Redaktsioon: 3. oktoober 2016, kell 14:51 redigeeri Andrus Kallastu (arutelu \| kaastöö) 7286 muudatust →‎Kirjandus ← Vanem muudatus		Redaktsioon: 3. oktoober 2016, kell 15:03 redigeeri eemalda Andrus Kallastu (arutelu \| kaastöö) 7286 muudatust →‎Dave Benson. Music, a mathematical offering Uuem muudatus →
315. rida: ===Dave Benson. Music, a mathematical offering=== :Sisukord ::Preface ix ::Introduction ix ::Books xii :::Acknowledgements xiii :::Chapter 1. Waves and harmonics 1 :::1.1. What is sound? 1 :::1.2. The human ear 3 :::1.3. Limitations of the ear 8 :::1.4. Why sine waves? 13 :::1.5. Harmonic motion 14 :::1.6. Vibrating strings 15 :::1.7. Sine waves and frequency spectrum 16 :::1.8. Trigonometric identities and beats 18 :::1.9. Superposition 21 :::1.10. Damped harmonic motion 23 :::1.11. Resonance 26 :::Chapter 2. Fourier theory 30 :::2.1. Introduction 31 :::2.2. Fourier coefficients 31 :::2.3. Even and odd functions 37 :::2.4. Conditions for convergence 39 :::2.5. The Gibbs phenomenon 43 :::2.6. Complex coefficients 47 :::2.7. Proof of Fej´er’s Theorem 48 :::2.8. Bessel functions 50 :::2.9. Properties of Bessel functions 54 :::2.10. Bessel’s equation and power series 55 :::2.11. Fourier series for FM feedback and planetary motion 60 :::2.12. Pulse streams 63 :::2.13. The Fourier transform 64 :::2.14. Proof of the inversion formula 68 :::2.15. Spectrum 70 :::2.16. The Poisson summation formula 72 :::2.17. The Dirac delta function 73 :::2.18. Convolution 77 :::2.19. Cepstrum 79 :::2.20. The Hilbert transform and instantaneous frequency 80 :::2.21. Wavelets 81 :::Chapter 3. A mathematician’s guide to the orchestra 83 :::3.1. Introduction 83 :::3.2. The wave equation for strings 85 :::3.3. Initial conditions 91 :::3.4. The bowed string 94 :::3.5. Wind instruments 99 :::3.6. The drum 103 :::3.7. Eigenvalues of the Laplace operator 109 :::3.8. The horn 113 :::3.9. Xylophones and tubular bells 114 :::3.10. The mbira 122 :::3.11. The gong 124 :::3.12. The bell 129 :::3.13. Acoustics 133 :::Chapter 4. Consonance and dissonance 136 :::4.1. Harmonics 136 :::4.2. Simple integer ratios 137 :::4.3. History of consonance and dissonance 139 :::4.4. Critical bandwidth 142 :::4.5. Complex tones 143 :::4.6. Artificial spectra 144 :::4.7. Combination tones 147 :::4.8. Musical paradoxes 150 :::Chapter 5. Scales and temperaments: the fivefold way 153 :::5.1. Introduction 154 :::5.2. Pythagorean scale 154 :::5.3. The cycle of fifths 155 :::5.4. Cents 157 :::5.5. Just intonation 159 :::5.6. Major and minor 160 :::5.7. The dominant seventh 161 :::5.8. Commas and schismas 162 :::5.9. Eitz’s notation 164 :::5.10. Examples of just scales 165 :::5.11. Classical harmony 173 :::5.12. Meantone scale 176 :::5.13. Irregular temperaments 181 :::5.14. Equal temperament 190 :::5.15. Historical remarks 193 :::Chapter 6. More scales and temperaments 200 :::6.1. Harry Partch’s 43 tone and other just scales 200 :::6.2. Continued fractions 204 :::6.3. Fifty-three tone scale 213 :::6.4. Other equal tempered scales 217 :::6.5. Thirty-one tone scale 219 :::6.6. The scales of Wendy Carlos 221 :::6.7. The Bohlen–Pierce scale 224 :::6.8. Unison vectors and periodicity blocks 227 :::6.9. Septimal harmony 232 :::Chapter 7. Digital music 235 :::7.1. Digital signals 235 :::7.2. Dithering 237 :::7.3. WAV and MP3 files 238 :::7.4. MIDI 241 :::7.5. Delta functions and sampling 242 :::7.6. Nyquist’s theorem 244 :::7.7. The z-transform 246 :::7.8. Digital filters 247 :::7.9. The discrete Fourier transform 250 :::7.10. The fast Fourier transform 253 :::Chapter 8. Synthesis 255 :::8.1. Introduction 255 :::8.2. Envelopes and LFOs 256 :::8.3. Additive Synthesis 258 :::8.4. Physical modeling 260 :::8.5. The Karplus–Strong algorithm 262 :::8.6. Filter analysis for the Karplus–Strong algorithm 264 :::8.7. Amplitude and frequency modulation 265 :::8.8. The Yamaha DX7 and FM synthesis 268 :::8.9. Feedback, or self-modulation 274 :::8.10. CSound 278 :::8.11. FM synthesis using CSound 284 :::8.12. Simple FM instruments 286 :::8.13. Further techniques in CSound 290 :::8.14. Other methods of synthesis 292 :::8.15. The phase vocoder 293 :::8.16. Chebyshev polynomials 293 :::Chapter 9. Symmetry in music 296 :::9.1. Symmetries 296 :::9.2. The harp of the Nzakara 307 :::9.3. Sets and groups 310 :::9.4. Change ringing 314 :::9.5. Cayley’s theorem 317 :::9.6. Clock arithmetic and octave equivalence 319 :::9.7. Generators 320 :::9.8. Tone rows 322 :::9.9. Cartesian products 324 :::9.10. Dihedral groups 325 :::9.11. Orbits and cosets 327 :::9.12. Normal subgroups and quotients 328 :::9.13. Burnside’s lemma 330 :::9.14. Pitch class sets 332 :::9.15. P´olya’s enumeration theorem 336 :::9.16. The Mathieu group M12 341 :::Appendix A. Answers to almost all exercises 344 :::Appendix B. Bessel functions 360 :::Appendix C. Complex numbers 369 :::Appendix D. Dictionary 372 :::Appendix E. Equal tempered scales 377 :::Appendix F. Frequency and MIDI chart 379 :::Appendix I. Intervals 380 :::Appendix J. Just, equal and meantone scales compared 383 :::Appendix L. Logarithms 385 :::Appendix M. Music theory 389 :::Appendix O. Online papers 396 :::Appendix P. Partial derivatives 443 :::Appendix R. Recordings 446 :::Appendix W. The wave equation 451 :::Green’s identities 452 :::Gauss’ formula 452 :::Green’s functions 454 :::Hilbert space 455 :::The Fredholm alternative 457 :::Solving Laplace’s equation 459 :::Conservation of energy 462 :::Uniqueness of solutions 463 :::Eigenvalues are nonnegative and real 463 :::Orthogonality :::Inverting the Laplace operator 464 :::Compact operators 466 :::The inverse of the Laplace operator is compact 467 :::Eigenvalue stripping 468 :::Solving the wave equation 469 :::Polyhedra and finite groups 470 :::An example 471 :::Bibliography 477 :::Index 493 ===John Fauvel, Raymond Floods, Robin Wilson (ed.). Music and Mathematics. From Pythagoras to Fractals===

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