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Gareth Loy

Musimathics, Volume 1: The Mathematical Foundations of Music (The MIT Press) Paperback – August 19, 2011

by Gareth Loy (Author)

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A commonsense, self-contained introduction to the mathematics and physics of music; essential reading for musicians, music engineers, and anyone interested in the intersection of art and science.

“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.

Print length

504 pages
Language

English
Publisher

The MIT Press
Publication date

August 19, 2011
Grade level

12 and up
Reading age

18 years and up
Dimensions

8.6 x 6.7 x 0.9 inches
ISBN-10

9780262516556
ISBN-13

978-0262516556
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Editorial Reviews

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Musimathics is destined to be required reading and a valued reference for every composer, music researcher, multimedia engineer, and anyone else interested in the interplay between acoustics and music theory. This is truly a landmark work of scholarship and pedagogy, and Gareth Loy presents it with quite remarkable rigor and humor.

―Stephen Travis Pope, CREATE Lab, Department of Music, University of California, Santa Barbara

About the Author

Gareth Loy is a musician and award-winning composer. He has published widely and, during a long and successful career at the cutting edge of multimedia computing, has worked as a researcher, lecturer, programmer, software architect, and digital systems engineer. He is President of Gareth, Inc., a provider of software engineering and consulting services internationally.

Product details

ASIN ‏ : ‎ 0262516551
Publisher ‏ : ‎ The MIT Press; Reprint edition (August 19, 2011)
Language ‏ : ‎ English
Paperback ‏ : ‎ 504 pages
ISBN-10 ‏ : ‎ 9780262516556
ISBN-13 ‏ : ‎ 978-0262516556
Reading age ‏ : ‎ 18 years and up
Grade level ‏ : ‎ 12 and up
Item Weight ‏ : ‎ 1.7 pounds
Dimensions ‏ : ‎ 8.6 x 6.7 x 0.9 inches

Best Sellers Rank: #791,353 in Books (See Top 100 in Books)
- #541 in Mathematics History
- #852 in Music Theory (Books)
- #1,871 in Music History & Criticism (Books)

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Excellent book combines music, math, and programming

Reviewed in the United States 🇺🇸 on January 10, 2007

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After about a ten year hiatus on books of this type being published, this is one of several new books combining mathematics, music, and programming aimed at musicians who want to know more about the math behind their musical compositions and are not content to just know what drop-down windows to click on using the latest musical software. The book starts with the basics of music and sound and works up to basic music theory, physics and sound, and acoustics and psychoacoustics. The final chapter of the book is the most interesting, since it concerns mathematics and composition techniques using the author's C++ based library "Musimat". Both this book and Musimat have companion websites, although the Musimat site is the most interesting with plenty of downloads in case you are interested in how to use this compositional library. There is a volume two scheduled for release in Spring 2007 that gets into signal processing, the role of digital signals, and the wave equation, so together they are a very complete treatise on math, music, and programming aimed at the musical composer. I highly recommend it. Of course, if you want to dig deep into individual subjects such as acoustics and psychoacoustics, you are going to need additional references. But this text is clear enough to get you started. The following is the table of contents:

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

64 people found this helpful

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Tommy Chess Coach

Extraordinary Beyond the Title, a must for all Math Lovers

Reviewed in the United States 🇺🇸 on June 17, 2010

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The sad thing about this series is that the keywords that invite readers to stop by, hide the fact that these texts go far beyond music, to USE music as a gentle introduction to extremely complex, relevant and timely math concepts. The best teachers use four paths to explain a math concept: verbal, formulaic, algorithmic and pictographic. These help the brain comprehend the topic regardless of our learning modality. The authors here are simply MASTERFUL math teachers, and clarify everything from Eulers Law (relation of e, the base of the natural logarithms to pi, the base of the trig functions) to Fourier Transforms, in a way that a bright High School student will get. If you've been out of math (any math) for a long time, and want a masterful review of math concepts and techniques, this series is THE place to start. You can then extend that foundation to many other applied areas, from signal processing to physics, voice recognition, etc. Fourier transforms (and their more recent spin off in Cepstrums) are being used in too many fields to list today, from radar and electronic engineering, to whale songs.

In every section, the author's excitement is contagious. Rather than give a bunch of dry proofs that reek of hubris and disregard for the reader, Gareth uses a "curious mind" tone, as if he were just learning and discovering this too, like a kind of puzzle or murder mystery. Loy is Monk, Holmes and Columbo combined. For example, he gives a few expansion series for e, then says: "Wow, there seems to be a striking and beautiful pattern here, doesn't there? Wonder what it can be?" Leave it to a guy into both math and music to see the wonder in a time series!

