1.1 What is sound?
1.2 The human ear
1.3 Limitations of the ear
1.4 Why sine waves
1.5 Harmonic motion
1.6 Vibrating strings
1.7 Trigonometric identities
1.8 Superposition
1.9 Damped harmonic motion
1.10 Resonance
2.1 Introduction
2.2 Fourier coefficients
2.3 Even and odd functions
2.4 Conditions for convergence
2.5 The Gibbs phenomenon
2.6 Complex coefficients
2.7 Proof of Fejˇr's theorem
2.8 Bessel functions
2.9 Properties of Bessel functions
2.10 Bessel's equation and power series
2.11 Fourier series for FM synthesis and planetary motion
2.12 Pulse streams
2.13 The Fourier transform
2.14 The Poisson summation formula
2.15 The Dirac delta function
2.16 Convolution
2.17 Cepstrum
2.18 The Hilbert transform and instantaneous frequency
2.19 Wavelets
3. A mathematician's guide to the orchestra
3.1 The wave equation for strings
3.2 Initial conditions
3.3 Wind instruments
3.4 The horn
3.5 The drum
3.6 Eigenvalues of the Laplace operator
3.7 Xylophones and tubular bells
3.8 The gong
3.9 Acoustics
4.1 Harmonics
4.2 Simple integer ratios
4.3 Historical explanations of consonance
4.4 Critical bandwidth
4.5 Complex tones
4.6 Artificial spectra
4.7 Combination tones
4.8 Musical paradoxes
5. Scales and temperaments: the fivefold way
5.1 Introduction
5.2 Pythagorean scale
5.3 The cycle of fifths
5.4 Cents
5.5 Just intonation
5.6 Major and minor
5.7 The dominant seventh
5.8 Commas and schismas
5.9 Eitz's notation
5.10 Examples of just scales
5.11 Classical harmony
5.12 Meantone scale
5.13 Irregular temparaments
5.14 Equal temperament
5.15 Historical remarks
5.16 The role of the synthesizer
6. More scales and temperaments
6.1 Harry Partch's 43 tone and other super just scales
6.2 Continued fractions
6.3 Fifty-three tone scale
6.4 Other equal tempered scales
6.5 Thirty-one tone scale
6.6 The scales of Wendy Carlos
6.7 The Bohlen-Pierce scale
6.8 Unison vectors and periodicity blocks
6.9 Septimal harmony
7.1 Digital signals
7.2 Dithering
7.3 WAV and MP3 files
7.4 MIDI
7.5 Delta functions and sampling
7.6 Nyquist's theorem
7.7 The z-transform
7.8 Digital filters
7.8 The discrete Fourier transform
8.1 Introduction
8.2 Envelopes and LFOs
8.3 Additive synthesis
8.4 Physical modeling
8.5 The Karplus-Strong algorithm
8.6 Filter analysis for the Karplus-Strong algorithm
8.7 Amplitude and frequency modulation
8.8 The Yamaha DX7 and FM synthesis
8.9 Feedback, or self-modulation
8.10 CSound
8.11 FM synthesis using CSound
8.12 Simple FM instruments
8.13 Further techniques in CSound
8.14 Other methods of synthesis
8.15 The phase vocoder
8.16 Chebychev polynomials
9.1 Symmetries
9.2 Sets and groups
9.3 Change ringing
9.4 Cayley's theorem
9.5 Clock arithmetic and octave equivalence
9.6 Generators
9.7 Tone rows
9.8 Cartesian products
9.9 Dihedral groups
9.10 Normal subgroups and quotients
9.11 Orbits and cosets
9.12 Burnside's lemma
9.13 P—lya's enumeration theorem
9.14 The Mathieu group M12
Appendix A: Answers to Almost All Exercices
Appendix B: Bessel functions
Appendix C: Complex numbers
Appendix D: Dictionary
Appendix E: Equal tempered scales
Appendix F: Frequency and MIDI chart
Appendix G: Getting stuff from the internet
Appendix I: Intervals
Appendix J: Just, equal and meantone scales compared
Appendix L: Logarithms
Appendix M: Music theory
Appendix O: Online papers
Appendix P: Partial derivatives
Appendix R: Recordings
Appendix W: The wave equation
Bibliography
Index