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Index

Bold page numbers indicate a main entry. = 1 2π θ θ θ am π 0 cos(m ) f ( )d ,40 amplifier, 14, 268 AAC, 250 response, 53 Aaron, Pietro (ca. 1480–1550) amplitude, 6, 7, 17, 25, 26, 27 meantone temperament, 186 instantaneous, 90 abelian group, 325 modulation, 276 abs (CSound), 307 peak, 21, 25 absolute RMS, 25 integrability, 74 AMSLATEX, xiii value, 49 analogue acceleration, 19 modelling synthesizer, 71 acoustic signal, 245 law, Ohm’s, 11 synthesizer, 71, 268, 385 pressure, 107, 119, 142 angular velocity ω = 2πν, 21, 33 acoustics, 117, 137, 142 animals, hearing range of, 13 nonlinear, 71 antanairesis, 163 additive synthesis, 268 anvil, 8 ADSR envelope, 266 apical, 9 adufe, 117 apotom¯e, Pythagorean, 164, 181, 369 Aeolian, 379 approximation aeolsklavier, 91 rational, 214–223 aerophones, 91 Archytas of Tarentum (428–365 bce), 203 Africa, 131 arctan function, 218 Agamemnon, 313 area Agricola, Martin (1486–1556) density, 112, 133 monochord, 175 in polar coordinates, 75 AIFF sound file, 249, 291 Aristotle (384–322 bce), 4 air, 5 Aristoxenus (ca. 364–304 bce), 208 bulk modulus, 108, 142 arithmetic, clock, 333 algorithm Aron, see Aaron DX7, 285, 288 artifacts, 255 Euclid’s, 163, 223, 336 Ascending and Descending (Escher), 158 inductive, 215 ascending node, 222 Karplus–Strong, 273, 291 ascii, 292, 295 aliasing, 254 associative law, 87, 325 alpha scale (Carlos), 207, 232, astronomy, 147, 148 369 atonal music, 205, 348 alternating group, 355 attack, 266 aluminium, 129 AU sound file, 249 Alves, Bill, 381 auditory canal, 7 AM radio, 276 augmented triad, 342 amplification, 258 aulos, 171

393

394 Index

auricle, 7 Berg, Alban (1885–1935), 320, 348 authentic mode, 379 Bernoulli, Daniel (1700–1782), 37, 126 auxiliary equation, 28 solution of wave equation, 98 AVI movie file, 248 Bessel, F. W. (1784–1846) equation, 64 = 1 2π θ θ θ function, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, bm π 0 sin(m ) f ( )d ,40 Babbitt, Milton, 348 69, 282, 361 Bach, J. S. (1685–1750) computation of, 363 Italian Concerto, etc. (Rousset), 198, 381 graph of, 60 Jesu, der du meine Seele, 184 hyperbolic, 134 Musical Offering, 314 Neumann’s, 65 Organ Music (Fagius), 198, 381 power series for, 65 Partitia no. 5, Gigue, 183 zeros of, 114, 361 Toccata and Fugue in D, 312 horn, 121 Toccata in F minor, 192 beta scale (Carlos), 207, 232, 369 Well Tempered Clavier, 189 Bewitched, The (Partch), 385 Bach, P. D. Q. (1807–1742)?, 322, 360 Biber, Heinrich Ignaz Franz von (1644–1704), 381 Badings, Henk, 229 bifurcation, 288 bagel, 226 bijective function, 332 bagpipe tuning, 212 binary representation, 245 balafon, 91 birnd (CSound), 307 balance, 8 Blackwood, Easley (1933–), 228, 381 Balinese gamelan, 207 Blake, William (1757–1827), 319 bamboo marimba (Partch), 210 blend factor, 274 band pass filter, 151 block bandwidth, 35 diagram (DX7), 282 critical, 150, 151 periodicity, 237, 240, 344 Barca, Alessandro, 194 Bôcher, Maxime (1867–1918), 53 Barker, Andrew, 208 Boethius, Anicius Manlius Severinus Barlow, Clarence, 228, 381 (ca. 480–524 ce), 171 Barnes, John, 196 Bohlen–Pierce scale, 162, 233, 368 baroque music, 181 Bologna State Museum, 230 Bartók, Béla (1881–1945) Bombelli, Rafael (1526–1572), 215 Fifth String Quartet, 312 Boo I (Partch), 210 Music for Strings, Percussion and Celeste, 319 Bosanquet, Robert H. M. (1841–1912), 224 basal end of cochlea, 9 bottle, plucked, 275 base of natural logarithms, 218 Boulanger, Nadia (1887–1979), 233 basilar membrane, 9, 19, 151 Boulanger, Richard Charles (1956–), 236, 308 basis for a lattice, 238 boundary conditions, 96 bass bowed string, 49, 103 clef, 367, 375 parabolic envelope, 104 singer, 15 BP intervals, 234, 235 bassoon, 304 BP-just scale, 235 baud rate, MIDI, 251 brass, 120, 288 bc -l, 364 brightness, 268 beam equation, Euler–Bernoulli, 126 Brombaugh, John (organ builder), 189 beat, dead, 29 Brouncker, William (1620–1684), 218 beats, 23, 200, 368 Brown, Colin, 180 Beauty in the Beast (Carlos), 207 brown noise, 81 Beethoven, Ludwig van (1770–1827), 205, 312, 384 Brownian motion, 47, 81 Békésy, Georg von (1899–1972), 11 Brun, Viggo (1885–1978), 223 bel (= 10 dB), 14 bulk modulus, 108, 142 bell, 138 bullroarer, 91 (FM & CSound), 300 Burnside, William (1852–1927) change ringing, 329 lemma, 345, 356 Chinese, 139 Buxtehude, Dietrich (ca. 1637–1707), 382 tubular, 122 Byrd, William (1543–1623), 188, 321, 382 Bendeler, P. (1654–1709) temperaments, 192 C1 function, 47 bending moment, 124 C++ Benedetti, Giovanni Battista (1530–1590), 171 Bessel calculator, 363

