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[1] Emmanuel Amiot’s Music through Fourier Space (MFS) is an impressive and important milestone in mathematical music theory, presenting in a succinct volume the many discoveries made and theoretical ideas generated based on the discrete Fourier transform (DFT) over the past two decades—developments in which Amiot himself was a central player.
[2] Amiot’s book amply demonstrates that the DFT is relevant to a wide variety of musical questions—about rhythm, about pitch-class sets and interval content, and about tonality and non-tonality—so many theorists will find something in the book relevant to questions in their own research. Within each of these topics, Amiot’s discourse varies between questions of primarily mathematical interest and those of more direct musical interest. Yet for a book that generally proceeds in the manner of mathematical writing, from definition to proposition and lemma to theorem, it laudably never strays far from immediately appreciable musical relevance.
[3] Nonetheless, for theorists who do not have extensive mathematical background, the methods and notation may seem to be a barrier to entry. Part of the goal of this review, then, is to explain the notations and why Amiot uses them. The tutorial below does this, preceded by an overview of the book’s content and followed by a novel analytical example from Ligeti’s Etude no. 8 to illustrate some of Amiot’s results in the rhythmic domain.
[4] Musicians are likely to be most familiar with the Fourier transform as an operation performed on sound signals to translate raw amplitude-over-time information into frequency information. That application of the Fourier transform is one that Amiot’s book does not address. Instead, he applies the DFT to pitch-class sets and rhythms, which exist in completely different kinds of spaces. Rhythms are in a temporal space like waveforms, but exist on a different timescale and at a higher cognitive level. Pitch-class sets are harmonic objects, but exist in pitch-class space, something many degrees removed and abstracted from anything like a sound signal.
[5] Yet this must seem awfully suspicious: Why should this same mathematical technique apply to so many apparently unrelated aspects of music? The question is not answered directly by Amiot, so I will venture an explanation here. Wherever we look in music, we find cyclic spaces—or, more precisely, music...