Consistency, Context and Symmetry in Alberto Ginastera’s String Quartets Nos. 1 (1948) and 2 (1958, First Version)
By
David L. Sommerville
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor Dave Headlam
Department of Music Theory Eastman School of Music
University of Rochester Rochester, New York
2009
Curriculum Vitae
The author was born in Philadelphia, Pennsylvania on February 15, 1969. He
attended Emory University from 1987 to 1991 and graduated with a Bachelor of Arts
degree in International Studies, focusing on Spanish and the history and politics of Latin
America. After several years of pursuing music performance, he began post-
baccalaureate studies in music at Georgia State University, eventually earning a Master
of Music degree in Music Theory from that institution in 1999 and was inducted into the
Pi Kappa Lambda music honor society. He enrolled in the Doctor of Philosophy program
in Music Theory at the Eastman School of Music, University of Rochester in the Fall of
1999, won the outstanding TA prize in 2001, and earned the Master of Arts degree in
Music Theory in 2002. In addition to teaching at Eastman, he also lectured in Music
Theory at Nazareth College. He pursued his research on the music of Alberto Ginastera
under the direction of Professor Dave Headlam.
ii
Acknowledgements
I wish to acknowledge several people whose influence and support of me during
my doctoral studies at Eastman has been absolutely vital, not only to the completion of
the program, but to the betterment of my life. First, I thank my parents David and Diane
Sommerville for working hard and sacrificing to afford me the numerous opportunities I
have received at every stage in my journey. Second, I thank my wife Maria, who has
delayed her dreams to allow me to pursue mine. Third, I wish to thank the teachers I have
had, from Ken Adkins at Albuquerque Academy so long ago, through Susan Tepping,
John Nelson and Ron Squibbs at Georgia State who introduced me to music theory, to
Dave Headlam (especially), Matthew Brown (especially), John Covach and Steve Laitz at
Eastman who saw me through the happy and the difficult times during my course of
study. Fourth, I wish to thank my other committee members Jeannie Guerrero and
Malena Kuss for their excellent help in the preparation of this document. Fifth, I thank
my friends Jocelyn Kovaleski (née Swigger), Josh Mailman, Sam Ng, Jeff Tucker and
Alfred Vitale for their friendship and support. Finally, I thank my precious daughter
Rory, who is and always will be the greatest part of my life.
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Abstract
Despite its richness, presence in the recital and concert hall, amenity to
established analytic avenues, and its potential to support new directions in music-
theoretic research, the music of Alberto Ginastera has received relatively little attention
among the music theory community. This dissertation attempts to remedy this by
selecting passages from the First and Second String Quartets (1948 and 1958) and
developing appropriate analytic tools rooted in the recent confluence of Perle-based and
transformation theories. The study of these important works in Ginastera’s career reveals
that some of the salient aspects of the musical surface, such as its emphasis on the
ensemble’s open strings, its cyclic pitch structure, and its frequent embrace of repetition
and patterns, are indeed revelatory of deeper structures in the music. In its approach, the
dissertation develops a simple specialized vocabulary to identify and contextualize
significant compositional constructions and relate them in various ways to a few basic
concepts: as a reflection of 1) a cyclic or transformational property of the OS set (open
string) or a significant conjunct subset called a shading or a dyad-space, 2) an interaction
of interval cycles called coloring, or 3) a type of generative motivic pattern called a
source trichord and its subsequent unfolding of larger cyclic pc collections called U-cells
and U-chains. Ultimately, the study uses subtle differences in the above constructions to
engage issues of stylistic development during the decade separating the quartets’
composition.
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After an introductory chapter that provides both a basic context for the analysis of
subsequent chapters and a literature review, the second chapter details the methodological
aspects of the dissertation, discussing not only the established music-theoretic and
analytic approaches the dissertation utilizes, but also the original contribution to
scholarship the dissertation offers. The third and fourth chapters contain analyses of
movements from the First and Second String Quartets, with the former focusing on U-
transformations and dyad-spaces in the first movements of both quartets and the latter
focusing on various internal movements of the Second Quartet. Finally, the fifth chapter
concludes the dissertation by summarizing the previous chapters and offering several
possible avenues of further research.
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Table of Contents
Chapter One Introduction, Review of the Literature, 1 Broad Remarks and Chapter Overviews I. Introduction 1 II. Review of the Literature 5 III. Broad Remarks 8 IV. Chapter Overviews 12 Chapter Two Methodology 14 I. Pitches, Pitch-classes and Pitch-class Collections 15 II. Contextual Transformations K(nm) and U(nm) 18 III. Tiles 28 IV. Crosscuts 29 V. Overview of Main Rhythmic Profiles 31 Chapter Three Dyad-Spaces, U-transformations and Cycles in the 33 First Movements of Quartets One and Two I. Dyad-Spaces and Large-Scale Harmony 38 II. U-transformations and the Melodic/Motivic Aspect 45 Chapter Four Interval Cycles and Cyclic Collections in 60 Internal Movements I. (Mostly) Twelve-Tone Movements Two and Three 61 Of the Second String Quartet Chapter Five Conclusions and Directions for Further Research 86 I. Conclusions 86
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List of Examples (Chapters 1-5)
Example Title Page Example 2.1a The OS set 16 Example 2.1b Trichordal subset shadings of the OS set 16 Example 2.1c Musical example of shading Appendix 2 Example 2.2a Musical example of coloring Appendix 2 Example 2.2b Coloring of the 7-cycle dyad G-D Appendix 2 Example 2.3a K-transformations 19 Example 2.3b U-transformations 19 Example 2.4 Quartet No. I/I, mm. 1-2 Appendix 2 Example 2.5a Transformations of source trichord <265> Appendix 2 Example 2.5b Illustration of terminology 23 Example 2.5c Demonstration of labels 23 Example 2.6 Generation of octatonic and hexatonic collections 27 Example 2.7a Generation of pentatonic/diatonic collections 27 Example 2.7b Generation of extended tertian sonorities 27 Example 2.8a Depiction of Tile 1 in Quartet 2/III, mm. 16-19 Appendix 2 Example 2.8b Depiction of Tile 2 in Quartet 2/III, mm. 20-23 Appendix 2 Example 2.8c Depiction of Tile 7 in Quartet 2/III, mm. 44-47 Appendix 2 Example 2.9 Depiction of Crosscut in Quartet 2/I, mm. 32-37 Appendix 2 Example 3.1a Musical Example of Quartet 1/I, mm. 1-7 Appendix 2 Example 3.1b Chords in mm. 2 and 5 Appendix 2
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Example 3.1c Voice-leading between chords in mm. 2 and 5 Appendix 2 Example 3.2a Musical Example of First Theme of Quartet 2/I, Appendix 2 mm. 5-9 Example 3.2b Reduction of First Theme of Quartet 2/I Appendix 2 Example 3.3a Network of OS shadings and dyad-spaces in 42 Quartet 1/I Example 3.3b Network reinterpretation of outer voices of 43 Example 3.3a Example 3.4a Musical Example of Second Theme of Quartet 1/I Appendix 2 Example 3.4b Musical Example of Canon in Second Theme of Appendix 2 Quartet 1/I Example 3.4c Dyad-spaces in Recapitulation of Second Theme Appendix 2 Of Quartet 1/I Example 3.5a Musical Example of Quartet 1/I, mm. 16-24 Appendix 2 Example 3.5b Nested U-cells in First Theme of Quartet 1/I 48 Example 3.6 Five accompanimental chords in First Theme of Appendix 2 Quartet 1/I Example 3.7 Network of voice-leading in accompanimental chords 51 Example 3.8a Musical Example of Transition Theme One in Appendix 2 Quartet 1/I, mm. 60-61 Example 3.8b Musical Example of Transition Theme Two in Appendix 2 Quartet 1/I, mm. 72-73 Example 3.8c Musical Example of Transition Theme Two in Appendix 2 Quartet 1/I, mm. 76-77 Example 3.9a Musical Example of Quartet 2/I, mm. 1-4 Appendix 2 Example 3.9b U-cells and crosscuts in First Theme of Quartet 2/I Appendix 2 Example 3.9c Musical Example of First Theme of Quartet 2/I, Appendix 2 mm. 20-23
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Example 3.10a Musical Example of Second Theme of Quartet 2/I, Appendix 2 mm. 98-104 Example 3.10b Musical Example of Quartet 2/I, mm. 118-121 Appendix 2 Example 4.1a Rows 3-R1 and 3-R2 and their properties in 65 Quartet 2/III Example 4.1b Row 2-R1 and its properties in Quartet 2/III 65 Example 4.2 Row forms in Quartet 2/III 68 Example 4.3a Network of ops in Quartet 2/III 68 Example 4.3b Network of row forms in Quartet 2/III 68 Example 4.3c Network Q2/M3 68 Example 4.4a Musical Example of canon in Quartet 2/III, Appendix 2 mm. 165-167 Example 4.4b Musical Example of 3-cycle dyads in Quartet 2/III, Appendix 2 mm. 165-167 Example 4.5 Row Forms in B section of Quartet 2/III Appendix 2 Example 4.6 Musical Example of canon and interpretive network Appendix 2 In Quartet 2/II Example 4.7a Whole-tone collections in Quartet 2/II, mm. 23-24 Appendix 2 Example 4.7b Whole-tone collections in Quartet 2/II, mm. 27-28 Appendix 2 Example 4.8a Musical Example of hexatonic collections in 80 Quartet 2/III Example 4.8b Musical Example of Tile C1 in Quartet 2/III, Appendix 2 mm. 195-196 Example 4.8c Network interpreting Example 4.8b 80 Example 4.9a Musical Example of Tile C12 in Quartet 2/III, Appendix 2 mm. 202-204 Example 4.9b Network interpretation of Example 4.9a 82
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Example 4.10 B section of Quartet 2/II Appendix 2 Example 4.11 2-/4-cycle collections in Quartet 2/II Appendix 2 Example 5.1a Musical Example of Quartet 2/IV, mm. 1-4 Appendix 2 Example 5.1b Musical Example of Quartet 2/IV, m. 6 Appendix 2 Example 5.1c Musical Example of Quartet 2/IV, mm. 13-14 Appendix 2 Example 5.2 Musical Example of Quartet 3/III, mm. 1-7 Appendix 2 Example 5.3a Reduction of Sonata For Guitar Appendix 2 Example 5.3b K-nets in Sonata for Guitar Appendix 2 Example 5.3c Cycles in Sonata for Guitar Appendix 2 Example 5.3d “Hyper cycles” in Sonata for Guitar Appendix 2 Example 5.4 U-cells in Berg, Op. 4, No. 2 Appendix 2 Example 5.5 U-cells in Berg, Op. 2/III Appendix 2
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List of Tables Table Title Page Table 3.1 Form Chart of Quartet 1/I 34 Table 3.2 Form Chart of Quartet 2/I 35 Table 4.1 Form Chart of Quartet 2/II 62 Table 4.2 Form Chart of Quartet 2/III 63
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List of Examples (Appendix Two) Example Title Page Example 2.1c Musical Example of shading 1 Example 2.2a Musical Example of coloring 2 Example 2.2b Coloring of the 7-cycle dyad G-D 2 Example 2.4 Quartet No. I/I, mm. 1-2 3 Example 2.5a Transformations of source trichord <265> 3 Example 2.8a Depiction of Tile 1 in Quartet 2/iii, mm. 16-19 4 Example 2.8b Depiction of Tile 1 in Quartet 2/iii, mm. 20-23 4 Example 2.8c Depiction of Tile 7 in Quartet 2/iii, mm. 44-47 5 Example 2.9 Depiction of Crosscut in Quartet 2/I, mm. 32-37 6 Example 3.1a Musical Example of Quartet 1/I, mm. 1-7 7 Example 3.1b Chords in Quartet 1/I, mm. 2 and 5 7 Example 3.1c Voice-leading between chords in mm. 2 and 5 7 Example 3.2a First Theme of Quartet 2/I, mm. 5-9 8 Example 3.2b First Theme of Quartet 2/I throughout movement 8 Example 3.4a Second Theme of Quartet 1/I, mm. 85-88 9 Example 3.4b Canon in Second Theme of Quartet 1/I 9 Example 3.4c Dyad-spaces in Recapitulation in Quartet 2/I 10 Example 3.5a Musical Example of Quartet 1/I, mm. 16-24 11 Example 3.6 Five accompanimental chords in First Theme 12 Example 3.8a Transition Theme One in Quartet 1/I, mm. 60-61 12 Example 3.8b Transition Theme Two in Quartet 1/I, mm. 72-73 12
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Example 3.8a Transition Theme Two in Quartet 1/I, mm. 76-77 13 Example 3.9a Opening of Quartet 2/I, mm. 1-5 13 Example 3.9b U-cells in Crosscuts of First Theme in Quartet 2/I 14 Example 3.9c Closing of First Theme in Quartet 2/1, mm. 20-23 14 Example 3.10a Second Theme of Quartet 2/I, mm. 98-104 15 Example 3.10b Tile 15 in Quartet 2/I, mm. 118-121 15 Example 4.4a Canon in Tile A14 of Quartet 2/iii, mm. 165-167 16 Example 4.4b 3-cycle dyads in A14 canon 16 Example 4.5 Row Forms in B section of Quartet 2/iii 17 Example 4.6 Canon and interpretive network in Quartet 2/ii 17 Example 4.7a Whole-tone collections in Quartet 2/ii, mm. 23-24 18 Example 4.7b Whole-tone collections in Quartet 2/ii, mm. 27-28 18 Example 4.8b Tile C1 in Quartet 2/iii, mm. 195-196 19 Example 4.9a Tile C12 in Quartet 2/iii, mm. 202-204 19 Example 4.10 B section of Quartet 2/ii 20 Example 4.11 2-/4-cycle collections in Quartet 2/ii, mm. 13-14 21 Example 5.1a Opening of Quartet 2/iv, mm. 1-4 21 Example 5.1b Second thematic statement in Quartet 2/iv, m. 6 21 Example 5.1c End of Theme in Quartet 2/iv, mm. 13-14 22 Example 5.2 Opening of Quartet 3/iii, mm. 1-7 22 Example 5.3a Opening of Sonata for Guitar 23 Example 5.3b K-nets in Sonata for Guitar 23 Example 5.3c Cycles in Sonata for Guitar 24
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Example 5.3d “Hyper cycles” in Sonata for Guitar 24 Example 5.4 U-cells in Berg, Op. 4, No. 2 24 Example 5.5 U-cells in Berg, Op. 2, No. 3 25
1
Chapter One
Introduction, Review of the Literature, Broad Remarks, and Chapter Overviews
I. Introduction
Since the end of the 19th Century, composers working within the tradition of
Western art music have taken remarkably diverse approaches to composition. In works of
varying style and expression, among other developments, the language of tonality
becomes one choice among many, and formal as well as rhythmic musical elements
associated with regional traditions come to the forefront. In most cases, however, the
composers and specific works that tend to be the objects of music-theoretic research, as it
has developed in the latter half of the 20th century, continue to stem from the Western
European heritage. Moreover, much of the research done by theorists in particular often
disregards the panoply of issues that arises from the diverse national and cultural
influences associated with the language of new musical forms and expression; as
theorists, our approach to research tends to radiate from a Eurocentric nucleus, given that
many of our analytic tools were developed in response to the Western canon.1 Thus,
despite this variety of culturally encoded compositions, most music theorists have
focused their analytic and theory-building efforts on a very select group of works and
composers, relying upon a limited but highly developed set of analytic tools specifically
1 The Western canon is, of course, itself a concept that has been subjected to critical reassessment in recent years. See Patrick McCreless, “Rethinking Contemporary Music Theory,” in Keeping Score: Music, Disciplinarity, Culture, eds. David Schwartz, Anahid Kassabian and Lawrence Siegel. Charlottesville: University Press of Virginia, 1997, 13-52. For broader views on decolonializing discourse on music, see Kofi Agawu, Representing African Music: Postcolonial Notes, Queries, Positions. New York and London: Routledge Press, 2003; and Malena Kuss, “Prologue” to Music in Latin America and the Caribbean, ed. Malena Kuss. Austin: University of Texas Press, 2004, ix-xxvi.
2
designed to illustrate technical aspects of specific repertories. Regrettably, this focus has
resulted in the undue neglect of many worthy composers, as in the case of Argentine
composer Alberto Ginastera (1916-1983). Ginastera’s great admiration for and careful
study of the composers who tend to be the objects of study, such as Stravinsky, Bartók,
Schoenberg, Berg, Debussy, and Webern is well documented, not only by others but by
the composer himself.2 Yet only a few writers, principally among them Malena Kuss,
have availed themselves of the potential available for applying new directions in music-
theoretic research to his original approach to pitch, rhythm, and timbric organization.3
As Ginastera weaves into his music a rich web of associations, it is indeed
possible approach his works from many different perspectives. Accordingly, writers have
viewed Ginastera through a variety of lenses. One such focus views Ginastera in the
context of nationalist tendencies. Nationalist approaches tend to concentrate exclusively
on the presence or absence of folk references in the scores. For instance, Deborah
Schwartz-Kates relies on the three stylistic periods Ginastera himself formulated in
2 See Chase, Gilbert. “Alberto Ginastera: Argentine Composer,” The Musical Quarterly XLIII/4 (1957): 439-460; Wallace, David. “Alberto Ginastera: An Analysis of His Style and Techniques of Composition.” Ph.D. diss., Northwestern University, 1964; Suárez-Urtubey, Pola. Alberto Ginastera. Buenos Aires: Ediciones Culturales Argentinas, 1967, and Alberto Ginastera en cinco movimientos. Buenos Aires: Editorial Victor Lerú, 1972, Terrapon, Luc. “Gespräch mit Alberto Ginastera,” in Alberto Ginastera, ed. Friedrich Spangemacher, Bonn: Boosey & Hawkes, 1984 (original French, 1982); Tan, Lillian. “An Interview with Alberto Ginastera,” American Music Teacher XXXIII/3 (1984): 6-8; and Kuss, Malena, “Ginastera, Alberto” in Die Musik in Geschichte und Gegenwart: Allgemeine Enzyclopädie der Musik., 2nd revised edition, ed. Ludwig Finscher. Kassel: Bärenreiter Verlag; Stuttgart: Metzler Verlag, 2002, Personenteil, vol. 7, cols. 974-982. 3 An exception to this assertion of analytical neglect is to be found in the writings of Kuss, who has published numerous articles on Ginastera’s music, as will be discussed below.
3
interviews published by Pola Suárez Urtubey in 1967.4 These follow a rough trajectory,
from direct references to Argentinian folk materials through a sublimation of these
elements into a mature style and an engagement with contemporary techniques. As with
other composers, however, this stylistic division and the general application of
“nationalist” arguments is problematic in its application to Ginastera’s music. Writers
such as Kuss, for instance, have cautioned against employing the term “nationalism” to
Ginastera’s music, as indeed the composer himself has.5 Although the issues surrounding
these positions and the controversies found therein are of great interest and relevance in
understanding Ginastera’s music and its place in relation to other composers, such
concerns are not the focus of this dissertation. Instead, taking a more common music-
theoretic approach, the present work limits its scope to the application of recent advances
in the study of pitch organization, in particular those stemming from the writings of
George Perle, to Ginastera’s String Quartets nos. 1 (1948) and 2 (1958, first version).6
Although the opus numbers of Ginastera’s body of works identify a total of 54
beginning with the ballet Panambí, Op. 1 (1934-37) and ending with the one-movement
4 Deborah Schwartz-Kates. “Ginastera, Alberto,” in Grove Music Online. Oxford Music Online, http://www.oxfordmusiconline.com/subscriber/article/grove/music/11159 (accessed March 29, 2009). 5 In Kuss 2002, specific reference is made to Ginastera’s rejection of the application of the “musical nationalist” label to his music. 6 See Perle, George, The Listening Composer. Berkeley: University of California Press, 1990; Serial Composition and Atonality. 6th ed., rev. Berkeley: University of California Press, 1991; and Twelve-Tone Tonality. 2nd ed. Berkeley: University of California Press, 1996.
