Muusikamatemaatika

Seisuga 21. detsember 2024

Siin toimub Muusikamatemaatika käsiraamatu väljatöötamine.

Projekti tegevuste planeerimine toimub lehel http://et.parnu.wikia.com/wiki/Muusikamatemaatika

Muusikamatemaatika käsiraamatu eesmärk on kirjeldada, kuidas muusikalised struktuurid põhinevad matemaatilistel mudelitel ning kuidas matemaatilistele mudelitele tuginedes muusikalisi struktuure luua.

Muusika ja matemaatika ühendamisel tuleb siiski mõista nende eripära: kui matemaatika eesmärk on üldistamine, siis muusika eesmärk on unikaalsus.

Muusikamatemaatika teemad

Muusikamatemaatika teemadeks võivad olla näiteks

heli kui füüsikalise nähtuse matemaatiline modelleerimine, sealhulgas spektraalanalüüs ja helisüntees

heli ruumiakustiline modelleerimine

heli psühhoakustiline modelleerimine

heliridade ja häälestuse matemaatiline modelleerimine

muusikalise struktuuri matemaatiline modelleerimine, sealhulgas muusikaline hulgateooria

heliloomingu protsessi matemaatiline modelleerimine, sealhulgas algoritmiline komponeerimine

helilise analoogsignaali digitaliseerimine

Muusika matemaatilised mudelid kuuluvad kolme valdkonda:

heliallikas ja matemaatika, mille puhul modelleeritakse muusikainstrumendi füüsikalisi omadusi;

heli ja matemaatika, mille puhul modelleeritakse heli füüsikalisi omadusi;

muusikaline struktuur ja matemaatika, mille puhul uuritakse muusika kompositsiooni- ja analüüsimudelite matemaatilist olemust, käsitledes heliteost reeglipärase või juhusliku protsessi tulemusena.

Muusikalise objekti matemaatilise mudeli kirjelduse struktuur

Muusikalise objekti matemaatilise mudeli nimetus (versioon pp.kk.aaaa)

Sõnaline kirjeldus
Valem
Joonis
Simulatsioon
Rakendused
Seotud mõisted
Viited

Muusikaliste objektide matemaatiliste mudelite tähestikuline loend

Muusikalisi objekte on loendatud näiteks Vikipeedia loendites Muusika mõisteid, Nüüdismuusika mõisteid ja Elektroonilise muusika mõisteid.

Kirjandus

https://drive.google.com/drive/folders/0BzY5z6WLl9t-NC1YWUdYd05QZzQ?usp=sharing

Garet Loy. Musimatics

http://www.musimathics.com/

Sisukord:

