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Foreword
Max V. Mathews |
xiii |
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Preface |
xv |
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About the Author |
xvi |
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Acknowledgments |
xvii |
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1. |
Music and Sound |
1 |
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1.1 |
Basic Properties of Sound |
1 |
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1.2 |
Waves |
3 |
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1.3 |
Summary |
9 |
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2. |
Representing Music |
11 |
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2.1 |
Notation |
11 |
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2.2 |
Tones, Notes, and Scores |
12 |
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2.3 |
Pitch |
13 |
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2.4 |
Scales |
16 |
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2.5 |
Interval Sonorities |
18 |
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2.6 |
Onset and Duration |
26 |
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2.7 |
Musical Loudness |
27 |
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2.8 |
Timbre |
28 |
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2.9 |
Summary |
37 |
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3. |
Musical Scales, Tuning, and Intonation |
39 |
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3.1 |
Equal-Tempered Intervals |
39 |
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3.2 |
Equal-Tempered Scale |
40 |
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3.3 |
Just Intervals and Scales |
43 |
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3.4 |
The Cent Scale |
45 |
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3.5 |
A Taxonomy of Scales |
46 |
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3.6 |
Do Scales Come from Timbre or Proportion? |
47 |
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3.7 |
Harmonic Proportion |
48 |
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3.8 |
Pythagorean Diatonic Scale |
49 |
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3.9 |
The Problem of Transposing Just Scales |
51 |
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3.10 |
Consonance of Intervals |
56 |
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3.11 |
The Powers of the Fifth and the Octave Do Not Form a Closed System |
66 |
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3.12 |
Designing Useful Scales Requires Compromise |
67 |
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3.13 |
Tempered Tuning Systems |
68 |
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3.14 |
Microtonality |
72 |
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3.15 |
Rule of 18 |
82 |
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3.16 |
Deconstructing Tonal Harmony |
85 |
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3.17 |
Deconstructing the Octave |
86 |
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3.18 |
The Prospects for Alternative Tunings |
93 |
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3.19 |
Summary |
93 |
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3.20 |
Suggested Reading |
95 |
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4. |
Physical Basis of Sound |
97 |
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4.1 |
Distance |
97 |
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4.2 |
Dimension |
97 |
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4.3 |
Time |
98 |
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4.4 |
Mass |
99 |
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4.5 |
Density |
100 |
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4.6 |
Displacement |
100 |
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4.7 |
Speed |
101 |
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4.8 |
Velocity |
102 |
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4.9 |
Instantaneous Velocity |
102 |
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4.10 |
Acceleration |
104 |
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4.11 |
Relating Displacement, Velocity, Acceleration, and Time |
106 |
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4.12 |
Newton's Laws of Motion |
108 |
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4.13 |
Types of Force |
109 |
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4.14 |
Work and Energy |
110 |
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4.15 |
Internal and External Forces |
112 |
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4.16 |
The Work-Energy Theorem |
112 |
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4.17 |
Conservative and Nonconservative Forces |
113 |
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4.18 |
Power |
114 |
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4.19 |
Power of Vibrating Systems |
114 |
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4.20 |
Wave Propagation |
116 |
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4.21 |
Amplitude and Pressure |
117 |
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4.22 |
Intensity |
118 |
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4.23 |
Inverse Square Law |
118 |
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4.24 |
Measuring Sound Intensity |
119 |
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4.25 |
Summary |
125 |
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5. |
Geometrical Basis of Sound |
129 |
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5.1 |
Circular Motion and Simple Harmonic Motion |
129 |
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5.2 |
Rotational Motion |
129 |
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5.3 |
Projection of Circular Motion |
136 |
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5.4 |
Constructing a Sinusoid |
139 |
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5.5 |
Energy of Waveforms |
143 |
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5.6 |
Summary |
147 |
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6. |
Psychophysical Basis of Sound |
149 |
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6.1 |
Signaling Systems |
149 |
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6.2 |
The Ear |
150 |
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6.3 |
Psychoacoustics and Psychophysics |
154 |
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6.4 |
Pitch |
156 |
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6.5 |
Loudness |
166 |
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6.6 |
Frequency Domain Masking |
171 |
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6.7 |
Beats |
173 |
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6.8 |
Combination Tones |
175 |
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6.9 |
Critical Bands |
176 |
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6.10 |
Duration |
182 |
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6.11 |
