Free online version of the book Music: A Mathematical Offering. This work is
© Dave Benson 1995–2008
and is regularly updated since ‘95. Benson is a professor in the Department of Mathematics (of course) at the University of Aberdeen, Scotland.
List of contents (looooooong but cool)
Chapter 1. Waves and harmonics 1
1.1. What is sound? 1
1.2. The human ear 3
1.3. Limitations of the ear 8
1.4. Why sine waves? 13
1.5. Harmonic motion 14
1.6. Vibrating strings 15
1.7. Sine waves and frequency spectrum 16
1.8. Trigonometric identities and beats 18
1.9. Superposition 21
1.10. Damped harmonic motion 23
1.11. Resonance 26
Chapter 2. Fourier theory 30
2.1. Introduction 31
2.2. Fourier coefficients 31
2.3. Even and odd functions 37
2.4. Conditions for convergence 39
2.5. The Gibbs phenomenon 43
2.6. Complex coefficients 47
2.7. Proof of Fej´er’s Theorem 48
2.8. Bessel functions 50
2.9. Properties of Bessel functions 54
2.10. Bessel’s equation and power series 55
2.11. Fourier series for FM feedback and planetary motion 60
2.12. Pulse streams 63
2.13. The Fourier transform 64
2.14. Proof of the inversion formula 68
2.15. Spectrum 70
2.16. The Poisson summation formula 72
2.17. The Dirac delta function 73
2.18. Convolution 77
2.19. Cepstrum 78
2.20. The Hilbert transform and instantaneous frequency 79
2.21. Wavelets 81
Chapter 3. A mathematician’s guide to the orchestra 83
3.1. Introduction 83
3.2. The wave equation for strings 85
3.3. Initial conditions 91
3.4. The bowed string 94
3.5. Wind instruments 99
3.6. The drum 103
3.7. Eigenvalues of the Laplace operator 109
3.8. The horn 113
3.9. Xylophones and tubular bells 114
3.10. The mbira 122
3.11. The gong 124
3.12. The bell 129
3.13. Acoustics 133
Chapter 4. Consonance and dissonance 136
4.1. Harmonics 136
4.2. Simple integer ratios 137
4.3. History of consonance and dissonance 139
4.4. Critical bandwidth 142
4.5. Complex tones 143
4.6. Artificial spectra 144
4.7. Combination tones 147
4.8. Musical paradoxes 150
Chapter 5. Scales and temperaments: the fivefold way 153
5.1. Introduction 154
5.2. Pythagorean scale 154
5.3. The cycle of fifths 155
5.4. Cents 157
5.5. Just intonation 159
5.6. Major and minor 160
5.7. The dominant seventh 161
5.8. Commas and schismas 162
5.9. Eitz’s notation 164
5.10. Examples of just scales 165
5.11. Classical harmony 173
5.12. Meantone scale 176
5.13. Irregular temperaments 181
5.14. Equal temperament 189
5.15. Historical remarks 192
Chapter 6. More scales and temperaments 200
6.1. Harry Partch’s 43 tone and other just scales 200
6.2. Continued fractions 204
6.3. Fifty-three tone scale 213
6.4. Other equal tempered scales 217
6.5. Thirty-one tone scale 219
6.6. The scales of Wendy Carlos 221
6.7. The Bohlen–Pierce scale 224
6.8. Unison vectors and periodicity blocks 227
6.9. Septimal harmony 232
Chapter 7. Digital music 235
7.1. Digital signals 235
7.2. Dithering 237
7.3. WAV and MP3 files 238
7.4. MIDI 241
7.5. Delta functions and sampling 242
7.6. Nyquist’s theorem 244
7.7. The z-transform 246
7.8. Digital filters 247
7.9. The discrete Fourier transform 250
7.10. The fast Fourier transform 253
Chapter 8. Synthesis 255
8.1. Introduction 255
8.2. Envelopes and LFOs 256
8.3. Additive Synthesis 258
8.4. Physical modeling 260
8.5. The Karplus–Strong algorithm 262
8.6. Filter analysis for the Karplus–Strong algorithm 264
8.7. Amplitude and frequency modulation 265
8.8. The Yamaha DX7 and FM synthesis 268
8.9. Feedback, or self-modulation 274
8.10. CSound 278
8.11. FM synthesis using CSound 284
8.12. Simple FM instruments 286
8.13. Further techniques in CSound 290
8.14. Other methods of synthesis 292
8.15. The phase vocoder 293
8.16. Chebyshev polynomials 293
Chapter 9. Symmetry in music 296
9.1. Symmetries 296
9.2. The harp of the Nzakara 307
9.3. Sets and groups 309
9.4. Change ringing 313
9.5. Cayley’s theorem 316
9.6. Clock arithmetic and octave equivalence 318
9.7. Generators 319
9.8. Tone rows 321
9.9. Cartesian products 323
9.10. Dihedral groups 324
9.11. Orbits and cosets 326
9.12. Normal subgroups and quotients 327
9.13. Burnside’s lemma 329
9.14. Pitch class sets 331
9.15. P´olya’s enumeration theorem 335
9.16. The Mathieu group M12 340
