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.

本文（是由Springer International Publishing瑞士于2016年出版，并在此之前在Springer International Publishing Switzerland瑞士的允许下在《数学与音乐杂志》上转载的《通过傅立叶空间的音乐》的第一章）给出了该定义的基本定义和工具。循环群子集的离散傅里叶变换（DFT），可对例如音调类集或周期节律建模。我介绍分布的环境空间，其中pc-set（或周期性节奏）是值仅为零和一的元素，以及一些重要的运算，尤其是卷积，这导致“乘法d'accords”（移调组合），和弦/小数的代数组合，拼贴，区间功能和许多音乐概念。定义了所有内容，并希望本文是独立的，也许关于循环矩阵的部分除外，该部分使用了线性代数的一些概念：矩阵的特征值和对角化。确实，希望本文中的材料将用于教学目的，作为研究复数和指数，模数算法，代数结构等的动机。