Volume I

Foreword by Max Mathews xiii

Preface xv

About the Author xvi

Acknowledgments xvii

1 Music and Sound 1

1.1 Basic Properties of Sound 1

1.2 Waves 3

1.3 Summary 9

2 Representing Music 11

2.1 Notation 11

2.2 Tones, Notes, and Scores 12

2.3 Pitch 13

2.4 Scales 16

2.5 Interval Sonorities 18

2.6 Onset and Duration 26

2.7 Musical Loudness 27

2.8 Timbre 28

2.9 Summary 37

3 Musical Scales, Tuning, and Intonation 39

3.1 Equal-Tempered Intervals 39

3.2 Equal-Tempered Scale 40

3.3 Just Intervals and Scales 43

3.4 The Cent Scale 45

3.5 A Taxonomy of Scales 46

3.6 Do Scales Come from Timbre or Proportion? 47

3.7 Harmonic Proportion 48

3.8 Pythagorean Diatonic Scale 49

3.9 The Problem of Transposing Just Scales 51

3.10 Consonance of Intervals 56

3.11 The Powers of the Fifth and the Octave Do Not Form a Closed System 66

3.12 Designing Useful Scales Requires Compromise 67

3.13 Tempered Tuning Systems 68

3.14 Microtonality 72

3.15 Rule of 18 82

3.16 Deconstructing Tonal Harmony 85

3.17 Deconstructing the Octave 86

3.18 The Prospects for Alternative Tunings 93

3.19 Summary 93

3.20 Suggested Reading 95

4 Physical Basis of Sound 97

4.1 Distance 97

4.2 Dimension 97

4.3 Time 98

4.4 Mass 99

4.5 Density 100

4.6 Displacement 100

4.7 Speed 101

4.8 Velocity 102

4.9 Instantaneous Velocity 102

4.10 Acceleration 104

4.11 Relating Displacement,Velocity, Acceleration, and Time 106

4.12 Newton's Laws of Motion 108

4.13 Types of Force 109

4.14 Work and Energy 110

4.15 Internal and External Forces 112

4.16 The Work-Energy Theorem 112

4.17 Conservative and Nonconservative Forces 113

4.18 Power 114

4.19 Power of Vibrating Systems 114

4.20 Wave Propagation 116

4.21 Amplitude and Pressure 117

4.22 Intensity 118

4.23 Inverse Square Law 118

4.24 Measuring Sound Intensity 119

4.25 Summary 125

5 Geometrical Basis of Sound 129

5.1 Circular Motion and Simple Harmonic Motion 129

5.2 Rotational Motion 129

5.3 Projection of Circular Motion 136

5.4 Constructing a Sinusoid 139

5.5 Energy of Waveforms 143

5.6 Summary 147

6 Psychophysical Basis of Sound 149

6.1 Signaling Systems 149

6.2 The Ear 150

6.3 Psychoacoustics and Psychophysics 154

6.4 Pitch 156

6.5 Loudness 166

6.6 Frequency Domain Masking 171

6.7 Beats 173

6.8 Combination Tones 175

6.9 Critical Bands 176

6.10 Duration 182

6.11 Consonance and Dissonance 184

6.12 Localization 187

6.13 Externalization 191

6.14 Timbre 195

6.15 Summary 198

6.16 Suggested Reading 198

7 Introduction to Acoustics 199

7.1 Sound and Signal 199

7.2 A Simple Transmission Model 199

7.3 How Vibrations Travel in Air 200

7.4 Speed of Sound 202

7.5 Pressure Waves 207

7.6 Sound Radiation Models 208

7.7 Superposition and Interference 210

7.8 Reflection 210

7.9 Refraction 218

7.10 Absorption 221

7.11 Diffraction 222

7.12 Doppler Effect 228

7.13 Room Acoustics 233

7.14 Summary 238

7.15 Suggested Reading 238

8 Vibrating Systems 239

8.1 Simple Harmonic Motion Revisited 239

8.2 Frequency of Vibrating Systems 241

8.3 Some Simple Vibrating Systems 243

8.4 The Harmonic Oscillator 247

8.5 Modes of Vibration 249

8.6 A Taxonomy of Vibrating Systems 251

8.7 One-Dimensional Vibrating Systems 252

8.8 Two-Dimensional Vibrating Elements 266

8.9 Resonance (Continued) 270

8.10 Transiently Driven Vibrating Systems 278

8.11 Summary 282

8.12 Suggested Reading 283

9 Composition and Methodology 285

9.1 Guido's Method 285

9.2 Methodology and Composition 288

9.3 Musimat: A Simple Programming Language for Music 290

9.4 Program for Guido's Method 291

9.5 Other Music Representation Systems 292

9.6 Delegating Choice 293

9.7 Randomness 299

9.8 Chaos and Determinism 304

9.9 Combinatorics 306

9.10 Atonality 311

9.11 Composing Functions 317

9.12 Traversing and Manipulating Musical Materials 319

9.13 Stochastic Techniques 332

9.14 Probability 333

9.15 Information Theory and the Mathematics of Expectation 343

9.16 Music, Information, and Expectation 347

9.17 Form in Unpredictability 350

9.18 Monte Carlo Methods 360

9.19 Markov Chains 363

9.20 Causality and Composition 371

9.21 Learning 372

9.22 Music and Connectionism 376

9.23 Representing Musical Knowledge 390

9.24 Next-Generation Musikalische Würfelspiel 400

9.25 Calculating Beauty 406

Appendix A 409

A.1 Exponents 409

A.2 Logarithms 409

A.3 Series and Summations 410

A.4 About Trigonometry 411

A.5 Xeno's Paradox 414

A.6 Modulo Arithmetic and Congruence 414

A.7 Whence 0.161 in Sabine's Equation? 416

A.8 Excerpts from Pope John XXII's Bull Regarding Church Music 418

A.9 Greek Alphabet 419

Appendix B 421

B.1 Musimat 421

B.2 Music Datatypes in Musimat 439

B.3 Unicode (ASCII) Character Codes 450

B.4 Operator Associativity and Precedence in Musimat 450

Glossary 453

Notes 459

References 465

Equation Index 473

Subject Index

Volume II

Foreword by John Chowning

Preface

1 Digital Signals and Sampling