One more example. Any texts on waveforms have to involve deep calculus, especially PDE's. Unfortunately, deep PDE's don't happen until grad school. But, rather than assume the reader uses calculus all day long, Loy starts with the basics at "now let's see how the first derivative is actually slope finding and integration is the area covered by the moving curve..." including those perhaps more musically inclined who have forgotten what a derivative is. Astonishingly, Loy sneaks around the dry topic of limits to use MUSIC as a great practical refesher on calculus (p. 263 of the second volume, in the section that is the hottest topic in Physics today, from Astronomy to Medical Imaging to of course music: Resonance).

Gareth is one of the few mathematicians around who can relate math to the astonishment of life around us. After all, our brain is doing advanced Fourier Transforms every time we cross a street in traffic, and when we get an MRI, the Fourier Transforms that convert magnetic alignment to pictures are assuming that the atoms in our body are a song, which when pulsed with a radio wave, will sing the positions of their water molecules back to us in harmonics that can be seen as well as heard.

Highly recommend this series, not only for everyone interested in math and music, but math and life!

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MurrJamm

IN DEPTH LOOK AT ONE THING I KNOW WELL AND ANOTHER THING I DON'T KNOW AT ALL WELL ENOUGH

Reviewed in the United Kingdom 🇬🇧 on August 12, 2018

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I can see why there are many who love these books. . .unfortunately, they're over my head for the most part. (ESPECIALLY VOL 2). I've been a professional musician and composer for 30+ years, but having flunked mathematics at GCSE Level, (i.e. at 15-16 years old), I am having a GREAT deal of trouble comprehending much of this material. The books were recommended to me by a lecturer at Goldsmiths Uni, London; not his fault at all for not knowing that I didn't have a solid mathematical skillset. . .but I read about the books before purchasing, including the rear cover notes from Volume 1, and thought, "these are for me!".

This endorsement on back cover of Vol 1 from Stephen Travis, CREATE Lab, Dept of Music, University of California, Santa Barbara : "Musimathics is destined to be required reading and a valued reference for every composer, music researcher, multimedia engineer, and anyone else interested in the inter play between acoustics and music theory"

And this from the rear cover description of Vol 1: "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."

I feel that the cover notes are misleading; but, perhaps most music composers DO have the mustard to understand the contents of these books. . .poor fool me!

In conclusion, I would love to have known that a specific (GCSE/A Level) level of mathematical ability is advisable. At the very least, the book’s description could include a note detailing the level and particular fields within mathematics that one would do well to get familiar with, so as to truly engage with this incredible material; resulting in an enjoyable, thought-provoking and ultimately inspiring read.

With a sufficient level of mathematical prowess, and an understanding of the basics of music theory, this would probably be a real page turner. . .but for me. . .I've begun doing some veeery basic and rudimentary revision of early GCSE mathematics, with a view to, one day, actually reading and digesting what is written in Musimathics Vol I and Vol II.

One person found this helpful

qirong29

A great thick pleasant book to read

Reviewed in Australia 🇦🇺 on September 27, 2022

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This is a serious book on the mathematical foundation of music, but despite almost 500 pages, it is actually written in an easy style. I have only just recently got this book, and have not quite read it through yet, but I find it pleasant reading. It covers broad grounds, explains topics in clear and straight-forward terms, and is sufficiently grounded in maths that is comprehensible to most school leavers. It is a great reference book to keep, especially to lovers of music and its relationship to maths.

Antonio Ciampi

Great book for serious musicians

Reviewed in Canada 🇨🇦 on May 15, 2021

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I already know the book, but I needed to have a paper copy. The copy I received is in good condition...

illimitarte

Bellissimo!

Reviewed in Italy 🇮🇹 on April 8, 2019

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Questo testo è meraviglioso e dovrebbe essere letto da tutti i docenti di tecnologie musicali del liceo musicale PRIMA di mettere piede in classe. Per gli alunni sarebbe un notevole beneficio!

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hansi

A great read for the mathematically inclined

Reviewed in Germany 🇩🇪 on January 8, 2015

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I love this book for not messing around. Using mathematics allows Loy to formulate music theory and acoustics in a very concise way. As a student of mathematics I can't judge the difficulty, but it should be understandable for almost everybody (lots fractions, solving simple equations and a bit of differentiation is almost all that's needed). The range of topics covered in the two books together is breath taking. I'm currently half way through the first book and read it whenever I have a free minute, it's a great read!

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