
118
)2
(
3153
)2
(
3/2 24
41)
2(
82
140
)2
(
5899
)2
(
5187
)2
(
3 õô/2
øõ
1.48
3154
1.48
4375
1.48
5994
1.49
3759
1.49
3802
1.49
4686
1.49
5349
1.49
5518
1.49
5865
1.49
6278
1.49
6296
1.49
6898
1.49
8307
1.49
9639
1.49
9775
1.49
9941
1.50
0000
1.50
0419
1.50
0782
1.50
0932
1.50
1294
1.50
3135
388.
031
388.
350
388.
774
390.
806
390.
817
391.
048
391.
221
391.
266
391.
356
393.
415
391.
469
391.
627
391.
995
392.
344
392.
379
392.
423
392.
438
392.
548
392.
643
392.
682
392.
777
393.
259
682.
402
683.
827
685.
714
694.
737
694.
786
695.
810
696.
578
696.
774
697.
176
697.
654
697.
674
698.
371
700.
000
701.
538
701.
695
701.
887
701.
955
702.
439
702.
857
703.
030
703.
448
705.
570
332 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÔÖÒ
ÔÖÓ
ÔÖÔ
ÔÖÕ
ÔÖÖ
ÔÖ×
ÔÖØ
ÔÖÙ
Ô×Ð
Ô×Ñ
Ô×Ò
Ô×Ó
Ô×Ô
Ô×Õ
Ô×Ö
Ô××
Ô×Ø
Ô×Ù
ÔØÐ
ÔØÑ
ÔØÒ
ÔØÓ
17-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
22-e
t jus
t and
Pyt
hago
rean
per
fect
fifth
sep
tend
ecim
al s
uper
fifth
und
ecim
al g
rave
or s
mal
l aug
men
ted
fifth
97th
har
mon
ic
5-et
acu
te o
r lar
ge fi
fth
nona
dec
imal
sup
erfif
th
subd
imin
ished
six
th
acut
e or
larg
e fif
th
53-e
t acu
te o
r lar
ge fi
fth
Pyth
agor
ean
dou
ble
augm
ente
d fo
urth
, or c
yclic
acu
te o
r lar
ge fi
fth (A
) XI
II
43-e
t dim
inish
ed s
ixth
sep
timal
sup
erfif
th
23-e
t acu
te o
r lar
ge fi
fth
18-e
t sup
erfif
th
31-e
t sup
erfif
th, o
r dim
inish
ed s
ixth
mea
nton
e d
imin
ished
six
th a
nd w
olf f
ifth
(D) X
I4
32�
sep
timal
dim
inish
ed s
ixth
(49
th h
arm
onic
)
13-e
t sup
erfif
th
trivi
gesim
al s
uper
fifth
21-e
t sup
erfif
th
dim
inish
ed s
ixth
, or s
uper
fifth
1017
)2
(
1322
)2
(
68/4
5
50/3
3
97/6
4 35
)2
(
144/
95
1024
/675
243/
160
3253
)2
(
3 ñó/2
òð, o
r 159
4323
/104
8576
26
43)
2(
32/2
1 1423
)2
(
1118
)2
(
1931
)2
(
2621
44/1
7714
7×0.
3636
368081
49/3
2 813
)2
(
23/1
5 1321
)2
(
192/
125
1.50
3407
1.50
6196
1.51
1111
1.51
5152
1.51
5625
1.51
5717
1.51
5789
1.51
7037
1.51
8750
1.51
9686
1.52
0465
1.52
0611
1.52
3810
1.52
4880
1.52
7435
1.52
9334
1.53
1237
1.53
1250
1.53
1966
1.53
3333
1.53
5861
1.53
6000
393.
330
394.
059
395.
345
396.
402
396.
526
396.
550
396.
569
396.
896
397.
344
397.
589
397.
792
397.
831
398.
104
398.
948
399.
616
400.
113
400.
611
400.
614
400.
802
401.
159
401.
820
401.
857
705.
882
709.
091
714.
732
719.
354
719.
895
720.
000
720.
083
721.
508
723.
014
724.
528
725.
415
725.
581
729.
219
730.
435
733.
333
735.
484
737.
637
737.
652
738.
462
740.
006
742.
857
743.
014
Appendices 333
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÔØÔ
ÔØÕ
ÔØÖ
ÔØ×
ÔØØ
ÔØÙ
ÔÙÐ
ÔÙÑ
ÔÙÒ
ÔÙÓ
ÔÙÔ
ÔÙÕ
ÔÙÖ
ÔÙ×
ÔÙØ
ÔÙÙ
ÕÐÐ
ÕÐÑ
ÕÐÒ
ÕÐÓ
ÕÐÔ
ÕÐÕ
trid
ecim
al g
rave
or s
mal
l aug
men
ted
fifth
53-e
t sup
erfif
th
cycl
ic s
uper
fifth
(A)
XXV
fifte
en e
qua
l qua
rter-t
ones
43-e
t trip
le a
ugm
ente
d fo
urth
und
ecim
al s
uper
fifth
(99t
h ha
rmon
ic)
untri
gesim
al s
ubm
inor
six
th
19-e
t Pyt
hago
rean
min
or s
ixth
fifte
en q
uarte
r-ton
es, o
r unt
riges
imal
sup
erfif
th
nona
vige
simal
gra
ve o
r sm
all a
ugm
ente
d s
ixth
11-e
t fift
een
qua
rter-t
ones
sep
timal
sub
min
or s
ixth
53-e
t aug
men
ted
fifth
14-e
t aug
men
ted
fifth
cycl
ic a
ugm
ente
d fi
fth (A
) XX
XVII
mea
nton
e au
gmen
ted
and
aug
men
ted
fifth
(A) V
III2
ß (
25th
har
mon
ic)
31-e
t aug
men
ted
fifth
, or s
ubm
inor
six
th
trivi
gesim
al s
ubm
inor
six
th
17-e
t aug
men
ted
fifth
20-e
t aug
men
ted
fifth
43-e
t aug
men
ted
fifth
und
ecim
al a
ugm
ente
d fi
fth
20/1
3 3353
)2
(
3 òõ/2
óù 5
8)
2(
, or a
ppro
xim
atel
y 12
8/83
27
43)
2(
99/6
4
48/3
1 1219
)2
(
31/2
0
45/2
9 711
)2
(
14/9
3453
)2
(
914
)2
(
3 ó÷/2
õø
6561
/409
6×0.