Index 395

C programming language, 291 Chowning, John, 278 calculus Chowning, Maureen, 236 vector, 118 chromatic calm temperaments, 199 genus, 203 Calvin and Hobbes, scale, 207 campanology, 329 church modes, medieval, 379 canal circle of fifths, 164, 337 auditory, 7 circular motion, 26 semicircular, 8 circulating temperament, 189 Cancrizans, 314 clarinet, 15, 41, 91, 110, 234, 289, 304 canon classes, pitch, 300, 334 retrograde/crab, 314 classical harmony, 181 capacitor, 277 clef, 367, 374 cards, shuffling, 359 clipping, 292 Carlos, Wendy (1939–), 207, 212, 231, 369, 382 clock arithmetic, 333 Carpenter, Charles, 237, 383 closed carrier frequency, 277 bounded region, 118 carry feature (CSound), 299 interval, 47, 50 Carter, Elliott, 348 tube, 108 Cartesian cochlea, 8, 13, 19 product, 339 collision frequency, 5 cascade modulation, 285 coloratura soprano, 236 Cataldi, P. A. (1548–1623), 215 colour, 4, 146 cathode ray tube, 53 columnella, 9 Cauchy, Augustin Louis (1789–1857) combination tone, 155 integral formula, 67 comma, 163, 164, 166, 168, 171, 172, 181 principal value, 74 BP 7/3, 235, 369 Caus, Salomon de (ca. 1576–1626) ditonic, 163 monochord, 175 notation (superscript), 172 causality, 87 of Didymus, 168 cello, 186 ordinary, 168 centroid, 125 Ptolemaic, 168 cents, 16, 166, 369 Pythagorean, 163, 166, 190, 201, 223, 225, 369 cepstrum, 88 scale of, 226 Cesàro, Ernesto (1859–1906) septimal, 171, 369 sum, 47, 51, 55 syntonic, 168, 369, 377 chain commutative ossicular, 8 law, 87, 325 rule, multivariable, 96 ring, 355 Chalmers, John (1940–), 212 compact disc, 245 Chambonnières, Jacques Champion de Alves, 214, 381 (ca. 1601–1672), 383, 385 Bach/Christophe Rousset, 198, 381 change ringing, 329 Bach/Hans Fagius, 198, 381 Chao Jung-Tze, 217 Between the Keys, 228, 381 chaos, 47, 288, 291 Biber, 188, 381 character theory, 261, 337 Blackwood, 228, 381 Chaw, Ousainou, 103 Buxtehude, 382 Chebyshev, Pafnuti L. (1821–1894) Byrd, 321, 382 polynomials, 309 Carlos, 207, 214, 382 chimaeric sounds, 90 Carpenter, 237, 383 Chinese Chambonnières, 188, 383 bells, 139 Chapman, 189, 383 Lü scale, 211 Chemillier, 324, 383 Chladni, Ernst F. F. (1756–1827) Computer Music Journal, 381 drawings, 137 Cook, 237, 383 patterns, 115 d’Anglebert, 189, 383 Chopin, Frédéric Fran¸cois (1810–1849), 205, 384 Froberger, 189 Etude´ , Op. 25 No. 10, 338 Harrison, 198, 214, 383 Waltz, Op. 34 No. 2, 314 Harvey, 384 chordophones, 91 Haverstick, 228, 384 chorus, 251, 268 Haydn, 321, 384

396 Index

compact disc (cont.) cos, cosh, cosinv (CSound), 307 Houtsma, Rossing & Wagenaars, 160, 384 coset, 239, 342 Johnson, 384 representatives, 237, 343 Katahn/Foote, 189, 198, 384 counting problems, 353 Lagacé, Wellesley organ, 189, 384 Couperin, Fran¸cois (1668–1733), 198, 385 Machaut/Hilliard Ensemble, 165, 322, 384 Couperin, Louis (1626–1661), 385 Mathews & Pierce, 237, 384 coupled oscillators, 222 Microtonal Works, 384 crab canon, 314 Parmentier, 189, 385 critical bandwidth, 150, 151 Partch, 214, 385 critically damped system, 30 Rich, 214, 385 crystallography, 318 Sethares, 155, 385 Cscore, 307 Tomita/Mussorgsky, 266, 385 CSound, 291, 381 Winkelman, 189, 198, 385 AV, 308 Xentonality (Sethares), 228, 385 The CSound book, 308 complementary function, 32 curvature, radius of, 125 completeness, 119 cycle complex per second, 13 analysis, 67 index, 354 exponential, 54 notation, 327 numbers, 33, 54, 223 of fifths, 164 Composers Recordings Inc., 385 cyclic group, 312, 335 compression, data, 249 Cygwin, 364 concert hall acoustics, 142 cylinder of thirds and fifths, 186 concert pitch, 21 cymbal, 13, 91, 137 concertina, 204 conch shell, 91 d’Alembert, Jean-le-Rond (1717–1783), 37, 94, 143 concha, 7 d’Anglebert, Jean-Henri (1635–1691), 383, 385 configuration, 353 damped harmonic motion, 17, 28 counting series, 355 DAT, 245 congruence, 334 data conical tube, 110 compression, 249 conjunction of tetrachords, 202 transmission, MIDI, 251 consonance, 144, 146, 148 DATA chunk, 249 continued fractions, 214, 219, 230 dB, 14 for e, 218 dB SPL, 14 / for log2(3 2), 223 dBA, 14 / / / for log2(3 2) log2(5 4), 232 dead beat, 29 / for log3(7 3), 234 Debussy, Claude (1862–1918) for √π, 216 Reverieˆ , 320 for 2, 222 decay, 266 periodic, 222 of Fourier coefficients, 50 continued subtraction, 163 stretching, 275 continuous decibels, 14 dependence on initial conditions, 102 deciem (bell), 138 function, 46 delay, 257, 268, 271, 273 nowhere differentiable, 47 t, time between samples, 251 piecewise, 50 δ(t), Dirac delta function, 82, 251 control rate (CSound), 292 δs(t), sampling function, 252 convergence density, 93, 112, 124, 133 mean square, 48 derivative, 46 pointwise, 51, 53 partial, 92 uniform, 40, 42, 51, 53 descending node, 222 convergents, 217, 234 determinant, 239 convex drums, 120 Deutsch, Diana, 159 convolution, 86, 259 diaschisma, 171, 174, 369 convolve, 86 diatonic Cooley and Tukey, 263, 264 genus, 203 coordinates, polar, 75, 80, 113 syntonic scale, 202 Cordier, Serge, 201 Didymus ho mousicos (1st c. bce), 168 Corti, organ of, 10 comma of, 168

Index 397

diesis, 163, 171, 181, 225, 369 drum, 91, 112, 144, 274, 290 BP-minor, 235, 369 convex, 120 great, 171, 186, 197, 369 ear, 8 difference tones, 155 hearing the shape of, 119 differentiable periodic function, 50 kettle, 115 differential equation, 17, 19, 64 square, 117 linear second order, 28 wood (FM & CSound), 303 partial, 71, 92, 112 DuBois-Reymond, Paul (1831–1889), 47 digital Dufay, Guillaume (ca. 1400–1474), 204 audio tape (DAT), 245 Duke of Bedford, 204 delay, 257 Dunstable, John (ca. 1390–1453), 204 filters, 88, 257 Duphly, Jacques (1715–1789), 198, 385 music, 245 duration, 6, 7 representation of sound, 245 DX7, Yamaha, 280 signals, 245 dynamic friction, 49 synthesizer, 153, 207 dihedral group, 318, 340 E (Young’s modulus), 125 Diliges Dominum (Byrd), 321 e, continued fraction for, 218 diminished e (identity), 325 seventh chord, 342 ear, 7, 368 triad, 182, 379 drum, 8 diode, 277, 278 eccentricity of ellipse, 68 Dirac, Paul Adrien Maurice (1902–1984) echo, 268 delta function, 82, 251 eclipse, ecliptic, 222 direct product, 339 effective length, 110 Dirichlet, Peter Gustav Lejeune (1805–1859), effects unit, 268 46, 221 eigenfunction, 118 kernel, 57 eigenvalues, 117, 118 spectrum, 119 Eitz, Carl (1848–1924) disc, compact, 245 notation, 172, 235 discrete Fourier transform, 261, 337 elasticity, longitudinal, 125 discriminant, 29 electric guitar, 92 discrimination, limit of, 16 electroencephalogram, 13 disjunction of tetrachords, 202 electromagnetic wave, 6 dispfft, display (CSound), Electronic Musician (magazine), 308, 309 306 electrophones, 91 displacement, 107 Elektra (Strauss), 313 dissonance, 144, 146, 148 element, 324 dissonant octave, 153 elementary divisors, 345 distortion range variable, 71 11-limit, 211 distribution, 22, 84 eleventh harmonic, 370 Fourier transform of, 85 elliptic orbit, 68 tempered, 84 Ellis, Alexander J. (1814–1890), 166, 200, distributive law, 87 368 dithering, 246, 247 end correction, 110 ditonic comma, 163 energy density, 80 divergence theorem, 118 enharmonic divisors, elementary, 345 genus, 203 domain, fundamental, 237 notes, 164 dominant, 378 entropy, 250 seventh, 170 enumeration theorem, Pólya’s, 353 Doppelganger¨ (J. A. Lyndon), 317 envelope, 266, 268, 297, 308 Dorian, 379 epimorphism, 332 tetrachord, 202 equal temperament, 190, 199, 214, 368, 374 dot notation, 19 Cordier’s, for piano, 201 double equation angle formula, 25 auxiliary, 28 flat, 164 Bessel’s, 64 integral, 75 differential, 17, 19, 64 sharp, 164 partial differential, 71, 92, 112 draconic month, 222 quadratic, 29, 222

