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Index

(C,O), 261(G : H), 1540, 622, 642a, 43<, 57=, 41AC, 177AllOrd, 73@X, 252C-major

scale, 175Cyc(Γ, x), 143D, 177DX , 145Df(x), 264Dia, 193Dia(3), 193Euler, 227, 262EulerSpace, 227, 233, 241, 262Fin(a), 58G/H, 153GL(M), 229Gopp, 148H\G, 154I, 149IV LS, 152Ida, 52Im(x), 115J℘(X), 270K, 177K4, 149, 154Ker(f), 155

M∗, 147Mor(C), 249Motk, 261N(C), 129OFT , 241OP , 149Ob(C), 249PCChords, 157P12, 53, 65, 144Path(Γ ), 123R, 102, 149R/I, 210R?, 251RI, 149R[X], 206R〈A〉, 206Re(x), 115SX , 145Sn, 75, 76, 150Sym(OP ), 149Sym(X), 143Sym(n), 150T (E), 270TDS, 55TI, 153TIZ12, 157TZ∗

12, 173T t, 92, 153T t.s, 173T t+, 153

T t−, 92, 153

T t±, 153

TX , 145

303© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3

304 Index

U , 155, 156W+, 195W−, 195Walk(Δ), 125Word(A), 143X7, 183Xn/Sn, 76Xmaj7, 183Xm7, 183Z, 195[a, b], 258]a, b[, 258[x], 58[x] (orbit of x), 152AND , 37AffR(M,N), 230C, 113, 143, 210Δa, 50End(M), 145Γ@X, 283Γ0, 228Gest, 282Grp, 250Grp(G,H), 147IMPLIES , 37

LocR, 250M(R), 250Mm,n(R), 218ModR, 250ModR(M,N), 225Mon(M,N), 145N, 72, 74, 143NOT , 37Nat(C,D), 252OR , 37Φ(x), 43Q, 93, 143R, 100, 143R[X], 210REHLD, 226, 261Rehld, 226R+, 104R−, 104Set, 250Set(a, b), 61Top(X,Y ), 259Top, 259Z, 89, 143Z3 × Z4, 171

Zn, 156multiplication on -, 173

Z12

symmetry on -, 173Z24, 156Z7, 156x̄, 115⋃

, 42∪, 43∪ (poetry), 145∩, 44χ, 65cos(x), 259∅, 40�, 122, 269ba, 96

∈, 39〈S〉, 146, 148, 226, 260≤, 57log, 109C, 102, 249C@, 252Copp, 250O, 102�∈, 41−→X , 281π, 17∏

, 66�, 53∼→, 53�, 54, 122, 269∼, 57sin(x), 259n√a, 107

⊂, 41�, 53℘, 270℘E , 265℘H , 265RS, 226a− b, 45a+, 43a1/n, 107cadi, 195codom(f), 52d(z, w), 173dX , 145de/dE, 265deg(P ), 211

Index 305

det(M), 223dim(M), 233dom(f), 52e, 17f : a → b, 52gH, 153h ◦ g, 51i, 113int(r, s), 152n-simplex, 129nth

root, 107ord(x), 154presto�, 30, 234xRy, 57— (poetry), 145RUBATO�, 30, 234, 260sX , 145sup(A), 105, 107tX , 1454-group

Klein -, 149, 1547th

diminished -, 162dominant -, 161major -, 161minor

major -, 162minor -, 162

12-temperedtuning, 99, 108, 111, 121, 242

12-tonetheory, 26series, 53

AA Love Supreme, 185–187A Musical Offering, 12abelian

group, 147absolute

value, 90, 104accelerando, 208acoustical

argument, 176acoustics

musical -, 269action, 151

group -, 29, 152

simplytransitive -, 152

transitive -, 152addition

of natural numbers, 83monotony of -, 104of integers, 90of rational numbers, 96

additiveinverse element, 97monotony, 97neutral element, 97

address, 252, 253addressed

object, 231adic

normal form, 86representation, 111

adjunctionnatural -, 64

Adorno, Theodor Wiesengrund, 268Aeneid, 145affine

function, 259homomorphism, 230

agogics, 265Airegin Club, 183algebra

Boolean -, 45monoid -, 206polynomial -, 206

algebraicgeometry, 30

Allais, Alphonse, 39Allegro Barbaro, 79alphabet, 143alteration, 134, 175alternative, 72Alunni, Charles, 279, 281American Set Theory (AST), 25, 28, 29Amuedo, John, 29An der schönen blauen Donau, 95analysis