4
Sonata No. 3 for piano, Op. 54 (1982), 53 exist as authorized works.7 Among these
works, several emerge as significant. Foremost of these are his three operas Don Rodrigo,
Op. 31 (1964), Bomarzo, Op. 34 (1967), and Beatrix Cenci, Op. 38 (1971); the Piano
Concerto No. 1, Op. 28 (1961) and Piano Concerto No. 2, Op. 39 (1972); the Sonata for
Piano, Op. 22 (1952) and Sonata for Piano No. 2, Op. 53 (1981); the String Quartet No.
1, Op. 20 (1948), String Quartet No. 2, Op. 26 (1958, revised in 1967) and String Quartet
No. 3, Op. 40 (1973); the symphonic fresco Popol Vuh, Op. 44 (1975-1982), and the
Cantata para América mágica, Op. 27 (1960). The significance of these works stems
from their position in Ginastera’s development as a composer, and, in the case of the
String Quartet No.2 and the opera Don Rodrigo, from critically acclaimed performances
in the United States that helped shape Ginastera’s international recognition. Among these
works, the first and second string quartets suit the present purpose of exploring
Ginastera’s musical language in the context of theoretical methods developed from
studies of European composers. In response to the musical surfaces in selected
movements of the first and second string quartets, this dissertation will develop an idea
rooted in transformation theory that models a process by which a trichordal motive
unfolds systematically into larger symmetrical pitch-class constructions. My analysis of
the quartets, partially based on this process, reveals striking relationships between the
quartets, providing one type of evidence supporting Kuss’ view of Ginastera’s work as a
unified body across which multiple manifestations of musical ideas create a “network of
7 Kuss 2002 (MGG entry) provides the most comprehensive, detailed, and accurate overview of Ginastera’s life and works. Also see Kuss, Malena, Alberto Ginastera: Musikmanuskripte. Winterthur, Schweiz: Amadeus Verlag, 1990, 32 pages (Inventare der Paul Sacher Stiftung, No.8) for additional discussion the issue of authorized and unauthorized works.
5
intratextual relationships” (Netz intratextueller Verbindungen) which problematizes the
categorization of the composer’s work into “style periods” as several authors have
attempted to do.8
II. Review of the Literature
In 1942 Aaron Copland brought attention to a young Ginastera in his well-known
article published in Modern Music.9 Gilbert Chase, writing in 1957, established a context
for the composers works written up to the Harp Concerto of 1956, surveying relevant
aspects of Argentina’s history and geography as well as the composer’s family,
education, and works up to 1956, with examples of his music.10 Pola Suárez Urtubey
authored two Spanish-language books (1967 and 1972) containing interviews with the
composer, published critical reactions to specific works, and brief analytical references
with musical examples.11 Malena Kuss’ numerous publications over a 40-year span begin
with an article in 1970 detailing the composer’s approach to teaching.12 Her analytic
writings on Ginastera’s works begin with her 1976 dissertation on Argentine operas at the
8 Based on the comprehensive sketch study of Ginastera’s works, Kuss 2002 recognizes within them Schoenbergs’s concept of the Entelechie relevant, a single energy or impulse whose various musical manifestations recur in new forms to create this intratextual network and identifies numerous examples of reappearing musical ideas within the composer’s oeuvre; see also Kuss 1990. 9 Copland, Aaron. “The Composers of South America.” Modern Music 19/2 (1942): 75-82. 10 Chase, Gilbert. “Alberto Ginastera: Argentine Composer,” The Musical Quarterly XLIII/4 (1957): 439-460. 11 Suárez Urtubey, Pola. Alberto Ginastera. Buenos Aires: Ediciones Culturales Argentinas, 1967 and Alberto Ginastera en cinco movimientos. Buenos Aires: Editorial Víctor Lerú, 1972. 12 Kuss, Malena. “Alberto Ginastera and the Early Training of the Composer.” Heterofonía 3/1 (Enero-Febrero 1970, 13-17 (abstracted in Melos [September 1970], 350).
6
Teatro Colón (1908-1972), a work that establishes one of the major themes in her work,
namely musical structure and dramaturgy in the composer’s three operas.13 Articles on
this specific subject of opera analysis include studies of the encoding of a native idiom in
Don Rodrigo’s (1964) basic twelve-tone row and its variety of structural roles in 1980,
dramaturgic and musical structure in Bomarzo (1967) in 1984 and 2002, and a study of
the idiosyncratic assimilation of nativistic strands in (among others) Don Rodrigo in 1990
and 1992.14 Outside of the operas, Kuss has also discussed structural elements of many of
the composer’s other works, including the Second String Quartet (1958) in 1989 and
2002, the Piano Sonata No. 1 (1952) in 2002, organized the inventory of the composer’s
collected works for the Paul Sacher Stiftung in 1990, authored an extensive entry in Die
13 Ibid, Nativistic Strains in Argentine Operas Premiered at the Teatro Colón (1908-1972). Ph.D. dissertation, Historical Musicology, University of California at Los Angeles, 1976. Ann Arbor, Michigan: University Microfilms International, 76-28570, 533 pages. 14 Ibid, “Type, Derivation, and Use of Folk Idioms in Ginastera’s Don Rodrigo (1964),” Latin American Music Review 1/2 (Fall-Winter 1980), 176-196; “Symbol und Phantasie in Ginasteras Bomarzo (1967)” in Alberto Ginastera, edited by Friedrich Spangemacher. Bonn: Boosey & Hawkes, 1984, 88-102 (Series Musik der Zeit: Dokumentationen und Studien, No. 4); “Identity and change: Nativism in operas from Argentina, Brazil, and Mexico” in Musical Repercussions of 1492: Encounters in Text and Performance, edited by Carol E. Robertson. Washington, D.C.: The Smithsonian Institution Press, 1992, 299-335 (Proceedings of the symposium on Musical Repercussions of 1492, Quincentennary Program, Smithsonian Institution, 1988); “The structural role of folk elements in 20th-century art music” in Proceedings of the XIVth Congress, International Musicological Society: Transmission and Reception of Musical Culture, edited by Lorenzo Bianconi, F. Alberto Gallo, Angelo Pompilio, and Donatella Restanti. Torino: EDT/Musica, 1990, vol. III, 99-120; and “’Si quieres saber de mí, te lo dirán unas piedras’: Alberto Ginastera, autor de Bomarzo” in Ópera en España e Hispanoamérica, 2 vols., edited by Emilio Casares Rodicio and Álvaro Torrente. Madrid: ICCMU (Instituto Complutense de Ciencias Musicales), 2002, vol. II, 393-411.
7
Musik in Geschichte und Gegenwart in 2002.15 The composer’s death in 1983 inspired a
spate of articles and interviews appearing in Alberto Ginastera (1984) and the Latin
American Music Review 6/1 (1985), the latter edited by Kuss. The former includes
German translations of essays by Ginastera, a definitive analysis of the Cantata para
América mágica (1960) by Hans-Werner Heister, and an interview with the composer by
Luc Terrapon, while the latter contains articles by Chase, Stuart St. Pope, Carleton
Sprague Smith, and Robert Stevenson.16 In addition, dissertations by Wallace in 1964,
Richards in 1985, and Campbell in 1991 provide analyses ranging from broad treatments
of the composer’s early work (Wallace), pitch-structure in Don Rodrigo (Richards), and
the three piano sonatas (Campbell).17 The most recent research on the composer is a book
by Antonieta Sottile in 2007.18
15 Ibid, “Alberto Ginastera,” Mitteilungen der Paul Sacher Stiftung (Basel, Switzerland), No. 2 (Januar 1989), 17-18; “Ginastera, Alberto” in Die Musik in Geschichte und Gegenwart: Allgemeine Enzyclopädie der Musik., 2nd revised edition, ed. Ludwig Finscher. Kassel: Bärenreiter Verlag; Stuttgart: Metzler Verlag, 2002, Personenteil vol. 7, cols. 974-982; “La poética referencial de Astor Piazzolla,” in Estudios sobre la obra de Astor Piazzolla, edited by Omar García Brunelli. Buenos Aires: Gourmet Musical Ediciones, 2007, 57-76 and Revista del Instituto Superior de Música (Santa Fe, Argentina, Universidad Nacional del Litoral), No. 9 (2002), 11-29; “La certidumbre de la utopía: Estrategias interpretativas para una historia musical americana,” Música (La Habana, Boletín de Casa de las Américas), Nueva Época, No. 4 (2000), 4-24; and Alberto Ginastera: Musikmanuskripte. Winterthur, Schweiz: Amadeus Verlag, 1990, 32 pages (Inventare der Paul Sacher Stiftung, No.8). 16 See Alberto Ginastera, edited by Friedrich Spangemacher. Bonn: Boosey & Hawkes, 1984, 88-102 (Series Musik der Zeit: Dokumentationen und Studien, No. 4). 17 Wallace, David. “Alberto Ginastera: An Analysis of His Style and Techniques of Composition.” Ph.D. diss., Northwestern University, 1964; Richards, James Edward, Jr. “Pitch Structure in the Opera Don Rodrigo.” Ph.D. diss., University of Rochester, 1985; and Campbell, Grace M. “Evolution, Symmetrization, and Synthesis: The Piano Sonatas of Alberto Ginastera.” D.M.A. diss., University of North Texas, 1991 (advised by Kuss). 18 Sottile, Antonieta. Alberto Ginastera: Le(s) style(s) d’un compositeur argentin. Paris: L’Harmattan, 2007.
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III. Broad Remarks
As mentioned previously, this dissertation is narrowly focused to analyses of
Ginastera’s first and second string quartets and their engagement with the European
tradition.19 Despite the pivotal roles the first and second string quartets play in
understanding the evolution of Ginastera’s compositional career, little analytic work has
been done on these quartets outside of Kuss 1989 and 2002. In published works, only five
authors have engaged the quartets from an analytic perspective, but in most cases, the
analyses are used in broad discussions of large issues such as musical style, nationalism
and 12-tone techniques.20 Of these authors and works, only Wallace 1964 addresses
specific issues of form, pitch and rhythm that arise in the quartets as autonomous works.
Although the First String Quartet (1948) and the Second String Quartet (1958)
are ten years apart, the works share many striking similarities. Both quartets reflect
neoclassic approaches to form, texture, phrasing, and tonal centricity. Formally, the
quartets contain four and five movements respectively, employing thematic sonata
forms21 (first movement of both quartets), three-part Rondo forms (ABA in movements
two and three of the first quartet and movements two and four of the second), five-part
19 Two versions of the second quartet exist. The earlier version (1958) is the one this dissertation will address. The revised version (1967) omits both a reference to a folk rhythm in its third movement and a quotation from “Triste” in the fourth movement of the composer’s Cinco canciones populares argentines, Op. 10 (1943). These changes were first introduced in the Concerto per corde, Op. 33 (1965), an arrangement of the last four movements of the quartet for string orchestra (see Kuss 2002). 20 See Chase 1957, pp. 441, 450-451; Wallace 1964, pp. 144-152, 219-245; Suárez Urtubey 1967, pp. 27-29 and 1972, pp. 44-46, 58-60, pp. 875 and 877; and Kuss 2002, cols. 976, 980. Parts of the second and third movements of the second quartet are 12-tone. Chapter four of this dissertation analyzes these movements in detail. 21 In the present case, “thematic” sonata form refers to the use of two contrasting themes within a three-part, exposition-development-recapitulation rhetorical plan. Unlike traditional sonata forms, however, the first movements do not feature the transposition of exposition themes in the recapitulation.
9
Rondo forms (ABABA in movement four of the first quartet and ABACA in movement
three of the second quartet), and Theme and Variations (movement four of the second
quartet).22 Texturally, the pieces most often employ melody and accompaniment,
imitative counterpoint, and pointillistic approaches.
In addition to broad ideas of form and texture, issues of pitch are of paramount
importance in the quartets. Significant pitches generally appear with salience on the
musical surface as extended pedal points, registral boundaries, points of imitation or
initiation of themes and other significant constructions; a study of the various analytic
contexts surrounding the salient pitch (and more generally pitch-class) appearance is the
central aim of this dissertation.23
One such analytic context is found in the frequent appearance of salient pcs as
subsets of familiar referential collections.24 A quintessential example of this type of
context appears in the first two measures of the first movement of the String Quartet No.
1, in which the pc D, the central pc of the movement, initiates a six eighth-note octatonic
22 Kuss 2002 in particular notes the connection between the second quartet and Bartók’s Fourth String Quartet (1928) and Berg’s Lyric Suite (1926). The connection to the Lyric Suite is also made explicit in Irving Lowens’ article “Current Chronicle,” MQ 44 (1958), p. 378. Also see Ginastera, “Hommage à Bartók” and Terrapon 1982/4 for references to other composers Ginastera admired. 23 A list of ten criteria for salience in atonal music appear in Lerdahl, Fred. “Atonal Prolongational Structure,” Contemporary Music Review 4 (1989): 64-87. Lerdahl’s study determines that an event is salient if it is: a) attacked within a particular time span, b) in a relatively strong metrical position, c) relatively loud, d) relatively prominent timbrally, e) in an extreme registral position, f) relatively dense, g) relatively long in duration, h) relatively important motivically, i) next to a relatively large grouping boundary, and j) parallel to a choice made elsewhere in the analysis. This dissertation does not rigorously reconcile the musical surfaces studied with these exact criteria, but its claims regarding the salience of the events comport comfortably with them. These criteria are tested and refined in Dibben, Nicola. “The Perception of Structural Stability in Atonal Music: The Influence of Salience, Stability, Horizontal Motion, Pitch Commonality, and Dissonance,” Music Perception 16/3 (1999): 265-294. 24 These include pentatonic, diatonic, hexatonic, octatonic, and whole-tone collections.
10
gesture that unfolds from what might be termed a three-note Grundgestalt <D-F#-F>.25
Several authors above refer to various pc collections, establishing a precedent for delving
into the issue of octatonicism in the composer’s music.26 Beginning with Wallace 1964,
some of these authors refer to the multiple salient appearances of different combinations
of major and minor triads in a larger sample of pieces as “polytonality.”27 Since the mid-
1980s, but formulating it in print in 2002, Kuss has described Ginastera’s pre-12-tone
pitch language as “octatonic with modal interaction.”28 The present dissertation develops
the issue of the interaction between motive and collection in terms of interval cycles
(cyclic pc collections) that often result from a systematic transformation of motives.29
25 This example illustrates the multifaceted role D plays in the movement, as D 1) establishes itself as a focal pc, 2) initiates the preeminent motive of the movement, and 3) locates that motive within a familiar pc collection. 26 See Wallace 1964, Kuss 1980 and 2002; Heister, “Trauer eines Halbkontinents und Vergegenwärtigung von Geschichte: Ginasteras ‘Cantata para América mágica, Op. 27,” in Alberto Ginastera, edited by Friedrich Spangemacher. Bonn: Boosey & Hawkes, 1984, 45-75 (Series Musik der Zeit: Dokumentationen und Studien, No. 4); Campbell, Grace M. “Evolution, Symmetrization, and Synthesis: The Piano Sonatas of Alberto Ginastera,” D.M.A. diss., University of North Texas, 1991; and Tabor, Michelle, “Alberto Ginastera’s Late Instrumental Style,” Latin American Music Review 15/1 (1994): 1-31. 27 See Berger, Arthur. “Problems of Pitch Organization in Stravinsky,” Perspectives of New Music 2/1 (1963): 11-42; and Lendvai, Ernö. Béla Bartók: an analysis of his music. London: Kahn & Averill, 1973 for seminal music-theoretic work on the interactions of diatonic and octatonic collections in Bartók’s music. 28 See “oktatonisch mit modaler Interaktion,” col. 979. Although Kuss’ use of the term in this case is technically limited to the Piano Sonata (1952), the term is generally applicable to a noteworthy amount of Ginastera’s music. The original concept of octatonic-diatonic interaction was first introduced in such a form by Pieter C.van den Toorn in The Music of Igor Stravinsky. New Haven, Connecticut: Yale University Press, 1983. 29 Composer and musicologist George Perle was the first to develop a theory based on interval cycles (see Perle 1962, 1977, and 1990 for a comprehensive exposition of interval cycles). Also see Antokoletz, Elliott. The Music of Béla Bartók. Berkeley: University of California Press, 1984, and Headlam, Dave. The Music of Alban Berg. New Haven: Yale University Press, 1996.
11
Salient pitches, pcs, collections and motives are often concrete manifestations of
the more general idea of patterning in Ginastera’s music. The use of patterns of notes
and/or rhythms is a hallmark of the composer’s style and can be found readily in many of
the composer’s works. In the present quartets, the aforementioned salient musical
features are in many cases manifestations of patterns, and an understanding of the
mechanism behind the patterns helps to provide a more complete view of a major source
of consistency between the quartets. In its analysis of the quartets, this dissertation offers
one possible transformational model of the above unfolding process, the pattern’s source,
which links salient pc and motive to referential collection.
The use of cyclic, hence symmetrical, pitch and pc collections invites inquiry into
potential ways various types of symmetry manifest themselves on the musical surface and
structure. However, the music-theoretic literature on symmetry within the context of
music analysis is far too vast to discuss presently. In literature on Ginastera’s work,
Chase 1957 discusses formal symmetry, as does Suárez Urtubey 1967, Kuss 1980, 1984,
2000 and 2002, and Heister 1984; Kuss 1980, 1984, 1989, and 2002, and Campbell 1991
discuss symmetry in pitch. The present dissertation discusses various formal and
pitch/pc-based symmetries in the quartets. The latter receives special attention due to the
ubiquity of interval cycles in the music.
Much like the presence of interval cycles in music invites symmetry-based
analytic approaches, the presence of formal symmetry in both the scholarly literature and
the music on which that literature focuses invites a more detailed investigation into any
role symmetry may play in the music’s rhetoric and form. In its approach of this subject,
the present dissertation develops a simple concept that incorporates the above notion of
12
mechanism into a small-scale formal apparatus that comports with aspects of pitch/pc
language and hypermeter. Specifically, the device provides a facile vocabulary for
discussing the issue of seemingly discontinuous rhetorical elements and their relationship
to the pitch/pc element of the music.
Despite various attempts over the years to divide Ginastera’s compositional
output into labeled style periods, the picture that emerges from the present study of the
first two quartets is one of tremendous continuity that transcends the traditional labels
often applied to the composer’s music.
IV. Chapter Overviews
To accomplish the goals stated above, the dissertation offers five chapters. The
present chapter provides a cursory overview of the composer’s works and summarizes
some of the scholarly community’s reception of them. In this initial Chapter One, we
have learned about Ginastera scholarship, significant broad aspects of the composer’s
style, the pivotal roles the first two string quartets play in stylistic assessments of the
composer’s works, and the present author’s basic intuitions about the musical contents of
the quartets. Chapter Two discusses the basic methodology behind the present approach,
reviewing applicable existing analytic approaches and providing several original
contributions stemming from the present author’s experience with the quartets. Chapter
Three presents detailed analytic commentary on the first movements of the quartets based
on the new methodology discussed in Chapter Two, while Chapter Four engages in a
more traditional cyclic analysis of two internal movements of the latter quartet. Finally,
13
Chapter Five offers conclusions based upon the previous chapters and proposes several
avenues for further research.