398 Index

equation (cont.) finalis, 379 Sturm–Liouville, 121 finality, 182 wave, 92, 108 first isomorphism theorem, 345 Webster’s horn, 121 fish, proving the existence of, 81 equilibrium position, 18 Fisk organ, Wellesley, 189 equivalence, 239 5-limit, 211 octave, 146, 237, 333 fixed point, 327 pitch class sets, 348 flare parameter, 121 Erlangen monochord, 174 flat, double, 164 Erlich, Paul, xiii, 213, 236, 242 Fletcher–Munson curves, 15, 250 error, mean square, 48, 188 flute, 91, 108, 111, 289 escape (diesis), 171 FM Escher, Maurits Cornelius (1898–1972) instruments in CSound, 300 Ascending and Descending, 158 radio, 276 ethnomusicology, 91 synthesis, 60, 68, 278, 280, 291, 298 Euclid (ca. 330–275 bce) focus of ellipse, 68 algorithm, 163, 223, 336 focusing of sound, 7 Euler, Leonhard (1707–1783), 37, 47 Fogliano, Lodovico (late 1400s–ca. 1539) Bernoulli beam equation, 126 monochord, 175 continued fraction for e, 218 Fokker, Adriaan D. (1887–1972), 229, 237 joke, 256 folk music, 181 monochord, 178, 240 British, 203 phi function, 336, 354 Foote, Edward (piano technician), 384 even function, 44 force, shearing, 123 expansion, Laurent, 67 forced harmonic motion, 31 exponential form, 339 exp (CSound), 307 formants, 89 function, complex, 54 FORMAT chunk, 248 interpolation, 301 formula extension, 125 Cauchy’s integral, 67 extraduction, 359 double angle, 25 Forte, Allen, 348 ∞ −2πiνt fˆ(ν) = −∞ f (t)e dt,73 Forty-Eight Preludes and Fugues (J. S. Bach), 189 Fagius, Hans (1951–), 198, 381 forty-three tone scale, 211 Farey, John (1766–1826), 200 four group, Klein, 340 fast Fourier transform, 263, 306 Fourier, Jean Baptiste Joseph, Baron de Faux Bourdon, 204 (1768–1830), 37 Fay, R., 13 coefficients, 38, 40 feedback, 68, 287 bounded, 49 Fejér, Lipot (1880–1959), 47 rapid decay of, 50 kernel, 56 series, 21, 36,58 fenestra transform, 73 ovalis, 10 discrete, 261, 337 rotunda, 10 fast, 263 FFT, 263 of f (t), 75 FFTea, 76 of a distribution, 85 Fibonacci (= Leonardo of Pisa, ca. 1180–1250) fourth, perfect, 152, 161, 167, 202 series, 222 frac (CSound), 307 fifteenth harmonic, 371 fractal fifth waveform, 47 circle/cycle of, 164, 337 fractions, 214 harmonic, 167, 370 continued, 214, 219, 230 perfect, 146, 152, 227, 375 partial, 222, 260 Cordier’s equal temperament, 201 Frazer, J., 161 sequence of, 162 Frederick the Great, 314 spiral of, 164 French horn, 91 fifty-three tone scale, 224 French Revolution, 38 filter, 87, 268 frequency, 6, 7, 145, 149, 155 band pass, 151 chart, 367 digital, 88, 257 cochlea, 10 low pass, 245, 255 collision, 5

Index 399

combination tone, 155 Gaudi (Rich), 385 combined, 24 Gauss, Johann Carl Friedrich (1777–1855), 58, Hertz, 13 220, 222 instantaneous, 89 Gaussian integers, 223 limit of discrimination, 16 GEN05 (CSound), 301 missing fundamental, 157 GEN07 (CSound), 297 modulation, 276 GEN10 (CSound), 294 multiples of fundamental, 37 generalized nominal, 117 function, 84 Nyquist, 254 keyboard harmonium, 224 octave, 20 generators piano strings, 25 for a group, 335 ratio, 166 for a sublattice, 239 resonant, 18, 34, 35 genus, 202 sine wave, 21 geometric series, 57 spectrum, 14, 80 geometry, 147 standard, 21 Germain, Sophie (1776–1831), 133 friction, 28, 49 Gibbs, Josiah Willard (1839–1903) frieze patterns, 318 phenomenon, 42, 50 Froberger, Johann Jakob (1616–1667), 198, 383, 385 Gibelius’ monochord, 186 Fubini, Eugene (1913–1997) Gilbert and Sullivan, 36 solutions, 71 gitangi, 323 Fubini, Guido (1879–1943), 71 Glareanus, 379 function, 332 glide reflection, 314 arctan, 218 global variables (CSound), 297 Bessel, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, golden ratio, 218, 222, 290, 301, 282, 361 303 complementary, 32 gong, 133, 144 cosine, 23 Gordon, Webb and Wolpert, 119 Dirac delta, 82, 251 gourd, 131 Euler’s phi, 336, 354 grains, 308 even, 44 granular synthesis, 291, 308 generalized, 84 graphicx, xiii Heaviside, 86 Gray’s Anatomy, 7 inverse, 326, 332 great diesis, 171, 186, 197, 369 kernel, 85 Great Highland Bagpipe, 212 L1,74 greatest common divisor, 163, 336 logarithm, 166, 368 Greek periodic, 37, 39, 150 music, 202 rational, 260 scales, 202 sampling, 252 group, 312, 324 sawtooth, 47, 49, 50 abelian, 325 sine, 23 alternating, 355 square wave, 41, 47, 50, 71, 267 cyclic, 335 tangent, 218 dihedral, 340 test, 84 infinite cyclic, 312 transfer, 258 infinite dihedral, 318 triangular, 49 Klein four, 340 Weber, 65 Mathieu M12, 358 fundamental, 144, 152 permutation, 328 domain, 237 simple, 359 missing, 6, 117, 157 sporadic, 359 fundamental root (Rameau), 148 symmetric, 328 Gruytters, Joannes de (1709–1772), Gaffurius, Franchinus (1451–1522), 147 198, 385 Galilei, Galileo (1564–1642) gu, 141 experiment, 5 guitar, electric, 92 pitch and frequency, 149 gamelan, 2, 144, 207 H(t), 86 gamma scale (Carlos), 233, 369 ,73 gas, 5 Hába, Alois (1893–1973), 228

