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Musimathics: v. 1: The Mathematical Foundations of Music Hardcover – 14 Jul 2006

4.5 out of 5 stars 4 customer reviews

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Product details

  • Hardcover: 608 pages
  • Publisher: MIT Press (14 July 2006)
  • Language: English
  • ISBN-10: 0262122820
  • ISBN-13: 978-0262122825
  • Product Dimensions: 17.8 x 2.1 x 22.9 cm
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Bestsellers Rank: 2,186,288 in Books (See Top 100 in Books)

Product Description

Review

"It will be obvious by now that this is a book wholeheartedly to be recommended for a wide readership. As definitive as is possible, certainly the leading resource in the field, it will meet many needs. Those with school algebra and physics who are performers, engineers, composers, listeners will almost inevitably get much from Musimathics...The two volumes of this book, then, must be considered as the place to start any exploration of this field. Entertaining, secure, comprehensive, clear, authoritative, timely, up-to-date, well wrought in every way, these are for every music lover's shelves." -- Mark Sealey, Classical Net: The Internet's Premier Classical Music Source "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 "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 Mecanique et d'Acoustique, CNRS, France "Volume 1 of Musimathics is the ideal introduction to the science of musical acoustics and composition theory, and volume 2 succeeds as no other tutorial does in making the theory of computer music and digital signal processing accessible to a broad audience. Loy's typically careful treatment leads to a book that combines readability and fun with exhaustive and meticulous coverage of each of the topics he addresses. It can serve equally well as an introduction and as a desk reference for experts." --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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Format: Hardcover Verified Purchase
I was critical of the first edition (2006) because of the large number of mistakes in the printing, for which Goolgle "musimathics errata".

However, I'm delighted to see a second edition, not only with corrections but with new material and positively the best layout I've seen in an academic book - it's lovely just to look at. It's also much cheaper, which is a rarity.

This two-volume set (oddly they are different heights) is the best reference on Maths and Music that I know.

Very highly recommended.
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Format: Paperback Verified Purchase
I was critical of the first edition back in 2007 because of the misprints. Well, this new edition is fantastic. Not only is it hugely authoritative but the layout is beautiful. The typesetting of the maths, the clarity of the diagrams, the weight of the paper all say this is a quality product. I've only got the second volume so far but I'm so pleased that I bought it. There is also a very clear structure to the material and it's a tribute to the beauty of publishing on paper. Well done to Dr Loy - this is an excellent book and I can't wait to get the first volume.

This second volume, especially with Boulanger's Audio Programming book are the new frontrunners in the theory and execution of digital music making.
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I thought I was getting the complete work but there is lots of reference to a Vol.2 which I find disappointing
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Great book would highly recommend
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Most Helpful Customer Reviews on Amazon.com (beta)

Amazon.com: 4.6 out of 5 stars 30 reviews
60 of 64 people found the following review helpful
5.0 out of 5 stars Excellent book combines music, math, and programming 10 Jan. 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
5.0 out of 5 stars Extraordinary Beyond the Title, a must for all Math Lovers 17 Jun. 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!
30 of 34 people found the following review helpful
5.0 out of 5 stars A good book on musical signal processing concepts 13 Jun. 2007
By calvinnme - Published on Amazon.com
Format: Hardcover
If you are to really understand what is going on in this book you need volume one where the foundations are discussed. Likewise, volume one of Musimathics will often stop short of a truly complete explanation and say that further study will be picked up in volume two. Thus, these two volumes are actually just the halves of one book. However, if you are interested in musical signal processing, you probably need to read volume two. It covers much ground in depth, and gives numerous examples that are very practical and accessible for people who are working with musical and audio signals. The appendix has some useful tutorials and tables involving mathematics if you happen to be rusty. The following is the table of contents:

1 Digital Signals and Sampling 1
1.1 Measuring the Ephemeral 1
1.2 Analog-to-Digital Conversion 9
1.3 Aliasing 11
1.4 Digital-to-Analog Conversion 20
1.5 Binary Numbers 22
1.6 Synchronization 28
1.7 Discretization 28
1.8 Precision and Accuracy 29
1.9 Quantization 29
1.10 Noise and Distortion 33
1.11 Information Density of Digital Audio 38
1.12 Codecs 40
1.13 Further Refinements 42
1.14 Cultural Impact of Digital Audio 46

2 Musical Signals 49
2.1 Why Imaginary Numbers? 49
2.2 Operating with Imaginary Numbers 51
2.3 Complex Numbers 52
2.4 de Moivre's Theorem 62
2.5 Euler's Formula 64
2.6 Phasors 68

2.7 Graphing Comlpex Signals 86
2.8 Spectra of Complex Sampled Signals 87
2.9 Multiplying Phasors 89
2.10 Graphing Complex Spectra 92
2.11 Analytic Signals 95

