
Foreword
Max V. 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 
EqualTempered Intervals 
39 

3.2 
EqualTempered 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 WorkEnergy 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 
OneDimensional Vibrating Systems 
252 

8.8 
TwoDimensional 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 
NextGeneration 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 
475 
