Musimathics: The Mathematical Foundations of Music: 2 [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

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!

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..