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Music: A Mathematical Offering

Book Description

Since the time of the Ancient Greeks, much has been written about the relation between mathematics and music: from harmony and number theory, to musical patterns and group theory. Benson provides a wealth of information here to enable the teacher, the student, or the interested amateur to understand, at varying levels of technicality, the real interplay between these two ancient disciplines. The story is long as well as broad and involves physics, biology, psycho acoustics, the history of science, and digital technology as well as, of course, mathematics and music. Starting with the structure of the human ear and its relationship with Fourier analysis, the story proceeds via the mathematics of musical instruments to the ideas of consonance and dissonance, and then to scales and temperaments. This is a must-have book if you want to know about the music of the spheres or digital music and many things in between.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Preface
  6. Acknowledgements
  7. Introduction
  8. 1. Waves and harmonics
    1. 1.1 What is sound?
    2. 1.2 The human ear
    3. 1.3 Limitations of the ear
    4. 1.4 Why sine waves?
    5. 1.5 Harmonic motion
    6. 1.6 Vibrating strings
    7. 1.7 Sine waves and frequency spectrum
    8. 1.8 Trigonometric identities and beats
    9. 1.9 Superposition
    10. 1.10 Damped harmonic motion
    11. 1.11 Resonance
  9. 2. Fourier theory
    1. 2.1 Introduction
    2. 2.2 Fourier coefficients
    3. 2.3 Even and odd functions
    4. 2.4 Conditions for convergence
    5. 2.5 The Gibbs phenomenon
    6. 2.6 Complex coefficients
    7. 2.7 Proof of Fejér’s theorem
    8. 2.8 Bessel functions
    9. 2.9 Properties of Bessel functions
    10. 2.10 Bessel’s equation and power series
    11. 2.11 Fourier series for FM feedback and planetary motion
    12. 2.12 Pulse streams
    13. 2.13 The Fourier transform
    14. 2.14 Proof of the inversion formula
    15. 2.15 Spectrum
    16. 2.16 The Poisson summation formula
    17. 2.17 The Dirac delta function
    18. 2.18 Convolution
    19. 2.19 Cepstrum
    20. 2.20 The Hilbert transform and instantaneous frequency
  10. 3. A mathematician’s guide to the orchestra
    1. 3.1 Introduction
    2. 3.2 The wave equation for strings
    3. 3.3 Initial conditions
    4. 3.4 The bowed string
    5. 3.5 Wind instruments
    6. 3.6 The drum
    7. 3.7 Eigenvalues of the Laplace operator
    8. 3.8 The horn
    9. 3.9 Xylophones and tubular bells
    10. 3.10 The mbira
    11. 3.11 The gong
    12. 3.12 The bell
    13. 3.13 Acoustics
  11. 4. Consonance and dissonance
    1. 4.1 Harmonics
    2. 4.2 Simple integer ratios
    3. 4.3 History of consonance and dissonance
    4. 4.4 Critical bandwidth
    5. 4.5 Complex tones
    6. 4.6 Artificial spectra
    7. 4.7 Combination tones
    8. 4.8 Musical paradoxes
  12. 5. Scales and temperaments: the fivefold way
    1. 5.1 Introduction
    2. 5.2 Pythagorean scale
    3. 5.3 The cycle of fifths
    4. 5.4 Cents
    5. 5.5 Just intonation
    6. 5.6 Major and minor
    7. 5.7 The dominant seventh
    8. 5.8 Commas and schismas
    9. 5.9 Eitz’s notation
    10. 5.10 Examples of just scales
    11. 5.11 Classical harmony
    12. 5.12 Meantone scale
    13. 5.13 Irregular temperaments
    14. 5.14 Equal temperament
    15. 5.15 Historical remarks
  13. 6. More scales and temperaments
    1. 6.1 Harry Partch’s 43 tone and other just scales
    2. 6.2 Continued fractions
    3. 6.3 Fifty-three tone scale
    4. 6.4 Other equal tempered scales
    5. 6.5 Thirty-one tone scale
    6. 6.6 The scales of Wendy Carlos
    7. 6.7 The Bohlen–Pierce scale
    8. 6.8 Unison vectors and periodicity blocks
    9. 6.9 Septimal harmony
  14. 7. Digital music
    1. 7.1 Digital signals
    2. 7.2 Dithering
    3. 7.3 WAV and MP3 files
    4. 7.4 MIDI
    5. 7.5 Delta functions and sampling
    6. 7.6 Nyquist’s theorem
    7. 7.7 The z-transform
    8. 7.8 Digital filters
    9. 7.9 The discrete Fourier transform
    10. 7.10 The fast Fourier transform
  15. 8. Synthesis
    1. 8.1 Introduction
    2. 8.2 Envelopes and LFOs
    3. 8.3 Additive synthesis
    4. 8.4 Physical modelling
    5. 8.5 The Karplus–Strong algorithm
    6. 8.6 Filter analysis for the Karplus–Strong algorithm
    7. 8.7 Amplitude and frequency modulation
    8. 8.8 The Yamaha DX7 and FM synthesis
    9. 8.9 Feedback, or self-modulation
    10. 8.10 CSound
    11. 8.11 FM synthesis using CSound
    12. 8.12 Simple FM instruments
    13. 8.13 Further techniques in CSound
    14. 8.14 Other methods of synthesis
    15. 8.15 The phase vocoder
    16. 8.16 Chebyshev polynomials
  16. 9. Symmetry in music
    1. 9.1 Symmetries
    2. 9.2 The harp of the Nzakara
    3. 9.3 Sets and groups
    4. 9.4 Change ringing
    5. 9.5 Cayley’s theorem
    6. 9.6 Clock arithmetic and octave equivalence
    7. 9.7 Generators
    8. 9.8 Tone rows
    9. 9.9 Cartesian products
    10. 9.10 Dihedral groups
    11. 9.11 Orbits and cosets
    12. 9.12 Normal subgroups and quotients
    13. 9.13 Burnside’s lemma
    14. 9.14 Pitch class sets
    15. 9.15 Pólya’s enumeration theorem
    16. 9.16 The Mathieu group M12
  17. Appendix A Bessel functions
  18. Appendix B Equal tempered scales
  19. Appendix C Frequency and MIDI chart
  20. Appendix D Intervals
  21. Appendix E Just, equal and meantone scales compared
  22. Appendix F Music theory
  23. Appendix G Recordings
  24. References
  25. Bibliography
  26. Index