Discrete Fourier transform (DFT) has emerged as a powerful and popular tool in mathematical music theory. Amiot’s nice book gives the state of the art in the usage of DFT of abstract musical structures such as rhythms, scales, chords, pitch class distributions, and so on. In the author’s own words, this is a book that is “not about harmonics” but one on “harmonic analysis” (of musical structures).

Chapter 1 is introductory in nature and deals with DFT of distributions. The author provides the reader with the mathematical basics of the subject. Homometry and the phase retrieval problem are addressed in chapter 2. This includes spectral units and their extensions and generalizations. Chapter 3 is on nil Fourier coefficients and tilings, which also contains algorithms for computing a DFT. The next chapter is on saliency, which covers topics such as generated scales, maximal evenness, and PC-sets with large Fourier coefficients. Chapter 5 takes up continuous spaces and continuous FT. The last chapter is on phases of Fourier coefficients, which includes topics such as how to define the torus of phases, phases between tonal and atonal music, and so on.

This is possibly the first textbook on the topic. The author provides useful exercises and solutions to some of them. Handy supplementary material is also made available online. The target readership includes graduates and advanced undergraduates of computational music science and engineering. Researchers in music should also find it of interest.

There are a few minor errors, however. For example, the year of publication of Ian Quinn’s paper in the journal Perspectives of New Music in the reference section is quite correctly given as 2006-2007, but it is wrongly referred to as Quinn (2005) in the “Introduction” on page V.