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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