creative -, 133, 183, 186Andreatta, Moreno, 25, 33, 34Anglo-Saxon

gesture theory, 279antinomy

Russell’s, 56

306 Index

Russell’s -, 40antisymmetric

relation, 57Appassionata

Sonata, 253approximate

solution, 99archicembalo, 108Archimedean

ordering, 98property, 104

Archita, 245Arezzo, Guido d’, 269argument

acoustical -, 176Aristotle, 279arpeggio, 185, 222arrow, 29, 122artes liberales, 7Ascension, 185Assayag, Gérard, 25associativity, 45AST (American Set Theory), 25, 28, 29atonal

music, 27attack, 231attribute, 43augmented

triad, 134, 160, 179, 185Auhagen, Wolfgang, 31autocomplementarity

function, 177axiom, 40

of choice, 44, 58axioms

Peano -, 74ZFC -, 41

BBabbitt, Milton, 25–27Bach, Johann Sebastian, 10, 12, 21, 26ball

open, 261band

harmonic -, 130, 139, 198Moebius -, 131, 135

Bardez, Jean-Michel, 33Bartók, Béla, 3, 79base, 260

basis, 232beat, 74

theory, 21Beethoven, Ludwig van, 134, 149, 191,

192, 199, 236, 253Benjamin, Walter, 268Beran, Jan, 31Bernoulli, Johann, 17Bernstein, Leonard, 95Bernstein-Schröder

Theorem of -, 56Betta, Marco, 235Big Science in Music, 31bijective, 53binary, 86

relation, 57black

key, 175Blue Monk, 80body

gesture -, 282Boolean

algebra, 45boson, 195Boulanger, Nadia, 23Boulez, Pierre, 3, 24, 133, 183, 186, 231,

280Braitenberg, Valentin, 25, 31

Ccadence, 192–194cadential

sequence, 55Cage, John, 41, 235calculus, 268Campanella, Tommaso, 279canon

retrograde -, 12Cantor, Georg, 38cantus firmus (CF), 284cantus firmus (CF), 13, 44, 175Capella, Martianus, 279Carmina Burana, 80cartesian, 29, 116

dualism, 277product, 50, 62

of a family of sets, 65of functions, 55topology, 258

Index 307

categoriestheory of -, 28

category, 226, 249of gestures, 282of groups, 250of local compositions, 250of matrices, 250of modules, 250of sets, 250of topological spaces, 259opposite -, 250theory, 29, 30, 121

Cauchysequence, 100, 104

Cavaillés, Jean, 280CEMAMU, 23Cent (Ct), 110center, 219CERN, 6CF (cantus firmus), 13, 44, 175Châtelet, Gilles, 279chain

digraph, 123length of -, 123

rule, 264undirected -, 125

chamber a, 110changes

chord -, 182chant

Gregorian -, 273characteristic

function, 65cheironomy, 275Cherubini, Luigi, 11Chew, Elaine, 30chin, 275Chinese

drum notation, 276gestural music notation, 275language, 275

choiceaxiom of -, 44, 58

chord, 58changes, 182class, 28classification, 65, 174diminished

seventh -, 133

inversion, 76major -, 246mystic -, 179of pitch classes, 157prime form of -, 67, 158seventh -, 183transposition

class of a -, 157transposition-inversion

class of a -, 157Cicero, 279class

chord -, 28equivalence -, 58pitch -, 86, 156transposition -, 153

Classical Symphony, 237classification

chord -, 174of chords, 65

classifiersubobject -, 65

clefs, 12clivis, 275closed

interval, 258set, 257

closure, 262coda, 145codomain, 52, 249coefficient, 218cogitans

res -, 116Coltrane, John, 181combinatorial

musictheory, 16

combinatorics, 16commutative

diagram, 62group, 147monoid, 143ring, 46

commutativity, 44compact, 260

locally -, 261compact-open

topology, 261complementation, 157

308 Index

complexnumber, 113

complexitymusical -, 137

compositionlocal -, 250musical -, 64of functions, 53of graphs, 51of morphisms, 249

compoundmeter, 94

concatenationof paths, 123principle

general - -, 150conchoid, 238

equation, 238conductor

orchestral -, 277conjugation, 115consonance, 6, 18, 177, 228, 245, 262consonant

interval, 176continuity, 257continuous

function, 259contravariant

functor, 251convergence, 102, 104, 263coproduct, 63corollary, 40coset, 153

left -, 153right -, 154

counterpoint, 10, 175, 284florid -, 176worlds, 179

covariantfunctor, 251

creativeanalysis, 133, 183, 186

creativitymusical -, 151

crescendo, 208, 227, 259Ct (Cent), 110Cubharmonic, 167curve, 281

integral -, 271

cycle, 126, 143Euler -, 126Hamilton -, 126

cyclicgroup, 156

DD (discantus), 44, 175dactyl, 145Dahlhaus, Carl, 176dance, 276de Bruijn, Nicolaas Govert, 28de la Motte, Helga, 31decimal, 86