400 Index

half-period symmetry, 45 homogeneity, 26 Hall, Donald E., 196 homomorphism, 332 hammer, 8 Hooke, Robert (1635–1703) Hammond organ, 269 law, 108, 125, 142 Han dynasty (206 bce–221 ce), 211 horn, Bessel, 121 Hanson, Howard (1896–1981), 228 horn equation, Webster’s, 121 Hardy and Wright, 219, 220, 222 Hornbostel, Erich Moritz von (1877–1935), harmonic, 5, 167, 370 91 ﬁfth, 167 Houtsma, Rossing and Wagenaars, 384 law (Kepler), 148 Hua Loo Keng, 222 motion, 18 Huai-nan-dsi, 211 damped, 28 Hubei province, China, 139 forced, 31 Huffman coding, 250 mth, 144 hum (bell), 138 odd, 41 human ear, 7, 368 piano (Harrison), 212 Huygens, Christiaan (1629–1695), 227, scale (Carlos), 212 228 second, 371 hyperbolic Bessel functions, 134 series, 144, 296 Hypoaeolian, 380 seventh, 144, 167, 232, 233, 370 Hypodorian, 379 third, 167 Hypoionian, 380 Harmonices Mundi (Kepler), 148 Hypolydian, 380 harmonium, 224 Hypomixolydian, 380 voice (Colin Brown), 180 Hypophrygian, 380 harmony, 16, 181 Hz, 13 septimal, 242 harp, 91, 322, 342 In(z), 134 harpsichord, 91, 198, 385 I (inversion), 338 Harris, Sidney, identities, trigonometric, 23 Harrison, John (1693–1776), 188 identity Harrison, Lou (1917–), 212, 383 element, 87, 325 Harrison, Michael, 212 Parseval’s, 80, 85, 253 Hauptmann, Moritz (1792–1868), 172 idiophones, 91 Haverstick, Neil, 384 Ile de feu 2 (Messiaen), 358 Haydn, Franz Joseph (1732–1809), 321, illusion, visual, 161 384 image, 332 header (CSound), 292 I’m Old Fashioned (Kern/Mercer), 184 hearing Impromptu No. 3 (Schubert), 190 range (frequency), 13 impulse response, 87, 258 threshold of, 14 incus, 8 Heaviside, Oliver (1850–1925) index function, 86 of modulation, 281 Heisenberg, Werner Karl (1901–1976) of unison sublattice, 239 uncertainty principle, 73, 143 Indian helioctrema, 9 drum, 117 helix, 7 Sruti scale, 213 Helmholtz, Hermann (1821–1894), 11, 103, inductive algorithm, 215 148, 172 inﬁnite Hertz, Gustav Ludwig (1887–1975) cyclic group, 312 unit of frequency, 13 dihedral group, 318, 341 hexachord, 351, 356 order, 328 hexadecimal, 248 information theory, 250 , 223 inharmonic spectrum, 284, 290 Hilbert, David (1862–1943) initial conditions, 100 transform, 89 injective function, 332 Hilliard Ensemble, 384 inner Hindemith, Paul (1895–1963), 148, 321 ear, 7 history of temperament, 202 product, 40, 54 Hoffnung, Gerard (1925–1959), 233 instantaneous Hofstadter, Douglas R., 223 amplitude, 90 hole, 111 frequency, 89

Index 401

instrument major bowed string, 49 scale, 168, 173, 181 percussion, 91, 273, 289 sixth, 152, 168, 370 wind, 107 third, 152, 168, 370 int (CSound), 307 triad, 180 integer, 37, 39, 144, 147–168, 326 minor Gaussian, 223 semitone, 369 part, 52, 215 sixth, 168 ratio, 145 third, 152, 168, 370 integral tone, 369 double, 75 triad, 173, 181 22 − π for 7 , 222 noticeable difference, 15 formula, Cauchy’s, 67 super just scale, 211 particular, 32 intensity, sound, 14, 15 Kac, Mark (1914–1984), 119 internal direct product, 340 kalimba, 131 Interplanetary Music Festival, 233 Karplus–Strong algorithm, 273, 291 interpolation Katahn, Enid (pianist), 384 exponential, 301 kazoo, 91 linear, 245, 297 Kelletat, Herbert, 196 interval Kellner, Herbert Anton, 196, 198, 385 major Kepler, Johannes (1571–1630) sixth, 227 laws, 68, 148 third, 168, 227 monochord, 175 minor Kern, Jerome (1885–1945), 184 seventh, 163 kernel sixth, 163, 227 Dirichlet, 57 third, 163, 227 Fejér, 56 perfect fifth, 146, 227 functions, 85 table of, 368 of a homomorphism, 345 vector, 350 kettledrum, 115, 116 wolf, 186 key intonation, just, 16, 167, 210, 237 characteristics, 191 inverse, 353 off, key on, 267 element, 325 signature, 378 Fourier transform, 76 split, 204 function, 326, 332 Keyboard (magazine), 308, 309 inversion, 312, 338, 376 King Fâng (3rd c. bce), 227 Ionian, 379 Kirchoff, Gustav (1824–1887), 136 irrational numbers, 150, 214 Kirnberger, Johann Philipp (1721–1783) irregular temperament, 189, 190 approximation of, 200 isomorphism, 332 scales of, 179, 193, 194 isophon, 15 Klavierstuck¨ (Schoenberg), 320 isospectral plane domains, 119 Klein four group, 340 iTunes, 77, 250 Kliban, B., 81 Ives, Charles (1874–1954), 228, 381 Klytemnestra, 313 Körner, T. W. (1946–), 37 Jn(z), 58, 69, 361 Jaja, Bruno Heinz, 233 L1 function, 74 Jaramillo, Herman, xiii labyrinth, 7 jazz, 181 membranous, 8 Jesu, der du meine Seele (J. S. Bach), 184 osseous, 8 Johnson, Ben (1926–), 384 Lagacé, Bernard, 189, 384 joke, Euler’s, 256 Lambda scale, 235 Jones, Lindley Armstrong ‘Spike’ (1911–1965), Lambert, Johann Heinrich, 194 339 lamellophone, 91, 131 Josquin Desprez (ca. 1440–1521), 204 lamina spiralis JP-8000/JP-8080, 71 ossea, 9 Jupiter, orbit of, 222 secundaria, 9 just Laplace, Pierre-Simon (1749–1827) intonation, 16, 167, 210, 237 operator, 117

402 Index

lattice, 213, 237 Ludus Tonalis (Hindemith), 321 Laurent, Pierre-Alphonse (1813–1854) lunar eclipse, 222 expansion, 67 lute tunings, Mersenne’s, 177 law Lydian, 379 associative, 325 Lyndon, J. A., 317 commutative, 325 lyre, 95 Hooke’s, 108, 125, 142 Lyric Suite (Berg), 320 Kepler’s, 68, 148 Mersenne’s, of stretched strings, 99 m4a, 250 Newton’s, of motion, 18, 94, 108, 112, MacCsound, 291 124, 142 Machaut, Guillaume de (1300–1377), 165, leak (diesis), 171 322, 384 leger line, 367, 375 magazine Lehman, Bradley, 197 Electronic Musician, 308 lemma, Burnside’s, 345, 356 Keyboard, 308, 309 length, effective, 110 major Leonardo da Vinci (1452–1519), 204 scale, 163 LFO (low frequency oscillator), 71, 266, 267 seventh, 371 Licklider, J. C. R., 157 sixth, 227 Liebestraum (Spike Jones), 339 just, 152, 168 liftering, 88 third, 16, 168, 202, 227 light, 146 just, 152, 168 likembe, 131 tone, 163 limanza, 323 triad, 168 limen, 15 Malamini organ, 204 limit, 53, 74, 211 Malcolm’s monochord, 178 left/right, 50 malleus, 8 of discrimination, 16 mammoth, woolly, 108 limitations of the ear, 13 Mandelbaum, M. Joel, 229 limma, 163, 369 marimba, 129 Linderholm, Carl E., 359 bamboo (Partch), 210 linear Marpurg, Friedrich Wilhelm (1718–1795) algebra, 118 monochord, 177 density, 93, 124 temperament I, 194 interpolation, 245, 297 Marquis Yi, 139 linearity, 26 masking, 12, 155, 157, 250 lineseg (CSound), 305 master volume, 251 Lissajous figures, 103 Mathieu, E. (1835–1900) little endian, 248 group M12, 358 local variables (CSound), 297 mathlib, 364 logarithm, 368 matrix, 239 log2(3) is irrational, 214 Mattheson, Johann (1681–1764), 198, 385 log, log10 (CSound), 307 mbira, 91, 130 natural base of (e), 218 mean logarithmic scale free path, 5 for cepstrum, 88 square convergence, 48 of centi, 166 square error, 48, 188 of decibels, 14 velocity of air molecules, 5 long division, 222 meantone scale, 162, 185, 186, 187, 230, 368, 377, longitudinal 383 elasticity, 125 meatus auditorius externus, 7 wave,6,92 Media Lab, MIT, 291 lookup table, 281 medieval church modes, 379 loop, 308 membrana lossless compression, 250 basilaris, 9 lossy compression, 250 tympani secundaria, 10 loudness, 6, 7 membrane low frequency oscillator (LFO), 71, 267 basilar, 9, 151 low pass filter, 245, 255 tympanic, 8 Lü scale, Chinese, 211 membranophones, 91 Lucy, Charles, 188 membranous labyrinth, 8