14
Chapter Two
Methodology
As discussed in the preceding chapter, Ginastera’s First and Second String
Quartets play significant roles in his oeuvre in a number of contexts. This chapter
presents a methodology for the detailed analyses to follow in chapters 3 and 4. The main
features of the music to be examined are its melodic and rhythmic patterns, symmetry,
and overt repetition of focal pitches. In addition, the relationship of motivic elements to
larger octatonic and other symmetrical collections will be articulated. Building upon
well-established ways to relate individual pitch-classes (pcs) to small cyclic and non-
cyclic pitch-class sets (pcsets as articulated in pcset theory, transformation theory, and
cyclic theory), the methodology introduces seven analytic/interpretive devices which
highlight the roles of the above features in specific ways, allowing the idiosyncrasies of
the quartets to participate meaningfully in this interpretation of their musical processes.30
These seven devices fall into three broad categories. The first is based on pc- and
pc-collections and includes definitions of “dyad-space,” as well as “coloring” and
“shading” basic collections. The second stems from transformational and interpretive
devices which demonstrate order and invariant relationships, given below as K(nm), U(nm),
and J00. Finally, the third emerges from formal/rhetorical devices described as “tiles” and
“crosscuts.” The seven devices in these categories combine to present a more
comprehensive picture of the musical processes in the first two quartets.
30 Large-scale representatives of these approaches are contained in Forte 1973, Lewin 1987, and Perle 1962, 1977 and 1990.
15
I. Pitches, Pitch-classes and Pitch-Class Collections
[Examples 2.1a, b and c]
Each of the nine movements of the first two quartets significantly incorporates the
ensemble’s open strings, either as a referential collection or specified subset.31 Example
2.1a illustrates the basic 7-cycle pentatonic collection {CDEGA} of the string quartet’s
component instruments; each instrument contributes a conjunct subset of the overall
collection. Contextual emphasis on a specific conjunct subset to the overall collection
will be referred to as a “shading” of the basic collection. For example, the leftmost arrow
in Example 2.1a indicates a cello/viola “shading” of the basic collection, emphasizing the
open string tetrachord {CGDA} within the overall pentachord. The rightmost depicts a
violin shading {GDAE}. Example 2.1b shows two different shadings of the overall
collection focused on conjunct trichords {CDG} and {DAE}. Example 2.1c provides an
example of how such shadings could be represented on a musical surface, showing a
passage from the first theme from the Second String Quartet in which the sonority on the
downbeats of mm. 16, 18 and 20 is considered a {CGD} trichordal shading of the OS
set.32 Owing to the ubiquity of such collections in the music, the notion of shading proves
useful in discussing connections between formal and tonal components of the musical
surface.
[Examples 2.2a and b]
Within collections associated with shading, pairs of pcs may emerge as structural
units; these pairs also extend beyond the shading note groups. Any pair of pcs interpreted
31 This is reminiscent of Berg’s Violin Concerto, which is discussed at length in Headlam 1996. 32 The shaded subset can be merely a pc (and not necessarily pitch) subset. In addition, shading does not have to be associated with a specific instrument.
17
as either a transposition or an inversion could be considered to establish or participate in
an established “dyad-space.” In general, a dyad-space involving pcs q and r is formally
defined as Tn/In(q) = r (and expressed by the argument q/r-(Tn/In)), meaning pcs q and r
are Tn- or In-related. Thus, a dyad-space is considered to be a transformational space
where the path from one pc to the other is an active transformation, rather than viewing
the dyad as a harmonic entity. An example of a dyad-space is given in Examples 2.2a and
b; Example 2.2a provides the score of the violin I melody from mm. 5-16 of the first
movement of the Second Quartet, while Example 2.2b depicts a reduction of the part.33
Pitch-classes G and D, the boundary pcs of the passage, are familiar from the previous
figure as members of the open string pc collection {CDEGA}. The two networks below
Example 2.2b provide two transformational interpretations of the passage. The leftmost
network interprets the reduction with two transpositions and two inversions, while the
rightmost interprets it as four transpositions.34 In the present context, salient pcs G and D
form two dyad-spaces, G/D-(I9) and G/D-(T7) depending upon the preferred
interpretation of the passage. Herein lies the advantage of the concept of the dyad-space:
various musical situations could highlight these two pcs in different ways. Interpreting
them as dyad-spaces with specific components (pcs and a transformation) facilitates the
many comparisons of a limited number of privileged pcs and/or transformations the
music frequently features.
Examples 2.2a and b also provide an illustration of the “coloring” of the
fundamental 7-cycle with a 6-cycle. As a member of set-class [0167], pc set <G-Ab-Db-
33 This passage is associated with the First Theme of the movement’s sonata form. 34 These are indeed well-formed K-net and L-net interpretations, but this fact is not germane to the present discussion since the networks are not included in a larger network demonstrating hyper-transformations.
18
D> exhibits both 5-/7-cycle and 6-cycle properties. Since G and D recur throughout the
quartets as privileged members of the fundamental open string collection while Ab and
Db do not, it becomes advantageous to refer to the pc set as projecting a primary 7-cycle
<G-D> colored by secondary 6-cycles, since one cycle is central to the quartets’ tonal
language and the other is not, thereby implying a relative hierarchic superiority of the
dyad <G-D>. The quartets very commonly feature passages in which multiple cycles and
cyclic processes occur simultaneously, and the ability to characterize these passages
accurately and easily is of value.
II. Contextual Transformations K(nm) and U(nm)
[Examples 2.3a and b]
The contextual transformations K(nm) and U(nm) are I and RI operators performed
on ordered trichords. Let both n and m be one of three order-positions f(0), s(1), or t(2); f,
s, and t stand for the first, second, and third terms in an ordered trichord <fst> called a
“source trichord.” For example, let pcset <265> be the source trichord for a particular
passage with f=2, s=6, and t=5. K(ff) performed on <265> inverts it via the inversion
operator that maps the first pc, in this case 2, into itself. This inversional index (in this
case I4) is then applied to the entire trichord, yielding <2te> (sum 4). Thus,
K(ff)(<265>)=<2te>.35 The K transformation can be applied to ff, fs, ft, ss, st and tt for a
35 As defined, K(ff) is limited to ordered trichords. The label J00 will be used for the same essential operation (inversion mapping the first element of an ordered pc set onto itself) performed on larger sets. For example, the tone-row <05e2819t7436> (the first row in the third movement of the second quartet) is J00-related to <071t4e325896>. Thus, in the present case, J00 is I0.
20
total of six possible transformations. The six K-transformations for the trichord <265>
are given in Example 2.3a.
Since K(nm) is an inversion operation, it is an involution; successive applications
of the identical transformation will toggle back and forth between the domain and its
range. The last column (I index) gives the familiar operation that would produce the
identical range for each K-transform. Again, since K/U-transforms are based on order-
positions in a source trichord, the In index would change given a different source trichord.
U(nm) works similarly, except it generates a retrograde-inversion of the given
source trichord. For example, U(st) performed on source trichord <265> inverts it via the
inversion operation that maps the second term, in this case 6, into the third term, in this
case 5. Thus, U(st) in this case performs RIe on <265>, yielding <659>.36 Since U(nm)
generates a retrograde-inversion of the source trichord, multiple applications may not
result in mere back-and-forth toggling between two pc sets. Example 2.3b provides the
same information for U-transforms as Example 2.3a gave for K-transforms, but it
incorporates the necessary concept of a p-value, which reflects the number of subsequent
identical transformations (the “order” of the transformation) required to return to the
original trichord.37 For instance, U(ff) on <265> requires four transformations to return
<265> and thus has an order of 4: <265> to <et2> (inversional sum 4 with a retrograde),
<et2> to <80e> (inversional sum t with a retrograde), <80e> to <548> (inversional sum 4
with a retrograde), and finally <548> to <265> (inversional sum t with a retrograde). The
considerable difference in the amount of information contained in both tables reflects the
36 Herein lies an intimate connection with Lewin’s RICH transformation. See Generalized Musical Intervals and Transformations, pp.180-181. 37 The notational convention in the leftmost column is intended to parallel mathematical exponents; p stands for “power.”
21
appreciable effect of retrogression in U-transforms. Despite these differences, however,
one similarity persists: like all K-transforms, U(ft) and U(ss) are involutions (order 2)
regardless of the pc membership of the source trichord. However, the remaining four
transformations are not involutions, and their order is dependent upon the source
trichord’s pc membership. For example, given source trichord <265>, U(ff) and U(tt) return
to the original source trichord <265> after four transformations, or are of order 4 ((U(ff))4
and/or (U(tt))4 of <265> is <265>) while U(ft) and U(st) are of order 8.38
[Example 2.4]
Looking at the RIn indices in Example 2.3b, we notice the cyclic nature of the “n”
values: RI4/t in a 6-cycle, T/I8/5/2/e in a 9- or 3-cycle. We also notice a symmetry, where
U(st) has the retrograde of the “n” values of U(fs), and U(tt) has the retrograde U(ff). Aside
from their symmetry, the “n” values are in interval cycles, and thus examples 2.3a and b
readily underscore the great relevance of interval cycles to a passage of music modeled
by K/U-transformations, especially U-transforms with p-values greater than 1. Example
2.4 provides such a passage, taken from the first two measures of the first quartet. Here
the initial forceful gesture is <D-F#-F-A-Ab> in violin I and cello, while violin II and
viola punctuate the 3-cycle <D-F-Ab>. The addition of violin I’s C5 on the downbeat of
m.2 finishes the gesture and extends the leftover dyad <F#-A> into another 3-cycle <F#-
A-C>. The held chord in m.2 is comprised entirely of the notes from m.1 (plus the violin
C), symmetrically registered <D2-A2-F#3-Ab3-F4-C5> around G3. In addition to its
cyclic interpretation as two partial 3-cycle collections {D,F,Ab} and {F#,A,C}, the chord
could also be considered as interlocking D major and F minor triads, an interpretation
38 A more detailed account of each step of the six U-transformations appears after in Appendix One, which appears after Chapter 5.
22
bolstered by the conspicuous placement of the D major triad (with the P5 D2-A2) in the
lower voices and F minor triad (with the P5 F4-C5) in the upper. Despite these tonal
readings, however, the current methodology sheds light on the generative process lurking
beneath the passage; indeed, the theme of surface convention and underlying generation
will recur throughout this dissertation. Figures 2.3a and b illustrate each step of the
process.
[Examples 2.5a, b, and c]
Example 2.5a depicts the initial three-note gesture <D-F#-F> as a pc motive
<265> that is transformed first into <659>, and then <598> and <980> by RIe, RI2, and
RI5, respectively, providing the individual “frames” in a larger “film” showing <265>
transforming into <980> via RI2. This succession of imbricated trichords extends one
pitch-class at a time. Ultimately, the film created by the chaining of individual frames is,
in a general sense, well-formed; each successive frame is generated by the same
homomorphism.
The passage is modeled in a more organic sense in Example 2.5b, which
interprets the same passage as three individual U(st) transformations ((U(st))3).39 The
source trichord is indicated on the bottom left of the figure, the dotted box highlights the
distinct four-note U-cell, and the dashed box highlights the six-note U-chain. Below the
figure appears a chart that provides the p-value (the order, or the number of times the
transformation is applied), the nomenclature for the pc set that successive transformations
39 The use of the term organic is intended to convey that the operations that create U-cells and U-chains from a source trichord derive from the trichord itself; a property of the source trichord determines the transformational path of the chain. In the example, U(st) is chosen because it is the only one of the twelve contextual transformations that replicates the int<4e4e4> in the chain.
24
generate, followed by the respective pc sets, Tn-types, and Tn/TnI-types. Example 2.5c
depicts how the objects appear in subsequent examples or prose. For example, the whole
incipit can now be meaningfully discussed as the U-chain <265>-<980>, a pc
construction made by unfolding the first term, source trichord <265>, into the second
term, final trichord <980> by means of U(st)3.
The network of Example 2.5b demonstrates two of the basic advantages of
contextual K-/U-transformations. First, they locate the origin of an ordered trichord’s
transformational path within the trichord itself. Thus, inversional sums become less
abstract and thereby enhance their role in a piece of music that unequivocally
demonstrates a clear context for inversional symmetry.40 Second, by utilizing only one
transformation, they underscore the potential for maximal transformational efficiency.
As the above discussion demonstrates, K-/U-transformations systematically
unfold IS pcsets from a single ordered trichord, and each unfolded set is a member of the
same set-class as the source trichord. However, this dissertation will distinguish between
I-related Tn-types, thus the source trichord and its first K-/U-transform are considered as
members of Tn-types [034] ({256}) and [014] ({569}). The union of a source trichord
and its first K/U-transform constitute a “K-cell” or “U-cell” (p=1, see Table 2.1). In the
above example, the tetrachord <2659>, the first four notes of the gesture, is a U-cell
<256>-<9>.41 The cardinality of K/U-cells is defined by numbers of distinct notes, and
ranges from three (all common tones between the source trichord and its K/U-transform)
40 The analyses in Chapters 3 and 4 attest to the context for inversional symmetry in Ginastera’s first and second quartets. 41 Technically, they form a “U(st)-cell,” but the informal name suffices for prose. Initially, “U” stood for “union.” Cells of this particular type are especially germane to Ginastera’s quartets, thus the remainder of the discussion will focus on this type exclusively.
25
to five (one common tone) depending on the particularities of the source trichord and its
first transform. In the above example, U(st) mapped <265> into <659>, but the common
tones F# and F were not articulated a second time.42 A second transformation results in
an “extended U-cell” (<265>-<98>, p=2), and a third (see Figure 2.1b) yields a “U-
chain” (<265>-<980>, p=3) in which each member of the source trichord has been
mapped once (9=(6+5)-2, 8=(5+9)-6, and 0=(9+8)-5). Further extensions of the chain (p
= or >4) are “extended
U-chains.”
The cyclic nature of U-cells, extended U-cells, U-chains and extended U-chains
causes them to produce familiar symmetrical pc collections. For example, the chromatic
scale (1-cycle) is an extended U-chain on a source trichord <pc, pc+1, pc+2>; both
whole-tone scales (2-cycles) are U-chains of source trichord <pc, pc+2, pc+4>, the three
3-cycle collections are U-cells of source trichord <pc, pc+3, pc+6>, etc.43 The string
quartet’s open string collection itself is an extended U-chain of source trichord <CGD>.
In the abstract, such “purely cyclic” collections are best considered as T-cycles, although
it is possible to consider such collections in a piece of music as U-transformations given
sufficient context through the overt presence of IS in p-space, a cyclic motive, or some
connection or pattern in the (R)In indices implied by significant pairs of pcs. However,
42 Strictly speaking, the “real” transformation in this case is, given <fst>, let the next note be (s+t)-f mod12. Each of the twelve K(nm)/U(nm) transforms can be defined similarly. 43 The initial ideas on cycles are explicitly discussed by Perle, but the present discussion has more direct roots in Headlam 1996, pp.14-17. Collections comprised of aligned interval cycles, such as the octatonic and hexatonic, are formed by interpolating another pc from outside the basic cycle, as seen with the trichord <265> in Example 2.3b.
26
non-symmetrical source trichords produce other familiar IS collections in surprising
ways.44
[Example 2.6]
Contextual K-/U-transformations can also be intimately linked with familiar
cyclic pc collections such as the octatonic [0134679t] and hexatonic [014589]. Example
2.6 depicts two such cases in which a subtle re-ordering of a source trichord’s elements
generates different pc collections. In the top example, source trichord <265> generates
the C/D octatonic collection as an extended U-chain made by five U(st) transformations
(imbricated trichords <265>-<659>-<598>-<980>-<80e>-<0e3>); the bottom example
re-orders the source trichord from <265> to <256> and generates the C#/D hexatonic
collection as a U-chain via three U(st) transformations (imbricated trichords <256>-
<569>-<69t>-<9t1>).45 Thus, a link between cycles and U-transforms is established: the
interval from f to t in a source trichord determines the fundamental cyclic bent of the pc
collection its (extended) U-chain generates. In the example, the equations “t-f” appearing
below the identification of the common collections indicate that the intervals from the
first to the third elements in the source trichords are 3 and 4 respectively. These intervals
in turn give rise to the 3-cycle octatonic and 4-cycle hexatonic collections.
[Examples 2.7a and b]
The chromatic, whole-tone, octatonic and hexatonic are not the only familiar
collections that unfold via U-transforms. Extended U-chains from source trichords of Tn-
44 For example, the pentatonic collection {CDEGA} can be generated as an extended U-cell (U(st)2) of source trichords <CGD> ([027]) or <EGA> ([025]); the former yields the ordered collection <C-G-D-A-E>, while the latter yields <E-G-A-C-D>. 45 K-/U-transformations cannot link the two source trichords, as the latter is r2R0 of the former.
28
types [025] and [035] eventually unfold the pc aggregate, but the initial extensions of the
given source trichords can unfold various pentatonic 5-cycle chords, given that the
trichord’s ic5 appears in f and t. Example 2.7a illustrates the unfolding of source trichord
<247> (Tn [025]) via U-transform into U-cell <247>-<9> ([0257]), and extended U-cell
<247>-<90> (pentatonic [02479]). At this point in the chain, the next transformation
yields trichord <902>, which returns pc 2 before continuing on to trichord <025> and
completing the diatonic hexachord [024579]. Thus, the unfolding has created a multi-set
U-chain <247>-<902> (still [02479]) on its way to a multi-set extended U-chain (and
diatonic hexachord [024579]) <247>-<9025>. Example 2.7b depicts the unfolding
extended U-cell, with some tonal labels, from source trichords <269> (D major) and
<259> (D minor) and interprets these unfoldings in staff notation and provides common
nomenclature for the extended tertian sonorities they create (up to seven members).
III. Tiles
In general, much of the quartets’ musical material is composite in nature, often
resembling a type of musical mosaic in which (roughly) 4-measure blocks of one type of
gesture dominates the musical surface. These concatenated blocks are hereafter referred
to as “tiles,” in deference to the overall aesthetic associated with visual and material art
bearing the moniker “mosaic.”46 Generally, a tile is roughly analogous to a musical
phrase due to its size and focus on one particular musical aspect, but the inherent
46 The meaning of the term “mosaic” here is not to be confused with that associated with Martino (1961), Morris and Alegant (1998) and Mead (1998). In general, the term connotes a series of discrete events in the present context, and while I will be discussing 12-tone rows here, I will not utilize the aggregate-partitioning methodology and terminology of the above authors.
29
implications of the term “phrase” complicate its broad application to many of the musical
situations that arise in the quartets. Ultimately, the definition of a tile is intentionally
flexible.
[Examples 2.8a, b and c]
Examples 2.8a, b and c each present four-measure tiles from the third movement
of the second quartet. Tile 1 (Example 2.8a) presents the first complete iteration (mm. 16-
20) of the movement’s theme, which is in essence two statements of the movement’s first
row (3-R1) as a stream of staccato eighth notes orchestrating row A’s discrete trichords
in each instrument. The same description applies to the musical surfaces of Tiles 2
(Example 2.8b) and 7 (Example 2.8c); only the row-forms change. The uniformity in the
presentation of each four-measure unit challenges the efficacy of the term “phrase” for
each, yet each maintains an individual identity within the overall formal section. Indeed,
it is possible to discuss these groups of measures without labeling them as tiles. However,
the concept will greatly streamline such discussions.
