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Musimathics: The Mathematical Foundations of Music, Volume 1

Gareth Loy
Publisher: 
MIT Press
Publication Date: 
2006
Number of Pages: 
482
Format: 
Hardcover
Price: 
50.00
ISBN: 
0262122820
Category: 
Monograph
[Reviewed by
Sarah Boslaugh
, on
11/27/2006
]

Musimathics presents a thorough discussion of the mathematical and physical bases of music, from the basic properties of sounds through advanced topics such as acoustics, psychophysics, and mathematical systems of composition. The author, Gareth Loy, states that he once considered subtitling this book “Everything I wanted to know about music when I was eleven” because at that age he was frustrated by his inability to understand the mathematics which lie behind the phenomenon of music. So in one sense Musimathics is written for all the people who have also wished (even if they are no longer 11 years old!) that they could understand the technical bases of music but lack the necessary mathematical background: it meets them halfway by presenting the necessary technical information along with its specific application to music, and conveys that information without assuming a background beyond advanced high school algebra and a basic knowledge of music.

Loy is an admirably clear writer who has organized his material in a logical and systematic manner. He clearly believes, as he states in the preface, that “Mathematics can be as effortless as humming a tune, if you know the tune” and does not talk down to his readers or oversimplify his material. Although advanced mathematical knowledge is not necessary to understand the material presented in Musimathics, a healthy degree of persistence and self-confidence are required. This is a serious book which demands that the reader pay attention and put forth some intellectual effort if they want to understand the content: the casual reader may soon find themselves skipping large chunks of text as the topics become more arcane, the explanations increasingly technical and the formulas and diagrams increasingly complex. Readers seeking a more popular presentation of the relationship between mathematics and music would be better served by a book such as Harkleroad’s The Math Behind the Music.

However, readers who want a technical understanding of music and are willing to devote some effort to that topic will find that the time invested in working through Musimathics will be rewarded with increased understanding and appreciation of both music and mathematics. People working in electronic music will find it particularly useful. Students of mathematics with an interest in music will also find much to interest them within Musimathics. Additional material related to Musimathics, including a downloadable version of Loy’s programming language Musimat (discussed in chapter 9), is available from http://www.musimathics.com/.

Gareth Loy is a performing musician, composer, software architect and digital audio systems engineer who received his DMA in composition from Stanford University in 1980. Further information about his musical and engineering activities is available from Loy’s personal web page, http://www.garethloy.com/ and the web page for his consulting company, http://www.garethinc.com/.


Sarah Boslaugh is a Performance Analyst for BJC HealthCare and an adjunct professor at the Washington University School of Medicine in St. Louis, Missouri. She has written two books, An Intermediate Guide to SPSS Programming: Using Syntax for Data Management (SAGE, 2005) and Secondary Data Sources for Public Health: A Practical Guide (Cambridge University Press, forthcoming May 2007) and is editor-in-chief of The Encyclopedia of Epidemiology (forthcoming from Sage, November 2007).

 


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 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 475
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