Music A Mathematical Offering 1st Edition by Dave Benson ISBN 0521619998 9780521619998

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ISBN 10: 0521619998
ISBN 13: 9780521619998
Author: Dave Benson

Music A Mathematical Offering 1st Table of contents:

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 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

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 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
 5.9 Eitz’s notation
 5.10 Examples of just scales
 5.11 Classical harmony
 5.12 Meantone scale
 5.13 Irregular temperaments
 5.14 Equal temperament
 5.15 Historical remarks

6. More Scales and Temperaments

 6.1 Harry Partch’s 43 tone and other 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 modelling
 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 Chebyshev 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 Orbits and cosets
 9.12 Normal subgroups and quotients
 9.13 Burnside’s lemma
 9.14 Pitch class sets
 9.15 Pólya’s enumeration theorem
 9.16 The Mathieu group M12

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