Spis treści: Musimathics: (Volume 1)


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 239

8 Vibrating Systems 239

  • 8.1 Simple Harmonic Motion Revisited  241
  • 8.2 Frequency of Vibrating Systems  243
  • 8.3 Some Simple Vibrating Systems 247
  • 8.4 The Harmonic Oscillator  249
  • 8.5 Modes of Vibration 251
  • 8.6 A Taxonomy of Vibrating Systems 252
  • 8.7 One-Dimensional Vibrating Systems  266
  • 8.8 Two-Dimensional Vibrating Elements  270
  • 8.9 Resonance (Continued)  278
  • 8.10 Transiently Driven Vibrating Systems 282
  • 8.11 Summary 283
  • 8.12 Suggested Reading 285

9 Composition and Methodology 285

  • 9.1 Guido's Method  288
  • 9.2 Methodology and Composition 290
  • 9.3 MUSIMAT: A Simple Programming Language for Music 291
  • 9.4 Program for Guido's Method 292
  • 9.5 Other Music Representation Systems 293
  • 9.6 Delegating Choice 299
  • 9.7 Randomness 304
  • 9.8 Chaos and Determinism 306
  • 9.9 Combinatorics 311
  • 9.10 Atonality 317
  • 9.11 Composing Functions 319
  • 9.12 Traversing and Manipulating Musical Materials 332
  • 9.13 Stochastic Techniques 333
  • 9.14 Probability  343
  • 9.15 Information Theory and the Mathematics of Expectation
  • 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 Wurfelspiel 400
  • 9.25 Calculating Beauty 406

Appendix A 409

  • A.l   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. l MUSIMAT 421 B.2 Music Datatypes in MUSIMAT 439 B.3 Unicode (ASCII) Character Codes 450 B.4   Operator Associativity an Precedence in MUSIMAT 450

Glossary 453
Notes 459
References 465
Equation Index 473
Subject Index 475

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