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| Preface Acknowledgements Introduction Waves and harmonics 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 Sine waves and frequency spectrum 1.8 Trigonometric identities and beats 1.9 Superposition 1.10 Damped harmonic motion 1.11 Resonance Fourier theory 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 Fej6r'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 feedback and planetary motion 2.12 Pulse streams 2.13 The Fourier transform 2.14 Proof of the inversion formula 2.15 Spectrum 2.16 The Poisson summation formula 2.17 The Dirac delta function 2.18 Convolution 2.19 Cepstrum 2.20 The Hilbert transform and instantaneous frequency A mathematician's guide to the orchestra 3.1 Introduction 3.2 The wave equation for strings 3.3 Initial conditions 3.4 The bowed string 3.5 Wind instruments 3.6 The drum 3.7 Eigenvalues of the Laplace operator 3.8 The horn 3.9 Xylophones and tubular bells 3.10 The mbira 3.11 The gong 3.12 The bell 3.13 Acoustics 4 Consonance and dissonance 4.1 Harmonics 4.2 Simple integer ratios 4.3 History of consonance and dissonance 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 …… 6 More scales and temperaments 7 Digital music 8 Synthesis 9 Symmetry in music Appendix A Bessel functions Appendix B Equal temperde scales Appendix C Frequency and MIDI chart Appendix D Intervals Appendix E Just，equal and meantone scales comparde Appendix F Music theory Appendix G Recordings References Bibliography Index |
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