Review of Emmanuel Amiot, Music through Fourier Space: Discrete Fourier Transform in Music Theory (Springer, 2016)

Abstract

[...]the Fourier transform: the basic mathematical procedure that allows us to view an object in a cyclic space as a collection of periodic elements.At some point, these functions are needed for computation, but when doing mathematics at a more abstract level, they are excessively cumbersome, requiring complex trigonometric identities that clutter up proofs.[...]Amiot generally takes advantage of the Euler identity, the beautiful discovery of Leonard Euler, which is: \[ e^{\phi{}i} = \cos \phi + i \sin \phi \] Example 1.The number of coinciding events (onsets or rests) between the hands per measure [30] Of course, because the left-hand and right-hand rhythms exist in different cyclic universes, they can be compared only by adapting them to a common cycle.Since Ligeti repeats them strictly over 24-plus measures, we can bring the rhythms into a common cycle by “oversampling”—repeating them nine and eight times over 144 s. Here, Amiot’s Theorem 1.21 tells us what we need to know: the DFT of...