Link to excellent free book from the author: "Music - A Mathematical Offering"

[zorro]

Link to excellent free book from the author: "Music - A Mathematical Offering"

Hi all,

For all interested in mathematical aspects of music, I recommend this excellent book:

Music - A Mathematical Offering, by David Benson

An online version is freely available for download from the author's page:

**broken link removed**

Enjoy!

Z

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

 

[albbg]

Re: Link to excellent free book from the author: "Music - A Mathematical Offering"

Very interesting. Thanks

 

[yugal]

Re: Link to excellent free book from the author: "Music - A Mathematical Offering"

thanks zorro, very good book.
thanks again
yugal

 

[debjit625]

Re: Link to excellent free book from the author: "Music - A Mathematical Offering"

Thanks,and yes very interesting.