Music a Mathematical Offering
https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-music.html
The current online version (14 December 2008) is available FREE in pdf format
https://homepages.abdn.ac.uk/d.j.benson/pages/html/music.pdf
Current Chapter Headings (online version):
1. 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
2. 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 Fejér’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 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
2.21 Wavelets
3. 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 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
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. Digital music
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
7.9 The discrete Fourier transform
7.10 The fast Fourier transform
8. Synthesis
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. Symmetry in music
9.1 Symmetries
9.2 The harp of the Nzakara
9.3 Sets and groups
9.4 Change ringing
9.5 Cayley’s theorem
9.6 Clock arithmetic and octave equivalence
9.7 Generators
9.8 Tone rows
9.9 Cartesian products
9.10 Dihedral groups
9.11 Normal subgroups and quotients
9.12 Orbits and cosets
9.13 Burnside’s lemma
9.14 Pitch class sets
9.15 Pólya’s enumeration theorem
9.16 The Mathieu group M12
Appendices
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
Green’s identities
Gauss’ formula
Green’s functions
Hilbert space
The Fredholm alternative
Solving Laplace’s equation
Conservation of energy
Uniqueness of solutions
Eigenvalues are nonnegative and real
Orthogonality
Inverting the Laplace operator
Compact operators
The inverse of the Laplace operator is compact
Eigenvalue stripping
Solving the wave equation
Polyhedra and finite groups
An example
Bibliography
Index