58180
, or 2
5/16
20
31)
2(
36/2
3 1117
)2
(
1320
)2
(
2843
)2
(
11/7
1.53
8462
1.53
9692
1.54
1209
1.54
2211
1.54
5321
1.54
6875
1.54
8387
1.54
9260
1.55
0000
1.55
1724
1.55
4406
1.55
5556
1.55
9960
1.56
1418
1.56
2236
1.56
2500
1.56
3914
1.56
5217
1.56
5972
1.56
9168
1.57
0433
1.57
1429
402.
501
402.
823
403.
220
403.
482
404.
296
404.
702
405.
098
405.
326
405.
520
405.
971
406.
672
406.
973
408.
126
408.
507
408.
721
408.
790
409.
160
409.
501
409.
698
410.
535
410.
866
411.
126
745.
786
747.
170
748.
875
750.
000
753.
488
755.
228
756.
919
757.
895
758.
722
760.
647
763.
636
764.
916
769.
811
771.
429
772.
335
772.
627
774.
194
775.
636
776.
471
780.
000
781.
395
782.
492
334 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÕÐÖ
ÕÐ×
ÕÐØ
ÕÐÙ
ÕÑÐ
ÕÑÑ
ÕÑÒ
ÕÑÓ
ÕÑÔ
ÕÑÕ
ÕÑÖ
ÕÑ×
ÕÑØ
ÕÑÙ
ÕÒÐ
ÕÒÑ
ÕÒÒ
ÕÒÓ
ÕÒÔ
ÕÒÕ
ÕÒÖ
ÕÒ×
23-e
t Pyt
hago
rean
min
or s
ixth
sep
tend
ecim
al s
uper
augm
ente
d fi
fth
101s
t har
mon
ic
nona
dec
imal
sup
erau
gmen
ted
fifth
Pyth
agor
ean
min
or s
ixth
(D)
IV
53-e
t Pyt
hago
rean
min
or s
ixth
acut
e or
larg
e au
gmen
ted
fifth
nona
dec
imal
sub
min
or s
ixth
cycl
ic m
inor
six
th (A
) XL
IX
equa
l min
or s
ixth
sep
tend
ecim
al s
ubm
inor
six
th
sep
tend
ecim
al s
uper
augm
ente
d fi
fth (5
1st h
arm
onic
)
43-e
t jus
t min
or s
ixth
trivi
gesim
al s
uper
augm
ente
d fi
fth
31-e
t sup
erau
gmen
ted
fifth
, or j
ust m
inor
six
th
mea
nton
e an
d ju
st m
inor
six
th (D
) IV
1�
53-e
t jus
t min
or s
ixth
Pyth
agor
ean
augm
ente
d fi
fth (A
) VIII
22-e
t jus
t min
or s
ixth
19-e
t sev
ente
en q
uarte
r-ton
es
sep
timal
sup
erau
gmen
ted
six
th
103r
d ha
rmon
ic
1523
)2
(
85/5
4
101/
64
30/1
9
128/
81
3553
)2
(
405/
256
19/1
2
3 ôù/2
÷÷ 2
3)
2(
, or a
ppro
xim
atel
y 10
0/63
27/1
7
51/3
2 2943
)2
(
115/
72
2131
)2
(
128/
81×
81/8
0, o
r 8/5
36
53)
2(
3 ø/2
ñò, o
r 656
1/40
96
1522
)2
(
1319
)2
(
45/2
8
103/
64
1.57
1534
1.57
4074
1.57
8125
1.57
8947
1.58
0247
1.58
0496
1.58
2031
1.58
3333
1.58
3550
1.58
7401
1.58
8235
1.59
3750
1.59
5953
1.59
7222
1.59
9276
1.60
0000
1.60
1302
1.60
1807
1.60
4160
1.60
6822
1.60
7143
1.60
9375
411.
154
411.
818
412.
878
413.
093
413.
433
413.
498
413.
900
414.
240
414.
297
415.
305
415.
523
416.
966
417.
542
417.
874
418.
412
418.
601
418.
942
419.
074
419.
689
420.
386
420.
470
421.
054
782.
609
785.
404
789.
854
790.
756
792.
180
792.
453
794.
134
795.
558
795.
795
800.
000
800.
910
806.
910
809.
302
810.
678
812.
903
813.
686
815.
094
815.
640
818.
182
821.
053
821.
398
823.