Index 403

Menuetto al rovescio (Haydn), 321 de Caus’s, 175 Mercer, Johnny (1909–1976), 184 Erlangen, 174 Mercury, rotation of, 222 Euler’s, 178, 240 Mersenne, Marin (1588–1648), 37 Fogliano’s, 175 improved meantone temperament, 192 Gibelius’, 186 law of stretched strings, 99 Kepler’s, 175 picture, 99 Malcolm’s, 178 pitch and frequency, 149 Marpurg’s, 177 spinet/lute tunings, 176 Montvallon’s, 178 message, system exclusive, 251 Ramis’, 174 Messiaen, Olivier (1908–1992), 358 Romieu’s, 178 Metamagical Themas (Hofstadter), 223 Rousseau’s, 179 MetaPost, xiii monomorphism, 332 Meyer, Alfred, 12 Monteverdi, Claudio (1567–1643), 204 middle ear, 7 Montvallon, André Barrigue de MIDI, 251, 291, 367 monochord, 178 baud rate, 251 Moog, Robert A. (1934–) files, 188 synthesizer, 207, 382 to CSound, 291 Moon, 222 MIDI2CS, 291 Moonlight Sonata (Beethoven), 312 Miller, James Charles Percy motion algorithm, 363 Brownian, 47, 81 minor circular, 26 scale, 163 damped harmonic, 17, 28 semitone, 163, 226 harmonic, 18 just, 369 planetary, 68, 147 seventh, 163 simple harmonic, 17 sixth, 163, 227 Mozart, Wolfgang Amadeus (1756–1791) just, 168 Fantasie (K. 397), 184, 189, 198, 384 third, 163, 227 pitch, 21 just, 152, 168 Sinfonia Concertante, 190 tone, just, 369 Sonata (K. 333), 183 triad, 169, 377 Spiegel, 313 just, 173 MP3 sound file, 248 Mirror Duet (attr. Mozart), 315 MPEG, 249 missing fundamental, 6, 117, 157 MPEG 4 Audio, 250 MIT Media Lab, 291 MS-DOS, 291 Mixolydian, 379 Muffat, Gottlieb (1690–1770), 189 mode, 379 multiplication table, 325 Dorian, 202 Musæ Sioniæ (M. Praetorius), 16 vibrational, 20 music modelling, physical, 270 atonal, 205, 348 Modern Major General, 36 baroque, 181 modification, 170 digital, 245 modiolus, 9 folk, 181 modulation Greek, 202 amplitude, 276 of the spheres, 147 frequency, 276 polyphonic, 203 index of, 281 rock, 181 pulse width, 71, 268 romantic, 181 ring, 277 theory, 374 modulus twelve tone, 205 bulk, 108, 142 Musical Offering (J. S. Bach), 314 Young’s, 125, 133 Musical World, 161 moment MusicTEX, xiii bending, 124 Mussorgsky, Modest (1839–1881), 266, sectional, 126, 133 385 Mongean shuffle, 359 monochord N (sample rate), 251 Agricola’s, 175 nabla squared (∇2), 118 BP, 236 Nachbaur, Fred, 313

404 Index

natural opcode (CSound), 293 logarithms open tube, 108 base of (e), 218 operator, 281 minor scale, 377 Laplace, 117 pitch, 20 orbit, 342 Nature,52 orchestra, 91 necklace, 353 file (CSound), 291 Neidhardt, Johann Georg (1685–1739), 193, 198, 381 order, 328 Neumann, Carl Gottfried (1832–1925) ordered pairs, 339 Bessel function, 65 ordinary spectrum, 119 comma, 168 neutral surface, 125 differential equation, 17, 19, 28, 64 new moon, 222 organ, 269 newton (unit of force), 93 Duke/Brombaugh, 189 Newton, Sir Isaac (1642–1727) Knox/Toronto/Wollf, 189 laws of motion, 18, 94, 108, 112, 124, 142 Malamini, 204 ngbaki` a`, 323 of Corti, 10 Nicomachus (ca. 60–120 ce), 208 stops, 269 Nine Taylors (Dorothy Sayers), 329 Tröchtelborn, 198, 385 nineteen tone scale, 212, 227 Wahlberg, 381 ninth harmonic, 370 Wellesley/Fisk, 189 node, 222 organum, parallel, 203 noise, 47, 288 orientation, 240 white, pink, brown, 81 origin, 237 nominal orthogonality relation, 40, 54, 261 (bell), 138 oscil, oscili (CSound), 293 frequency, 117 oscillator, 268 nonlinear acoustics, 71 coupled, 222 nonlinearity, 156 low frequency (LFO), 267 normal subgroup, 344 oscilloscope, 77 notation osseous labyrinth, 8 cycle, 327 ossicular chain, 8 dot (derivative), 19 outer ear, 7 Eitz’s, 172, 235 oval window, 8, 10 roman numeral, 182 overblowing, 110 notepad, 295 overdamped system, 29 nu (ν, frequency), 21 overtone, 144 numbers, 147 complex, 33, 54, 223 p-limit, 211 irrational, 150, 214 Palestrina, Giovanni Pierluigi da (ca. 1526–1594), 204 rational, 214, 216 palindrome, 314, 321 Nyquist, Harold (1889–1976) Pallas, orbit of, 222 frequency, 254 panning, 251, 305 theorem, 254 parabolic envelope (bowed string), 104 Nzakara, 322, 342 paradox Russel’s, 324 oboe, 91, 110 Shepard’s, 158 Ockeghem, Johannes (ca. 1415–1497), 204 tritone, 159 octahedron, 213 parallel organum, 203 octave, 16, 20, 145, 152, 374 parallelogram, 27, 240 dissonant, 153 parameter, flare, 121 equivalence, 146, 237, 333 paranoia in the music business, 245 stretched, 188, 201 Parmentier, Edward, 385 odd Parseval, Marc Antoine (1755–1836) function, 44 identity, 80, 85, 253 harmonics, 41 Partch, Harry (1901–1974), 184, 211, 385 Odington, Walter (fl. 1298–1316), 203 partial, 144, 145, 157, 269 Ohm’s acoustic law, 11 derivative, 92 omega ω = 2πν,21 differential equation, 71, 92, 112 one-to-one correspondence, 332 fractions, 222, 260 Op de Coul, Manuel, xiii particular integral, 32