representation, 112definition, 40

of a gesture, 279degree, 55, 130, 193, 194

fundamental -, 196of a polynomial, 211parallel -, 131

Deliège, Célestin, 33delta

Kronecker -, 218denotator, 30dense, 262density, 104dependent

linearly -, 232derivative, 264Descartes, René, 8determinant, 222development, 145diagonal, 50diagram, 29

commutative -, 62diameter, 178diapason, 269diastematic

notation, 275dichotomy

Fux -, 178interval -, 177major -, 177strong -, 177

differenceof sets, 45

differentiablefunction, 264

Index 309

differentialquotient, 265

digraph, 122, 143chain -, 123final -, 122morphism, 123spatial -, 281

dimension, 232diminished

7th, 162seventh

chord, 133triad, 160, 179

diminuendo, 208direct sum, 226direct sums

universal property of -, 228directed

graph, 122Dirichlet, Peter Gustav Lejeune, 19discantus, 284discantus (D), 44, 175disjoint, 44dissonance, 177, 245dissonant

interval, 176distance

function, 260Euclidean - -, 260metrical -, 173

distributive, 219law, 205

distributivity, 45, 225Division

Theorem, 85, 211divisor, 213dodecaphonic

series, 75, 126, 148dodecaphonism, 53domain, 52, 249

integral -, 214dominant

7th, 161Dominant (D), 55, 145dualism

cartesian -, 277duality, 64Dubiel, Joseph, 33duration, 231

keyboard -, 53

EEckardt, Jason, 33Ehrenfels, Christian von, 261Eigen, Manfred, 31Eilenberg, Samuel, 29, 251Eimert, Herbert, 3, 28, 222Eine kleine Nachtmusik, 237electromagnetic

force, 195element

minimal -, 58neutral -, 45, 143order of -, 154

embeddingEuler -, 227

emptyset, 41

Encarnaçao, José, 31epi, 53equal temperament, 9equal-tempered

tuning, 247equation, 46, 89

conchoid -, 238Lagrange -, 154linear -, 223, 234Pythagorean -, 99

equipollent, 56equivalence

class, 58relation, 57

erhu, 108Escher

Theorem, 284Euclid, 85, 87Euclidean

distance function, 260length, 115

Eulercycle, 126embedding, 227number, 100space, 227

Euler space, 18Euler’s substitution theory, 262Euler, Leonhard, 17, 227, 231, 262Examples

310 Index

Mathematical, viiiMusical, viii

Exercise, Solution of -, ixExercises

Mathematical, viiiMusical, viii

exercisessolutions of the -, 289

experimentalmusic

theory, 191exponential, 64

function, 259exponentiation

of natural numbers, 83exposition, 145expressive

performance, 267research, 267

extensares -, 116

Fface

of a simplex, 129factorization

prime -, 214family

of elements, 66Fast Fourier Transform (FFT), 20, 117Ferneyhough, Brian, 97FFT (Fast Fourier Transform), 20, 117fiber

of a function, 64Fibonacci, 79

numbers, 3, 101field, 211

performance -, 270tempo-articulation -, 270vector -, 270

fifth, 176, 246interval, 75

Fifth Symphony, 80fifths

parallel -, 10sequence, 131sequence of -, 175

finaldigraph, 122

set, 61finite

group, 150set, 58, 75

fixpoint, 92group, 153

Flanagan, Tommy, 181flat �, 269florid

counterpoint, 176Flusser, Vilem, 279Fontana, Lucio, 239Footprints, 97forbidden

parallel fifths, 179force

electromagnetic, 195gravitational, 195strong, 195weak, 195

forgetfulfunctor, 251

form, 30linear -, 228sonata -, 145song -, 145

formula, 2Forte, Allen, 26, 27, 33founded, 72Fourier

Theorem, 19, 21Fourier, Jean-Baptiste Joseph, 19, 117fourth, 176, 246

interval, 75fourths

sequence of -, 175Fraenkel, Abraham, 40frame

of the performance, 270free

module, 232freezing

gestures, 275Frege, Gottlob, 280French

gesture theory, 280frequency, 110Fripertinger, Harald, 28function, 52, 280

Index 311

affine -, 259autocomplementarity -, 177characteristic -, 65continuous -, 259differentiable -, 264distance -, 260Euclidean distance -, 260exponential -, 259fiber of a -, 64gradus suavitatis -, 18identity -, 52logical -, 37periodic -, 19polynomial -, 208, 259Riemann -, 56sinusoidal -, 110theory, 55tonality -, 130trigonometric -, 259