801
Appendices 335
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÕÒØ
ÕÒÙ
ÕÓÐ
ÕÓÑ
ÕÓÒ
ÕÓÓ
ÕÓÔ
ÕÓÕ
ÕÓÖ
ÕÓ×
ÕÓØ
ÕÓÙ
ÕÔÐ
ÕÔÑ
ÕÔÒ
ÕÔÓ
ÕÔÔ
ÕÔÕ
ÕÔÖ
ÕÔ×
ÕÔØ
ÕÔÙ
16-e
t sev
ente
en q
uarte
r-ton
es
nona
vige
simal
gra
ve o
r sm
all n
eutra
l six
th
seve
ntee
n q
uarte
r-ton
es
untri
gesim
al s
uper
augm
ente
d fi
fth
13-e
t sev
ente
en q
uarte
r-ton
es
23-e
t neu
tral s
ixth
neut
ral s
ixth
43-e
t dou
ble
dim
inish
ed s
even
th
53-e
t neu
tral s
ixth
cycl
ic n
eutra
l six
th (A
) XX
10-e
t neu
tral s
ixth
trid
ecim
al g
rave
or s
mal
l neu
tral,
or o
verto
ne s
ixth
(13
th h
arm
onic
)
dou
ble
augm
ente
d fi
fth
17-e
t neu
tral s
ixth
mea
nton
e d
oubl
e au
gmen
ted
fifth
(A) X
V4
33
ß
seve
ntee
n eq
ual q
uarte
r-ton
es
31-e
t dou
ble
augm
ente
d fi
fth, o
r neu
tral s
ixth
und
ecim
al g
rave
or s
mal
l neu
tral s
ixth
sept
imal
neu
tral s
ixth
(10
5th
harm
onic
)
7-et
gra
ve o
r sm
all m
ajor
six
th
trid
ecim
al a
cute
or l
arge
neu
tral s
ixth
53-e
t gra
ve o
r sm
all m
ajor
six
th
1116
)2
(
29/1
8
50/3
1
155/
96
913
)2
(
1623
)2
(
81/5
0 3043
)2
(
3753
)2
(
3 òð/2
óñ 710
)2
(
13/8
625/
384
1217
)2
(
1434
8907
/838
8608
×0.
2666
678180
17
24)
2(
, or a
pp
roxi
mat
ely
49/3
0 22
31)
2(
18/1
1
105/
64
57
)2
(
64/3
9 3853
)2
(
1.61
0490
1.61
1111
1.61
2903
1.61
4583
1.61
5866
1.61
9616
1.62
0000
1.62
1888
1.62
2382
1.62
3661
1.62
4505
1.62
5000
1.62
7604
1.63
1142
1.63
2667
1.63
3915
1.63
5438
1.63
6364
1.64
0625
1.64
0671
1.64
1026
1.64
3739
421.
345
421.
508
421.
977
422.
416
422.
752
423.
733
423.
833
424.
327
424.
457
424.
791
425.
012
425.
142
425.
823
426.
748
427.
147
427.
474
427.
872
428.
115
429.
229
429.
241
429.
334
430.
044
825.
000
825.
667
827.
592
829.
394
830.
769
834.
783
835.
193
837.
209
837.
736
839.
100
840.
000
840.
528
843.
300
847.
059
848.
676
850.
000
851.
613
852.
592
857.
095
857.
143
857.
517
860.
377
336 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÕÕÐ
ÕÕÑ
ÕÕÒ
ÕÕÓ
ÕÕÔ
ÕÕÕ
ÕÕÖ
ÕÕ×
ÕÕØ
ÕÕÙ
ÕÖÐ
ÕÖÑ
ÕÖÒ
ÕÖÓ
ÕÖÔ
ÕÖÕ
ÕÖÖ
ÕÖ×
ÕÖØ
ÕÖÙ
Õ×Ð
Õ×Ñ
cycl
ic g
rave
or s
mal
l maj
or s
ixth
(A)
XXXI
I
grav
e or
sm
all m
ajor
six
th
43-e
t dou
ble
augm
ente
d fi
fth
18-e
t gra
ve o
r sm
all m
ajor
six
th
und
ecim
al a
cute
or l
arge
neu
tral s
ixth
nona
vige
simal
acu
te o
r lar
ge n
eutra
l six
th
11-e
t gra
ve o
r sm
all m
ajor
six
th
53rd
har
mon
ic
15-e
t jus
t maj
or s
ixth
Pyth
agor
ean
dim
inish
ed s
even
th (D
) IX
53-
et ju
st m
ajor
six
th
19-e
t jus
t maj
or s
ixth
just
maj
or s
ixth
cycl
ic m
ajor
six
th (A
) XL
IV
23-e
t jus
t maj
or s
ixth
mea
nton
e m
ajor
six
th (A
) III
43
ß
107t
h ha
rmon
ic
31-e
t jus
t maj
or s
ixth
43-e
t jus
t maj
or s
ixth
equa
l maj
or s
ixth
nona
dec
imal
sup
erm
ajor
six
th
grav
e or
sm
all d
imin
ished
sev
enth
3 óò/2
õð
400/
243 31
43)
2(
13
18)
2(
33/2
0
48/2
9 811
)2
(
53/3
2 1115
)2
(
3276
8/19
683
3953
)2
(
1419
)2
(
5/3
3 ôô/2
öù 1723
)2
(
27/1
6×1.
