The discrete Fourier transform of distributions

Abstract

This article (being the first chapter of the book Music Through Fourier Space published by Springer International Publishing Switzerland in 2016 and reprinted here in the Journal of Mathematics and Music with the kind permission of Springer International Publishing Switzerland) gives the basic definitions and tools for the discrete Fourier transform (DFT) of subsets of a cyclic group, which can model for instance pitch-class sets or periodic rhythms. I introduce the ambient space of distributions, where pc-sets (or periodic rhythms) are the elements whose values are only zeros and ones, and several important operations, most notably convolution, which leads to “multiplication d'accords” (transpositional combination), algebraic combinations of chords/scales, tiling, intervallic functions, and many musical concepts. Everything is defined and the article is hopefully self-contained, except perhaps for the section on circulant matrices, which uses some notions from linear algebra: eigenvalues of matrices and diagonalisation. Indeed, it is hoped that the material in this article will be used for pedagogical purposes, as a motivation for studying complex numbers and exponentials, modular arithmetic, algebraic structures, and so forth. The proof of an important theorem demonstrates that the DFT is the only transform that simplifies the convolution product into the ordinary, termwise product.