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Musimathics: The Mathematical Foundations of Music Hardcover – Jun 16 2006

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Product Details

  • Hardcover: 504 pages
  • Publisher: The MIT Press; 1 edition (June 16 2006)
  • Language: English
  • ISBN-10: 0262122820
  • ISBN-13: 978-0262122825
  • Product Dimensions: 17.8 x 2.1 x 22.9 cm
  • Shipping Weight: 953 g
  • Average Customer Review: Be the first to review this item
  • Amazon Bestsellers Rank: #927,041 in Books (See Top 100 in Books)
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Product Description


From his long and successful experience as a composer and computer-music researcher, Gareth Loy knows what is challenging and what is important. That comprehensiveness makes Musimathics both exciting and enlightening. The book is crystal clear, so that even advanced issues appear simple. Musimathics will be essential for those who want to understand the scientific foundations of music, and for anyone wishing to create or process musical sounds with computers.

(Jean-Claude Risset, Laboratoire de Mécanique et d'Acoustique, CNRS, France)

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.

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Amazon.com: HASH(0x9dc30c18) out of 5 stars 18 reviews
60 of 64 people found the following review helpful
HASH(0x9e463660) out of 5 stars Excellent book combines music, math, and programming Jan. 10 2007
By calvinnme - Published on Amazon.com
Format: Hardcover Verified Purchase
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
28 of 29 people found the following review helpful
HASH(0x9db29b40) out of 5 stars Extraordinary Beyond the Title, a must for all Math Lovers June 17 2010
By Let's Compare Options Preptorial - Published on Amazon.com
Format: Hardcover Verified Purchase
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!
13 of 15 people found the following review helpful
HASH(0x9db37fe4) out of 5 stars Excellent March 7 2010
By Dr. Lee D. Carlson - Published on Amazon.com
Format: Hardcover Verified Purchase
Music and mathematics are not quite synonymous, in spite of what the ancient Greeks asserted. This book however convinces the reader that music is a close approximation of mathematics, and vice versa, and that a purely mathematical formulation of music is possible, this formalism encompassing all conceivable compositional techniques and able to capture completely the aural sensibility of any composer. Whatever their background, the author has done an excellent job of presenting to readers the role that mathematics can play in musical theory. Best of all, one need not even have a background in music to understand the book, since the author presents this background as the text proceeds.

Indeed in chapters 1-3 the author gives a detailed overview of basic music theory, including scales, equal-tempered intervals, and tempering. For non-musicians, such as this reviewer, there is much to absorb, but the author's presentation is lucid enough as to make the material easy to digest and remember. The most amazing thing that is brought out from the study of these chapters is the relative paucity of scales compared to what could be obtained from the entire range of human hearing. The most interesting discussion from a mathematical standpoint is that of the use of recursion and continued fractions in fret calculations.

The most interesting chapter of the book is the one on composition and methodology, for it is here that the author touches on the subject of automated musical composition. Markov chains, Petri nets, neural networks, fractal geometry, genetic programming, and predicate transition nets are all discussed in terms of their efficacy in generating musical compositions that emulate a particular composer's musical style and their creative musical ability. Even more interesting is that the author addresses the question of whether the machines that compose music are indeed actually intelligent. He briefly describes an experiment that would test whether the EMI (Experiments in Musical Intelligence) would indeed capture the "aural sensibility" of Mozart. If EMI "drifts off" in some direction that is not "Mozartian", then one might say that EMI has failed in this regard. But this discussion is very powerful in that it points to the next grand project in machine intelligence. The "drifting" of the machine from a certain context or body of knowledge may in fact be a sign of its need for exploring new frontiers of knowledge. The machine would be exhibiting curiosity, a trait that has not yet been exhibited by the machines to this date. Not today, but tomorrow yes.

Note: This review is based on a reading of chapters 1-3 and chapter 9.
2 of 2 people found the following review helpful
HASH(0x9e3ca630) out of 5 stars Math and music Dec 31 2012
By David Lynch - Published on Amazon.com
Format: Paperback Verified Purchase
I have always wondered about the differences between a note played on different instruments, and how this affects the sound wave, and this book explained this well. I am a computer geek considering a programming project involving music. This book gave me the background I was looking form, but does not cover any of the programming aspects of real time midi input and sound output.

There is quite a bit of math and physics in the book and I did skip quite a bit of it.
There is also quite a lot of information on different scales and changes in the frequency of notes through history.
I took piano lessons as a kid, but don't have a good understanding of the different keys and scales so this was tough for me, but also very interesting.
4 of 5 people found the following review helpful
HASH(0x9dcd0bd0) out of 5 stars Required reading for every music student June 11 2009
By Charles Madden - Published on Amazon.com
Format: Hardcover
This book is nothing short of excellent. It is well-written and thorough, by a musician, for musicians. Every music student should take a course with this book as its text.