function theoryRiemann -, 130

functionalgraph, 52

functionscomposition of -, 53

functorcontravariant -, 251covariant -, 251forgetful -, 251

functorial, 125, 133, 220, 226fundamental

degree, 196step, 192

Fuxdichotomy, 178

Fux, Johann Joseph, 176

GGötze, Heinz, 31Gaudí, Antoni, 50, 239Gauss

plane, 114Gauss, Carl Friedrich, 113general

concatenationprinciple, 150

Generalized Interval Systems (GIS), 29,152

geometric representation of

a simplex, 130geometry

algebraic -, 30metrical -, 173

gestalt, 261gestural

performance, 267gesture, 2, 29, 117, 273, 282

body, 282definition of a -, 279physical -, 277skeleton, 282symbolic -, 277tamed -, 281theory

Anglo-Saxon - -, 279French - -, 280

wild -, 281gestures

category of -, 282freezing -, 275morphism of -, 282philosophy of -, 279

Giant Steps, 181Giant’s Steps, 183GIS (Generalized Interval Systems), 29,

152Glasperlenspiel, 3, 32glissando, 100, 208, 227, 260global structure in music, 22gluon, 195golden

ratio, 101Gould, Glenn, 253gradus suavitatis, 228, 231, 262

function, 18Graeser, Wolfgang, 21, 26graph, 121, 124

directed -, 122functional -, 52inverse -, 50undirected -, 124

graphsmorphism of -, 125

gravitationalforce, 195

graviton, 195Gregorian

chant, 273

312 Index

musical figure, 275Gregorian notation, 274Grisey, Gérard, 3, 101Grothendieck, Alexander, 1, 30group, 147

abelian -, 147action, 28, 29, 152commutative -, 147cyclic -, 156finite -, 150fixpoint -, 153homomorphism, 147of permutations, 76opposite -, 148quotient -, 155symmetric -, 150theory, 28, 75torus -, 171trivial -, 150unitary -, 155, 156

Guérin, Michel, 279

HHába, Alois, 108Hamilton

cycle, 126Hammerklavier

Sonata, 133, 191, 199, 277, 285Hanslick, Eduard, 2, 40harmonic

band, 130, 139, 198retrograde

inversion, 183syntagmatics, 55syntax, 144

harmony, 58, 245jazz -, 183Riemann -, 130

Hascher, Xavier, 33Hauer, Josef Matthias, 3Hausdorff, 261head, 122Helmholtz resonator, 20Henry, Pierre, 24Herma, 46Hertz (Hz), 110Hesse, Hermann, 3, 32hexadecimal, 86hexameter, 145

Hindemith, Paul, 243Hofmann, Georg Rainer, 31homeomorphism, 259homogeneity, 225homogeneous, 219homomorphism

affine -, 230group -, 147linear -, 225ring -, 207

homotopytheory, 30

Honegger, Arthur, 23Horowitz, Vladimir, 253Husmann, Heinrich, 21hypergesture, 277Hz (Hertz), 110

Iideal, 102, 209

maximal -, 211principal -, 210

idempotency, 45identity, 143, 145, 147, 207, 225, 250,

259, 282function, 52matrix, 218morphism, 123

IFM (Institut für Grundlagenforschungin der Musik), 31

imaginarynumber, 113time, 116, 277unit, 114

improvisation, 284independent

linearly -, 232indeterminate, 206index, 154inequality

triangular -, 91, 98, 104infinity, 43initial

set, 61injective, 53Institut für Grundlagenforschung in der

Musik (IFM), 31integer, 89

negative -, 90

Index 313

positive -, 90integers

addition of -, 90multiplication of -, 91

integral, 270curve, 271domain, 214

interpretationtriadic -, 194

intersection, 44interval, 58, 62, 75, 90, 144, 152, 176

closed -, 258consonant -, 176dichotomy, 177dissonant -, 176fifth -, 75fourth -, 75major second -, 75major seventh -, 75major sixth -, 75major third -, 75minor second -, 75minor seventh -, 75minor sixth -, 75minor third -, 75open -, 258prime -, 75tritone -, 75, 232

intonation, 260, 265inverse

graph, 50relation, 57

inverse elementadditive -, 97multiplicative -, 97

inversion, 26, 28, 92, 153, 222chord -, 76retrograde -, 222

invertible, 147involution, 46IRCAM, 3, 24, 32, 33irreducible

polynomial, 213iso, 53isomorphism, 123, 145

JJacobian

matrix, 270

jazzharmony, 183

justtuning, 8, 111, 176, 215, 228, 241, 262

KKandisky, Wassily, 235Kant, Immanuel, 281Kendon, Adam, 279Kepler, Johannes, 101kernel, 154, 208key

