Enhanced descriptions from Syndetics:

"Mathematics can be as effortless as humming a tune, if you know the tune," writes Gareth Loy. In Musimathics , Loy teaches us the tune, providing a friendly and spirited tour of the mathematics of music -- a commonsense, self-contained introduction for the nonspecialist reader. It is designed for musicians who find their art increasingly mediated by technology, and for anyone who is interested in the intersection of art and science.

In Volume 1, Loy presents the materials of music (notes, intervals, and scales); the physical properties of music (frequency, amplitude, duration, and timbre); the perception of music and sound (how we hear); and music composition. Calling himself "a composer seduced into mathematics," Loy provides answers to foundational questions about the mathematics of music accessibly yet rigorously. The examples given are all practical problems in music and audio.

Additional material can be found at http://www.musimathics.com.

Table of contents provided by Syndetics

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

Reviews provided by Syndetics

CHOICE Review

Music and mathematics have a history of interaction; their roots run as deep as each separately. Of music's many dimensions (pitch, rhythm, timbre, form), each has a distinctive mathematical profile. A treatise on music-mathematics connections should embrace the ferocious diversity of the world's music without lapsing into empty generalities. Basic topics (arithmetic of tuning, physical modeling of musical instruments, psychophysics of auditory perception, propagation of sound through space) receive a clean treatment here, trenchant and widely applicable; but existing, albeit scattered, literature already covers similar material. The most interesting and distinctive part is in chapter 9, "Composition and Methodology," a third of the whole. Here, composer Loy sets out a variety of tools, many from research in artificial intelligence, for discussing music per se (and not merely sound in general). Computer models that can produce, e.g., credible chorale harmonization in the style of Bach must constitute a very concrete consolidation of music-theoretical knowledge, and yet these methods use general AI techniques not specific to Bach or even European music. Loy only surveys this material--readers wanting details will need the citations--but he develops the mathematics necessary to make this exciting literature universally accessible. Summing Up: Highly recommended. Upper-division undergraduates through faculty. D. V. Feldman University of New Hampshire