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Musimathics: The Mathematical Foundations of Music, Volume 2

Gareth Loy
Publisher: 
MIT Press
Publication Date: 
2007
Number of Pages: 
562
Format: 
Hardcover
Price: 
50.00
ISBN: 
9780262122856
Category: 
Monograph
[Reviewed by
Sarah Boslaugh
, on
09/1/2007
]

Musimathics: the mathematical foundations of music vol. 2 is a continuation of volume 1 of the same title, focusing in even greater depth on the mathematics behind music and sound, with particular emphasis on digital and computer-based music. Although volume 1 is intended as preparation for volume 2, either can be read independently.

This is not a book for the mathematically faint-of-heart, but interested musicians (or mathematicians interested in music) will find it rewarding if they are willing to put forth the effort to work through the explanations provided.

Loy writes with the assumption that, as he states in the preface, “enlightened common sense and inference are the whole of mathematics” and that “the cure for any lack of mathematical preparation on the reader’s part is simply to focus on what makes the most sense, and the rest will follow.” Fair enough: Loy is a remarkably clear writer and if you don’t understand, say, sampling and aliasing after diligent study of his text, perhaps you never will. Additionally, it’s not necessary to read Musimathics from cover to cover: individual chapters can be read as the interest and needs of the reader dictates, and each chapter includes the necessary mathematics to understand the content of that chapter.

Still, this is a very technical text which demands the reader be willing to put forth a fair amount of intellectual effort to get any benefit from reading it. Readers, particularly those working in electronic music, who are willing to put in the necessary time will be amply rewarded with increased understanding and appreciation of both music and mathematics. Additional material related to Musimathics vols 1 and 2 is available from http://www.musimathics.com/.

Gareth Loy is a performing musician, composer, software architect and digital audio systems engineer. He received his DMA in composition from Stanford University in 1980. Further information about his musical and engineering activities is available from Loy’s personal web page, http://www.garethloy.com/ and the web page for his consulting company, http://www.garethinc.com/.


Sarah Boslaugh (seb5632@bjc.org) is a Performance Analyst for BJC HealthCare and an adjunct professor at the Washington University School of Medicine in St. Louis, Missouri. She has written two books, An Intermediate Guide to SPSS Programming: Using Syntax for Data Management (SAGE, 2005) and Secondary Data Sources for Public Health: A Practical Guide (Cambridge University Press, forthcoming May 2007) and is editor-in-chief of The Encyclopedia of Epidemiology (forthcoming from Sage, November 2007).

 

Foreword
John Chowning
xiii
Preface xv
Acknowledgments xvi
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
Summary 47
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
Summary 100
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
Summary 156
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
Summary 193
Suggested Reading 194
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
Summary 261
6 Resonance 263
6.1 The Derivative 263
6.2 Differential Equations 276
6.3 Mathematics of Resonance 280
Summary 297
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
Summary 324
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
Summary 360
Suggested Reading 362
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
Summary 450
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
Summary 507
Suggested Reading 509
Epilogue 511
Appendix 513
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
Notes 533
Glossary 539
References 543
Equation Index 547

Dummy View - NOT TO BE DELETED