Music: A Mathematical Offering
Since the time of the Ancient Greeks, much has been written about the relation between mathematics and music: from harmony and number theory, to musical patterns and group theory. Benson provides a wealth of information here to enable the teacher, the student, or the interested amateur to understand, at varying levels of technicality, the real interplay between these two ancient disciplines. The story is long as well as broad and involves physics, biology, psycho acoustics, the history of...
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Explores interaction between music and mathematics including harmony, symmetry, digital music and perception of sound.
Preface xiAcknowledgements xiiiIntroduction 1Waves and harmonics 5What is sound? 5The human ear 7Limitations of the ear 13Why sine waves? 17Harmonic motion 18Vibrating strings 19Sine waves and frequency spectrum 21Trigonometric identities and beats 23Superposition 26Damped harmonic motion 28Resonance 31Fourier theory 36Introduction 37Fourier coefficients 38Even and odd functions 44Conditions for convergence 46The Gibbs phenomenon 50Complex coefficients 54Proof of Fejer's theorem 55Bessel functions 58Properties of Bessel functions 61Bessel's equation and power series 63Fourier series for FM, feedback and planetary motion 68Pulse streams 71The Fourier transform 73Proof of the inversion formula 77Spectrum 80The Poisson summation formula 81The Dirac delta function 82Convolution 86Cepstrum 88The Hilbert transform and instantaneous frequency 89A mathematician's guide to the orchestra 91Introduction 91The wave equation for strings 92Initial conditions 100The bowed string 103Wind instruments 107The drum 112Eigenvalues of the Laplace operator 117The horn 120Xylophones and tubular bells 122Thembira 130The gong 133The bell 138Acoustics 142Consonance and dissonance 144Harmonics 144Simple integer ratios 145History of consonance and dissonance 148Critical bandwidth 151Complex tones 152Artificial spectra 153Combination tones 155Musical paradoxes 158Scales and temperaments: the fivefold way 161Introduction 162Pythagorean scale 162The cycle of fifths 164Cents 165Just intonation 167Major and minor 168The dominant seventh 170Commas and schismas 171Eitz's notation 172Examples of just scales 174Classical harmony 181Meantone scale 185Irregular temperaments 189Equal temperament 198Historical remarks 202More scales and temperaments 210Harry Partch's 43 tone and other just scales 210Continued fractions 214Fifty-three tone scale 223Other equal tempered scales 227Thirty-one tone scale 228The scales of Wendy Carlos 231The Bohlen-Pierce scale 233Unison vectors and periodicity blocks 237Septimal harmony 242Digital music 245Digital signals 245Dithering 247WAV and MP3 files 248MIDI 251Delta functions and sampling 251Nyquist's theorem 254The z-transform 256Digital filters 257The discrete Fourier transform 261The fast Fourier transform 263Synthesis 265Introduction 265Envelopes and LFOs 266Additive synthesis / 268Physical modelling 270The Karplus-Strong algorithm 273Filter analysis for the Karplus-Strong algorithm 275Amplitude and frequency modulation 276The Yamaha DX7 and FM synthesis 280Feedback, or self-modulation 287CSound 291FM synthesis using CSound 298Simple FM instruments 300Further techniques in CSound 304Other methods of synthesis 308The phase vocoder 309Chebyshev polynomials 309Symmetry in music 312Symmetries 312The harp of the Nzakara 322Sets and groups 324Change ringing 329Cayley's theorem 331Clock arithmetic and octave equivalence 333Generators 335Tone rows 337Cartesian products 339Dihedral groups 340Orbits and cosets 342Normal subgroups and quotients 343Burnside's lemma 345Pitch class sets 348Polya's enumeration theorem 353The Mathieu group M12 358Bessel functions 361Equal tempered scales 365Frequency and MIDI chart 367Intervals 368Just, equal and meantone scales compared 372Music theory 374Recordings 381References 386Bibliography 389Index 393
\ From the Publisher"Perhaps our children will one day remark on the group symmetries in their favorite music in the way that we now simply note a beautiful tune. They, no less than we, will have much to learn from this delightful book, which sets a new standard of excellence and inclusiveness. Anyone who knows some college-level mathematics and is curious about how it can illuminate music will be richly rewarded by reading Benson's outstanding book." \ Peter Pesic, Tutor and Musician-in-Residence at St. John's College, Santa Fe\ "... A precise selection of solutions..."\ Luigi Carlo Berselli, Mathematical Reviews\ "… an excellent introduction to the interdisciplinary subject of music and mathematics (which also involves physics, biology, psycho-acoustics, and the history of science and digital technology). The book can easily be used as the text for undergraduate courses."\ The Mathematical Intelligencer\ \ \