black -, 175keyboard -, 53signature, 175white -, 175

keyboard, 111duration, 53key, 53note, 53

King, Emily, ixKircher, Athanasius, 12, 14Klein 4-group, 149, 154Klumpenhouwer

network, 122Klumpenhouwer net (K-net), 29Klumpenhouwer, Henry, 29, 122knowledge

science, 15Kronecker

delta, 218Kronecker, Leopold, 72, 73Kunst der Fuge, 21, 26

LL’essence du bleu, 133, 191La Bohème, 10Lagrange

equation, 154Lagrange, Joseph-Louis, 19Lakoff, George, 280language

Chinese -, 275law

distributive -, 205lazy

path, 123Le Corbusier, 23lead sheet, 182

314 Index

leadingvoice -, 198

leftcoset, 153

legato, 270Leibniz, Gottfried Wilhelm, 16Lemma

Steinitz -, 233Yoneda’s -, 252

lemma, 40length

Euclidean -, 115of a chain digraph, 123

Lewin, David, 26, 29, 122, 152lexicographic

ordering, 58, 59, 66, 157Leyton, Michael, 31Lichtenhahn, Ernst, 31Lied of dem Wasser zu singen, 145Ligeti, György, 235limit, 100linear

equation, 223, 234form, 228homomorphism, 225ordering, 94

of real numbers, 104relation, 58

linearlydependent, 232independent, 232

Lluis Puebla, Emilio, 33local

composition, 250locally

compact, 261logarithm, 109, 227, 259logic, 37logical

function, 37loudness, 231, 269Lullus, 16

MM.M. (Maelzel Metronome), 265Mac Lane, Sounders, 29, 251Maelzel Metronome (M.M.), 265major

7th, 161

chord, 246dichotomy, 177scale, 243second, 176, 246seventh, 176, 246sixth, 176, 246third, 176, 246triad, 8, 160, 179

major secondinterval, 75

major seventhinterval, 75

major sixthinterval, 75

major thirdinterval, 75

Mannone, Maria, vii, 33, 167, 238marche funèbre, 39mathematical

music theory, 29, 144precision, 268technique, 10theory

of gestures, 273mathematics

statements in -, 40matrices

product of -, 218sum of -, 218

matrix, 11, 218identity -, 218Jacobian -, 270transposition, 218

Matterhorn, 253Max, 25maximal

ideal, 211Mazzola, Christina, 31Mazzola, Guerino, vii, 24, 29, 30, 33, 34,

76, 121, 133, 183, 234, 276MCM (Society of Mathematics and

Computation in Music), 30, 33, 34McNeil, David, 279Mead, Andrew, 33measure, 74, 94mechanism

modulation -, 193, 195melody, 123mental

Index 315

space, 116Mersenne, Père, 9Messiaen

scale, 144, 185Messiaen, Olivier, 3, 23meter

compound -, 94simple -, 94

metricaldistance function, 173geometry, 173

Mi votu e mi rivotu, 95Michelangelo, 273microtiming, 265microtonal

tuning, 108MIDI, 74minimal

element, 58minor

7th, 162major

7th, 162scale, 243second, 176seventh, 176sixth, 176third, 176, 246triad, 8, 161, 179

minor secondinterval, 75

minor seventhinterval, 75

minor sixthinterval, 75

minor thirdinterval, 75

modulation, 133, 145, 191, 192, 195, 246mechanism, 193, 195quantized -, 196quantum, 196Theorem, 197theory, 58

modulator, 196module, 225

free -, 232generated by S, 226quotient -, 229theory, 30

Moebiusband, 131, 135

mono, 53monochord, 6, 52monoid, 29, 143

algebra, 206commutative -, 143morphism, 145word -, 143

monotonyadditive -, 97multiplicative -, 98of addition, 104of multiplication, 104

morphism, 249digraph -, 123identity -, 123monoid -, 145of gestures, 282of graphs, 125simplicial -, 132

morphismscomposition of -, 249

Morris, Robert, 26–28, 33most packed to the left, 67, 158motive, 261motivic

similarity, 261work, 133zigzag, 133

Mozart, Wolfgang Amadeus, 3, 17, 192,236

multiplicationmonotony of -, 104of integers, 91of natural numbers, 83of rational numbers, 96on Zn, 173scalar -, 218, 225table, 147

multiplicativeinverse element, 97monotony, 98neutral element, 97

Murail, Tristan, 3music

atonal -, 27Chinese gestural - notation, 275global structure in -, 22

316 Index

quartertone -, 156serial -, 55symbolic -, 46technology, 2theory, 109, 150, 171, 213, 230

combinatorial -, 16experimental - -, 191mathematical - -, 29, 144

musicalacoustics, 269complexity, 137composition, 64creativity, 151dice game, 17figure

