Musimathics: The Mathematical Foundations of Music, Volume 2 (review)

Since the advent of publications like Computer Music Journal, writers have attempted to democratize the mysteries of digital signal processing (DSP) to musicians. It is an abstract subject, and each writer uses his or her wisdom and unique understanding of it to suggest intuitive approaches, bypassing much of the background and theory with which electrical engineers are more intimately familiar. Although a number of effective texts have come out in recent years, now Gareth Loy’s Musimathics, Volume 2 may take its place as one of the essential DSP texts for musicians.

Chapter 1, “Digital Signals and Sampling,” begins with an example of tidewater levels. The simple problem of measuring them leads to a discussion of continuous vs. discrete measurements, lowpass filtering, sample and hold, and slew rates. The concept of aliasing is explained with another intuitive example: observing a marked spoke on a spinning bicycle wheel illuminated with a strobe light. From this familiar and intuitive analogy, readers are quickly treated to the level of thoroughness and precision that characterizes the book throughout as an equation is derived that relates actual and apparent frequencies for any sampled waveform.

A few points may alarm novice readers. Functional notation, f (x), is relied upon but not explained. At the same time, the floor function appears, but is explained only in a footnote in the back of the book. Lastly, when aliasing is introduced, it is given the synonymous term “foldover,” but the reason is not given for another six pages. The intimidation factor could have been lessened easily enough with a simple reassurance like “for reasons that will be explained presently. . . ”

Thus, the same caveat from my review of Musimathics, Volume 1 (in CMJ 31:4) applies: This is not a casual read for novices; beginners will likely benefit from guidance from someone with more experience. Still, this book is unmatched for wonderful real-world analogies that instantly put things into perspective. A few examples are included here by way of illustration:

If we think of a sample sequence as a “dehydrated” version of the original analog signal, we “rehydrate” it to recover the original analog waveform using digitalto- analog conversion.

(p. 20)

Distortion, like a fun house mirror, disarranges the proportions of the signal being recorded.

(p. 35)

After another familiar analogy of mixing apples and oranges (p. 51), Chapter 2, “Musical Signals,” introduces readers to the world of complex numbers. The unit circle, Taylor series, and Euler’s formula are introduced one by one. This series of introductions leads to circular motion, phasors, and projections. Anyone who has not had the “wow” moment of seeing a helix advancing like a propeller, with orthogonal projections creating real and imaginary functions, will experience this moment of enlightenment here. The point is further reinforced by illustrations of a hypothetical sine and cosine machine consisting of a motor, two arms, and a rotor, each arm tracing the motion of a phasor. These graphical illustrations are brought full circle (as it were) when they are shown to illustrate Euler’s identity. Through some asides, Mr. Loy references demodulation as a result of multiplying signals, and shows how the Hilbert transform is at the basis of envelope followers and frequency detectors. These asides are not as detailed as the rest of the text, begging the questions of whether it is worth it to bring these topics up if they’re not to be covered as thoroughly as everything else. In the same way, linear function graphs could be better explained, though they are examined again in the next chapter.

Chapter 3, “Spectral Analysis,” explores the Fourier transform. In one...