Introduction to the special issue on perfect balance and the discrete Fourier transform

The five articles featured in this special issue rapidly bring the reader from the fundamentals to the forefront of knowledge concerning the discrete Fourier transform and perfect balance, and their application to music analysis. A scale is perfectly balanced if the first Fourier coefficient of its indicator vector is zero, or equivalently, if the zeroth Fourier coefficient of its Argand vector is zero. The first article Amiot (2017a), which has minimal prerequisites, is a delightful, elegant, and short invitation that motivates the reader to delve deeper into the two subjects. This first article Amiot (2017a) is an adapted translation of Amiot (2010). The second article in this special issue, Amiot (2017b), methodically develops the mathematical framework of the discrete Fourier transform in music theory, and connects it to prominent music-theoretical notions and questions. This second article Amiot (2017b) is a republication of the first chapter of the milestone book Amiot (2016). The centerpiece of the special issue, Milne, David, and Steffen (2017), studies the mathematical ramifications of perfectly balanced scales and rhythms via the discrete Fourier transform, and musically explores the smooth manifold of such in the authors’ software XronoMorph. Carey (2017) studies perfectly balanced scales from a word-theoretic perspective, focusing on those perfectly balanced scales with circular palindromic “richness” and relatively few step “differences.” Yust (2017) analyzes Debussy’s prelude “Les sons et les parfums tournent dans l’ air du soir” from a Fourier perspective, highlighting perfectly balanced pitch-class sets. The papers can be read in any order. Each is as self-contained as possible in order to welcome newcomers to the field. Scientific progress on the discrete Fourier transform in music spans decades, countries, and continents. This special issue is a testament to the vitality and resiliency of international scholarly collaboration, and the joy this cooperative, creative endeavor brings.