3 Spectral Analysis and Synthesis 103
3.1 Introduction to the Fourier Transform 103
3.2 Discrete Fourier Transform 103
3.3 Discrete Fourier Transform in Action 125
3.4 Inverse Discrete Fourier Transform 134
3.5 Analyzing Real-World Signals 138
3.6 Windowing 141
3.7 Fast Fourier Transform 145
3.8 Properties of the Discrete Fourier Transform 147
3.9 A Practical Hilbert Transform 154

4 Convolution 159
4.1 Rolling Shutter Camera 159
4.2 Defining Convolution 161
4.3 Numerical Examples of Convolution 163
4.4 Convolving Spectra 168
4.5 Convolving Sigals 172
4.6 Convolution and the Fourier Transform 180
4.7 Domain Symmetry between Signals and Spectra 180
4.8 Convolution and Sampling Theory 187
4.9 Convolution and Windowing 187
4.10 Correlation Functions 191

5 Filtering 195
5.1 Tape Recorder as a Model of Filtering 195
5.2 Introduction to Filtering 199
5.3 A Sample Filter 201
5.4 Finding the Frequency Response 203
5.5 Linearity and Time Invariance of Filters 217
5.6 FIR Filters 218
5.7 IIR Filters 218
5.8 Canonical Filter 219
5.9 Time Domain Behavior of Filters 219
5.10 Filtering as Convolution 222
5.11 Z Transform 224
5.12 Z Transform of the General Difference Equation 232
5.13 Filter Families 244

6 Resonance 263
6.1 The Derivative 263
6.2 Differential Equations 276
6.3 Mathematics of Resonance 280

7 The Wave Equation 299
7.1 One-Dimensional Wave Equation and String Motion 299
7.2 An Example 307
7.3 Modeling Vibration with Finite Difference Equations 310
7.4 Striking Points, Plucking Points, and Spectra 319

8 Acoustical Systems 325
8.1 Dissipation and Radiation 325
8.2 Acoustical Current 326
8.3 Linearity of Frictional Force 329
8.4 Inertance, Inductive Reactance 332
8.5 Compliance, Capacitive Reactance 333
8.6 Reactance and Alternating Current 334
8.7 Capacitive Reactance and Frequency 335
8.8 Inductive Reactance and Frequency 336
8.9 Combining Resistance, Reactance, and Alternating Current 336
8.10 Resistance and Alternating Current 337
8.11 Capacitance and Alternating Current 337
8.12 Acoustical Impedance 339
8.13 Sound Propagation and Sound Transmission 344
8.14 Input Impedance: Fingerprinting a Resonant System 351
8.15 Scattering Junctions 357

9 Sound Synthesis 363
9.1 Forms of Synthesis 363
9.2 A Graphical Patch Language for Synthesis 365
9.3 Amplitude Modulation 384
9.4 Frequency Modulation 389
9.5 Vocal Synthesis 409
9.6 Synthesizing Concert Hall Acoustics 425
9.7 Physical Modeling 433
9.8 Source Models and Receiver Models 449

10 Dynamic Spectra 453
10.1 Gabor's Elementary Signal 454
10.2 The Short-Time Fourier Transform 459
10.3 Phase Vocoder 486
10.4 Improving on the Fourier Transform 496
10.5 Psychoacoustic Audio Encoding 502

A.1 About Algebra 513
A.2 About Trigonometry 514
A.3 Series and Summations 517
A.4 Trigonometric Identities 518
A.5 Modulo Arithmetic and Congruence 522
A.6 Finite Difference Approximations 523
A.7 Walsh-Hadamard Transform 525
A.8 Sampling, Reconstruction, and Sinc Function 526
A.9 Fourier Shift Theorem 528
A.10 Spectral Effects of Ring Modulation 529
A.11 Derivation of the Reflection Coefficient 530
13 of 13 people found the following review helpful
5.0 out of 5 stars Extraordinary Beyond the Title, a must for all Math Lovers 17 Jun. 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!
11 of 11 people found the following review helpful
5.0 out of 5 stars Solid intro to DSP concepts for musicians 27 Dec. 2009
By tangent - Published on Amazon.com
Format: Hardcover
The first volume of Musicmathics is primarily an intro to the mathematical aspects of music theory, harmonics, scale construction, music perceptions, etc.. The second volume is basically an introduction to Digital Signal Processing (DSP) with discussion of how it applies to music.

My background is in Electrical Engineering so I am well versed in the basic DSP concepts outlined in this book. Gareth Loy has done a fantastic job of 'gently' presenting this material so that even musicians without extensive advanced mathematical training should be able to grasp it. I have seen these concepts presented in a number of different textbooks and this book is far more straightforward than many of the EE signal processing books. Loy goes out of his way to highlight which concepts are the most important and often gives multiple illustrations to highlight the implications of these key concepts. I wish I had this book when I was first learning DSP!

The one complaint I have is that too much attention is given to Fourier techniques and not enough attention paid to Wavelet based methods which are increasingly replacing windowed fourier variants like STFT in many real world applications. However, with the background material presented here the interested reader should be able to quickly grasp the fundamentals of wavelets.

Highly recommended for anyone interested in DSP, music synthesis/analysis, sound modeling, etc..
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