1.1 Measuring the Ephemeral

1.2 Analog-to-digital Conversion

1.3 Aliasing

1.4 Digital-to-analog Conversion

1.5 Binary Numbers

1.6 Synchronization

1.7 Discretization

1.8 Precision and Accuracy

1.9 Quantization

1.10 Noise and Distortion

1.11 Information Density of Digital Audio

1.12 Codecs

1.13 Further Refinements

1.14 Cultural Impact of Digital Audio

1.15 Summary

2 Musical Signals

2.1 Why Imaginary Numbers?

2.2 Operating with Imaginary Numbers

2.3 Complex Numbers

2.4 de Moivre's Theorem

2.5 Euler's Formula

2.6 Phasors

2.7 Graphing Complex Signals

2.8 Spectra of Complex Sampled Signals

2.9 Multiplying Phasors

2.10 Graphing Complex Spectra

2.11 Analytic Signals

2.12 Summary

3 Spectral Analysis and Synthesis

3.1 Introduction to the Fourier Transform

3.2 Discrete Fourier Transform

3.3 The DFT in Action

3.4 The Inverse Discrete Fourier Transform

3.5 Analyzing Real-world Signals

3.6 Windowing

3.7 Fast Fourier Transform

3.8 Properties of the Discrete Fourier Transform

3.9 A Practical Hilbert Transform

3.10 Summary

4 Convolution

4.1 The Rolling Shutter Camera

4.2 Defining Convolution

4.3 Numerical Examples of Convolution

4.4 Convolving Spectra

4.5 Convolving Signals

4.6 Convolution and the Fourier Transform

4.7 Using the FFT for Convolution

4.8 The Domain Symmetry between Signals and Spectra

4.9 Convolution and Sampling Theory

4.10 Convolution and Windowing

4.11 Correlation Functions

4.12 Summary

4.13 Suggested Reading

5 Filtering

5.1 Tape Recorder as a Model of Filtering

5.2 Introduction to Filtering

5.3 A Simple Filter

5.4 Finding the Frequency Response

5.5 Linearity and Time Invariance of Filters

5.6 FIR Filters

5.7 IIR Filters

5.8 Canonical Filter

5.9 Time-Domain Behavior of Filters

5.10 Filtering as Convolution

5.11 The Z Transform

5.12 The Z Transform of the General Difference Equation

5.13 Filter Families

5.14 Summary

6 Resonance

6.1 The Derivative

6.2 Differential Equations

6.3 Transient Vibrations

6.4 Mathematics of Resonance

6.5 Summary

7 Wave Equation

7.1 One-dimensional Wave Equation and String Motion

7.2 An Example

7.3 Modeling Vibration with Finite Difference Equations

7.4 Striking Points, Plucking Points, and Spectra

7.5 Summary

8 Acoustical Systems

8.1 Dissipation and Radiation

8.2 Acoustical Current

8.3 Linearity of Frictional Force

8.4 Inertance, Inductive Reactance

8.5 Compliance, Capacitive Reactance

8.6 Reactance and Alternating Current

8.7 Capacitive Reactance and Frequency

8.8 Inductive Reactance and Frequency

8.9 Combining Resistance, Reactance and Alternating Current

8.10 Resistance and Alternating Current

8.11 Capacitance and alternating current

8.12 Acoustical Impedance

8.13 Sound Propagation and Sound Transmission

8.14 Input Impedance: Fingerprinting a Resonant System

8.15 Scattering Junctions

8.16 Summary

8.17 Suggested Reading

9 Sound Synthesis

9.1 Forms of Synthesis

9.2 A Graphical Patch Language for Synthesis

9.3 Amplitude Modulation

9.4 Frequency Modulation

9.5 Vocal Synthesis

9.6 Synthesizing Concert Hall Acoustics

9.7 Physical Modeling

9.8 Source Models and Receiver Models

9.9 Summary

10 Dynamic Spectra

10.1 Gabor's Elementary Signal

10.2 The Short-time Fourier Transform

10.3 Phase Vocoder

10.4 Improving on the Fourier Transform

10.5 Summary

10.6 Suggested Reading

10.7 Foundations

11 Epilogue

Appendix

A.1 About Algebra

A.2 About Trigonometry

A.3 Series and Summations

A.4 Trigonometric Identities

A.5 Modulo Arithmetic And Congruence

A.6 Finite Difference Approximations

A.7 Walsh-Hadamard Transform

A.8 Sampling, Reconstruction, and the Sinc Function

A.9 Fourier Shift Theorem

A.10 Spectral Effects of Ring Modulation

Glossary

Equation Index

Subject Index

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

http://www.ester.ee/record=b2448376*est

Sisukord:

Part I: Music and mathematis through histroy

Part II: The mathematics of musical sound

Part III: Mathematical structure in music

Part IV: The composer speaks (mikrotonaalsus ja fraktalid muusikas)

Leon Harkleroad. The Math behind the Music

http://www.ester.ee/record=b2198763*est

Sisukord:

1 Mathematics and Music, a Duet

2 Pitch: The Ground of Music

3 Tuning Up

4 How to Vary a Theme Mathematically

5 Bells and Groups

6 Music by Chance

7 Pattern, Pattern, Pattern

8 Sight Meets Sound

9 How NOT to Mix Mathematics and M;usic

Raammatu lisa: näidete CD

David Wright. Mathematics and Music

http://www.ester.ee/record=b2749919*est

Sisukord:

1 Basic Mathematical and Musical Concepts

2 Horizontal Structures

3 Harmony and Related Numerolgy

4 Ratios and Musical Intervals

5 Logarithms and Musical Intervals

6 Chromatic Scales

7 Octave Idendification and Modular Arithmetic

8 Algebraic Properties of the Integers

9 The Integers as Intervals

10 Timbre and Periodic Functions

11 The Rational Numbers as Musical Intervals

12 Tuning the Scale to Obtain Rational Intervals