IV. Crosscuts
[Example 2.9]
In addition to tiles, the quartets contain an intriguing musical analogue to a film
editing technique, which also has its roots in Stravinsky’s music.47 Crosscutting is a
technique in film editing which describes the “[alternation] of [movie] shots from one
47 See Cone, Edward T. Musical Form and Musical Performance, Perspectives on Schoenberg and Stravinsky, ed. Benjamin Boretz and Edward T. Cone. Princeton: Princeton University Press, 1968, for a relevant discussion of stratification, interlock, and synthesis.
30
line of action in one place with shots of other events in other places.”48 In film studies
and filmmaking, it is a narrative device designed to “tie together…different lines of
action,” allowing spatially distinct events to exist together within one brief time span.49
The quartets occasionally yet significantly feature a musical analogue to the technique.50
Example 2.9 presents a six-measure passage from the Second quartet in which two
distinct gestures are temporally linked like the different lines of action discussed above.
The first gesture, appearing in mm.32-33 and 35-36, is referred to beneath the example as
“downbeat chords” in each of the instruments. The second appears in the rising canonic
figure in all voices in m. 34 and m. 37.
The networks appearing below Example 2.9 show two distinct processes at work
in the passage. The upper network interprets the downbeat chords as four members of
[0347] articulated as two pairs of T6-related tetrachords in dashed boxes occurring on
beats one and two of the aforementioned measures; the T7 arrow interprets the
relationship between the pairs.51 The lower network models the two canonic crosscuts,
again in two pairs, as four U-chains of source trichords <69e>, <t13> (twice), and <257>,
members of Tn [035] unfolding diatonic hexachords [024579]. Each pair of U-chains,
again appearing in dashed boxes, is T4-related, while the pairs themselves are T8-related.
Thus, the two distinct gestures feature their own transformational process; the downbeat
48 Bordwell, David and Thompson, Kristin. Film Art: An Introduction, 5th ed. McGraw-Hill, 1997, pp. 297-298. Significant examples of early uses of the technique occur in D.W. Griffith’s The Battle at Elderbrush Gulch (1913) and Sergei Eisenstein’s Battleship Potenmkin (1925). 49 Ibid. 50 This temporal juxtaposition of material from seemingly disparate sources is reminiscent of the discussion of “event time” versus “gestural time” in Kramer, Jonathan. “Questions of Time in the Music of Beethoven,” Perspectives of New Music XI/2 (1973): [pages]. 51 The two pairs of [0347]s result in C/C# and C#/D octatonic collections.
31
chords of [0347] not only incorporate interval 7s in each, but also reflect a 7-cycle in
their transpositional index, while the U-chains are based on a 4-cycle. The initiating pc of
each U-chain, F#, Bb (twice) and D, extends the pronounced influence of these three pcs
established in the immediately preceding section and on a larger level, reflect the coloring
of a fundamental 7-cycle with a 4-cycle.
Ultimately, crosscuts are local phenomena and do not significantly impact large-
scale formal design, nor do they appear in every movement this dissertation discusses.
Rather, they are a technique the composer uses to temporarily interrupt a musical flow
with either related or contrasting material. Crosscuts enrich the musical passage in which
they appear in diverse ways, often through the development of a predominant motive, the
introduction of a new motive or compositional technique (such as an inverted canon), or
the invigoration of a static ostinato with elements of surprise.
V. Overview of Main Rhythmic Profiles
In general, surface rhythms engage the notated meters in the nine movements of
the quartets, demonstrating a high degree of consistency through their frequent
employment of motives and ostinati. In a very broad sense, three basic rhythmic profiles
emerge: 1) uniform, in which the predominant rhythmic figures reinforce the notated
meter, 2) free contrapuntal, in which the figures are in tension with the meter, and 3)
rhapsodic, where the figures negate or suspend the meter. Uniform rhythmic profiles,
which dominate movements 1, 2 and 4 of the First Quartet and movements 1, 3 and 5 of
the Second Quartet, can be characterized as textures in which rhythmic and melodic
events unfold with temporal regularity. In such textures, the predominant rhythmic and
32
melodic events such as accompanimental or motivic ostinati and canonic thematic
entrances generally confirm the notated meter. Free contrapuntal textures, which appear
in movement 3 of the First Quartet and movement 2 of the Second Quartet, feature
significant thematic and motivic material that neither contradicts nor overtly supports the
notated meter. Rhapsodic rhythmic profiles, which occur significantly in movements 2, 3
and 4 of the Second Quartet, are the freest of the three and feature solo passages, either
alone or accompanied, that temporarily suspend the notated meter. The passage in
Example 2.9 depicts a uniform rhythmic profile, employing pronounced chords on both
downbeats of the 6/8 meter in mm. 32-33 and 35-36; the crosscuts in mm. 34 and 37,
while representing a clear interruption of the prevailing chordal punctuations, maintain
the overall rhythmic flow of the excerpt.
33
Chapter Three: Dyad-Spaces, U-transformations and Cycles in the First Movements of Quartets One and Two
Despite the divergent opinions of scholars regarding the respective roles the first
and second quartets play in Ginastera’s compositional career, the pieces themselves
present remarkably similar musical surfaces and structures. The respective first
movements in particular appear more alike than any other pair of movements. Both are,
in essence, thematic sonata forms that feature traditionally aggressive First Themes and
placid Second Themes. Both quartets present salient, central pitch-classes (pcs) as
touchstones that structure the movements through their pairing with other pcs in “dyad-
spaces.” In addition to their role in these spaces, the salient pcs also engage thematic
elements as initiation points of U-cells, U-chains, wedges and other symmetric
formations. These salient pcs also play fundamental roles in the harmonic dimension of
the movements, figuring prominently as members of interval cycles and familiar cyclic
pc collections (diatonic/pentatonic, octatonic, hexatonic and whole-tone) that appear with
regularity on the musical surface. The present chapter analyzes the first movements of
both quartets within the context of these essential constructs, juxtaposing specific
passages from both movements to investigate the various how pcs interact with
formal/rhetoric-based elements in an effort to understand and answer the many intriguing
questions the movements engage.
[Tables 3.1, 3.2]
36
In a broad sense, the thematic sonata form of both first movements is articulated
in part by salient presentations of each movement’s central pcs, D (first quartet) and G
(second quartet). Tables 3.1 and 3.2 provide form charts for both movements
respectively, detailing specific measure numbers for both the large overarching formal
features of exposition, development, recapitulation, etc., and the smaller formal features
of individual tiles within the larger sections.52 Some of the larger sections have salient
pcs in parentheses appearing after their formal designation, indicating particular emphasis
on these pcs in conjunction with the overall central pcs. A casual observance of the
parenthetic pcs on the tables reveals that as a whole, the pcs associated with the open
strings of the string quartet as a musical ensemble are of particular importance in the
music. In general, the central pcs G and D, along with the other open string pcs, become a
referential collection for both movements; the influence of this referential collection is
not limited exclusively to the pitch and pc arenas in the pieces, as it has ramifications for
their form as well.
As a referential pc set, the open string pcset (hereafter “OS set”) {C-G-D-A-E}
contains the potential for tonal and cyclic (7-cycle) associations, and its influence
52 Tiles are defined in Chapter 2; briefly, a tile is flexibly defined as a small formal unit, roughly the size of a phrase, which expresses one basic idea. The charts also include parenthetic descriptions of the general contents of each constituent tile. For example, tile 3 in the FTA of the first quartet (Table 3.1) contains the ascent of the first main theme, tile 4 contains the descent of the theme, tile 5 contains material that reworks and intensifies previous material, etc. On occasion, the qualifying parentheses contain numbers referring to previous tiles. For example, tile 15, which appears in the recapitulation of the STA in the same quartet, returns material previously heard in tile 9.
37
is unmistakable in both quartets’ first movements.53 The first movement of the first
quartet provides many instances of salient OS pcs: the establishment of fundamental pc D
in the Introduction, the FTA in both Exposition and Recapitulation, and the incorporation
of pcs A and E in the STA in the same formal sections are just a few noteworthy
examples. The second quartet also prominently features its fundamental pc G along with
other OS set pcs C and D in its FTA and pcs D, A and E in its STA, broadly projecting
{C-G-D} and {D-A-E} shadings as local referential collections. Despite both
movements’ significant reliance upon subsets of the OS set, however, the first quartet’s
movement employs the set and its subset shadings more consistently throughout the
movement than does the second quartet’s movement.
The remainder of the present chapter provides some analytic details about the first
movements of both quartets. It divides into two basic sections. The first provides a
general discussion of the large-scale harmonic (vertical) dimension of the quartets,
identifying significant dyad-spaces and demonstrating their structuring of the music.
After the broad framework established in the first section, the second section focuses on
U-transformations and their articulations of dyad-spaces in the thematic and motivic
dimensions of the pieces. The second section also incorporates significant harmonic
constructions local to the melody and motive under discussion.
53 Whether considered as a pentatonic collection, a partial diatonic collection, a 5/7-cycle collection, or a stack of conjunct T7 transformations, sonorities of this type are very common occurrences in all of Ginastera’s work and understanding the various contexts surrounding their use is essential to gaining insight into large amounts of his work. Ginastera also frequently employs the ordered pitch collection <E2-A2-D3-G3-B3-E4>, also known as the “guitar chord” (the standard-tuned guitar’s OS set), in pieces for other instruments and ensembles (see Chase 1957, Wallace 1964, Suárez Urtubey 1967 and 1972, and Kuss 1980 and 2002). The view of the close associations between Ginastera’s use of the chord and Argentine nationalism is often discussed by the above authors and the others discussed in Chapter One.
38
I. Dyad-spaces and large-scale harmony
While a particular shading of the 7-cycle OS set can suggest various relationships
among pcs on a musical surface, the interpretive aspect of a dyad-space demonstrates one
way OS subsets create harmonic structures beneath the musical surface. The introduction
of the first quartet’s movement provides a prime example of how a structural dyad-space
can emerge from an OS set shading.
[Examples 3.1a, b and c]
Example 3.1a, from mm. 1-7 (tile 1 and the beginning of tile 2), features three rising
gestures <D-F#-F-A-Ab-C> in mm. 1, 4 and 7 punctuated by two vertical hexachords
(low to high) <D-A-F#-Ab-F-C> and <G-D-Bb-E-G#-B-(C)> in mm. 2 and 5 (C5 is
considered a non-chord tone).54 These gestures and the punctuating chords that follow
establish many of the movement’s essential features: the salience of D as the gestures’
initiation points and as the lowest cello note of the chord of m.2, the movement’s main
motive (source trichord) <D-F#-F> and its contextual transformation, the movement’s
texture and tessitura, and transformational relationships that form an essential part of the
movement’s structure. Significantly, the first chord is a verticalization of the rising
gestures orchestrated as a D major triad beneath an F minor triad, while the second chord
is a G minor triad beneath an E major triad; both strongly affirm the chord’s 7-cycle
54 Chapter Two discussed this passage in the context of U-transformations.
39
element.55 Example 3.1b isolates both chords and indicates their membership in the
octatonic subset [013479]. The latter example also highlights the lowest voices in the
cello and their pc multiset c.o.=<D-A-G-D> (see the dashed box), a tetrachord that recurs
throughout the movement as an agent of harmonic balance.56 For instance, this tetrachord
accompanies the closure of the FTA in the exposition (tile 6), recapitulation (tile 20) and
the end of the Coda (tile 25). Although the two chords are not immediately adjacent,
Example 3.1c provides two models for the voice-leading between these chords; the
leftmost diagram treats all motions from the first to the second chord as T2, while the
rightmost treats all as I4. The smoothness of the I4 voice-leading in the rightmost model
(compare the arrow lines connecting the two chords in Ex. 3.1c) makes it the more
attractive of the two interpretations. Thus, given the musical context for the “balance”
tetrachord <D-A-G-D>, the two Ds are I4-related, as are the notes A and the G. Such an
interpretation suggests that within each of the two chords, the intervals from D2 to A2 in
the first chord and G2 to D3 in the second are T7, reinforcing the “conventional wisdom”
that the open strings of the cello are T7-related and affording the balance tetrachord the
common tonal suggestiveness associated with 7-cycle sonorities. The inversional sum 4
pairing of D with itself is further supported by the music in tile 2, the first measure of
55 The use of superimposed triads here is reminiscent of Berg’s Op.2/IV, which prominently features members of [014589] expressed as concomitant E minor and Ab major triads in m. 22 and A minor and Db major triads in m. 23. 56 The abbreviation “c.o.” stands for “canonic ordering,” which, given an inversionally symmetrical pc set, locates the sum pairs in the first/fourth and second/third positions within the angle brackets customarily used for ordered sets. In the present example, c.o.=<D-A-G-D> locates the sum 4 pc pair D/D in the first and fourth slots and the other sum 4 pair A/G in the second and third. The c.o. can be generalized as follows: given pcs <wxyz>, w+z=x+y. The c.o. translates easily into a 2T/2I K-net with the intervals from w to z and x to y labeled with I-arrows and the intervals from either w to x or x to w and y to z or z to y labeled with T-arrows.
40
which (m.7) appears at the end of Example 3.1a. By featuring the salient pc D as the
origin of the pitch wedge made by inversionally-related instances of the basic gesture, the
continuation of the first several measures fortifies D’s role as the initiator of the dyad-
space D/D-(I4), itself a structural entity which defines the transformational relationships
involving the {G-D-A} shading of the OS collection.
[Examples 3.2a and b]
Like the first quartet’s opening movement, the second quartet’s movement also
features a “balance tetrachord” based upon a trichordal shading of the OS set. The left of
Example 3.2a depicts the beginning of the movement’s first theme (tile 2, mm. 5-7),
while Example 3.2b presents a reduction of the entire theme (tile 2, mm. 5-19).57 The
theme itself begins as a focal OS pc G and unfolds a wedge through violin I pcs Ab
(m.7), Db (m.9) to OS pc D (m.16) and viola pcs F# (m.7), C# (m.9) to OS pc C (m.16).
The culmination of the theme occurs simultaneously with the arrival of the balance
tetrachord c.o.=<G-D-C-G>, whose shading {C-G-D} sounds on the downbeat of m.
16.58 In analogy to the pairing of D with itself at sum 4 in the previous example, the
present pairs G with itself in the dyad-space G/G-(I2), grouping the other OS pcs C and D
as a sum 2 pair within the multiset tetrachord. As is the case in the first quartet, the
activation of harmonic dyad-spaces in first theme resonates throughout the movement,
thereby extending the dyad-space’s structuring capacity throughout the entire movement.
The use of structuring dyad-spaces to create and interpret referential sonorities in both
quartets illustrates a point of consistency between the quartets and supports a recent trend
57 This theme recurs at pitch in mm. 55-97 and 302-319. 58 The details of this unfolding are discussed as resulting from U-transformations later in the present chapter.
41
in Ginastera scholarship to challenge the stringent period-based understanding of his
compositional career.59
Of the two movements in question, the earlier quartet’s movement places greater
emphasis on harmonic dyad-spaces than the later quartet. As established above, the
Introduction, important elements of the FTA, and the Coda significantly employ the
balance tetrachord <D-A-G-D>, a 7-cycle pc set structured by a single dyad-space D/D-
(I4). The movement’s remaining formal sections are largely structured by three additional
harmonic dyad-spaces: A/E-(I1) in the STA (tiles 9-11), C/G-(I7) in the recapitulation of
the FTA (tile 19), and D/Ab-(It) (tile 8) in the transition. Of the three remaining structural
dyad-spaces, two, A/E-(I1) and C/G-(I7), incorporate pairs of OS pcs, while the third,
D/Ab-(It), does not. The following paragraphs present additional analytic details
regarding the musical context in which these harmonic dyad-spaces function.
[Examples 3.3a and b]
In addition to possessing distinct themes, the earlier quartet’s first movement
generally features a distinct OS set shading and dyad-space. Example 3.3a provides a
broad overview of the OS shadings, dyad-spaces, and some small transformation
networks that relate salient pcs in each of the above components. The selection and
presentation of elements in the figure highlights several important features of the music
and implies some inherent methodological assumptions. First, as can be seen in the
horizontal space identifying the movement’s formal sections, the movement features a
59 See Tabor 1994.
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
reverse recapitulation, returning the STA before the FTA in the form.60 Second, as
implied by the list of OS shadings, motion between shadings does not result from pc
transformations, but from a change in perspective toward the a priori OS set. Third,
multiple dyad-spaces can be projected within one section, but the pcs involved in that
projection do not owe exclusive allegiance to one particular space. For example, the STA
in both exposition and recapitulation projects two dyad-spaces, sometimes
simultaneously. In addition, both spaces within this formal section involve pc A, placing
it in two distinct relationships: as the sum 1 partner of E and the sum e partner of D.
Finally, the transformation network atop the figure suggests that the movement’s focal pc
D is an active agent throughout the movement. As D “moves” throughout the piece via
T0, it appears in a number of distinct transformational relationships with other pcs. The
60 Additional discussion of this phenomenon appears later in the present chapter.
44
following analysis discusses each of these features and assumptions, demonstrating their
significance in their respective movements.
Despite myriad local relationships entered into by the fundamental pc D, one
particular set of relationships unifies the various dyad-spaces, OS shadings, and the
transformation network depicted in Figure 3.3, linking them all to the formal structure of
the movement. The transformation network in Figure 3.3a interprets the various
relationships between the fundamental D, pictured along the bottom horizontal as
resulting from four T0 transformations, and the four basic dyad-spaces that help to define
structure in the movement. As the movement progresses chronologically (reading left-to-
right on the Figure), the fundamental D enters four basic relationships. The first
relationship pairs D with D via I4 in the balance tetrachord at the beginning and end of
the movement. The transition section between the two main themes pairs D with Ab via
It; this change from D/D to D/Ab is interpreted in the figure as the upper left D moving to
Ab via T6 while the fundamental D moves via T0. The STA precipitates another
inversional pairing of D with B via I1 within the dyad-space A/E-(I1). In the figure, the
motion from Ab to B is interpreted as T3. The recapitulation of the FTA begins with a
reharmonization (c.o.=<C-D-F-G>) of the balance tetrachord, articulating the dyad-space
C/G-(I7). Within this dyad-space, D appears paired with F at I7, while the motion from the
STA D/B pair to the FTA D/F pair is interpreted as B moving to F via T6. The final
motion from D/F back to the balance tetrachord D/D at the end of the movement is
interpreted as F moving to D via T9. The network of Example 3.3b is a reinterpretation of
the topmost and bottommost horizontals in Example 3.2a. Example 3.3b depicts the
fundamental D as the lower of the two Ds linked by the vertical arrow labeled I4. The
45
upper D is a subset of the 3-cycle {D-F-Ab-B} that appears in the square L-net atop
Example 3.3.61 This upper square is a reinterpretation of the top line of Example 3.3a that
highlights the cyclic underpinnings of the set of fundamental D partners. Thus, the
unifying device connecting the various pc constructions to musical structure results from
the inversional pairing of the fundamental pc D with each member of the complete 3-
cycle collection {D-F-Ab-B}.
II. U-transformations and the melodic/motivic aspect
In addition to dyad-spaces and their connection to the large-scale harmonic
dimension of the pieces, U-transformations and their connection to the melodic and
motivic aspects of the pieces play a central role as well. As opposed to the earlier
movement’s emphasis on vertical harmonies, the later movement’s emphasis on motivic
connections and processes affords greater opportunity for the interaction of motivic and
harmonic dimensions. Indeed, much of both movements’ richness results from the
blending of the music’s harmonic and melodic flavors. As the following analysis reveals,
the gentle receding of the OS set from the first quartet to the second diminishes, but does
not destroy, the pronounced influence of the 7-cycle associated with the first quartet.
