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Music : a mathematical offering / Dave Benson.

By: Benson, D. J. (David J.), 1955- [author.].
Publisher: Cambridge : Cambridge University Press, 2007Description: 1 online resource (xiii, 411 pages) : digital, PDF file(s).Content type: text Media type: computer Carrier type: online resourceISBN: 9780511811722 (ebook).Subject(s): Music -- Acoustics and physics | Music theory -- MathematicsAdditional physical formats: Print version: : No titleDDC classification: 781.2 Online resources: Click here to access online
Contents:
Waves and harmonics -- What is sound? -- The human ear -- Limitations of the ear -- Why sine waves? -- Harmonic motion -- Vibrating strings -- Sine waves and frequency spectrum -- Trigonometric identities and beats -- Superposition -- Damped harmonic motion -- Resonance -- Fourier theory -- Introduction -- Fourier coefficients -- Even and odd unctions -- Conditions for convergence -- The Gibbs phenomenon -- Complex coefficients -- Proof of Fejér's theorem -- Bessel functions -- Properties of Bessel functions -- Bessel's equation and power series -- Fourier series for FM feedback and planetary motion -- Pulse streams -- The Fourier transform -- Proof of the inversion formula -- Spectrum -- The Poisson summation formula -- The Dirac delta function -- Convolution -- Cepstrum -- The Hilbert transform and instantaneous frequency -- A mathematician's guide to the orchestra -- Introduction -- The wave equation for strings -- Initial conditions -- The bowed string -- Wind instruments -- The drum -- Eigenvalues of the Laplace operator -- The horn -- Xylophones and tubular bells -- The mbira -- The gong -- The bell -- Acoustics.
Symmetry in music -- Symmetries -- The harp of the Nzakara -- Sets and groups -- Change ringing -- Cayley's theorem -- Clock arithmetic and octave equivalence -- Generators -- Tone rows -- Cartesian products -- Dihedral groups -- Orbits and cosets -- Normal subgroups and quotients -- Burnside's lemma -- Pitch class sets -- Pólya's enumeration theorem -- The Mathieu group M₁₂ -- Appendix A : Bessel functions -- Appendix B : Equal tempered scales -- Appendix C : Frequency and MIDI chart -- Appendix D : Intervals -- Appendix E : Just, equal and meantone scales compared -- Appendix F : Music theory -- Appendix G : Recordings --
Digital music -- Digital signals -- Dithering -- WAV and MP3 files -- MIDI -- Delta functions and sampling -- Nyquist's theorem -- The z-transform -- Digital filters -- The discrete Fourier transform -- The fast Fourier transform -- Synthesis -- Introduction -- Envelopes and LFOs -- Additive synthesis -- Physical modelling -- The Karplus-Strong algorithm -- Filter analysis for the Karplus-Strong algorithm -- Amplitude and frequency modulation -- The Yamaha DX7 and FM synthesis -- Feedback, or self-modulation -- CSound -- FM synthesis using CSound -- Simple FM instruments -- Further techniques in CSound -- Other methods of synthesis -- The phase vocoder -- Chebyshev polynomials.
Consonance and dissonance -- Harmonics -- Simple integer rations -- History of consonance and dissonance -- Critical bandwidth -- Complex tones -- Artificial spectra -- Combination tones -- Musical paradoxes -- Scales and temperaments : the fivefold way -- Introduction -- Pythagorean scale -- The cycle of fifths -- Cents -- Just intonation-- Major and minor -- The dominant seventh -- Commas and schismas -- Eitz's notation -- Examples of just scales -- Classical harmony -- Meantone scale -- Irregular temperaments -- Equal temperament -- Historical remarks -- More scales and temperaments -- Harry Partch's 43 tone and other just scales -- Continued fractions -- Fifty-three tempered scales -- Thirty-one tone scale -- The scales of Wendy Carlos -- The Bohlen-Pierce scale -- Union vectors and periodicity blocks -- Septimal harmony.
Summary: 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.
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Preface -- Acknowledgements -- Introduction -- 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 unctions -- 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 -- 1.11. 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.

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 -- Appendix A : Bessel functions -- Appendix B : Equal tempered scales -- Appendix C : Frequency and MIDI chart -- Appendix D : Intervals -- Appendix E : Just, equal and meantone scales compared -- Appendix F : Music theory -- Appendix G : Recordings -- References -- Bibliography -- Index.

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.

4. Consonance and dissonance -- 4.1. Harmonics -- 4.2. Simple integer rations -- 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 tempered scales -- 6.5. Thirty-one tone scale -- 6.6. The scales of Wendy Carlos -- 6.7. The Bohlen-Pierce scale -- 6.8. Union vectors and periodicity blocks -- 6.9. Septimal harmony.

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.

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