3333
338180
107/
64
2331
)2
(
3243
)2
(
34
)2
(, o
r ap
pro
xim
atel
y 37
/22
32/1
9
2048
/121
5
1.64
5813
1.64
6091
1.64
8244
1.64
9721
1.65
0000
1.65
5172
1.65
5507
1.65
6250
1.66
2476
1.66
4787
1.66
5377
1.66
6524
1.66
6667
1.66
8267
1.66
9169
1.67
1851
1.67
1875
1.67
2418
1.67
5029
1.68
1793
1.68
4211
1.68
5597
430.
587
430.
659
431.
223
431.
609
431.
682
433.
035
433.
123
433.
317
434.
946
435.
551
435.
705
436.
005
436.
043
436.
461
436.
697
437.
399
437.
405
437.
547
438.
230
440.
000
440.
633
440.
995
862.
560
862.
852
865.
116
866.
667
866.
959
872.
378
872.
727
873.
505
880.
000
882.
405
883.
019
884.
211
884.
359
886.
020
886.
957
889.
735
889.
760
890.
323
893.
023
900.
000
902.
487
903.
911
Appendices 337
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
Õ×Ò
Õ×Ó
Õ×Ô
Õ×Õ
Õ×Ö
Õ××
Õ×Ø
Õ×Ù
ÕØÐ
ÕØÑ
ÕØÒ
ÕØÓ
ÕØÔ
ÕØÕ
ÕØÖ
ÕØ×
ÕØØ
ÕØÙ
ÕÙÐ
ÕÙÑ
ÕÙÒ
ÕÙÓ
53-e
t Pyt
hago
rean
maj
or s
ixth
Pyth
agor
ean
maj
or s
ixth
(A) I
II (27
th h
arm
onic
)
56th
cyc
lic fi
fth (A
) LV
I
21-e
t Pyt
hago
rean
maj
or s
ixth
17-e
t acu
te o
r lar
ge m
ajor
six
th
sep
tend
ecim
al s
uper
maj
or s
ixth
43-e
t dim
inish
ed s
even
th
109t
h ha
rmon
ic
trivi
gesim
al s
uper
maj
or s
ixth
13-e
t acu
te o
r lar
ge m
ajor
six
th
und
ecim
al g
rave
or s
mal
l aug
men
ted
six
th
dim
inish
ed s
even
th, o
r acu
te o
r lar
ge m
ajor
six
th
22-e
t acu
te o
r lar
ge m
ajor
six
th
53-e
t acu
te o
r lar
ge m
ajor
six
th
31-e
t sup
erm
ajor
six
th, o
r dim
inish
ed s
even
th
Pyth
agor
ean
dou
ble
augm
ente
d fi
fth, o
r cyc
lic a
cute
or l
arge
maj
or s
ixth
(A) X
V
mea
nton
e d
imin
ished
sev
enth
(D)
IX41
2�
sep
timal
sup
erm
ajor
six
th
9-et
nin
etee
n q
uarte
r-ton
es
und
ecim
al s
uper
maj
or s
ixth
(55t
h ha
rmon
ic)
23-e
t nin
etee
n q
uarte
r-ton
es
nine
teen
qua
rter-t
ones
, or u
ntrig
esim
al s
uper
maj
or s
ixth
4053
)2
(
3 ó/2
ô, or
27/
16
3 õö/2
øø 1621
)2
(
1317
)2
(
17/1
0 3343
)2
(
109/
64
46/2
7 1013
)2
(
75/4
4
128/
75
1722
)2
(
4153
)2
(
2431
)2
(
3 ñõ/2
òó, o
r 143
4890
7/83
8860
8
3276
8/19
683×
0.44
4444
8081
12/7
79
)2
(
55/3
2 1823
)2
(
31/1
8
1.68
7301
1.68
7500
1.69
1027
1.69
5728
1.69
9024
1.70
0000
1.70
2249
1.70
3125
1.70
3704
1.70
4361
1.70
4545
1.70
6667
1.70
8496
1.70
9512
1.71
0234
1.71
0523
1.71
1975
1.71
4286
1.71
4488
1.71
8750
1.72
0239
1.72
2222
441.
441
441.
493
442.
416
443.
646
444.
508
444.
763
445.
352
445.
581
445.
732
445.
904
445.
953
446.
508
446.
986
447.
252
447.
441
447.
517
447.
896
448.
501
448.
554
449.
669
450.
058
450.
577
905.
660
905.
865
909.
480
914.
286
917.
647
918.
642
920.
930
921.
821
922.
409
923.
077
923.
264
925.
418
927.
273
928.
302
929.
032
929.
325
930.
794
933.
129
933.
333
937.
632
939.
130
941.