Although significant uses of the OS set still occur in the latter movement, most 7-cycle
elements that appear are often colored by their blending with 2-, 3-, 4- and 6-cycle
elements that result from numerous melodic constructions, which are modeled by U-
transformations. The promotion of the motive and the ability of U-transformations to
61 L-nets are “all T” networks discussed in O’Donnell 1998.
46
embed competing interval cycles again appear to distinguish between the earlier and
latter quartets. Ironically, however, it is the persistent presence of pitch constructions
easily modeled by U-transformations that ultimately challenges the efficacy of the
concept of distinct style periods in Ginastera’s music.
In the earlier quartet’s first movement, the specific cyclic coloring of the
structuring 7-cycle OS set shadings resonates within the movement’s formal layout. For
example, the FTA and Coda feature the 3-cycle coloring of the structural OS set shading
{G-D-A} by U-transformations of the main motive <D-F#-F> discussed above. OS set
shadings recede in the two main parts of the exposition’s transition, as the D whole-tone
collection and dyad-space D/Ab-(It) dominates tile 7 and the dyad-space F/F-(It)
dominates tile 8.62 However, the STA’s OS set shading {D-A-E} remains essentially
uncolored by a competing cycle, except for a brief 4-cycle U-chain of the motive <Bb-A-
F#> in tile 17, resulting in the C#/D hexatonic collection. The following paragraphs
provide analytic detail surrounding each of these sections.
[Examples 3.4a, b and c]
The STA in both exposition and recapitulation offers the clearest example of the
uncolored 7-cycle OS set and its constituent dyad-spaces. As discussed above, the STA is
based on the {DAE} shading of the OS set; both dyad-spaces A/E-(I1) and D/A-(Ie) fulfill
structural roles in the section. Example 3.4a depicts the beginning of tile 9, the first
measures of the Second Theme, which significantly features a 7-cycle pentachordal
descent in reference to Amerindian musical practice.63 In its affect, the theme counteracts
62 The OS shading {G-D-A} must recede, as the D whole-tone collection does not contain pcs A and G. 63 I thank Malena Kuss for informing me of this connection.
47
the aggression of the First Theme, exploring the metric fluidity of juxtaposed rhythmic
figures at home both in the notated 3/4 meter (m. 85) and in 6/8 (m. 86). The theme
begins with an initial definition of the dyad-space A/E-(I1), establishing sum 1 as
referential. The figuration associated with the theme reveals a pitch axis of C/C#
operating throughout the entire section; in tile 9, the axis lies at C4/C#4 when the theme
appears in viola and cello and migrates up an octave to C5/C#5 in tiles 10 and 11 as the
theme moves into the violins. The second OS set dyad-space D/A-(Ie) structures the
presentation of the theme in the lower voices in tile 11, which occur in canon with the
upper voice A/E-(I1) dyad-space, as shown in Example 3.4b. The recapitulation of the
STA in tile 15 recalls the canon of tile 11, featuring a different pairing of voices (dyad-
space D/A-(Ie) in violin I and viola and A/E-(I1) in violin II and cello) and a different
time interval (one measure in the exposition, one eighth note in the recapitulation), as
shown in Example 3.4c. In this final presentation, the canons are transposed up one
semitone, allowing a new high point Bb5 to emerge from the texture and initiate the
aforementioned 4-cycle-based ({D-F#-Bb}) U-chain from source trichord <Bb-A-F#> to
close the section. Although the movement’s fundamental D temporarily recedes from the
salient prominence it enjoyed in the FTA, it nonetheless maintains a vital role in the STA
through its I1 association with B.
[Examples 3.5a and b]
As discussed briefly in Chapters One and Two, Kuss’ description of
Ginastera’s tonal language as “octatonic with modal interaction” readily applies to the
First Theme, though the present dissertation would describe this tonal language as a
48
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
7-cycle with 3-cycle coloring. In general, the First Theme, the score of which appears in
Example 3.5a, is a large arch, ascending in violin I from the focal D4 to C5 in tile 3, and
descending back to its origin in tile 4, followed by an motivic intensification in tile 5 and
closing gesture in tile 6. The large, overarching slur in Example 3.5b summarizes the
broad ascent from the first theme’s beginning on D4 in m. 16 to its apex on C5 in m. 24,
outlining the first theme’s broad dyad-space D/C-(I2) in a type of “composing out” of the
opening gesture in m. 1. On a broad scale, a new trichord <D-F-A> is transformed into
U-cell <D-F-A>-<C> via one U(st) transformation, as highlighted by the large slur
connecting the dyad-space’s boundaries.64 However, this broad motion is actually
comprised of two smaller, nested U-cells within the larger motion, as highlighted by the
64 The source trichord <D-F-A> incorporates significant notes from the theme’s first four notes, U-cell <D-F-F#>-<A>.
49
two smaller slurs in Example 3.5 connecting D4 to A5 in mm. 16 and 19 and F4 to A5
beginning in m. 20. The first nested U-cell, shown under the first small slur, begins with a
reordering of the opening gesture’s source trichord <D-F-F#> (r2RT0 of <D-F#-F>) and
transforms it into U-cell <D-F-F#>-<A> via U(st) culminating on the local melodic apex
A4 in m. 19, spanning the OS dyad-space D/A-(Ie). The second nested U-cell, shown
under the second small slur, re-interprets the new second and third terms of the previous
source trichord <F-F#> as the first and second terms of the new source trichord <F-F#-
G#>. The new source is in turn transformed into U-cell <F-F#-G#>-<A>, re-engaging the
original local melodic apex A4. The emphasis on the musical surface of pcs D, F and A
articulates the new, overarching source trichord <D-F-A> for the entire ascent of the first
theme’s melodic arch, as discussed above. The U-chain melody throughout this section is
comprised exclusively of the C/D octatonic collection, the only version that contains the
vital OS set dyad D/A as a subset.
The D/A-(Ie) dyad-space that structures the opening gesture’s U-cell <D-F#-F>-
<A> also defines the space in which aspects of this cell are developed as a Grundgestalt.
The subtle re-ordering of the opening gesture’s ordering of the U-cell’s four elements
<D-F#-F>-<A> as <D-F-F#>-<A> establishes a dialogue between the two sets. The
former occurs twice throughout the piece in the Introduction (tiles 1 and 2) and the Coda
(tile 24), while the latter recurs with all instances of the first theme (tiles 3 and 19). The
tension between these orderings is reflected in subtly distinct versions of the “cadence
motive” appearing as section-closing motivic gestures (the Grundgestalt) in mm. 33-34,
mm. 46-47 (tile 5), 51-52 (tile 6) in the exposition and 183-184 (tile 20) and 200-201 (tile
21) in the recapitulation. The first, fourth and fifth of these are essentially identical and
50
appear in Example 3.3a. Thus, after the gesture’s original statement in mm. 33-34, in
which it closes the theme but not the section, it returns to close not only the theme but the
main body of the sonata form.65 The second and third versions (Example 3.3b) challenge
the ordering of the initial statement, affording its formal return at the end of the
recapitulation a sense of closure by returning a conclusive version.
[Examples 3.6 and 3.7]
While the theme in this section highlights the C/D octatonic collection, the
accompanimental chords in the lowest three voices maintain a general 7-cycle harmonic
profile. Within the context of cyclic shadings and colorings, the accompaniment,
expressed on the musical surface as eighth note block chords, provides the 7-cycle
material that the 3-cycle melody colors.66 Example 3.6 depicts the five exclusive chords
which accompany the complete melodic arch in tiles 3 and 4, indicating the measure of
the chord’s first appearance and the set-class membership of each chord to illustrate the
considerable degree to which 5-/7-cycles permeate this musical figure. The first two and
last chords of this group are purely 7-cycles, while the third and fourth are multi-cyclic as
shown by the p-space transformations modeled in Example 3.7. In the figure, the first
chord, a multiset pentachord {2449e}, is segmented into an upper trichord {249} ([027])
and a lower dyad {4e} (Tn[07]) based on register.67 The upper rank of [027] trichords
65 This is analogous to Warren Darcy and James Hepokoski’s “Essential Sonata Closure (ESC).” See Darcy and Hepokoski 1997. This is especially true because of the reverse recapitulation, which leaves the first theme for last. 66 The texture of the accompaniment is essentially identical to the famous accompanimental ostinato in The Augurs of Spring from Stravinsky’s Rite of Spring. 67 The essential difference between the exposition and recapitulation of the FTA is that the latter features a reharmonization of the first several accompanimental chords. Most notably, the initial multiset {2449e} reappears as {0257} (c.o.=<CDFG>), expressed
51
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
follows a p-space transformational path of <T(p)+2, T(p)-4, T(p)+6/+1, T(p)-6/-1> throughout the
section; the lower rank essentially mirrors the upper, exchanging + for – and vice-versa
within the individual transformations. The first three chords contain five pitches, thus the
motions between them are easily represented as single transformations. As the passage
moves into the fourth and fifth (final) chords, the texture initially thickens to a hexachord
in m. 24 containing the greatest registral span (C2 to F#4) before collapsing into the final,
most registrally limited chord in m. 31. From a transformational perspective, the motion
to and from these final chords involves multiple paths. The fourth chord is a hexachord in
which the top tetrachord {1368} is considered as two overlapping [027] trichords {138}
and {168}, necessitating the interpretation of two different T(p) transformations (+6/+1
into the chord and -6/-1 out of it) in the upper [027]-based rank. The first four maintain a
sum 6 dyad in the outer voices (D and E in the first and second chords, C and F# in the
third and fourth), suggesting an accompanimental OS set dyad-space D/E-(I6) throughout
this section. The common thread between melody and accompaniment in the FTA is the
within the OS set dyad-space C/G-(I7) (see Example 3.2a). However, both versions of the theme terminate with the balance tetrachord <DAGD>.
52
emphasis on the fundamental pc D and the contexts in which it appears: as the initiation
point of the melodic/thematic 3-cycle U-chain and as a member of dyad-space D/E-(I6)
that governs the 7-cycle accompanimental chords.
Both the exposition and the recapitulation feature transition sections between the
first and second themes. Of the two, the first transition section, tiles 7 and 8, significantly
explores the cyclic coloring of the OS set.68 The division of the section into two tiles
reflects the basic binary division of the section according to stylistic, motivic and tonal
associations. Stylistically, the section bridges the gap between the aggressive first theme
and placid second theme through two intermediary steps. The first step (tile 7) maintains
the former’s rhythmic intensity while softening its impact by thinning the ensemble
texture; the second step (tile 8) initiates the long, sweeping phrases that will characterize
the STA. A similar transformation occurs motivically as initially forceful rhythmic
motives also give way to smoother figuration, gradually relaxing the first theme’s tension
as the music moves into the serene second theme.
[Examples 3.8a, b and c]
The area of greatest significance to the large-scale processes of the movement,
however, encompasses tonal issues. Broadly, each of the transition’s two main sections
projects two different cyclic collections. The first is based on the D whole-tone
collection, while the second is based on 5-/7-cycle pc sets {C-F-Bb} and {F#-B-E},
which simultaneously recall familiar OS harmonies and depart from OS set shadings,
which intensifies the return of the OS shading {D-A-E} at the arrival of the second
theme. Example 3.8a presents the first two measures of tile 7, depicting both D whole-
68 The second transition section in tile 18 contains only developmental rhythmic figuration of the OS set.
53
tone-based passages (<Ab-F#> in violin 1 and <D-E> in vcl in mm. 60-61) and gapped 5-
cycle collections {E-F#-A} (m. 61). The figure in violin I, viola and cello in m. 60 is the
first of three similar statements appearing in tile 7. The outer-voice dyad initiates the
structural dyad-space D/Ab-(It), establishing sum t as the primary transformational
relationship across tile 7. The second and third statements (not pictured) allow the sum t
dyad Bb-C to emerge at the beginning of tile 8, where this dyad is incorporated into the
vertical trichord {F-Bb-C} in m. 73, as shown in Example 3.8b.69 This trichord is the first
of two salient sum t 5-cycle sonorities; the trichord {E-F#-B} (Example 3.8c) emerges in
m. 77, recalling the sum t dyad {E-F#} from m. 61 but placing it in a new, non-gapped 5-
cycle context {E-F#-B}. Thus, the significant sum t orientation of the transition is
initiated by the replacing of the D/D-(I4) and D/A-(Ie) dyad-spaces of the first theme with
D/Ab-(It), a dyad within the D whole-tone collection. Example 3.3a interprets this change
as a motion via T6 from the D atop the balance tetrachord in the FTA to the Ab paired
with D in the transition.
As discussed above, the later quartet’s first movement shares many similarities
with its earlier counterpart. Indeed, both first movements possess common attributes in
five general areas. First, both are thematic sonata forms featuring aggressive first themes
and tranquil second themes. Second, the OS set figures prominently as a referential set in
both movements’ thematic organization, with specific shadings of the OS set aiding in the
distinction between formal sections. Third, dyad-spaces, both inclusive and non-inclusive
of OS pcs, structure much of the musical surface. Fourth, the various dyad-spaces that
structure elements within sections are colored by cyclic pc constructions. Finally, U-
69 The C appears on beat two in the viola.
54
transformations effectively model many of both movements’ melodic and motivic
constructions, often resonating in significant ways with the above-enumerated
similarities.
Although both movements employ many of the same devices, however, the later
movement distinguishes itself from the earlier movement in two basic areas. The first
distinction results from the expansion in the number and role of motivic pc sets recurring
as fundamental thematic and developmental sets throughout the movement. For example,
as the above analysis demonstrates, members of Tn-type [034] play a pivotal role in the
movement’s introduction and first theme; in the later movement, members of Tn/TnI-
types [014] and [016] feature prominently in virtually all formal sections, inhabiting both
melodic and harmonic dimensions. The second distinction results from the deeper
coloring of the basic OS set by 3-cycle and 4-cycle collections, with the latter often
appearing as a foil to the former. For example, the earlier movement’s first theme
features the coloring of its fundamental 7-cycle OS shading {G-D-A} by the 3-cycle
collection {D-F-Ab} as a result of multiple U-transformations of the motivic source
trichord <D-F#-F>; the resulting C/D octatonicism permeated important sections of the
movement. As the analysis below demonstrates, octatonic sections resulting from the 3-
cycle coloring of 7-cycle sets also include significant, salient 4-cycle coloring as well,
enriching melodic and harmonic dimensions. Thus, the expanded number of motives and
their relation to larger structural devices result not only in an expanded cyclic harmonic
palette, which also is reflected in broader compositional areas. The following analytic
detail develops these various issues in an attempt to answer the fundamental questions
posed in the present dissertation’s early chapters.
55
Like the first quartet’s first movement, OS set shadings and the dyad-spaces they
imply play an important role in establishing local referential collections through their
association with the movement’s major formal components. As discussed above, the
second quartet’s first movement is based on a {C-G-D} shading of the OS set and is
articulated by the balance tetrachord c.o=<G-D-C-G>, which opens and closes the
movement (tiles 2 and 37). The exposition’s transition (tiles 4-9) is generally based upon
a {G-D-A} shading of the OS set, while the exposition’s second theme (tiles 12-17 in the
exposition and 31-34 in the recapitulation) is generally based upon a {D-A-E} shading.
However, the later quartet’s movement only features clear OS set shadings in the
exposition; the recapitulations of the first and second themes depart from the tight
harmonic organization of the exposition versions.
[Example 3.9a]
Each of these OS set shadings combines with motives to suggest several locally
structural dyad-spaces. In essence, most of the significant motivic elements of the
movement derive from the two salient trichordal motives appearing in the movement’s
first four measures (tile 1). Example 3.9a provides the first four measures of the
movement. This dramatic opening statement introduces two salient trichordal motives
<D-Bb-C#> ([014]) and <Bb-F-E> ([016]) which appear in boxes in the example.
Members of these Tn/TnI classes constitute an integral part of all major themes in the
movement, and in each case significantly involve pcs and dyad-spaces associated with
the trichordal shadings of the OS set.
[Example 3.9b]
56
Both the movement’s first and second themes significantly feature members of
both of the boxed motives’ Tn-types [016] and [014]. The diagram accompanying
Example 3.9b shows two different views of the first theme’s wedge construction (also
depicted in Example 3.2b) on two bass staves. The top staff illustrates the vn 1 and viola
wedge emerging from the focal OS pc G in m. 5. The upper and lower voice link the
focal G to D and C via single U(st) transformations of source trichords <G-Ab-Db>-<D>
and <G-F#-C#>-<C> in m. 16, where they sound together as the familiar OS set shading
{C-G-D} and participate in the movement’s structure as the balance tetrachord.70 The
lower staff presents a different layer of the thematic wedge, bounded by the same OS set
shading. In this layer, the source trichords are members of [014], referencing the first
three notes of the piece. The notes involving the source trichords’ distinctive ic 3, the
upper-voice A# and B and lower-voice E and D#, appear in a distinctive rhythmic pattern
on the musical surface, establishing a model for subsequent motivic interpolations
discussed in Chapter 2 as “crosscuts.” The union of both top and bottom staves, and
thereby the union of [014] and [016] motives within the dyad-spaces G/D-(I9) and G/C-
(I7), reveals that the upper and lower “ramps” of the wedge radiating outward from the
central G are indeed statements of octatonic collections <G-Ab-A#-B-Db-D> (C#/D
octatonic) and <G-F#-E-D#-C#-C> (C/C# octatonic). The endpoints of these collections
reaffirm the controlling OS shading {C-G-D} and balance tetrachord c.o.=<G-D-C-G>.
Thus, the main body of the first theme illustrates the startling interconnectedness of
motive, U-cell, dyad-space, and OS shading.
[Example 3.9c]
70 Octave doublings of the focal G are included in the multiset balance tetrachord.
57
While the first theme’s wedge engages the aforementioned first stylistic
distinction, the final measures of the first theme, a passage immediately following the
wedge, clearly illustrates the second distinction. The cyclic interaction associated
with the wedge transforms from a 7-/3-cycle interaction into a 7-/4-cycle interaction as
depicted in Example 3.9c. In the example, the central OS pc G initiates a rising figure
<G-A-Bb-B-C#-D> in violins I and II and the cello; this figure is then transposed via a T8
cycle from its initial G in m. 20 to Eb in mm. 21 and 22, to B in m. 23, conjuring a
contrasting 4-cycle {G-Eb-B} that replaces the embedded 3-cycle {G-A#-Db-E}
contained in upper and lower wedge ramps.71 Thus, the first theme casts G in three
distinct cyclic layers: as a member of the 7-cycle OS shading {C-G-D}, the coloring 3-
cycle {G-A#-Db-E} and the contrasting 4-cycle {G-Eb-B}.
[Example 3.10a]
Like the first theme, the second theme also significantly features members of
[014] and [016] within its OS shading {D-A-E}. Broadly, the second theme divides into
two large sections, tiles 12-14 and 15-18, which respectively present main thematic
material and develop main thematic material. Example 3.10a shows gestures 1 and 2 as
they appear in tile 12, the first presentation of the second theme. The excerpt depicts a 7-
measure melodic arch formed by rising gesture 1 and “cascading” gesture 2. Gesture 1 is
formed by a concatenation of members of [014] unfolding the dyad-space D/A-(Ie) in a
U-cell <D-Ab-Eb>-<A> as points of imitation; the C/D octatonic pc collection results
from the union of these points of imitation and the held notes F (vcl), B(vla), F# (vn2)
and C (vn1) appearing at the tail end of the three-note [014] gestures. Another rising
71 This cycle appears at the beginning of each transposition, which occurs on the second eighth note of each measure.
58
figure in vn 1 beginning in m. 99 further opens the tessitura until the attainment of salient
OS pc E on the downbeat of m. 101. This arrival on E fully articulates the second theme’s
{DAE} shading of the OS set and initiates a downward cascade featuring two [016]-
based U-cells <E-B-Bb>-<F> and <Gb-Db-C>-<G>; unlike the first ascent, however, the
union of these U-cells is not octatonic. The return of the second theme in the
recapitulation (tile 31) features pcs B, A, G and Eb as points of imitation, disavowing the
initial U-cell construction and embracing the 4-cycle collection {G-B-Eb} from the end
of the first theme as a structural component. Indeed, the later quartet’s main themes boast
similar types of constructions as the earlier quartet’s themes: a balance tetrachord and
melodic U-cells which articulate the sections’ structuring dyad-spaces and OS set
shadings. However, the significant incorporation of U-cells comprised of two source
trichords of differing set-class membership and the incorporation of a structural 4-cycle
“foil” represent a substantive development from earlier practice.