Gregorian - -, 275precision, 268

Musique Concrète, 24Muzzulini, Daniel, 31Mycenae Alpha, 235mystic

chord, 179

NNúñez, Rafael, 280name

note -, 268Nattiez, Jean-Jacques, 33natural, 125

adjunction, 64number, 72, 73transformation, 252

natural numbersaddition of -, 83exponentiation of -, 83multiplication of -, 83

Nauert, Paul, 33negative

integer, 90neighborhood

open -, 258nerve, 129, 134, 198network

Klumpenhouwer -, 122neume, 273neutral element, 45, 143

additive -, 97multiplicative -, 97

neutralization, 192

Nicolas, François, 25Nietzsche, Friedrich, 23Noll, Thomas, 30, 32, 33normal

subgroup, 155normal form

adic -, 86notation

Chinese drum -, 276Chinese gestural music notation, 275diastematic -, 275Gregorian, 274Western musical -, 275

notekeyboard -, 53name, 268

Novalis, vNugent, Ronan, ixnumber

complex -, 113Euler -, 100imaginary -, 113natural -, 72, 73ordinal-, 72prime -, 86, 91, 213, 228rational -, 93real -, 100

numbersFibonacci -, 101

Oobject, 249

addressed -, 231octave, 246octaves

parallel -, 10One for Ben, 81onset, 14, 52, 53, 149op. 17.2 (Webern), 54op. 30 (Webern), 54open

ball, 261interval, 258neighborhood, 258set, 257

OpenMusic, 25, 46operator

performance -, 272opposite

Index 317

category, 250group, 148

orbifold, 76orbit, 152orchestral

conductor, 277order of

element, 154ordered

pair, 49set, 26, 49

orderingArchimedean -, 98lexicographic -, 58, 59, 66, 157linear -, 94

ordinalnumber, 72

orientation, 131overtone, 19, 117, 121

PPólya, George, 28Pa (pascal), 110pair

ordered -, 49set, 43

Palestrina,Giovanni Pietro Aloisio Sante da,176

palindrome, 14Pang, Jin, 108Pang, Yan, vii, 33paradigmatic, 261parallel

degree, 131fifths, 10

forbidden -, 179octaves, 10

partial, 19, 21, 117relation, 58

pascal (Pa), 110path, 123

lazy -, 123paths

concatenaton of -, 123Peano

axioms, 74Peano, Giuseppe, 73Peck, Robert, 34

Penrose, Roger, 115pentagramma, 274pentatonic

scale, 247perception

pitch -, 110Performance

rubette, 260performance, 117, 208, 265, 267

expressive -, 267field, 270frame of the -, 270gestural -, 267operator, 272transformation, 270

period, 112periodic

function, 19periodicity, 110Perle, George, 26permutation, 11, 26, 126permutations

group of -, 76pes, 275Petsche, Hellmuth, 31philosophy

of gestures, 279photon, 195physical

gesture, 277space, 116time, 116, 265

pianoquarter-tone -, 108

pitch, 52, 54, 109, 149, 231class, 86, 156classes

chord of - -, 157multiplication, 25perception, 110

pitch-classset, 26transformation, 231

pizzicato, 269plane

Gauss -, 114Plato, 5, 279podatus, 275poetry, 145

318 Index

polynomial, 207algebra, 206degree of a -, 211function, 208, 259irreducible -, 213

polyrhythm, 97portamento, 278positive

integer, 90real number, 103

Posner, Roland, 31Pousseur, Henri, 3powers, 43pre-semiotic, 280precision

mathematical -, 268musical -, 268

pressure, 110prime, 176

factorization, 214interval, 75number, 86, 91, 213, 228

prime form ofchord, 67, 158

principalideal, 210

productcartesian -, 50, 62cartesian - of a family of sets, 65of matrices, 218

projection, 50Prokofiev, Sergei, 237property

Archimedean -, 104universal -, 61

proposition, 40psychoacoustics, 110Ptolemy, 245Puccini, Giacomo, 10punctum, 275punctus contra punctum, 284pure