Index 405

Partitia no. 5, Gigue (J. S. Bach), 183 Fibonacci, patch, 251, 288 Fourier, Joseph, 38 patterns, frieze, 318 Gaffurius’ Experiences of Pythagoras, peak gong, amplitude, 21, 25 Hammond B3 organ, 270 of consonance, 152 Italian clavecin with split keys, pelog scale, 207 Kepler, Johannes, percussion instruments, 91, 273, 289 lyre, 95 perfect Malamini organ (split keys), BP-tenth, 234 Marpurg, Wilhelm, fifth, 146, 152, 227, 375 mbira, 131 Cordier’s equal temperament, 201 Mersenne, Marin, 99 fourth, 152, 161, 167, 202 osseous labyrinth, periodic Partch, Harry, 210 continued fraction, 222 piano keyboard, 321 function, 37, 39, 150 poor acoustics, 142 Riemann integrable, 48 proving the existence of fish, 81 wave, 37 Pythagoras, 162 periodicity block, 237, 240, 344 riti, 103 permutation, 207 simplified version for public, group, 328 singing bowl, 139 perpendicular, 40 theorbo, 93 Perret, Wilfrid, 212 timpani (Hoffnung), phase, 21, 25, 26, 27, 80, 107, 269 Trasuntius’ 31 tone clavicord, vocoder, 291, 309 tuba curva, 121 phenomenon, Gibbs, 42, 50 Vallotti, Francescantonio, phi function, Euler’s, φ(n), 336, 354 visual illusion, Philolaus of Tarentum (d. ca. 390 bce), 171, 226 WABOT-2, 266 phon, 15 Webern, Op. 24/28, 341 Phrygian, 379 xylophone, 122 physical modelling, 270 Yamaha DX7, 281 pi Pictures at an Exhibition (Mussorgsky), 266, continued fraction for, 216 385 is irrational, 218 piecewise continuity, 50 22 is smaller than 7 , 222 √ Pierce, John R. (1910–), 16, 153 π meantone scale based on 2, 188 Pierrot Lunaire (Schoenberg), 207 2π radians in a circle, 21 pink noise, 81 piano, 91 pinna, 7 harmonic (Harrison), 212 pipe, 107 tuning, 24, 25 pitch, 6, 7, 21, 149 Cordier’s equal temperament, 201 class set, 348 Pleyel, 201 classes, 300, 334 pictures envelope, 291 bells, 138 in Tudor Britain, 21 Boole orders lunch, Mozart’s, 21 Bosanquet’s harmonium, natural, 20 Brown’s voice harmonium, perception, place theory of, 11 Calvin and Hobbes, virtual, 156 Carlos, Wendy, place theory, 11 Chinese bell, 140 plagal Chinese flute, 108 mode, 379 Chladni patterns on drum, 116 Plain Bob, 330 Chladni’s drawings, 137 Plain Hunt, 330 Chowning, John, 279 Planck, Max (1858–1947) cochlea, constant, 73 coneflower, plane domains, isospectral, 119 d’Alembert, Jean-le-Rond, 95 planetary motion, 68, 147 bce ear chart, Plato (427–347 ), 203 Escher, Ascending and Descending, Republic, 147 Euler, Leonhard, Pleyel, piano tuning, 201 feedback in the cochlea, 12 Plomp, R. and Levelt, W. J. M., 151

406 Index

plucked quint (bell), 138 bottle, 275 Quintilianus, Aristides, 208 string, 273 pointwise convergence, 51, 53 R (retrograde), 339 Poisson, Siméon Denis (1781–1840) radians, 21, 307 ratio, 133 per second, 27 summation formula, 81, 253 radio, AM and FM, 276 polar coordinates, 75, 80, 113 radius of curvature, 125 poles, 260 ragas, 213 Pólya, George (1887–1985) rahmonics, 88 enumeration theorem, 353 rainbow, 146 polynomials, Chebyshev, 309 Rainforest (Rich), 385 polyphonic music, 203 Raman, Chandrasekhra Venkata (1888–1970), Portuguese square drum, 117 107, 117 position, equilibrium, 18 Ramanujan, Srinivasa Aiyangar (1887–1920), power 214 gain, 14 Rameau, Jean-Philippe (1693–1764), 148, 149, 199, intensity, 14 205 series, 355 Ramis, Bartolomeus de Pareja (1440–ca. 1491) series for Jn(z), 66, 67 monochord, 174 Praetorius, Michael (1571–1621) random Musæ Sioniæ,16 wave, 267 predictability in music, 314 ratio, 14 Preludes and Fugues (J. S. Bach), 189 frequency, 166 pressure, acoustic, 107, 119, 142 golden, 218, 222, 290, 301, prime 303 (bell), 138 of integers, 145, 147, 167, 168 form, 339, 349 Poisson’s, 133 principal value, Cauchy, 74 rational principle of reflection, 97 approximation, 214–223 Pringsheim, Alfred (1850–1941), 218 function, 260 product numbers, 216 Cartesian, 339 Ravel, Maurice (1875–1937) direct, 339 Rhapsodie Espagnole, 317 inner, 40, 54 recordings, 381 programming language, C, 291 recurrence relation progression, 181 for Jn(z), 63 proving the existence of fish, 81 Karplus–Strong algorithm, 273 psychoacoustics, 6, 157, 250, 384 recursive index, 63 Ptolemy, Claudius (ca. 83–161 ce), reflection, principle of, 97 208 reflectional symmetry, 312 comma, 168 register stops (organ), 269 diatonic syntonon, 202 Reiner, David, 348 public domain, 291 relation pulse width modulation, 71, 268 orthogonality, 40, 54, 261 PWM, 71, 268 recurrence, 63 pyknon, 203 relative minor, 378 Pythagoras (ca. 569–500 bce), 147 release, 266 Pythagorean, 226 repetition, 312 apotom¯e, 164, 369 representation of sound, digital, 245 comma, 163, 166, 190, 201, 223, 225, representatives, coset, 237, 343 369 Republic (Plato), 147 minor semitone, 163 resonance, 31, 33, 260, 268 scale, 162, 368 resonant frequency, 18, 34, 35 response, impulse, 258 quadratic equation, 29, 222 retrograde, 339 quadrivium, 147 canon, 314 quantization, 245 reverberation, 251, 268 quantum mechanics, 73 Reverieˆ (Debussy), 320 quarter-tone scale, 227 Rhapsodie Espagnole (Ravel), 317 quefrency, 88 ρ (density), 93, 112, 124, 133