[Example 3.10b]
After the thematic presentation appearing in tile 12, the tile 15 developmental
subsection of the second theme further underscores the expanded role of the 4-cycle.
Example 3.10b depicts the first two (of four) developmental subphrases in tile 15. Each
subphrase is in essence an amalgam of two tile 12 elements. The first, beginning with the
cello Db3 in m. 118 and the violin II F4 in m. 119, is an extension of the three-note [014]
motive featured in Example 3.10a; the second, appearing after the extended motives,
recalls the longer rhythms appearing in the [016] cascade. The four developmental
subphrases present 4-cycle pcs <Db-F-A-F> in points of imitation, as opposed to the
unfolded U-cell points appearing in tile 12. The symmetrical properties of these new 4-
59
cycle points of imitation also figure prominently on the musical surface, as the third
subphrase (beginning on A) replicates the first It note-for-note (beginning on Db), and the
fourth (beginning on F) replicates the second is It (also beginning on F).72 Thus, the
expanding role of cyclic pc sets not only results in a more variegated tonal palette, but it
also facilitates the deeper exploration of inversional symmetry associated with the
stylistic change between first and second quartets.
72 The inversional symmetry of this passage is reflected in p- as well as pc-spaces.
60
Chapter Four: Interval Cycles and Cyclic Collections in Internal Movements
In Chapter 3, we employed the various analytic devices described in Chapter 2 to
elucidate connections among form, tonal structure, and motive in first movements of
Ginastera’s String Quartets Nos. 1 and 2; the present chapter analyzes two internal
movements from the second quartet with the same fundamental goal. Of the four
movements discussed in the present chapter, two (movements two and three of the second
quartet) exhibit significant interaction between 12-tone and free atonal material, calling to
mind Berg’s early twelve-tone works, the Chamber Concerto and Lyric Suite.
Accordingly, the present chapter divides into two basic parts, broadly reflecting the use
of serial and non-serial techniques in the above movements. However, as the chapter
demonstrates, the structuring of musical material based on the interactions of interval
cycles transcends modes of surface pc presentation, further strengthening the connections
between the present movements and Berg’s music as discussed by Headlam in The Music
of Alban Berg.
Like most 20th-Century composers who utilized twelve-tone rows in their
compositions, Ginastera employed the technique idiosyncratically.73 In discussing his
73 See Wallace 1964, Kuss 1976, 1980, 2000, 2002 (Bomarzo), and 2002 (MGG), and Schwartz-Kates 2001. In general, Ginastera’s approach is most reminiscent of Webern’s linear unfolding of a row’s pc content as demonstrated in works beginning with Op. 21. His approach also bears similarities with that of Berg in its exploitation of connections between interval cycles and order positions (hereafter referred to as ops and indicated with underlines such as 012 for the trichord in a row’s first, second and third ops) and its use of multiple rows associated with operatic characters (see Perle 1985, Headlam 1985, 1990, 1996), Kuss 1976, 1980, 2000, 2002 (Bomarzo), and 2002 (MGG), and Richards 1986).
61
approach to twelve-tone composition, the composer himself observed in 1959 that his
technique “is adapted to the expressive necessities of the music,” revealing a conscious
flexibility to depart from an inviolable ordering of pitch material to accommodate an
aesthetic impetus.74 The following analyses of the second and third movements of the
second quartet confirm the above assertion, demonstrating the frequent coexistence of
three basic approaches to musical material within a single movement: 1) twelve-tone
serialism, 2) unordered pitch-class (pc) aggregates, and 3) unordered, non-aggregate-
based constructions. Within the first of the above three approaches, we may superimpose
three distinct musical contexts: 1) concatenated twelve-tone rows, 2) canonic row
presentations, and 3) segmented/partitioned rows (rows divided into adjacent and non-
adjacent collections) and ordered sub-collections.
I. (Mostly) Twelve-Tone Movements Two and Three of the Second String Quartet
[Tables 4.1 and 4.2]
Although the rondo-like forms of the third (A-B-A1-C-A2) and second (ABA’)
movements of the latter quartet are reflected in their rhetorical and textural dimensions,
the presence and/or absence of tone rows also plays a vital role in the delineation of
formal sections.75 Neither of these movements is entirely twelve-tone. The third
movement largely reflects the aforementioned first basic approach (twelve-
74 Ginastera, Alberto, Boletín Interamericano de Música, No. 14, Nov., 1959, pp. 3-4. 75 I will discuss the third movement before the second due to its stricter adherence to traditional twelve-tone orthodoxy. See also Kuss 2002 (MGG).
64
tone serialism) by significantly employing two source rows labeled 3-R1 and 3-R2 in its
constituent A, A1 and A2 (hereafter referred to as “A-based”) and B sections (see Table
4.1). The movement’s C section reflects the aforementioned second basic approach,
unordered pc aggregates. The second movement employs a single tone row labeled 2-R1
(see Table 4.2) in its outer A-based sections, interspersing manifestations of the third
basic approach (unordered, non-aggregate-based constructions) with row statements,
while the B section employs material derived from the second and third basic approach
material.
In addition to indicating the large formal sections of both movements, Tables 4.1
and 4.2 delineate smaller formal units as well. Table 4.1 depicts the presence of a main
theme in each of the A-based sections in the third movement, whose outer sections (A
and A2) feature only thematic and codetta sections and whose inner section (A1)
alternates between main thematic and contrasting sections. The B section divides into
four smaller subsections, each featuring a rhapsodic instrumental solo by the four
members of the string quartet. The A-based sections and the C section divide comfortably
into tiles that predominantly reflect 4-bar hypermetric units; in the A-based sections,
these units often are comprised of a single row statement. Table 4.2 illustrates the second
movement’s three overarching sections and subdivides them into subsections labeled
according to criteria other than tiles. The analytic paragraphs in the following section
discuss each section in greater detail.
[Examples 4.1a and b]
66
Each of the three row forms appearing in the latter quartet embeds interval
cycles in particular ways; the composer often exploits these encoded cycles through
choices of row form and row presentation on the musical surface. Example 4.1a provides
a basic harmonic profile of rows 3-R1 and 3-R2 (appearing in the third movement), while
Example 4.1b does the same for row 2-R1 (second movement). The discrete hexachords
of rows 3-R1 and 2-R1 are members of different set-classes ([012369]/[013467] for row
3-R1 and [013478]/[012569] for row 2-R1), while the discrete hexachords of row 3-R2
are members of the same set-class (both [012346]).76 As illustrated by the interval-class
vectors in the figure, the row 3-R1 hexachords favor ic3, and the row 2-R1 hexachords
favors ic4; the row 3-R2 hexachords are nearly chromatic and favor both ics 1 and 2.
However, further analysis reveals members of 3-cycle [0369] and [016] appearing in
segmental 1234 and partitional 01e are of paramount importance in the A-based sections
of the third movement; the same is true of members of 4-cycle [048] in 023 and 69t (the
two combine to create a member of 2-cycle [02468t]) in the A-based sections of row 2-
R1 of the second movement. The discrete trichords of row A are members of [016] and
[013], with the pair of [016]s RI1-related and the [013]s I1-related; of these discrete
trichords, the first figures significantly in the music. The discrete trichords of row 3-R2
are all members of different set-classes [016], [012], [015], and [014], and as subsequent
analysis demonstrates, the manner in which these discrete trichords are presented is a
vital component in the music. The discrete trichords of row 2-R1 feature three members
of [037] at Te, Ie, and I0 (the remaining trichord is a member of [016]). Again, as is the
76 The hexachords for row 2-R1, members of [013478] and [012569], are the famous Z-related pair often used by Schoenberg, as the latter contains the six pcs in the composer’s last name. Allen Forte labels these as 6-z19 and 6-z44 respectively.
67
case with the discrete trichords of rows 3-R1 and 3-R2, the cyclic properties of these
discrete trichords are of great relevance to the musical surface of the A-based sections of
the second movement.
As introduced earlier, one of the main assertions about the later quartet’s third
movement is that while mostly twelve-tone, its fundamental structuring aspects are cyclic
in origin. While ideas and procedures typically associated with twelve-tone music, such
as the segmentation of the musical surface into ops and row-forms are useful and are
employed in subsequent analytic observations, these devices are interpreted here in their
role as demonstrating the underlying influence of cyclic thinking in the movement. The
fundamental aim of the following analysis is to reveal the types of connections between
cyclic composition and twelve-tone expression in the third movement. To this end, the
analysis begins by identifying one basic motive and two cyclic pc sets on the musical
surface and illustrating how twelve-tone compositional procedures participate in their
development into some of the movement’s essential structural features. Of paramount
importance to the movement is the establishment of a “signature” opening trichord <C-F-
B> ([016]) which emerges as a fundamental structural component.
[Example 4.2]
The fundamental cyclic element of the A-based sections derives from the pcs and
symmetric properties of one cyclic pc set (and its subsets), the 6-cycle tetrachord
69
{B-C-F-F#} (a member of [0167]). As indicated above in Table 4.1, the outer A sections
feature only two main forms of row A, P/I0(3-R1). Each of these row forms embeds the
tritone {C-F#}({06}) at 0e due to the dyad’s inversional symmetry at I0. The
incorporation of this dyad with the row’s “signature” trichord <C-F-B> (<05e>) in 012
completes the above 6-cycle tetrachord, which appears in row partition 012e. The
inversional symmetry of this fundamental tetrachord facilitates its role as an overarching
element of unity among all of the movement’s A-based sections; its isomorphic
partitioning in two of the row forms appears in the inner A1 section.77 As indicated in
Table 4.1, the inner A section features four row forms in two J00-related pairs, P/Ie(3-R1)
and P/I5(3-R1). Two of these row forms, Ie(3-R1) and I5(3-R1), contain different
orderings of this tetrachord in 012e. Thus, this tetrachord’s appearance in the above row
partition provides the common element in all A-based sections. Example 4.2 provides a
chart detailing each row form in the movement. The leftmost column indicates the formal
sections in which the middle column rows appear, while the rightmost column identifies
the abovementioned musical contexts in which the row forms appear. The middle column
provides the actual rows used, highlighting in bold type the members of the above 6-
cycle tetrachord. Again, their conspicuous placement at the beginning (or end) of the row
forms underscores the importance of this tetrachord or its subsets on the musical
surface.78
[Examples 4.3a, b and c]
77 For definitions and discussions of isomorphic partitions, see Haimo, Ethan and Paul Johnson. “Isomorphic Partitioning and Schoenberg’s Fourth String Quartet.” Journal of Music Theory 28 (1984): 47-72. 78 Members of this tetrachord ([0167]) feature prominently in Bartók’s String Quartet No. 4 and Berg’s Lulu (see Perle 1955, 1985, Antokoletz 1984 and Headlam 1996.)
70
The first discrete trichord of the first row-form P0(3-R1), pcset <C-F-B>, referred
to above as the “signature trichord,” establishes the main transformational relationships
explored in the A-sections of the piece. The pcs of this trichord occupy not only 012 of
the basic row-from, but they occupy 0 in each of the aforementioned J00-related pairs that
exclusively comprise the movement. Thus, the trichord in 012 of P0(3-R1) is “telescoped”
across the entire movement. Example 4.3a provides a transformation network interpreting
the various relationships between both the pc set <C-F-B> as 012 of P0(3-R1) and the
relationships between the six main row-forms of the movement, each of which contain C,
F or B at 0. The network begins with C and interprets its motion through F and B as T5
and T6 respectively, as represented in the leftmost “northeastern” arrows. While these
motions describe the transformations within 012 of P0(3-R1), they also describe the
transformations among the pcs at 0 of P0(3-R1), P5(3-R1) and Pe(3-R1). Each of these pcs
is also paired with itself via inversion, as interpreted on the figure in the three horizontal
arrows bearing I operators (bottom to top) I0, It and It. While these inversions are only
implicit in the pcs of 012 in P0(3-R1), they are explicit within the context of 0 in row-
forms I0(3-R1), I5(3-R1) and Ie(3-R1), as shown by the row-form pairs appearing to the
right of the figure. Example 4.3b depicts this explicit connection by showing a
transformation network isomorphic to that in Example 4.3a in which row-forms are
substituted for the initial pcs.79 The isomorphic networks of Examples 4.3a and b imply
the transformation graph appearing in Example 4.3c, which models the transformational
relationships among all basic row forms in the movements’ A-based sections.
[Example 4.4a]
79 The J00 appearing below the I operators are intended only to illustrate the connection and are not intended to operate along with the T operators.
71
While the emphasis of 0e in the concatenated, individually presented J00-related
row-pairs highlights the 6-cycle subsets of the outer A-sections’ main pc set {B-C-F-F#},
similar partitionings of canonic row pairs highlights the interval 5 (and potential 5-cycle)
subsets {C-F} and {B-F#} in the inner A1 section. Example 4.4a depicts the beginning of
Tile A14, which models the canonic passage within the movement’s middle A1 section.
The canonic passages of Tiles A14 and A15 present concatenated row-forms in seamless
running eighth notes in the upper three instruments. The canon arises from the
simultaneous statements of row-forms in staggered entrances five eighth notes apart
beginning with viola F4 (RPe(3-R1)) on the downbeat of m. 165, followed by violin II B4
(RP5(3-R1)) on the sixth eighth note of the same measure and the violin I F5 (RP5(3-R1))
on the fifth eighth note of the following measure.80 The pcs F and B, which return
conspicuously throughout the A1 section, are stated at e in each pc’s row-form, appearing
temporally first as a result of the retrograded row-forms. Each two-measure row
statement roughly unfolds a “pseudo-wedge” with e occupying a registral midpoint
between the highest pc at 5 seven semitones higher and the lowest pc at 0 six semitones
lower. For example, the viola’s F4 at e of RPe(3-R1) is a registral midpoint between the
viola’s highest pc C5 at 5 and B4 at 0.81 In the case of the viola (the canonic dux), the
placement of the registral high point on the downbeat of m. 166 further adds to the
connection between the initial F4 and registral apex C5, linking the two pcs in an interval
5 dyad {C-F}; the comes in violin II connects pcs {F#-B} in an identical fashion. Thus,
the fundamental pcset {B-C-F-F#} exists in Tile A14 as the union of the above interval 5
dyads that occur in 5e of the two main row-forms comprising the canonic row pair.
80 This is reminiscent of mm. 46-69 of Berg’s Lyric Suite, movement III. 81 These are indicated by small boxes on the example.
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[Example 4.4b]
Whereas all A sections are unified through their use of the isomorphic partitioning
of {B-C-F-F#} at 012e of critical row forms, the inner A section stands out due to its
additional development and salient projection of the 3-cycle pcset {D-F-G#-B}. In
addition to projecting 6- and 5-cycles, the canon of Tile A14 also explores the nascent 3-
cycle tetrachord {D-F-G#-B} initially appearing in the row segment at 1234 of the
original row-form P0(3-R1).82 In the row-class of row (3-R1), the three members of
[0369] appear as the row segment 1234 and row partitions 06te and 5789; in the two row-
forms in Tile A14, the latter segments contain {D-F-G#-B}. Example 4.4b shows the
same passage as appeared in the preceding example, giving the first three measures of
Tile A14. However, in the present example, the boxed notes highlight salient members of
{D-F-G#-B} from 06te in both row-forms. Both pcsets {B-C-F-F#} and {D-F-G#-B}
share the dyad {F-B} at 0e, constituting another salient appearance of these particular
pcs. The remaining 6-cycle dyad {D-G#} appears conspicuously highlighted in two basic
ways on the musical surface, the first melodic and the second harmonic. First, since pcs D
and G# appear at t of row-forms Pe(3-R1) and P5(3-R1), the first sounding dyads in each
instrument’s canonic entrance are <F-D> and <B-G#> (see viola and violins II and I),
emphasizing the “3-cycleness” within the staggered entrances. Second, the metric
alignments of the canonic entrances highlight three dyadic subsets of the basic 3-cycle
collection: 1) {G#-B} between viola and violin II on the sixth eighth note of m. 165
(boxed), 2) {D-F} between violins I and II on the fifth eighth note of m. 166, and 3){B-
D} between violin I and viola on the sixth eighth note of m. 166. The subsequent musical
82 The relevant passage is depicted in the second box in Example 4.1.
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passage (Tile A15, mm. 169-172, not pictured) essentially repeats the canons with J00-
related row-forms RIe(3-R1) and RI5(3-R1), which exchanges the members of {D-F-G#-
B} appearing in the above horizontal and vertical dyads: the horizontal dyads <F-D> and
<B-G#> become <F-G#> and <B-D> and the vertical dyads {Ab-B}, {D-F} and {F-G#}
become {B-D}, {F-G#} and {G#-B} since the contextual inversion J00 is isomorphic to
the non-contextual transformation It in the present case.
[Example 4.5]
The trio B section contrasts the frenetic and strongly metric A-based sections by
presenting four rhapsodic instrumental solos (see Table 4.2) harmonized by long, held
notes in the other voices.83 The solos essentially present concatenated versions of the J00-
related row-form pairing of P/I4(3-R2), while the long accompanimental chords state
discrete trichords from the solo instrument’s row. Example 4.5 summarizes the first
(cello) solo beginning in m. 67, indicating the ops of the main row P4(3-R2) in both the
solo and the high accompanimental chords. As evident from the passage beginning in m.
72 of the example, the solo features row segments in addition to the complete form. The
(unordered) first discrete trichord {E-F-Bb} in row B is identical to the (unordered) first
discrete trichord in row-form P5(3-R1) appearing in the canons in Tiles A14 and A15 in
section A1, connecting the movement’s two main row-classes. Like the row-forms of the
A-based sections, the B section row-forms are represented by the J00 pair P4(3-R2) and
I4(3-R2). As opposed to the tritone at 0e of row 3-R1, row 3-R2 features the interval 4
dyad <E-C> at 0e. The J00-transform (and its non-contextual analogue I8) preserves the
first pc of the dyad and maps the second onto pc 8, completing the 4-cycle collection {C-
83 This is reminiscent of Lyric Suite, movement 5.
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E-Ab}. The exchanging pcs C and Ab are foreshadowed in the introduction of the
movement, where they form salient registral boundaries of the slow, methodical assembly
of row A’s first discrete pentachord <C-F-B-D-Ab> that characterizes the section.
