Mathematical foundations of complex tonality

Abstract

Equal temperament, in which semitones are tuned in the irrational ratio of (Formula presented.), 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 (Formula presented.), (Formula presented.), (Formula presented.), (Formula presented.), none satisfactory, is in our scheme represented by the complex ratio (Formula presented.). The major and minor whole tones, given by intervals of (Formula presented.) and (Formula presented.), can each be factorized into products of complex semitones, giving us a major complex semitone (Formula presented.) and a minor complex semitone (Formula presented.). The perfect third, given by the interval (Formula presented.), factorizes into the product of a complex whole tone (Formula presented.) 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.