Mathematical Foundations of Complex Tonality
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Equal temperament, in which semitones are tuned in the irrational ratio of 2^1/12 : 1, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals 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 45:32, 64:45, 36:25, or 25:18, none satisfactory, is in our scheme represented by the complex ratio 1 + i : 1. The major and minor whole tones, given by intervals of 98 and 109, can each be factorized into products of complex semitones, giving us a major complex semitone 34(1 + i) and a minor complex semitone 13(3 + i). The perfect third, given by the interval 54, factorizes into the product of a complex whole tone 12(1 + 2i) and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of powers of Gaussian primes leads very naturally to the construction of a complete system of major and minor scales in all keys.
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