Index 407

Rich, Robert, xiii 167, 209, 385 scale, 161 Riemann, (Georg Friedrich) Bernhard (1826–1866) Aaron’s meantone, 186 integrable periodic function, 48 Agricola’s monochord, 175 sum, 53 alpha (Carlos), 207, 232, 369 Riemann, (Karl Wilhelm Julius) Hugo (1849–1919), 1 Barca’s 6 -comma, 194 173 Bendeler, 192 RIFF, 248 beta (Carlos), 207, 232, 369 ring Bohlen–Pierce, 162, 233, 368 commutative, 355 BP-just, 235 modulation, 277 Chalmers’ just, 212 Risset, Jean-Claude (1938–), 160 Chinese Lü, 211 riti, 103 chromatic, 207 RMS amplitude, 25 de Caus’s monochord, 175 rnd (CSound), 307 diatonic syntonic, 202 rock music, 181 equal tempered, 199, 368 rod, vibrating, 122 Erlangen monochord, 174 Roland, JP-8000/JP-8080, 71 Euler’s monochord, 178, 240 Roman Empire, decline of, 203 fifty-three tone, 224 roman numeral notation, 182, 378 Fogliano’s monochord, 175 romantic music, 181 forty-three tone, 211 Romieu, Jean Baptiste (1723–1766), 155, gamma (Carlos), 233, 369 187 Gibelius’ meantone, 186 monochord, 178 Greek, 202 root, 376 harmonic (Carlos), 212 mean square, 25 Indian Sruti, 213 rosewood, 125, 129 irregular, 179, 189, 190, 192, 193, 194 Rossi, Lemme just, 168, 174–179 2 9 -comma temperament, 187 Carlos, 212 rotational symmetry, 314 Chalmers, 212 roughness, 146, 149 Lou Harrison, 212 round window, 10 Michael Harrison, 212 Rousseau, Jean-Jacques (1712–1778) Perret, 212 monochord, 179 Kepler’s monochord, 175 Russel, Bertrand (1872–1970) Kirnberger I, 179 paradox, 324 Kirnberger II–III, 193 Lambda, 235 m Lambert’s 1 -comma, 194 = 1 + θ + θ 7 sm 2 a0 (an cos(n ) bn sin(n )), 47 logarithmic scale of cents, 166 = n 1 Lou Harrison’s just, 212 saccule, 13 Lü (Chinese), 211 Sachs, Curt (1881–1959), 91 major, 163 sadness, 170 Malcolm’s monochord, 178 Salinas, Francisco de (1513–1590) Marpurg’s 1 -comma temperament, 187 3 monochord, 177 sample temperament I, 194 and hold, 245 meantone, 162, 185, 186, 187, 230, 368, 377 dump, 251 Mersenne’s frames, 249 improved meantone, 192 rate, 245 lute tunings, 177 (CSound), 292 spinet tunings, 176 sampling, 308 Michael Harrison’s just, 212 function, 252 minor, 163 theorem, 245, 255 Montvallon’s monochord, 178 Sankey, John, 188 Neidhardt, 193, 198, 381 sanzhi, 131 nineteen tone, 212, 227 Savart, Félix (1791–1841), 166, 369 of commas, 226 saw, bowed, 91 Partch’s forty-three tone, 211 sawtooth function, 47, 49, 50, 267 pelog, 207 Sayers, Dorothy Leigh (1893–1957), 329 Perret’s just, 212 scala Pythagorean, 162, 368 tympani, 9 quarter-tone, 227 vestibuli, 9, 13

408 Index

scale (cont.) series Ramis’ monochord, 174 configuration counting, 355 Romieu’s monochord, 178 Fibonacci, 222 2 Rossi’s 9 -comma, 187 Fourier, 21, 36,58 Rousseau’s monochord, 179 geometric, 57 1 Salinas’ 3 -comma, 187 harmonic, 144, 296 665 tone, 227 power, 66, 67, 355 sixteen tone, 212 trigonometric, 39 slendro, 207 set, 324 Sruti (Indian), 213 pitch class, 348 super just, 211 Sethares, William A. (1955–), 155, 228, 385 tempered, 185 seventh thirty-one tone, 228 dominant, 170 twelve tone, 164, 199, 211, 212 harmonic, 144, 167, 232, 233, 370 twenty-four tone, 227 major, 371 twenty-four tone just, 212 minor, 163 twenty-two tone, 213 sharp, double, 164 Vallotti and Young, 195 shearing force, 123 well tempered, 189 Shepard scale, 158 Werckmeister Shona people, 131 I–II, 190 shuffle, Mongean, 359 III, 198, 381 side band, 280, 282 III–V (Correct Temperament No. 1–3), σm = (s0 +···+sm )/(m + 1), 47 190 signal VI (Septenarius), 190 analogue, 245 Young’s No. 1, 194 digital, 245 2 to noise ratio, 14 Zarlino’s 7 -comma, 187 Scarlatti, Domenico (1685–1757), 384 signature, key, 378 schisma, 171, 174, 181, 369 Silbermann, Gottfried (1683–1753), 187 Schoenberg, Arnold (1874–1951) simple Klavierstuck¨ Op. 33a, 320 group, 359 Pierrot Lunaire, 207 harmonic motion, 17 Schouten, J. F., 156 simply connected, 119 Schubart, Christian Friedrich Daniel (1739–1791), sin, sinh, sininv (CSound), 307 191 sine wave, 10, 17, 267 Schubert, Franz (1797–1828) Sinfonia Concertante (Mozart), 190 Impromptu No. 3, 190 singer, bass, 15 Schwartz, Laurent (1915–) singing bowl, 138 space, 84 sixteen tone scale, 212 scordatura, 190 sixth score file (CSound), 291 major, 152, 227 scot (CSound), 307 minor, 163, 227 Scotland, 212 slendro scale, 207 second harmonic, 371 Slonimsky, Nicolas (1894–1995), vi, 161 sectional moment, 126, 133 slur, 274 sections (CSound), 299 small semitone, 369 self-modulation, 287 smell, 4 self-reference, 287 snail, 8 self-similarity, 47 software semicircular canals, 8 CSound, 291 semitone, 166, 185, 369, 374 MetaPost, xiii minor, 163, 226 solar eclipse, 222 small, 369 solution separable solution, 113 separable, 113 separation, spatial, 7 steady state, 33 septenarius, 190 Sonata K. 333 (Mozart), 183 septimal soprano, coloratura, 236 comma, 171, 369 Sorge, Georg Andreas (1703–1778), 149, 155, 187 harmony, 242 sound sequence, 312 focusing of, 7 of fifths, 162 intensity, 14, 15