126
338 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÕÙÔ
ÕÙÕ
ÕÙÖ
ÕÙ×
ÕÙØ
ÕÙÙ
ÖÐÐ
ÖÐÑ
ÖÐÒ
ÖÐÓ
ÖÐÔ
ÖÐÕ
ÖÐÖ
ÖÐ×
ÖÐØ
ÖÐÙ
ÖÑÐ
ÖÑÑ
ÖÑÒ
ÖÑÓ
ÖÑÔ
ÖÑÕ
14-e
t aug
men
ted
six
th
nona
vige
simal
gra
ve o
r sm
all a
ugm
ente
d s
ixth
19-e
t nin
etee
n q
uarte
r-ton
es
43-e
t trip
le d
imin
ished
oct
ave
trid
ecim
al g
rave
or s
mal
l aug
men
ted
six
th
nine
teen
eq
ual q
uarte
r-ton
es
53-e
t aug
men
ted
six
th
cycl
ic a
ugm
ente
d six
th (A
) XX
VII
111t
h ha
rmon
ic
augm
ente
d s
ixth
trivi
gesim
al s
ubm
inor
sev
enth
5-et
aug
men
ted
six
th
untri
gesim
al s
ubm
inor
sev
enth
mea
nton
e au
gmen
ted
six
th (A
) X21
2ß
31-e
t aug
men
ted
six
th, o
r sub
min
or s
even
th
sep
timal
sub
min
or s
even
th (
7th
harm
onic
)
21-e
t gra
ve o
r sm
all m
inor
sev
enth
nona
dec
imal
sup
erau
gmen
ted
six
th
53-e
t gra
ve o
r sm
all m
inor
sev
enth
16-e
t gra
ve o
r sm
all m
inor
sev
enth
cycl
ic g
rave
or s
mal
l min
or s
even
th (A
) XX
XIX
acut
e or
larg
e au
gmen
ted
six
th, o
r gra
ve o
r sm
all m
inor
sev
enth
1114
)2
(
50/2
9 1519
)2
(
3443
)2
(
45/2
6 19
24)
2(
, or a
pp
roxi
mat
ely
45/2
6 42
53)
2(
3 ò÷/2
ôò
111/
64
125/
72
40/2
3 45
)2
(
54/3
1
5904
9/32
768×
0.4
8180
2531
)2
(
7/4 17
21)
2(
100/
57
4353
)2
(
1316
)2
(
3 óù/2
öñ
225/
128
1.72
3946
1.72
4138
1.72
8444
1.72
9911
1.73
0769
1.73
1073
1.73
2017
1.73
3860
1.73
4375
1.73
6111
1.73
9130
1.74
1101
1.74
1935
1.74
6928
1.74
8905
1.75
0000
1.75
2633
1.75
4386
1.75
4817
1.75
6252
1.75
7516
1.75
7813
451.
028
451.
079
452.
205
452.
589
452.
813
452.
893
453.
140
453.
622
453.
757
454.
211
455.
001
455.
517
455.
735
457.
041
457.
558
457.
845
458.
534
458.
992
459.
105
459.
480
459.
811
459.
889
942.
857
943.
050
947.
368
948.
837
949.
696
950.
000
950.
943
952.
785
953.
299
955.
031
958.
039
960.
000
960.
829
965.
784
967.
742
968.
826
971.
429
973.
159
973.
585
975.
000
976.
245
976.
537
Appendices 339
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÖÑÖ
ÖÑ×
ÖÑØ
ÖÑÙ
ÖÒÐ
ÖÒÑ
ÖÒÒ
ÖÒÓ
ÖÒÔ
ÖÒÕ
ÖÒÖ
ÖÒ×
ÖÒØ
ÖÒÙ
ÖÓÐ
ÖÓÑ
ÖÓÒ
ÖÓÓ
ÖÓÔ
ÖÓÕ
ÖÓÖ
ÖÓ×
43-e
t aug
men
ted
six
th
11-e
t gra
ve o
r sm
all m
inor
sev
enth
sep
tend
ecim
al s
ubm
inor
sev
enth
113t
h ha
rmon
ic
17-e
t Pyt
hago
rean
min
or s
even
th
sep
tend
ecim
al s
uper
augm
ente
d s
ixth
23-e
t Pyt
hago
rean
min
or s
even
th
Pyth
agor
ean
min
or s
even
th (D
) II
53-e
t Pyt
hago
rean
min
or s
even
th
sup
erau
gmen
ted
six
th
nona
dec
imal
sub
min
or s
even
th (
57th
har
mon
ic)
cycl
ic m
inor
sev
enth
(A)
LI
equa
l min
or s
even
th
sep
timal
sup
erau
gmen
ted
six
th
43-e
t Pyt
hago
rean
min
or s
even
th
31-e
t sup
erau
gmen
ted
six
th, o
r min
or s
even
th
mea
nton
e m
inor
sev
enth
(D)
21�
19-e
t acu
te o
r lar
ge m
inor
sev
enth
untri
gesim
al s
uper
augm
ente
d s
ixth
trivi
gesim
al s
uper
augm
ente
d six
th (
115t
h ha
rmon
ic)
13-e
t Pyt
hago
rean
min
or s
even
th
acut
e or
larg
e m
inor
sev
enth
3543
)2
(
911
)2
(
30/1
7
113/
64
1417
)2
(
85/4
8 1923
)2
(
16/9
4453
)2
(
3645
/204
8
57/3
2
3 õñ/2
øð 5
6)
2(
, or a
pp
roxi
mat
ely
98/5
5
25/1
4 3643
)2
(
2631
)2
(
16/9
×2
8081
1619
)2
(
775/
432
115/
64
1113
)2
(
9/5
1.75
8022
1.76
3183
1.76
4706
1.76
5625
1.76
9730
1.77
0833
1.77
2870
1.77
7778
1.77
7918
1.77
9785
1.78
1250
1.78
1494
1.78
1797
1.78
5714
1.78
6591
1.78
8450
1.78
8854
1.79
2664
1.79
3981
1.79
6875
1.79
7702
1.80
0000
459.
944
461.
294
461.
692
461.
933
463.
007
463.
295
463.
828
465.
112
465.
149
465.
637
466.
021
466.
084
466.
164
467.
885
467.
418
467.
904
468.
010
469.
007
469.
351
470.
108
470.
325
470.
926
976.
744
981.
818
983.
313
984.
215
988.
235
989.
314
991.
304
996.
090
996.
226
998.
044
999.
468
999.
