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The Online Guitar Store - Musimathics, Volume 2: The Mathematical Foundations of Music
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List Price: $52.00
Our Price: $41.60
Your Save: $ 10.40 ( 20% )
Availability: Usually ships in 24 hours
Manufacturer: The MIT Press
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Average Customer Rating: 
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Binding: Hardcover
Dewey Decimal Number: 781.2
EAN: 9780262122856
ISBN: 0262122855
Label: The MIT Press
Manufacturer: The MIT Press
Number Of Items: 1
Number Of Pages: 576
Publication Date: 2007-06-30
Publisher: The MIT Press
Studio: The MIT Press
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Editorial Reviews:
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"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.
Volume 2 of Musimathics continues the story of music engineering begun in volume 1, focusing on the digital and computational domain. Loy goes deeper into the mathematics of music and sound, beginning with digital audio, sampling, and binary numbers, as well as complex numbers and how they simplify representation of musical signals. Chapters cover the Fourier transform, convolution, filtering, resonance, the wave equation, acoustical systems, sound synthesis, the short-time Fourier transform, and the wavelet transform. These subjects provide the theoretical underpinnings of today's music technology. The material in volume 1 is all preparatory to the subjects presented in this volume, although either volume can be read independently. Cross-references to volume 1 are provided for concepts introduced in the earlier volume, and additional mathematical orientation is offered where necessary. The topics are all subjects that contemporary composers, musicians, and music engineers have found to be important. The examples given are all practical problems in music and audio. The level of scholarship and the pedagogical approach also make Musimathics ideal for classroom use. Additional material can be found at a companion web site.
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Spotlight customer reviews:
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Customer Rating:
Summary: Excellent
Comment: The book is simply useful. Musicians and teachers can review their knowledge base. Novices have the possibility to learn the math-based laws of music by means of a non-academic language. Above all, the book is extended into a second volume for an in-depth learning.
Customer Rating:
Summary: Excellent
Comment: The book is simply useful. Musicians and teachers can review their knowledge base. Novices have the possibility to learn the math-based laws of music by means of a non-academic language. Above all, the book is extended into a second volume for an in-depth learning.
Customer Rating:
Summary: A good book on musical signal processing concepts
Comment: 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
Customer Rating:
Summary: Excellent book combines music, math, and programming
Comment: 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
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