Consonance and Dissonance |
184 |
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6.12 |
Localization |
187 |
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6.13 |
Externalization |
191 |
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6.14 |
Timbre |
195 |
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6.15 |
Summary |
198 |
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6.16 |
Suggested Reading |
198 |
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7. |
Introduction to Acoustics |
199 |
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7.1 |
Sound and Signal |
199 |
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7.2 |
A Simple Transmission Model |
199 |
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7.3 |
How Vibrations Travel in Air |
200 |
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7.4 |
Speed of Sound |
202 |
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7.5 |
Pressure Waves |
207 |
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7.6 |
Sound Radiation Models |
208 |
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7.7 |
Superposition and Interference |
210 |
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7.8 |
Reflection |
210 |
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7.9 |
Refraction |
218 |
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7.10 |
Absorption |
221 |
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7.11 |
Diffraction |
222 |
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7.12 |
Doppler Effect |
228 |
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7.13 |
Room Acoustics |
233 |
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7.14 |
Summary |
238 |
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7.15 |
Suggested Reading |
238 |
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8. |
Vibrating Systems |
239 |
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8.1 |
Simple Harmonic Motion Revisited |
239 |
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8.2 |
Frequency of Vibrating Systems |
241 |
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8.3 |
Some Simple Vibrating Systems |
243 |
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8.4 |
The Harmonic Oscillator |
247 |
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8.5 |
Modes of Vibration |
249 |
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8.6 |
A Taxonomy of Vibrating Systems |
251 |
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8.7 |
One-Dimensional Vibrating Systems |
252 |
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8.8 |
Two-Dimensional Vibrating Elements |
266 |
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8.9 |
Resonance (Continued) |
270 |
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8.10 |
Transiently Driven Vibrating Systems |
278 |
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8.11 |
Summary |
282 |
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8.12 |
Suggested Reading |
283 |
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9. |
Composition and Methodology |
285 |
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9.1 |
Guido's Method |
285 |
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9.2 |
Methodology and Composition |
288 |
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9.3 |
MUSIMAT: A Simple Programming Language for Music |
290 |
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9.4 |
Program for Guido's Method |
291 |
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9.5 |
Other Music Representation Systems |
292 |
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9.6 |
Delegating Choice |
293 |
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9.7 |
Randomness |
299 |
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9.8 |
Chaos and Determinism |
304 |
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9.9 |
Combinatorics |
306 |
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9.10 |
Atonality |
311 |
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9.11 |
Composing Functions |
317 |
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9.12 |
Traversing and Manipulating Musical Materials |
319 |
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9.13 |
Stochastic Techniques |
332 |
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9.14 |
Probability |
333 |
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9.15 |
Information Theory and the Mathematics of Expectation |
343 |
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9.16 |
Music, Information, and Expectation |
347 |
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9.17 |
Form in Unpredictability |
350 |
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9.18 |
Monte Carlo Methods |
360 |
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9.19 |
Markov Chains |
363 |
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9.20 |
Causality and Composition |
371 |
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9.21 |
Learning |
372 |
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9.22 |
Music and Connectionism |
376 |
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9.23 |
Representing Musical Knowledge |
390 |
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9.24 |
Next-Generation Musikalische Würfelspiel |
400 |
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9.25 |
Calculating Beauty |
406 |
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Appendix A |
409 |
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A.1 |
Exponents |
409 |
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A.2 |
Logarithms |
409 |
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A.3 |
Series and Summations |
410 |
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A.4 |
About Trigonometry |
411 |
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A.5 |
Xeno's Paradox |
414 |
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A.6 |
Modulo Arithmetic and Congruence |
414 |
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A.7 |
Whence 0.161 in Sabine's Equation? |
416 |
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A.8 |
Excerpts from Pope John XXII's Bull Regarding Church Music |
418 |
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A.9 |
Greek Alphabet |
419 |
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Appendix B |
421 |
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B.1 |
MUSIMAT |
421 |
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B.2 |
Music Datatypes in MUSIMAT |
439 |
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B.3 |
Unicode (ASCII) Character Codes |
450 |
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B.4 |
Operator Associativity and Precedence in MUSIMAT |
450 |
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Glossary |
453 |
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Notes |
459 |
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References |
465 |
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Equation Index |
473 |
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Subject Index |
475 |
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