705
1000
.000
1003
.802
1004
.651
1006
.452
1006
.843
1010
.526
1011
.798
1014
.588
1015
.385
1017
.596
340 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÖÓØ
ÖÓÙ
ÖÔÐ
ÖÔÑ
ÖÔÒ
ÖÔÓ
ÖÔÔ
ÖÔÕ
ÖÔÖ
ÖÔ×
ÖÔØ
ÖÔÙ
ÖÕÐ
ÖÕÑ
ÖÕÒ
ÖÕÓ
ÖÕÔ
ÖÕÕ
ÖÕÖ
ÖÕ×
ÖÕØ
ÖÕÙ
53-e
t acu
te o
r lar
ge m
inor
sev
enth
Pyth
agor
ean
augm
ente
d s
ixth
, or c
yclic
acu
te o
r lar
ge m
inor
sev
enth
(A) X
20-e
t acu
te o
r lar
ge m
inor
sev
enth
trid
ecim
al g
rave
or s
mal
l neu
tral s
even
th
7-et
twen
ty-o
ne q
uarte
r-ton
es
nona
vige
simal
gra
ve o
r sm
all n
eutra
l sev
enth
(29
th h
arm
onic
)
43-e
t dou
ble
dim
inish
ed o
ctav
e
und
ecim
al g
rave
or s
mal
l neu
tral s
even
th
22-e
t tw
enty
-one
qua
rter-t
ones
neut
ral s
even
th
15-e
t neu
tral s
even
th
53-e
t neu
tral s
even
th
cycl
ic n
eutra
l sev
enth
(A)
XXII
23-e
t neu
tral s
even
th
trid
ecim
al n
eutra
l sev
enth
(117
th h
arm
onic
)
31-e
t dou
ble
augm
ente
d s
ixth
, or n
eutra
l sev
enth
neut
ral s
even
th
acut
e or
larg
e d
oubl
e au
gmen
ted
six
th
und
ecim
al a
cute
or l
arge
neu
tral s
even
th
twen
ty-o
ne e
qua
l qua
rter-t
ones
17-e
t gra
ve o
r sm
all m
ajor
sev
enth
59th
har
mon
ic
4553
)2
(
3 ñð/2
ñõ, o
r 590
49/3
2768
17
20)
2(
65/3
6 67
)2
(
29/1
6 3743
)2
(
20/1
1 1922
)2
(
729/
400
1315
)2
(
4653
)2
(
3 òò/2
óô 2023
)2
(
117/
64
2731
)2
(
4000
/218
7
1875
/102
4
11/6
7
8)
2(
, or a
ppro
xim
atel
y 93
9/51
2 15
17)
2(
59/3
2
1.80
1323
1.80
2032
1.80
2501
1.80
5556
1.81
1447
1.81
2500
1.81
5624
1.81
8181
1.81
9619
1.82
2500
1.82
3445
1.82
5036
1.82
6618
1.82
7112
1.82
8125
1.82
8889
1.82
8989
1.83
1055
1.83
3333
1.83
4008
1.84
3379
1.84
3750
471.
272
471.
458
471.
580
472.
379
473.
921
474.
196
475.
014
475.
683
476.
059
476.
813
477.
060
477.
476
477.
890
478.
019
478.
284
478.
484
478.
510
479.
051
479.
647
479.
823
482.
275
482.
372
1018
.868
1019
.550
1020
.000
1022
.931
1028
.571
1029
.577
1032
.558
1034
.996
1036
.364
1039
.103
1040
.000
1041
.509
1043
.010
1043
.478
1044
.438
1045
.161
1045
.256
1047
.210
1049
.363
1050
.000
1058
.824
1059
.172
Appendices 341
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÖÖÐ
ÖÖÑ
ÖÖÒ
ÖÖÓ
ÖÖÔ
ÖÖÕ
ÖÖÖ
ÖÖ×
ÖÖØ
ÖÖÙ
Ö×Ð
Ö×Ñ
Ö×Ò
Ö×Ó
Ö×Ô
Ö×Õ
Ö×Ö
Ö××
Ö×Ø
Ö×Ù
ÖØÐ
ÖØÑ
43-e
t dou
ble
augm
ente
d s
ixth
trid
ecim
al a
cute
or l
arge
neu
tral s
even
th
53-e
t gra
ve o
r sm
all m
ajor
sev
enth
cycl
ic g
rave
or s
mal
l maj
or s
even
th (A
) XX
XIV
9-et
gra
ve o
r sm
all m
ajor
sev
enth
grav
e or
sm
all m
ajor
sev
enth
19-e
t gra
ve o
r sm
all m
ajor
sev
enth
sept
ende
cim
al n
eutra
l sev
enth
(11
9th
harm
onic
)
nona
vige
simal
acu
te o
r lar
ge n
eutra
l sev
enth
10-e
t jus
t dia
toni
c m
ajor
sev
enth
mea
nton
e m
ajor
sev
enth
(A) V
411ß
31-e
t jus
t dia
toni
c m
ajor
sev
enth
21-e
t jus
t dia
toni
c m
ajor
sev
enth
Pyth
agor
ean
dim
inish
ed o
ctav
e (D
) VII
53-e
t jus
t dia
toni
c m
ajor
sev
enth
just
dia
toni
c m
ajor
sev
enth
(15
th h
arm
onic
)
43-e
t jus
t dia
toni
c m
ajor
sev
enth
cycl
ic d
iato
nic
maj
or s
even
th (A
) XLV
I
11-e
t jus
t dia
toni
c m
ajor
sev
enth
23-e
t jus
t dia
toni
c m
ajor
sev
enth
equa
l maj
or s
even
th
sep
tend
ecim
al s
uper
maj
or s
even
th
3843
)2
(
24/1
3 4753
)2
(
3 óô/2
õó 89
)2
(
50/2
7 1719
)2
(
119/
64
54/2
9 910
)2
(
243/
128×
0.8
8180
2831
)2
(
1921
)2
(
4096
/218
7 48
53)
2(
15/8
3943
)2
(
3 ôö/2
÷ò 1011
)2
(
2123
)2
(
1112
)2
(, o
r app
roxi
mat
ely
967/
512
17/9
1.84
5128
1.84
6154
1.84
9061
1.85
1539
1.85
1749
1.85
1852
1.85
9271
1.85
9375
1.86
2069
1.86
6066
1.86
9186
1.87
0243
1.87
2235
1.87
2885
1.87
3402
1.87
5000
1.87
5112
1.87
6800
1.87
7862
1.88
3014
1.88
7749
1.88
8889
482.