set, 38Pythagoras, 1, 5, 111Pythagorean

equation, 99tuning, 8, 176, 245, 247

Qquantized

modulation, 196quantum

modulation -, 196quarter tone

music, 156quarter-tone

piano, 108quilisma, 275Quintilian, 279quotient

differential -, 265group, 155module, 229ring, 210

RRahn, John, 26, 27, 33ratio

golden -, 101rational

number, 93numbers

addition of -, 96multiplication of -, 96

realnumber, 100

positive -, 103numbers

linear ordering of - -, 104time, 277

recapitulation, 145reflection, 221reflexive

relation, 57reflexivity, 56relation

antisymmetric -, 57binary -, 57equivalence -, 57inverse -, 57linear -, 58partial -, 58reflexive -, 57symmetric -, 57total -, 57transitive -, 57

relative

Index 319

topology, 259representation

adic -, 111decimal -, 112

rescogitans, 116extensa, 116

researchexpressive performance -, 267

resonatorHelmholtz -, 20

retrograde, 149canon, 12inversion, 149, 222

harmonic - -, 183rhythm, 145Riemann

function, 56function theory, 130harmony, 130

Riemann, Bernhard, 55Riemann, Hugo, 55right

coset, 154ring, 205

commutative -, 46homomorphism, 207quotient -, 210

Riotte, André, 33ritardando, 208root

nth -, 107rotation, 221roughness, 21rubette

Performance -, 260Rubik cube, 167Rubik, Ernö, 167rule

chain -, 264Russell’s

antinomy -, 56Russell’s antinomy, 40Russell, Bertrand, 39

SSagrada Família, 50, 239Saint-Victor, Hugues de, 279, 281Saussure, Ferdinand de, 280

scalarmultiplication, 218, 225

scale, 144C-major -, 175major -, 243Messiaen -, 144, 185minor -, 243pentatonic -, 247whole-tone -, 144word, 144

scandicus, 275Schönberg, Arnold, 3, 26, 53, 55, 191,

192, 197, 202Schaeffer, Pierre, 24Schenker, Heinrich, 192Schmitt, Jean-Claude, 279Schmitt-Biggeman, Wilhelm, 16Schubert, Franz, 145Sciarrino, Salvatore, 235science

knowledge -, 15score, 116, 268Scriabin, Alexander, 179Sebestény, Péter, 167second

major -, 176, 246minor -, 176

semiotic, 279semitone, 110, 144, 156

step, 99sequence

cadential -, 55Cauchy -, 100, 104fifths -, 131of length n, 76zero -, 102

sequence offifths, 175fourths, 175

serialmusic, 55

serialism, 231series

12-tone -, 53dodecaphonic -, 75, 126, 148

setclosed -, 257complex, 27

theory, 27

320 Index

difference, 45empty -, 41final -, 61finite -, 58, 75initial -, 61open -, 257ordered -, 26, 49pair -, 43pitch-class -, 26pure -, 38theory, 38

seventhchord, 183major -, 176, 246minor -, 176

sharp �, 269shearing, 221Shorter, Wayne, 97sign, 273, 279signature

key -, 175time -, 94

signification, 50similarity

motivic, 261simple

meter, 94simplex, 129

face of a -, 129geometric representation of a -, 130

simplicialmorphism, 132

simplytransitive

action, 152sinusoidal

function, 110sixth

major -, 176, 246minor -, 176

skeletongesture -, 282

snail, 239Society of Mathematics and Compu-

tation in Music (MCM), 30, 33,34

solutionapproximate -, 99

Solution of Exercise, ix

solutionsof the exercises, 289

sonata, 133Appassionata -, 253form, 145Hammerklavier -, 133, 191, 199, 277,

285song

form, 145Song of Yi II—A Se, 124Song, Mingzhu, 80, 124sonification, 234sort, 40sound

technology, 117space

Euler -, 18, 227mental -, 116physical -, 116vector -, 232

span, 178spatial

digraph, 281species

fifth -, 176first -, 176fourth -, 176second -, 176third -, 176

Spectralism, 3spiral, 238spondee, 145staccato, 270Stange-Elbe, Joachim, 32statements in mathematics, 40Steibelt, Daniel, 149Steinitz

Lemma, 233step

fundamental -, 192semitone -, 99

Stockhausen, Karlheinz, 3, 101, 222Stolberg, Leopold, 145Strauss, Johann, 95Stravinsky, Igor, 95string

theory, 246vibrating -, 2, 246

strong

Index 321

dichotomy, 177force, 195

structuretheory, 267

Structures pour deux pianos, 231Stucki, Peter, 31subbase, 260Subdominant (S), 55, 145subgroup, 147

generated by S, 148normal -, 155

submonoid, 146generated by S, 146

subobjectclassifier, 65

subring, 205successor, 43, 74sum

direct, 226of matrices, 218

supersummativity, 261supremum, 105surjective, 53syllable, 145symbolic

gesture, 277music, 46time, 94, 265

symmetricgroup, 150relation, 57

symmetry, 21, 56, 183on Z12, 173

syntagmaticsharmonic -, 55

syntaxharmonic -, 144

Synthesis, 30, 234

Ttable

multiplication -, 147tacet, 41tai chi, 276tail, 122tamed

gesture, 281technique

mathematical -, 10

technologymusic -, 2sound -, 117

tempo, 94, 208, 265tempo-articulation

field, 270Terhardt, Ernst, 31tetractys, 6, 8, 111, 245tetrade, 183tetragramma, 274tetrahedron, 130, 132The Rite of Spring, 95The Scene of Sichuan Opera, 79Theodul, 253Theorem