The second quartet’s second movement presents a lyrical, contrapuntal adagio in
which twelve-tone and free atonal compositional strata interact freely. As indicated in
Table 4.1, the movement is an ABA’ form in which the A-based sections contain all
twelve-tone material (along with non-twelve-tone material), and the B section is freely
atonal.84 The basic compositional scheme in the A-based sections involves two
introductory individual row statements followed by five canonic pairings of
transpositionally-related row forms at ever-decreasing time intervals. Table 4.1 indicates
that the two thematic row statements occur in mm. 1 and 9, followed by three canonic
pairs labeled as Pairs 1-5 in mm. 13-20, 21-24, 25-26, 27 and 28 (labeled as “All
Voices”). The first two thematic statements of the A section present complete row
statements in one of the string quartet’s voices (viola in m. 1 and cello in m. 9), while the
following five stretto pairings stack the canonic entrances at a distance of two measures
(violins I and II in mm. 13-20 and viola and cello in mm. 21-24), one measure (viola in
mm. 25-26), one half note (violins I and II in m. 27), and one quarter note (all four voices
in m. 28); the final A’ section features one individual statement (m. 48) and a final
canonic pairing at a two bar interval (mm. 54-56). The compositional material
accompanying each of these thematic row presentations is freely atonal in style.
However, as the following analyses demonstrate, both twelve-tone and free atonal
accompanimental material are united in their use of interval cycles.
84 See Lyric Suite, movement 3.
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[Example 4.6]
The fundamental cyclic component of the thematic row forms in this movement is
based on whole-tone 2-cycle and 4-cycle collections. Example 4.6 presents the first
canonic pairing beginning in m. 13, providing the clearest example of the complete row,
its thematic expression, and its ramifications when presented in canon with other row
forms. Row form P9(2-R1) unfolds in violin I over the four measures of the excerpt
beginning in m. 13, while P7(2-R1) enters two bars later in violin II, providing a
counterpoint to the second hexachord of violin I’s row. From a cyclic perspective, this
example depicts two main ideas. The first underscores the essential 4-cycle+ component
of the row’s first discrete tetrachord <A-D-F-C#>, in which the pure 4-cycle collection
{C#FA} combines with “outside” pc D in an expression of [0148] from the excerpt’s first
measure to the downbeat of the second. The “flag” network appearing below the music
interprets this tetrachord, attaching the gesture’s lowest point D to its initial pc A via T7
and treating this as distinct from the 4-cycle upper tones. The second main idea
incorporates the gesture’s 4-cycle into the larger 2-cycle C# whole-tone collection in
02369t. Members of this collection appear saliently in the example on the downbeats of
each measure in violin I (<A-C#-B-Eb>) and in the registral extreme point of G6 on beat
three of the example’s final measure. This collection is reinforced on the musical surface
as the second canonic voice enters in m. 15. Its members G and B appear aligned with the
B and Eb on the downbeats of mm. 15 and 16 that result from the particular alignment of
6-e of P9(2-R1) in violin I and 0-5 of P7(2-R1) in violin II. Thus, both the melody and its
canonic pairing at the temporal interval of two measures firmly emphasizes the salience
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and structural importance of the C# whole-tone collection via its two component 4-cycle
subsets.
[Examples 4.7a and b]
The preservation of salient whole-tone sonorities remains throughout the next
passage despite the addition of non-twelve-tone material and the re-ordering of
material within a row form. Example 4.7a presents a reduction of the musical surface of
the beginnings of measures 23 and 24, in which the canonic pairing of P0(2-R1) and Pt(2-
R1) in the viola and cello are accompanied by a second stratum of non-twelve-tone
material in violins I and II. While the previous example features the projection of the C#
whole-tone collection through its alignment of P9(2-R1) and P7(2-R1), the present
example features the projection of the C whole-tone collection through its alignment of
P0(2-R1) and Pt(2-R1), resulting in the D5 and Bb3 appearing at the beginning of m. 23
and Gb5 and D4 appearing at the beginning of m. 24. In the example, the inner ic4s
between the outer registral extremes derive from an aligned triplet figure in vns I and II,
which emphasize dyads D4/F#4 in m. 23 and E4/G#4 and C5/Ab4 in salient locations in
m. 24. Thus, the resulting sonorities in both twelve-tone and non-twelve-tone strata, {Bb-
D-F#} and {D-E-Gb-Ab-C}, reinforce the local dominance of the C whole-tone
collection.
The localized projection of whole-tone collections persists throughout the next
passage in mm. 27 and 28, though it is accomplished through different means. At this
point in the piece, the gradual thickening of the musical texture combined with the
methodical reduction in the time interval between canonic voices dramatically increases
the tumultuousness of the end of the A section as it heightens in intensity going into the
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movement’s climactic B section. One of the mechanisms by which the intensity increases
is the rapid toggling between the C and C# whole-tone collections, as depicted in
Example 4.7b. As illustrated by the example, the quartet’s upper voices significantly
project pentachordal subsets of the C whole-tone collections {D-E-F#-Ab-Bb} and {D-E-
F#-G#-Bb} in beats one and two of mm. 27 and 28, and the C# whole-tone collection
{Db-D#-F-G-A} in beats three and four of m. 27. While these beats do not exclusively
contain pcs from these collections, constituent members of these collections appear on
metric downbeats of these measures, thereby gaining emphasis. As indicated below the
staff, the instruments contain hexachords of four different row forms, P6(2-R1), P7(2-R1),
P5(2-R1) and P2(2-R1). Three of these row forms, P7(2-R1) and P6(2-R1) and P5(2-R1),
feature retrograded second hexachords, which facilitates the downbeat projection of the
corresponding whole-tone collections. Thus, the uniform maintenance of whole-tone
collections at structurally significant points in the present case necessitates the
manipulation of the order of pcs within row forms, providing a concrete example of the
composer’s adaptation of strict twelve-tone procedures to accommodate the expressive
necessities of the music.
Since the juxtaposition between the two whole-tone collections is one of the
prime agents of tension within the outer A sections of the movement, it follows that the
absence of one of the collections could be interpreted as a relaxation of this tension. For
example, the initial statement of P9(2-R1) as depicted in Example 4.6 and its canonic
pairing with P7(2-R1) reinforced the C# whole-tone collection nascent in the row form.
Thus, any odd T-level row form will project the C# whole-tone collection. As illustrated
in Table 4.1, the piece does not begin with the above canonic pairing, but with T6-related
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P3(2-R1) and P9(2-R1) row forms. However, the odd T-levels tether these initial row
forms to the family of row forms that, given requisite compositional emphasis, project the
C# whole-tone collection. Although the movement contains ten of the twelve available
transpositions of the fundamental row class, the odd transpositions enjoy an elevated
status resulting from their privilege-of-place appearance as the first four presentations in
the A section and final three presentations, which occur as an initial solo statement of
Pe(2-R1) beginning in m. 48 and a final canonic pairing of P7(2-R1) and P9(2-R1) in mm.
54-59, similar to their appearance in mm. 13-17.85 Thus, from a cyclic harmonic
perspective, the C# whole tone collection initially establishes itself as referential, is
challenged through various juxtapositions with the C whole-tone collection, and
ultimately emerges to close the piece and resolve compositional tension.
At the core of twelve-tone composition is the idea that transformations of a
complete, ordered pc aggregate constitute a major structuring principle of a piece.
However, the utilization of a complete, unordered pc aggregate may also be a central
feature of music untethered to a specified ordering of the pc aggregate.86 Such is the case
with parts of the second quartet’s second and third movements. Specifically, the C
section of the third movement and the B section of the second movement significantly
feature localized pc aggregates without 12-tone rows. The following analysis provides a
detailed examination of the variety of ways pc aggregates operate in these sections,
describing, where applicable, issues of simple aggregate partitioning, interval cycles, and
operator cycles.
85 No I-forms appear in the movement. 86 Morris (2001) describes such constructions as “free arrays” (p. 181). The passages in question are brief and do not require most of the “aggregate partition” apparatus discussed by Morris (1987, 2001) and Mead (1988).
79
In the (non-twelve-tone) C section of the third movement, aggregate completion is
fundamentally achieved via the use of the four 4-cycle pcs sets and the 4-cycle T- and I-
cycles often associated with them. The analyses below demonstrate two distinct uses of
4-cycle-based, aggregate-producing processes in the two basic subsections within the
larger C section (Tiles C1-C3 and C4-C8). In general, this emphasis on 4-cycles within
the C section allows it to contrast the 6-, 5- and 3-cycles of the A-based sections while
linking it with the 4-cycle collection {048} that figures prominently in the B section.
[Examples 4.8a, b and c]
In the first subsection (see tile C1 in Table 4.1), pc aggregates are created by
uniting pairs of trichordal members of Tn-types [037] and [047] within complementary
hexatonic collections. Example 4.8a depicts the pc aggregate divided into two members
of the 4-cycle-based C#/D and B/C hexatonic collections, both of which are members of
[014589]. Both of these pc collections are further subdivided into the I-related pairs of
Tn-types [037] and [047] whose union is the complete hexatonic collection. For example,
the C#/D hexatonic collection {C#-D-F-F#-A-Bb} can be “disunited” into the I3-related
pair {D-F-A} ([037]) and {F#-Bb-C#} ([047]), the I7-related pair {Bb-C#-F} and{D-F#-
A}, or the Ie-related pair {F#-A-C#} and {Bb-D-F}, as appears on the left of the diagram.
An analogous situation obtains via the same operators within the B/C hexatonic
collection {Eb-E-G-Ab-B-C}, as shown on the right of the figure. The following analysis
confirms the relevance of these hexatonic collections and their disunification into minor
and major triads within the C section.
The presentation of members of [037] and [047] in Tiles C1-C3 (mm. 195-211)
highlights the transformational connection between the I-related trichords via the metric
80
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placement of representative I-related pc pairs. Example 4.8b depicts the first two
measures of Tile C1, while Figure 4.8c interprets them. As evident in Example 4.8b, two
complete pc aggregates, one per measure, are segmented into members of [037] and
[047] appearing as vertical triads in m. 195 and small arpeggios in m. 196. The diagram
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to the left of the example indicates that the verticalities expressed in the instrumental
pairings of the upper and lower voices on the first and second eighth notes of m. 195 do
not voice hexatonic collections, but rather pairs of I5-related triads which unite into
members of [023679] and [014679]; the I5 transformations between the individual voices
within each pair are depicted on the left column in Example 4.8c. However, the arpeggios
of m. 196 accompany a migration of the trichords within the upper three voices, as
indicated by the solid lines linking violin 1 to the viola, violin 2 to violin 1, viola to violin
2 and cello to itself from mm. 195 to 196. This re-orchestration precipitates a change in
the set-class membership of the upper and lower voice pairs from the above non-
hexatonic collections illustrated to the left of the example to the hexatonic collections on
the right. This migration and the ensuing emphasis on hexatonic collections appears in
the right column in Example 4.8c, which shows the I3- and I7-related pairings familiar
from Example 4.8a moving from outer and inner voice pairs that are displaced in time to
upper and lower voice pairs sounding together. Significantly, the cello’s arpeggio in m.
196 is distinct from the remaining voices in its presentation of a rotated (“second
inversion”) trichord, which aligns the I3 dyad {Gb-A} on the downbeat of the measure,
underscoring the presence of I3 through its metric placement.
[Examples 4.9a and b]
The combinatorial potential of [014589] is further explored in Tile C3 (mm. 202-
211) as a result of canons featuring 4-cycle operators. Example 4.9a provides the first
three measures of Tile C12, which maintain the instrumental figuration of m. 196 in Tile
C1 above. In addition to the familiar segmentation of members of [014589] into the I-
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related pairing of [037]/[047], Tile C12 incorporates I-related [014589] subsets
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[015]/[045], using them in tandem with the [037]/[047] segments to provide contrast
within the broader context of aggregate completion. The three boxes appearing in the
example each contain the pc aggregate; Example 4.9b illustrates the manner in which the
pc sets within the three boxes create and maintain aggregates throughout the entire
passage. The example begins by defining two pc sets A <Bb-D-F> ([047]) and B <Ab-E-
Eb> ([015]) which act as the excerpt’s most basic building blocks. The leftmost part of
the example depicts the C#/D hexatonic collection resulting from the familiar, metrically
supported pairing of T0(3-R1) and Ie(3-R1) appearing in the violins at the beginning of m.
202. The T8 arrows along the top rank of hexachords show the cyclic transposition of the
I-related pairs of trichord A that appear in beat one of each measure, from the sum e
83
pairing above through pairings of sums 3 (T8(3-R1)/I7(3-R1)) and 7 (T4(3-R1)/I3(3-R1)).
The “southeast” T8 arrow initiates a second cycle of T8 transpositions that anticipate the
upper rank pairings by one beat. A virtually identical situation obtains with the pairing of
T0(3-R2) and I3(3-R2) in the violins on the second beat of m. 202 and their subsequent
transformation via a T8-cycle through the same sums. The resulting texture is one
comprised of two interlocking T8-cycles featuring pc aggregates expressed in three
dimensions. The first exists in the violins within each measure, as beat one contains the
C#/D hexatonic collection in an [037]/[047] pairing and beat two contains the
complementary B/C hexatonic collection in an [015]/[045] pairing on beat two. The
second occurs in canon with the first in the viola and cello, with the [037]/[047] and
[015]/[045] pairings occurring on opposite beats. The third appears in all four voices on
each beat beginning on beat two of m. 202, as the necessary transpositions and inversions
of both trichords A and B align as a result of the canon between upper and lower voices.
In Example 4.9b, any pair of hexachords connected by a dashed line indicates a pc
aggregate. The first dimension appears in the top horizontal, the second in the bottom,
and the third in the vertical slices.
[Example 4.10]
Like the Second String Quartet’s third movement, the second movement also
integrates twelve-tone and free atonal material. The latter movement’s B section, itself
the center section of an ABA’ form, is not based upon a twelve-tone row. However, the
main thematic idea, a series of six hexachordal descents occurring within a very dense
musical texture, dialogues directly with the main twelve-tone theme of the outer A
sections through associations based upon membership in the same set-class. Example
84
4.10 presents a detailed diagram of these descents, depicting them as noteheads on the
staff and indicating the measures in which they occur. Below the staff is another detailed
diagram demonstrating the two transformational paths the initial row’s hexachords
traverse within the B section.87 The leftmost section of the staff, corresponding to mm.
13-16, presents the row form P9(2-R1) (discussed above as the first row of the first
canonic pairing) divided into its constituent hexachords. These hexachords are also
interpreted below the staff divided in two ways: above and below the boxes as members
of [013478] (above) and [012569] (below) and broken into the row’s discrete trichords as
members of Tn-types [037]. The modest transformation network above the boxed figures
depicts the motion of the row’s first hexachord through the upper register hexachords
appearing on the musical surface as quarter-note triplet descents in mm. 36, 38 and 44.
As illustrated by the arrows connecting them, these hexachords are all transpositionally-
related, moving via T5, Tt and T5.88 The bottom network depicts the motion of the row’s
second hexachord through the lower register hexachords appearing on the musical
surface in a variety of rhythmic presentations in mm. 37, 39 and 45 (immediately
following their upper register presentation). As illustrated by their arrow labels, the first
of the B section’s lower hexachords is inversionally-related to its A section analogue,
appearing in its Ie form. However, a second inversion I2 returns it to its original form
before its final motion via T5, underscoring the role of the m. 39 presentation as
restorative.89
[Example 4.11]
87 The diagram uses the standard mod12 integers to represent pcs. 88 On the musical surface, these figures all appear in violin I. 89 The first two of these hexachords appear in the cello, while the last appears in the viola.
85
While the hexachordal aspect of the B section provides the strongest link between
the twelve-tone A sections and the free atonal B section, the trichordal aspect is revealing
as well. The above analysis of the row and its cyclic structure places a premium on the
first hexachord’s whole-tone elements, especially emphasizing the 4-cycle {F-A-C#} as
its fundamental structural component. When viewed from a cyclic perspective, P9(2-
R1)’s first hexachord disunites into two whole-tone trichords {F-A-C#} [048] and {D-E-
G#} [026], from the opposing C# and C whole-tone collections. The final hexachord of
the B section, appearing in the rightmost box in Example 4.10, reveals that this initial
abstract segmentation is articulated concretely on the musical surface in the closing
melodic gesture of the B section. The viola statement <B-G-Eb-Gb-Ab-D>, pictured to
the right of the double bar in Example 4.11, presents members of set-classes [048] and
[026], stating overtly an initially abstract yet fundamental relationship of the row’s
signature gesture.
86
Chapter Five: Conclusions and Implications for Further Research
I. Conclusions
The methodology developed and applied through the analysis in this dissertation
reveals consistency among the diverse musical surfaces presented in Ginastera’s First and
Second String Quartets. In particular, three main avenues of compositional consistency
emerge as relevant in the chosen quartets. The first avenue addresses general issues of
tonality, such as the existence of and compositional contexts for referential collections
and their relationships to the various structural tonal centers and axes of symmetry
appearing in the music. The second avenue engages motivic issues and the consistent
manner in which pc motives emerge from the quartets’ structural cyclic tonal background
to comprise significant thematic and developmental constructions. The third avenue lies
in consistencies in the rhetorical unfolding of material in specific movements and how
that material is linked to the movements’ tonal, motivic, and formal constructions. As the
analytic chapters demonstrate, the fundamental precept supporting each of these avenues
of consistency is the interval cycle.
From a tonal perspective, Kuss’ ascription of the Berger/Van Den Toorn concept
of octatonic-diatonic interaction to Ginastera’s music is indeed borne out in the quartets
and provides a general framework for understanding the various pitch structures that
appear consistently and saliently in the music. As established by the detailed analyses of
recurring pitch structures in Chapters 3 and 4, the basic octatonic-diatonic frame can be
defined in terms of interval cycles and cyclic collections undergirding the diatonic and
octatonic collections. When these 3-cycle and 5-cycle collections combine with hexatonic
87
and whole-tone collections created by aligned 4-cycle collections (which initially appear
in the First Quartet and recur ubiquitously in the Second), a more complete and nuanced
picture of Ginastera’s tonal language emerges. For example, in the first movement of
both quartets, the 5-cycle pentachord {C-G-D-A-E} formed by the open strings of the
violin, viola and cello provides an a priori referential set whose constituent conjunct
trichords, referred to as shadings of the overall collection, play a critical role in defining
local referential collections. Both First Themes in the quartets’ first movements are
significantly octatonic, yet the particular 3-cycle-based collections within the themes are
ultimately controlled by the 5-cycle dyad spaces, resulting in passages in which the 3-
cycle melodic material colors the overarching referential 5-cycle. Ultimately, the analyses
of Chapters 3 and 4 demonstrate this nuanced cyclic approach in both free atonal and
twelve-tone music, strengthening the connection between Ginastera and major
proponents of cyclic organization, Bartòk and (above all) Berg.
In addition to tonal issues, motivic issues also comprise a major component in the
quartets under study. Specifically, the main thematic construction of the quartets’ first
movements can be modeled by a systematic unfolding of a single ordered, motivic
trichord via multiple U(st)-transformations, thereby locating within the motive and its
intervallic properties the potential for significant musical expression. For example, the
Chapter 3 analysis of the First Theme of the First Quartet’s first movement considers the
entire theme as a U-chain formed by three successive transformations of the movement’s
first three notes <D-F#-F>. Three of the four major thematic elements of the four main
themes of both quartets’ first movements feature U-transformations at their very core,
and the terminology developed in Chapter 2 greatly facilitates the discussions of the
88
movements’ themes and their roles within their respective movements. The analyses
based on U-transformations not only value highly the role of the motive in the quartets,
but they also participate significantly in the cyclic aspect of the music, as multiple U(st)-
transformations of a source set result in cyclic pc collections.