733
483.
001
483.
762
484.
410
484.
465
484.
492
486.
433
486.
460
487.
165
488.
211
489.
027
489.
303
489.
825
489.
995
490.
130
490.
548
490.
577
491.
019
491.
297
492.
645
493.
883
494.
182
1060
.465
1061
.427
1064
.151
1066
.470
1066
.667
1066
.762
1073
.684
1073
.781
1076
.288
1080
.000
1082
.892
1083
.871
1085
.714
1086
.315
1086
.792
1088
.269
1088
.372
1089
.930
1090
.909
1095
.652
1100
.000
1101
.045
342 Appendices
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
ÖØÒ
ÖØÓ
ÖØÔ
ÖØÕ
ÖØÖ
ÖØ×
ÖØØ
ÖØÙ
ÖÙÐ
ÖÙÑ
ÖÙÒ
ÖÙÓ
ÖÙÔ
ÖÙÕ
ÖÙÖ
ÖÙ×
ÖÙØ
ÖÙÙ
×ÐÐ
×ÐÑ
×ÐÒ
×ÐÓ
grav
e or
sm
all n
eutra
l sev
enth
(12
1st h
arm
onic
)
nona
dec
imal
sup
erm
ajor
sev
enth
13-e
t Pyt
hago
rean
maj
or s
even
th
grav
e or
sm
all d
imin
ished
oct
ave
53-e
t Pyt
hago
rean
maj
or s
even
th
Pyth
agor
ean
maj
or s
even
th (A
) V
58th
cyc
lic fi
fth (A
) LV
III
14-e
t Pyt
hago
rean
maj
or s
even
th
sep
timal
sup
erm
ajor
sev
enth
43-e
t dim
inish
ed o
ctav
e
61st
har
mon
ic
15-e
t jus
t dia
toni
c m
ajor
sev
enth
31-e
t sup
erm
ajor
sev
enth
, or d
imin
ished
oct
ave
mea
nton
e d
imin
ished
oct
ave
(D) V
II4
3 1�
16-e
t acu
te o
r lar
ge m
ajor
sev
enth
trivi
gesim
al s
uper
maj
or s
even
th
dim
inish
ed o
ctav
e, o
r acu
te o
r lar
ge m
ajor
sev
enth
17-e
t acu
te o
r lar
ge m
ajor
sev
enth
123r
d ha
rmon
ic
53-e
t acu
te o
r lar
ge m
ajor
sev
enth
cycl
ic a
cute
or l
arge
maj
or s
even
th (A
) XV
II
18-e
t acu
te o
r lar
ge m
ajor
sev
enth
121/
64
36/1
9 1213
)2
(
256/
135
4953
)2
(
3 õ/2
÷, or
243
/128
3 õø/2
ùñ 1314
)2
(
40/2
1 4043
)2
(
61/3
2 1415
)2
(
2931
)2
(
4096
/218
7×0.
5714
298081
15
16)
2(
23/1
2
48/2
5 1617
)2
(
123/
64
5053
)2
(
3 ñ÷/2
òö 1718
)2
(
1.89
0625
1.89
4737
1.89
6155
1.89
6296
1.89
8064
1.89
8438
1.90
2406
1.90
3390
1.90
4762
1.90
5583
1.90
6250
1.90
9683
1.91
2532
1.91
4046
1.91
5207
1.91
6667
1.92
0000
1.92
0093
1.92
1875
1.92
3050
1.92
4338
1.92
4448
494.
636
495.
712
496.
083
496.
120
496.
582
496.
680
497.
718
497.
976
498.
334
498.
549
498.
724
499.
622
500.
367
500.
763
501.
067
501.
449
502.
321
502.
346
502.
812
503.
119
503.
456
503.
