Mathematical foundations of complex tonality

Abstract

Equal temperament, in which semitones are tuned in the irrational ratio of 21/12:1$2^{1/12}:1$$2^{1/12}:1$, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of 2, 3, and 5, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system. The tritone, so problematic in just temperament, given ambiguously by the ratios 4532$\frac{45}{32}$, $\frac{64}{45}$, $\frac{36}{25}$, $\frac{25}{18}$, none satisfactory, is in our scheme represented by the complex ratio $1+\mathrm{i}:1$. The major and minor whole tones, given by intervals of $\frac{9}{8}$ and $\frac{10}{9}$, can each be factorized into products of complex semitones, giving us a major complex semitone $\frac{3}{4}(1+\mathrm{i})$ and a minor complex semitone $\frac{1}{3}(3+\mathrm{i})$. The perfect third, given by the interval $\frac{5}{4}$, factorizes into the product of a complex whole tone $\frac{1}{2}(1+2\mathrm{i})$ and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of low powers of Gaussian primes leads to the construction of a complete system of major and minor scales in all keys.