Index 409

spectrum, 6, 14, 73 summation formula, Poisson’s, 81, 253 what is it?, 5 super just scale, 211 Sound Frequency Analyzer (freeware), 76 superposition, 26 Sparschuh, Andreas, 196 superscript notation, Eitz’s, 172 spatial separation, 7 surface, neutral, 125 spectral display (CSound), 306 surjective function, 332 SpectroGraph, 76 surprise in music, 314 spectrum, 7, 14, 22, 80, 80, 118, 153 sustain, 266 Dirichlet, 119 Switched on Bach (Carlos), 207 inharmonic, 284, 290 Symm(X), 328 Neumann, 119 symmetric group, 328 spherical symmetry, 143 symmetry, 45, 312 Spiegel (attr. Mozart), 313 spherical, 143 spinet, 189 synodic month, 222 tunings, Mersenne’s, 176 synthesis, 265, 266 spiral of fifths, 164 additive, 268 split keys, 204 FM, 60, 68, 278, 280, 291, 298 sporadic group, 359 granular, 291, 308 sqrt (CSound), 307 software, 291 square wavetable, 308 drum, 117 synthesizer, 153, 207 wave, 41, 47, 50, 71, 267 analogue, 71, 268, 385 Sruti scale (Indian), 213 analogue modeling, 71 stability, 259 Moog, 207 stabilizer, 343 Yamaha DX7, 280 staircase, 245 syntonic comma, 168, 181, 369, 377 stapes, 8 system exclusive messages, 251 star sphere, 222 static friction, 49 Tn(x) (Chebyshev polynomials), 309 steady state solution, 33 T (transposition), 338 Stein, Richard Heinrich (1882–1942), table of intervals, 368 228 tan, tanh, taninv (CSound), 307 stereo, 305 tangent function, 218 Stevin, Simon (1548–1620), 187 Tartini, Giuseppe (1692–1770) stirrup, 8 tones, 155 Stockhausen, Karlheinz (1928–), 172 taste, 4 stops (organ), 269 Tavener, John Kenneth (1944–), 319 Strähle, Daniel, 202 Tchebycheff, 309 strain, tension, 125 temperament Strauss, Richard, 313 Aaron’s meantone, 186 1 Stravinsky, Igor (1882–1971), 233 Barca’s 6 -comma, 194 stress, tension, 124 Bendeler, 192 stretch factor, 275 calm, 199 stretched strings, laws of, 99 circulating, 189 strike points, 141 equal, 190, 199, 214, 368 string Cordier’s, for piano, 201 bowed, 49, 103 irregular, 189, 190 plucked, 273 Kirnberger I, 179 vibrating, 19, 92, 270 Kirnberger II–III, 193 stroboscopic tuning, 130 1 Lambert’s 7 -comma, 194 Sturm–Liouville equation, 121 Marpurg I, 194 subgroup, 328 Mersenne’s improved meantone, 192 normal, 344 Neidhardt’s, 193, 198, 381 sublattice, 239 2 Rossi’s 9 -comma, 187 subtraction, continued, 163 1 Salinas’ 3 -comma, 187 Sudan, 95 Vallotti and Young, 195 sui, 141 Werckmeister III, 198, 381 sum Werckmeister III–V (Correct Temperament Cesàro, 47, 51, 55 No. 1–3), 190 Riemann, 53 Young’s No. 1, 194 sumer is icumen in, 203 2 Zarlino’s 7 -comma, 187

410 Index

tempered transitive action, 342 distributions, 84 translational symmetry, 312 scale, 185 transposition, 338 tempo (CSound), 305 transverse wave, 6, 92 tension, 93, 112, 124 Trasuntinis, Vitus, 229 strain, 125 treble clef, 367, 374 stress, 124 Treitler, Leo, 208 test function, 84 tremolo, 268, 304 tetrachord, 202 triad, 182 tetrahedron, 213 augmented, 342 Theinred of Dover (12th c.), 203 diminished, 182 theorbo, 93 just major, 168 theorem just minor, 173 divergence, 118 minor, 169 Fejér’s, 48 triangular wave, 49, 267 Fermat’s last, 133 trigonometric first isomorphism, 345 identities, 23 Nyquist’s, 254 series, 39 Pólya’s enumeration, 353 tritave (BP), 234 sampling, 245, 255 tritone, 370 therapy, 359 paradox, 159 third Tröchtelborn organ, 198, 385 harmonic, 167, 370 trombone, 15, 91 major, 16, 152, 168, 202, 227 trumpet, 91, 288 minor, 152, 163, 227 Tsu Ch’ung-Chi, 217 thirteen tone scale, 234 tube, 107 thirteenth harmonic, 370 tubular bells, 122, 129 thirty-one tone scale, 228 Tudor pitch (Britain), 21 3-limit, 211 tuning threshold Mersenne’s lute, 177 of hearing, 14 Mersenne’s spinet, 176 of pain, 14 piano, 24 Tibetan singing bowl, 138 stroboscopic, 130 tie, 274 twelve tone tierce (bell), 138 music, 205 timbre, 6, 7, 267, 283 row, 337 timpani, scale, 164, 199, 211, 212 tinnitus, 12 twenty-four tone scale, 227 Toccata and Fugue in D (J. S. Bach), 312 twenty-two tone scale, 213 Toccata in F minor (J. S. Bach), 192 two’s complement, 249 Tomita, Isao (synthesist), 266, 385 tympanic membrane, 8 tone, 16, 374 tympanum, 7, 8 combination, 155 tyre, 226 control, 268 difference, 155 uncertainty principle, 73, 143 major, 163 underdamped system, 29 Tartini’s, 155 uniform convergence, 40, 42, 51, 53 tonic, 378 unison, 146 tonoi, 379 sublattice, 239 torque, 123 vector, 237, 344 torus of thirds and fifths, 226 Unix, 364 transfer function, 258 UnxUtils.zip, 364 transform unzip.exe, 292 discrete Fourier, 261, 337 fast Fourier, 263 Vallotti, Francescantonio (1697–1780), 195 Fourier, 73 variables (CSound), 297 of f (t), 75 vector of a distribution, 85 calculus, 118 Hilbert, 89 interval, 350 z, 256, 258, 273 space, 40 transients, 269 unison, 237, 344

Index 411

velocity, angular ω = 2πν, 21, 33 Weierstrass, Karl (1815–1897), 47 Vercoe, Barry, 291 Well Tempered Verheijen, Abraham, 187 Clavier (J. S. Bach), 189 vestibule, 8 Synthesizer (Carlos), 207 vibe, 129 well tempered scale, 189 vibrating Werckmeister, Andreas (1645–1706) drum, 112 temperaments, 190, 198, 377, 381 rod, 122 Wheatstone, Sir Charles (1802–1875) string, 19, 92, 270 concertina, 204 vibration microscope, 103 white noise, 22, 81 vibrational modes, 20, 115 Wilbraham, Henry, 52 vibrato, 268 Wilson, Ervin, 213 Vicentino, Nicola (1511–1576), 228 wind instruments, 107 violin, 49, 91 window virtual pitch, 156 oval, 8, 10 visual illusion, 161 phase vocoder, 309 vocoder, phase, 291, 309 round, 10 Vogel, Harald, 382 windowing, 73 voice, 15, 251 Winkelman, Aldert, 198, 385 DX7, 288 Winzip, 292 harmonium (Colin Brown), 180 wolf interval, 186 vortices, 111 wood drum (FM & CSound), 303 Vos, J., 16 woodwind, 289 Vyshnegradsky, Ivan Alexandrovich woolly mammoth, 108 (1893–1979), 228, 381 Xentonality (Sethares), 228, 385 WABOT-2, 266 xylophone, 91, 122 Wahlberg organ, 381 Wall, Hubert Stanley, 223 Yn(z), 65 Walliser, K., 157 Yamaha, 288 Walther, Johann Gottfried (1684–1748), 198, 385 DX7, 280 watts, 15 Yasser, Joseph (1893–1981), 227 per square meter, 14 Young, Thomas (1773–1829) WAV sound ﬁle, 248, 291 modulus, 125, 133 wave, 5 temperament No. 1, 194 electromagnetic, 6 equation, 92, 108, 112, 142 Z (integers), 326 fractal, 47 Z/n, 335 longitudinal, 6, 92 Z1, 237 periodic, 37 Z2, 238, 340 random, 267 Z3, 242, 340 sawtooth, 49, 50, 267 z-transform, 256, 258, 273 sine, 10, 17, 267 z = e2πiνt , 256 square, 41, 47, 50, 71, 267 z−1 (delay), 257, 271 transverse, 6, 92 Zande, 322 triangular, 49, 267 Zapf, Michael, 196 wavetable synthesis, 308 Zarlino, Gioseffo (1517–1590), 172 2 Weber, Heinrich F. (1842–1913) 7 -comma temperament, 187 function, 65 zeros of Bessel functions, 114, 361 Webern, Anton (1883–1945), 341 Zimbabwe, 131 Webster, Arthur Gordon (1863–1923) ZIP ﬁle, 250 horn equation, 121 Zwei Konzertstucke¨ (Richard Stein), 228