Division -, 85, 211Escher -, 284Fourier -, 19, 21modulation -, 197of Bernstein-Schröder, 56

theorem, 40theory

Anglo-Saxon gesture -, 279beat -, 21category -, 29, 30, 121Euler’s substitution -, 262French gesture -, 280function -, 55group -, 75homotopy -, 30mathematical - of gestures, 273mathematical music -, 144modulation -, 58module -, 30music -, 109, 150, 171, 213, 230of categories, 28set -, 38string -, 246structure -, 267topos -, 30transformational -, 29twelve-tone -, 26

thirdmajor -, 176, 246minor -, 176, 246torus, 171

Third String Quartet, 97thirds-divide subtract-add, 247Thirring, Walter, 31

322 Index

timbre, 269time

imaginary -, 116, 277physical -, 116, 265real -, 277signature, 94symbolic -, 94, 265

tonality, 55, 191, 193, 246function, 130

Tonic (T), 55, 145topology, 257

cartesian product -, 258compact-open -, 261generated by S, 260relative -, 259

topostheory, 30

torusgroup, 171third -, 171

totalrelation, 57

trajectory, 193transformation

natural -, 252performance -, 270pitch-class -, 231

transformationaltheory, 29

transitive, 72action, 152relation, 57

transitivity, 56transposability, 261transposition, 28, 92, 153

class, 153class of a chord, 157matrix -, 218

transposition-inversionclass of a chord, 157

triad, 76augmented -, 134, 160, 179, 185diminished -, 160, 179major -, 8, 160, 179minor -, 8, 161, 179

triadicinterpretation, 194

triangularinequality, 91, 98, 104

trigonometricfunction, 259

tritone, 176interval, 75, 232

trivialgroup, 150

tuning, 10912-tempered -, 99, 108, 111, 121, 242equal-tempered -, 247just -, 8, 111, 176, 215, 228, 241, 262microtonal -, 108Pythagorean -, 8, 176, 245, 247systems

Western - -, 227, 241tuplet, 96Tyner, McCoy, 182Tzu, Kuan, 247

Uundirected

chain, 125graph, 124

union, 42unit

imaginary -, 114unitary

group, 155, 156universal

property, 61of cartesian product of a family of

sets, 66of direct sums, 228of word monoids, 145

UPIC, 24, 234

VValéry, Paul, 280value

absolute -, 90, 104vector, 225

field, 270space, 232

Verdi, Giuseppe, 11Verdi, Luigi, 33vertex, 122vibrating

string, 2, 246Vicentino, Nicola, 108Vieru, Anatol, 28

Index 323

Villa-Lobos, Hector, 235virga, 275Virgil, 145visualization, 50, 234Vogel, Martin, 241, 243voice

leading, 76, 198von Hahn, Walther, 31von Helmholtz, Hermann Ludwig

Ferdinand, 20von Karajan, Herbert, 31von Webern, Anton, 54Vuza, Dan Tudor, 28

Wwalk, 125Walton, Cedar, 181Walzer und Schleifer mit zwei Würfeln

zu componieren..., 166weak

force, 195Weber-Fechner law, 110well-ordering, 58, 73, 74West Side Story, 95Western

musicalnotation, 275

tuning systems, 227, 241white

key, 175whole-tone

scale, 144Whymper, Edward, 253

Wieser, Heinz-Gregor, 31wild

gesture, 281word

monoid, 143universal property of - -, 145

scale -, 144work

motivic -, 133world-sheet, 278, 285, 286worlds

counterpoint -, 179

XXenakis, Iannis, 23, 28, 45, 46, 234

YYoneda’s

Lemma, 252Yoneda, Nobuo, 252

ZZarlino, Gioseffo, 8, 10, 243, 245Zermelo, Ernst, 40, 58Zermelo-Fraenkel-Choice (ZFC), 40, 72zero

sequence, 102ZFC

axioms, 41ZFC (Zermelo-Fraenkel-Choice), 40, 72Zhu, Zaiyu, 8, 99, 247zigzag

motivic -, 133