485
1102
.636
1106
.397
1107
.692
1107
.821
1109
.434
1109
.775
1113
.390
1114
.286
1115
.533
1116
.279
1116
.885
1120
.000
1122
.581
1123
.951
1125
.000
1126
.319
1129
.328
1129
.412
1131
.017
1132
.075
1133
.235
1133
.333
Appendices 343
DEG
REE
NUM
BER
INTE
RVA
L FA
CTO
R RA
TIO
(DEC
IMA
L)
FREQ
UEN
CY
(HER
TZ)
CEN
TS
×ÐÔ
×ÐÕ
×ÐÖ
×Ð×
×ÐØ
×ÐÙ
×ÑÐ
×ÑÑ
×ÑÒ
×ÑÓ
×ÑÔ
×ÑÕ
×ÑÖ
×Ñ×
×ÑØ
×ÑÙ
×ÒÐ
×ÒÑ
×ÒÒ
×ÒÓ
×ÒÔ ÐÑ
19-e
t tw
enty
-thre
e q
uarte
r-ton
es
20-e
t tw
enty
-thre
e q
uarte
r-ton
es
21-e
t tw
enty
-thre
e q
uarte
r-ton
es
twen
ty-th
ree
qua
rter-t
ones
43-e
t dou
ble
dim
inish
ed s
econ
d
untri
gesim
al s
uper
maj
or s
even
th (
31st
har
mon
ic)
22-e
t tw
enty
-thre
e q
uarte
r-ton
es
und
ecim
al s
ubd
imin
ished
oct
ave
23-e
t tw
enty
-thre
e q
uarte
r-ton
es
twen
ty-th
ree
equa
l qua
rter-t
ones
53-e
t sub
octa
ve
cycl
ic s
uboc
tave
(A)
XXIX
mea
nton
e au
gmen
ted
sev
enth
, or s
uboc
tave
(A)
XII
3ß
(12
5th
harm
onic
)
31-e
t aug
men
ted
sev
enth
, sub
octa
ve
43-e
t aug
men
ted
sev
enth
sep
timal
sub
dim
inish
ed o
ctav
e (6
3rd
har
mon
ic)
53-e
t gra
ve o
r sm
all o
ctav
e
grav
e or
sm
all o
ctav
e
cycl
ic g
rave
or s
mal
l oct
ave
(A)
XLI
acut
e or
larg
e m
eant
one
augm
ente
d s
even
th
127t
h ha
rmon
ic
octa
ve (
2nd
har
mon
ic)
1819
)2
(
1920
)2
(
2021
)2
(
60/3
1 4143
)2
(
31/1
6 2122
)2
(
64/3
3 2223
)2
(
2324
)2
(, o
r app
roxi
mat
ely
1024
/527
51
53)
2(
3òù/2
ôõ
5314
41/5
2428
8×0.
3333
338180
, or 1
25/6
4 30
31)
2(
42
43)
2(
63/3
2 5253
)2
(
160/
81
3ôñ/2
öô
2025
/102
4
127/
64
2/1
1.92
8352
1.93
1873
1.93
5064
1.93
5484
1.93
6549
1.93
7500
1.93
7969
1.93
9394
1.94
0626
1.94
3064
1.94
8365
1.95
0593
1.95
3125
1.95
5777
1.96
8019
1.96
8750
1.97
4014
1.97
5309
1.97
7205
1.97
7539
1.98
4375
2.00
0000
504.
506
505.
427
506.
262
506.
372
506.
651
506.
900
507.
022
507.
395
507.
717
508.
355
509.
742
510.
325
510.
987
511.
681
514.
884
515.
075
516.
452
516.
791
517.
287
517.
375
519.
163
523.
251
1136
.842
1140
.000
1142
.857
1143
.233
1144
.186
1145
.036
1145
.455
1146
.727
1147
.826
1150
.000
1154
.717
1156
.695
1158
.941
1161
.290
1172
.093
1172
.736
1177
.358
1178
.494
1180
.155
1180
.447
1186
.422
1200
.000
344 Appendices
Appendix B
Microtonal Notation Font
! " # $ % & '
32 33 34 35 36 37 38 39
( ) * + , - . /
40 41 42 43 44 45 46 47
0 1 2 3 4 5 6 7
48 49 50 51 52 53 54 55
8 9 : ; < = > ? 56 57 58 59 60 61 62 63
@ A B C D E F G
64 65 66 67 68 69 70 71
H I J K L M N O
72 73 74 75 76 77 78 79
P Q R S T U V W
80 81 82 83 84 85 86 87
X Y Z [ \ ] ^ _
88 89 90 91 92 93 94 95
` a b c d e f g
96 97 98 99 100 101 102 103
h i j k l m n o
104 105 106 107 108 109 110 111
p q r s t u v w
112 113 114 115 116 117 118 119
x y z { | } ~
120 121 122 123 124 125 126 127
‚ ƒ „ … † ‡
128 129 130 131 132 133 134 135
ˆ ‰ Š ‹ Œ
136 137 138 139 140 141 142 143
346 Appendices
‘ ’ “ ” • – —
144 145 146 147 148 149 150 151
˜ ™ š › œ Ÿ
152 153 154 155 156 157 158 159
¡ ¢ £ ¤ ¥ ¦ §
160 161 162 163 164 165 166 167
¨ © ª « ® ¯
168 169 170 171 172 173 174 175
° ± ² ³ ´ μ ¶ ·
176 177 178 179 180 181 182 183
¸ ¹ º » ¼ ½ ¾ ¿
184 185 186 187 188 189 190 191
À Á Â Ã Ä Å Æ Ç
192 193 194 195 196 197 198 199
Appendices 347
È É Ê Ë Ì Í Î Ï
200 201 202 203 204 205 206 207
Ð Ñ Ò Ó Ô Õ Ö ×
208 209 210 211 212 213 214 215
Ø Ù Ú Û Ü Ý
216 217 218 219 220 221 222 223
à á â ã ä å æ ç
224 225 226 227 228 229 230 231
è é ê ë ì í î ï
232 233 234 235 236 237 238 239
ð ñ ò ó ô õ
240 241 242 243 244 245 246 247
ø ù ú û ü ý þ ÿ
248 249 250 251 252 253 254 255
348 Appendices