The unfolding of connected trichords also links Ginastera to Webern, who
famously works with small sts in such pieces as the Fünf Sätze für Streichquartett Op. 5,
no. 3 and the Concerto for Nine Instruments, Op. 24. It is well known that the former
piece and movement feature significant trichordal material as the basis of its composition;
the movement even features two inversionally related extended U-chains <C#-A-G#-E-
D#-B-Bb> and < C-E-F-A-Bb-D-Eb> in violin I and vcl respectively in m. 7. The row of
the concerto is a derived row, exclusively featuring members of [014] in each of its
discrete trichords, and while no overt connection exists between the contextual
transformations defined in Chapter 2 of this dissertation, it is plausible that similar
mechanisms could be defined for a detailed investigation of the work. The notion of
connected trichords also links Ginastera to Schoenberg, who in his well-known radio
address on the Op. 22 Orchestral Songs describes the “developing variation” style of
unfolding trichords and discusses the concepts of Grundgestalt and Gedanke. These play
critical roles not only in his works but also appear in the works of Berg, Webern, and, as
discussed in Chapter 3 of this dissertation, Ginastera.90
90 As Boss 1992 indicates, “the original typescript of the lecture, in German, is in the archive of the Arnold Schoenberg Institute in Los Angeles. An English translation by Claudio Spies was published as ‘Analysis of the Four Orchestral Songs Op. 22’ in Perspectives of New Music 3, no. 2 (1965) and appears in Perspectives on Schoenberg and Stravinsky 2nd ed., Benjamin Boretz and Edward T. Cone (New York: Norton, 1972.) Currently, the archive resides in Vienna.
89
Returning to the Ginastera quartets, each of the movements studied employs a
similar rhetorical structure at significant formal junctures of the pieces. Major formal
sections often begin with a salient motivic and/or harmonic statement that establishes a
context for subsequent compositional reinterpretation. In the First Quartet’s first
movement, the aforementioned motive (and potential Grundgestalt) <D-F#-F>
participates significantly in the movement’s first theme and its tonal center, while in the
Second Quartet’s first movement, the initial trichord <D-Bb-C#> [014] and salient
trichord <Bb-F-E> [016] in tile 1 establish contexts for their significant appearance in the
motivic and harmonic character of the main themes. In the Second Quartet’s mostly serial
second and third movements, initial motivic statements <Eb-Ab-Cb-G> [0148] and <C-F-
B> [016] and their internal cyclic intervallic properties provide not only the source of
subsequent melodic constructions, but the statements’ transformation establishes cyclic
harmonic constructions critical to the piece’s harmonic and formal structures. Ultimately,
such consistency in the movements’ rhetorical unfolding could provide a potent insight
into the composer’s practice and should continue to do so in other of his pieces.
II. Suggestions for Further Research
Despite the considerable amount of highly-focused analytic detail on significant
aspects of the quartets’ tonal language and thematic construction offered in the previous
chapters, it represents a mere fraction of the potential analytic and music-theoretic work
suggested not only in the pieces themselves, but in all of the composer’s work and
beyond. The continuation of the present chapter elucidates several avenues for further
research. In general, the suggestions divide into two main categories, analytic and
90
theoretic. The former category begins with the first steps of an analysis of the 14-bar
theme of the Second Quartet’s fourth movement, a theme and variations featuring
rhapsodic solo passages in each of the string quartet’s voices. This initial analytic
category continues in illustrations of brief passages from the composer’s Third String
Quartet Op. 40 (1973) and Sonata for Guitar Op. 47 (1976) that clearly demonstrate the
applicability and relevance of the dissertation’s methodology to a broader sample of the
composer’s work. It closes with a brief analytic demonstration of the general
methodology in a passage from Berg’s Op. 4, No. 2 in an effort to place Ginastera’s
works in dialogue with other music that has received considerable analytic attention. The
discussion of the latter category addresses some theoretic elements introduced in Chapter
2 and speculates on how they may be developed further.
[Examples 5.1a, b and c]
The theme of the Second Quartet’s fourth movement is a sparsely-accompanied
14-measure rhapsodic violin solo whose three basic thematic statements establish 3-, 4-
and 5-cycle contexts for the theme’s focal pc G. The first two statements, depicted in
Examples 5.1a and b respectively, present two versions of the movement’s main idea.
The former clearly illustrates the establishment of G4 as the gesture’s melodic touchstone
in mm. 1-4, while the latter depicts the note’s maintenance within the second statement
beginning in m. 6. Both of these beginnings show the placement of the central G within
two cyclic contexts: the former presents G within the pentachordal subset <G-F-E-D-C#>
of the C/C# octatonic collection, while the latter presents G within the movement’s main
4-cycle {G-Eb-B} emphasized on the offbeats of beats three and four. The final cyclic
contextualization of G within the 5-cycle OS-set is initially hinted at in the
91
accompanimental chords that enter in mm. 3 and 4, in which the violin I G occurs
harmonized in the later measure by {B-C-D#-E}. The resultant pentachord, a member of
[01458], presents the focal G both as a member of the main 4-cycle collection and paired
with the cello C3 in an initial expression of the OS-set. These two cyclic associations, G
within the main 5- and 4-cycle sets, form the basis of the theme’s closure in mm. 13-14,
which is depicted in Figure 5.1c. This chord, itself a member of [0124589], clearly arises
from the conglomeration of two members of [0148], the 4-cycle+ collections {C-G-Eb-
B} in the cello and viola and its T7 transposition {G-D-Bb-F#} in the viola. Thus, the
theme’s final chord expresses the 5-cycle OS shading {C-G-D} in the accompanimental
voices’ bottom strings and links it to 4-cycle sets {G-Eb-B} and {D-F#-Bb}. Ultimately,
the cyclic contexts for the focal G are a fundamental structural feature of the remaining
three variations and help to clarify the role the movement plays in the entire quartet.
[Example 5.2]
Although this dissertation focuses on Ginastera’s first and second quartets, his
(final) String Quartet No. 3 also occasionally features musical constructions that can be
92
modeled by the analytic techniques developed in Chapter 2.91 Example 5.2 presents the
first seven measures of Movement III, which slowly assembles the 3-cycle trichord {E-G-
Bb} in mm. 6-7 through the unfolding of source trichord <E-F#-G> in the viola (mm. 2-
4) into <G-Ab-Bb> (mm. 4-6) in violin II (both are sc[013], although they are
inversionally related) via a single U(ff) transformation. Ultimately, the techniques
developed in this dissertation are of limited utility in the third quartet, but they do reveal
similar constructions among all the quartets and potentially provide an analytic point of
departure for the final quartet.
[Examples 5.3a, b, c and d]
The methodology developed in this dissertation is also applicable to other of
Ginastera’s works. Examples 5.3a, b, c and d depict the beginnings of analyses of the first
four phrases of the composer’s Sonata for Guitar, illustrating significant passages
comprised of interval cycles (Example 5.3a), extended U-chains (Ex. 5.3b), TTO cycles
(Example 5.3c), and a device I will describe as K-net “hyper cycles” which derive from
K-net interpretations of extended passages featuring TTO cycles (Example 5.3d).
Example 5.3a depicts the First Movement’s first phrase, which begins with a presentation
of the guitar’s open strings (the “guitar chord”), followed by a rubato passage comprised
91 The third string quartet contains five movements for string quartet and soprano voice in a work reflecting the composer’s affinity for Schoenberg’s Op. 10. The texts for the quartet are by 20th-Century Spanish poets Juan Ramón Jiménez (La Música, Movement I and Ocaso, Movement V), Federico García Lorca (Canción de Belisa, Movement III), and Rafael Alberti (Morir al Sol, Movement IV). Whereas the first two quartets are neo-classic in style, the third clearly reflects the experimentalism associated with the 1960’s and thus lies outside the scope of this dissertation. Of the third quartet and its context within his contemporaneous works, the composer notes “As in my most recent works…I have made use of a technique based on the interplay of fixed and variable structures, and on the creation and organization of multidimensional space wherein develop infinite phenomena – and corresponding resonance – of the ever-changing universe of sound.” See “Composer’s Note” (1973) in Boosey & Hawkes score, copyright 1977.
93
exclusively of 5-cycles in an exposition of one of the sonata’s fundamental sounds.
Example 5.3b presents the movement’s second phrase, in which a second statement of the
guitar chord is followed by a series of ascending chords that alternate between members
of [015] and [037] respectively (excluding the pedal A); the topmost voice <F-E-G-F#-A-
G#-B-Bb-C#> is an extended U-chain, while the trichords including the notes of the
upper extended U-chain form a pattern well-modeled by a pattern of K-net hyper
transformations. Example 5.3c presents a similar passage in its depiction of phrase three,
which illustrates a T-cycle (T3 in the present case) arising from the patterned motion of
the passage’s upper voices. Finally, Example 5.3d presents the movement’s fourth
phrase, in which another replicated pattern results in an upper-voice U-chain atop a
succession of tetrachords which, when interpreted as K-nets in a manner similar to
Example 5.3b, form a predictable cycle of hyper T-transformations, resulting in a pattern
of hyper T-transformations that I will call “hyper cycles.” Certainly, Examples 5.3a
through 5.3d are very speculative and are included here to suggest potential applications
of this dissertation’s methodology.
[Example 5.4]
Although this dissertation’s methodology and analytic approach arise in response
to some particularities of the tonal language, thematic and motivic process in Ginastera’s
first two quartets, similar peculiarities exist in the music of other 20th-Century composers
whose music the composer is known to have appreciated. The second song of Berg’s Op.
4 (“Sahst du nach dem Gewitterregen” from the Altenberg Lieder) provides a prime
94
example of music literature that displays such particularities.92 Example 5.4 provides a
reduction of two significant passages from the beginning of the song. The leftmost
measure marked “Gesang” features the first two measures of the vocal part, while the
rightmost measure presents a rising figure in the horn and bassoons in mm. 3-4
immediately following the initial vocal intonation. Beneath the example appear three
analytic diagrams indicating the Tn-type ([034]) formed by the initial vocal gesture <Bb-
B-G>, the U-cell <Eb-B-A>-<E> ([0167]) formed by the trailing end of the initial vocal
gesture, and the extended U-chain <B-D#-D>-<F#-F-A-Ab> formed by four U(st)
transformations of the source trichord <B-D#-D> in the horn and bassoon ascent. As
demonstrated in Headlam 1996, members of [014] and [016] play a vital role in the song;
presently, passages making significant use of members of [014] and [016] are included
merely to identify some bedrock methodological constructions of this dissertation within
significant passages of Berg’s song and demonstrate the viability of the present
approach.93
In addition to demonstrating promising potential for further analytic research, the
methodology also suggests a few areas for further theoretic research as well. The
following section identifies three main ideas contained previously in this dissertation and
expounds briefly on how these ideas could potentially be developed into viable theoretic
research. The first discussion revisits the useful concept of shading and leads to a second
(and related) discussion of “contextual transposition.” The final suggestion revisits the
92 See Headlam 1996, pp. 86-94 for a detailed analysis of the song from a cyclic perspective. 93 A second [016]-based U-cell occurs at the climax of the song in m.8, where it appears in a dramatically high and soft vocal subphrase on the text “Siehe Fraue.”
95
notion of extended tertian sonorities as verticalized U-cells and their extensions and
suggests some potentially relevant music literature for such an inquiry.
At its core, the concept of shading is designed to easily accommodate contextual
shifts among various subsets of a background cyclic collection without invoking a
specific TTO. For example, the shift in contextual emphasis from the subset {CGD} in
the First Theme of the Second Quartet (see Chapter 4) to the subset {DAE} in the Second
Theme is most simply understood as a change of focus within a sui generis referential 5-
/7-cycle {CGDAE} formed by the open strings of a string quartet rather than the result of
either T2 or I4 of the initial collection. Since such an understanding privileges the notes of
the cyclic collection over the repeated transposition operation that creates the collection,
shading could prove a useful concept in the analysis of any music that features prominent
background cyclic collections, such as the music of Berg, Stravinsky, Bartòk, and
Scriabin.
In a sense, a change in the shading of a particular cyclic pc collection is a type of
contextual transformation performed on the larger collection’s subsets. Since the shaded
subcollection is both inversionally and transpositionally symmetrical, a change in shading
could be considered either a contextual inversion or transposition operation. In
transformational music-analytic literature, especially in the works of David Lewin, the
concept of contextual inversion is well established and employed. However, the notion of
contextual transposition is at present unexplored. Presently, the motion from {CGD} to
{DAE} in the above example can be understood in two basic ways; the motion either
arises from shading of the OS set or results from performing a TTO (T2 or I4). If the a
priori construct of the governing 5-/7-cycle OS set is accepted as a structural entity, then
96
the motion from the former to the latter trichordal subset could be considered a contextual
transposition T2n, with n representing the cyclic interval.94 Ultimately, the use of both
contextual and non-contextual transpositions could enrich the analyst’s understanding of
a cyclic piece’s structure by separating transpositions within a cyclic structural entity
from transpositions outside that entity, potentially providing a more nuanced analysis.
[Example 5.5]
Finally, Example 2.7b in Chapter 2 illustrates a remarkable connection between
collections created by U-transformations of triadic source trichords and extended tertian
“third stacks” common to jazz, popular music, and occasionally Western music theory
and repertoire. Example 5.5 recreates part of Headlam’s (1996) Example 1.13b, which
depicts the beginning of a chord progression appearing in a famous passage in the first
several measures of Berg’s Op. 2, no. 2. Headlam’s point in the example is to illustrate
how a specific alignment of descending 1-cycles and an ascending 5-cycle creates a
musical surface in which the vestiges of late tonal practice (the ideas of falling fifths and
“Fr+6” chords, members of [0268]) and an emergent cyclic atonal language coexist
almost indistinguishably. The present example reinterprets the first three chords of
Headlam’s example with standard jazz/pop symbols for each chord. Appearing below the
chord symbols are diagrams indicating the potential interpretation of each of these chords
as U-cells of the given source trichords. Although the Berg passage is unequivocally in
the Western Art tradition, it employs chords found in jazz and popular traditions. Thus,
the potential for a cyclic-based understanding of jazz and popular music theory exists,
94 In the present formulation, the collection is considered a 7-cycle, with 2n equaling 2 (14 mod12).
97
and aspects of the methodology developed in this dissertation could prove useful in such
an inquiry.
Finally, the methodology employed is used to begin to answer a broader question
of the place and nature of Ginastera’s String Quartets in his oeuvre. In response to some
scholars problematic “style period-based” view of Ginastera’s work, this dissertation
engages the debate from two angles, one theoretic and one epistemic. In the former, the
emergence of the structural 4-cycle in the Second Quartet marks a distinct departure from
the octatonic/diatonic 3- and 5-cycle tonal language of the First Quartet, yet the
methodology demonstrates consistency in the use of 3-, 4- and 5-cycle elements. This
demonstrated consistency informs the latter, epistemic angle that questions the general
efficacy of a rigid period-based understanding of the composer’s work. Indeed, the
remarkable rhetorical and stylistic consistency across the composer’s work suggests that
a more comprehensive investigation of similarity, if not unity, in his output would
provide a clearer understanding of Ginastera’s music and its role in the 20th-Century. As
discussed previously, the “period-based” perspective on the composer’s music remains
problematic for the preeminent Ginastera scholar Kuss, and the approach developed in
this dissertation supports her general position by demonstrating an overarching
consistency between the first two quartets, works composed ten years apart.
Fundamentally, the broad question considered here lies at the intersection of a
20th-Century composer’s work and the issues it raises for the practice of music analysis.
As is the case with any music-analytic project, the careful study of the music leads the
analyst through a select group of concerns of seemingly divergent natures; the analyst
must define and contend with numerous questions regarding the composer’s background
98
and aesthetics, scholars’ engagement with the composer’s life and music, the suitability
of existing analytic tools and, as is the case in the present study, the panoply of issues that
arise in the building of novel theoretic and analytic tools in response to significant aspects
of the music’s construction. Scholars’ engagement of Ginastera has produced a body of
work marked by varying degrees of quality and success and has resulted in an appreciable
degree of understanding of the composer and his music. The analytical framework of this
dissertation does provide some insight into a few technical aspects of the composer’s
music and offers a viable path to an increased understanding of how the work of this
remarkable composer opens a dialogue with the work of other composers who receive
considerably more attention from the scholarly community.
Appendix One
Appendix One provides a detailed account of how each of the six U-
transformations transforms the given source trichord <265>, the first three notes of the
First Quartet’s first movement (see Example 2.3b). In so doing, Appendix One illustrates
two essential pieces of information: 1) the reason behind each operation’s order and 2)
the direct link between the interval from the first to the third terms in the source trichord
and the predominant interval cycle in the pc collection resulting from multiple
applications of the particular transformation. The overall progression of transformations
in Appendix One is from the smallest order (2) to the largest (8). It is essential to note
that the order of each transformation is dependent upon the interval between the first and
third terms of the source trichord.
Given source trichord <265>, f=2, s=6 and t=5; s-f=4, t-s=e, and t-f=3.
Order 2: U(ft)p and U(ss)
p
U(ft)p: <265> - <215> - <265>
U(ft) is an RI operation (sum 7) which maps <265> into <215> and back to <265>. By
definition, 2 maps into 5 (f into t) and 5 into 2; 6 maps into 1. <265> inverts to <512>,
which is in turn retrograded to <215>. Since the domain and the image have the same pcs
in the same ops, the second transformation returns the original source trichord, hence the
order is 2.
U(ss)p: <265> - <76t> - <265>
U(ss) is an RI operation (sum 0) which maps <265> into <76t> and back to <265>. By
definition, 6 maps into itself (s into s); 2 maps into t and 5 maps into 7. <265> inverts
into <t67>, which is in turn retrograded to <76t>. Since the domain and range have the
same pc in s, the operation must be an involution, hence the order is 2.
Order 4: U(ff)p and U(tt)
p
U(ff)p: <265> - <et2> - <80e> - <548> - <265>
U(ff) is an RI operation which in this case alternates between sums 4 and t, depending on
the identity of f in each trichord. The first operation maps 2 into itself (f into f) via sum 4;
6 maps to t and 5 maps to e. The retrograde of the image reverses the ops of the first and
third terms. Since the first term of the domain inverts into the last term of the image and
vice-versa, the ic between f and t in both domain and image is preserved. However, since
the relationship between the two is inversion, a partial interval cycle based on the ic from
f to t results; 2 and 5 (f and t of the domain) invert into 2 and e via sum 4, which is then
retrograded to e and 2. The resulting union of both domain and image f and t is {e25}, a
member of [036]. Since by definition s never becomes f or t, the chain returns the original
source trichord after the 3-cycle closes with the fourth transformation. The total pc
content of the chain, {024568te}, is a member of [0124678t], and the complete 3-cycle
subset is {258e}.
U(tt)p: <265> - <548> - <80e> - <et2> - <265>
U(tt) operates identically to U(ff), except it produces a retrograde.
Order 8: U(fs)p and U(st)
p
U(fs)p: <265> - <326> - <e32> - <0e3> - <80e> - <980> - <598> - <659> - <256>
U(fs) is an RI operation which is substantially different from the above in its exchanging
of outer and middle terms. The first transformation inverts <265> into <623> via I8
(f+s=8), which in turn is retrograded into <326>. Since the operation is a retrograde, the
dyad <26>, occupying fs of the first domain, moves into st of the image. The second
transformation completes the position shift of the initial pc 2 from f of the initial source
trichord, to s of the second, and finally to t of the third, taking a second step to
accomplish what U(ff) (order 4) did in one. However, like the U(ff) example, the partial 3-
cycle {e25} remains in f and t of the original source trichord and its second
transformation. The difference in the orders of both results from the additional
transformation required to map the pc in f into the pc in t. The total pc content of the
chain is {235689e0}, a member of [0134679t] (octatonic).
U(st)p: <265> - <659> - <598> - <980> - <80e> - <0e3> - <e32> - <326> - <256>
U(st) operates identically to U(fs), except it produces a retrograde.
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