My first guess was that this book would be about using Fourier transforms to winkle out what notes went into a musical recording, such as was famously done by Jason Brown in 2008 for the opening chord of the Beatles’ song “A Hard Day’s Night”. But it’s really something very different: applying Fourier analysis to various aspects of musical scores, such as the notes in a chord, the intervals between notes, the rhythms, and more. All of these have definite patterns or repetitions in musical theory, and so Fourier analysis can become a useful tool. The subject draws a great deal of inspiration from crystallography, which also has many repeated structures and where some of the Fourier ideas here have been used.

The Preface states (p. VIII), “This book aims at being self-contained, providing coherent definitions and properties of DFT [Discrete Fourier Transform] for the use of musicians (theorists and practitioners alike).” The term “Fourier space” in the title means “working with the Fourier coefficients from the DFT”. I think the book does a reasonable job of this, although I suspect it would still be beyond the interests and mathematical abilities of most of its target audience. The mathematics is explained quite well, but the notation is still overwhelming. The announced prerequisites are fairly high for what is essentially a popular-math audience; the author expects some knowledge of set theory and group theory as well as a good grasp of real and complex numbers.

As with many applied subjects, the tricky parts are deciding what to measure and how to interpret the results, rather than the mathematics itself. I wasn’t completely convinced by many of the examples, although I think they may be more meaningful to music theorists, for whom the book is written, than for mathematicians.

The focus of the book is on the analysis of music, but there is some far-out stuff too, such as “Fourier scratching”, an analog of the scratching that DJs use with vinyl records on turntables. In Fourier scratching the DJ starts with a rhythm loop, which being repetitive has a Fourier transform, and then manipulates the Fourier coefficients using a game controller to change the rhythm in real time.

Production quality of the book is generally good. In the proof of Theorem 1.11 an operator is defined that is sometimes typeset as italic and sometimes as script, which is confusing. The example on p. 44 seems to be garbled; we are asked to believe that if \(\cos \varphi = 1\) then \(\varphi = \pm 2 \pi/3\). The terminology is occasionally inconsistent; for example, a certain type of matrix is sometimes called circulant and sometimes circulating. The index is inadequate, and many frequently-used terms don’t appear in it. For example, nearly every page refers to pc-sets = pitch-class sets, but neither term is in the index and neither appears to be defined anywhere. There are many references to Babbitt’s hexachord theorem, which is stated on p. 41, but this too is not in the index.

Bottom line: first book in a cutting-edge subject, and although well-done may have trouble finding an audience.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.