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

With Musimathics, Gareth Loy solves a problem I have faced for a number of years now. As an instructor of a university Science of Music course, I have never found any single text that explains all necessary topics with equal depth and clarity, and thus have had to rely on course packs of photocopied compilations. In the future, this chore will be unnecessary, thanks to this book. Musimathics doesn't have absolutely everything, but it comes so very close, and then provides a wealth of bonuses. The book is singularly precise, thorough, and often very funny.

The first chapters are preliminary, providing basic vocabulary for concepts that are treated in greater depth later. Chapter 1, "Music and Sound," introduces air pressure, waves, and simple harmonic motion. Not a word is wasted anywhere. Students will likely be comforted by the low page count.

Chapter 2, "Representing Music," gives an overview of how music is translated into pictorial symbols, with explanations of what is being represented at each step. Pitch is explained as both a matter of frequency and of interval ratios, and is supported by time-domain graphs of air pressure changes, the amplitude envelope, staff notation, and how pitch subsets form various scales. Duration and loudness are covered in terms of time, tempo, musical dynamic notation, and time signature. Timbre is discussed in terms of spectral changes and time. Graphs illustrate a string's vibrational modes, plots of complex waves composed of harmonic partials, dynamic spectra, and sonograms.

Things take off with Chapter 3, "Musical Scales, Tuning, and Intonation." Here is where everything gets treated down to the fine details. The nature of scales and ratios is discussed as they pertain to equal temperament, just intonation, Pythagorean intonation (and the problems introduced by the syntonic comma), meantone temperament, well temperament, various ethnic and microtonal scales (Hindustani and Partch, to name just two), and fret calculations used by guitar makers. Anything left out of a reader's tuning background is likely to be covered here.

Chapter 4, "Physical Basis of Sound," is approximately two-thirds "physical basis" and one-third "sound." The majority of the chapter provides background in general physics, leading to musical considerations after covering underlying topics such as dimension, mass, density, velocity, Newton's laws of motion, work, and conservative versus non-conservative forces. There is a good balance of equations and text, providing both a conceptual and a quantitative context for all topics (I've found that most other books emphasize one or the other, but rarely both). Here the equations get denser, as more symbols are applied to more terms. This is the chapter where novice students will learn to slow down and absorb material, symbol by symbol. Mr. Loy is well aware of the intimidation factor that can present itself in material like this, as he makes clear in his Preface: "I know what it's like not to comprehend mathematics easily, and I also know what it's like not to give up" (p. xvii). This chapter is where the non-mathematically inclined are likely to face a test of will. But, as he describes, the answers will come to those who understand that learning may require some effort on their part. And they will appreciate that this treatment is probably the most unambiguous description they'll find anywhere.

Chapter 5, "Geometrical Basis of Sound," tidily covers the relationship of sinusoids to circles and rotating vectors. This material is straightforward, and gives readers what they need to learn concepts like radians and angles by rote, just as musicians learn scales.

Chapter 6, "Psychophysical Basis of Sound," goes inside of the human head, covering the auditory system...