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Equal temperament, in which semitones are tuned in the irrational ratio of $ 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 $ \tfrac {45}{32} $ 45 32 , $ \tfrac {64}{45} $ 64 45 , $ \tfrac {36}{25} $ 36 25 , $ \tfrac {25}{18} $ 25 18 , none satisfactory, is in our scheme represented by the complex ratio $ 1 + \rm {i} : 1 $ 1 + i : 1. The major and minor whole tones, given by intervals of $ \tfrac {9}{8} $ 9 8 and $ \tfrac {10}{9} $ 10 9 , can each be factorized into products of complex semitones, giving us a major complex semitone $ \tfrac {3}{4}(1 + \rm {i}) $ 3 4 (1 + i) and a minor complex semitone $ \tfrac {1}{3}(3 + \rm {i}) $ 1 3 (3 + i). The perfect third, given by the interval $ \tfrac {5}{4} $ 5 4 , factorizes into the product of a complex whole tone $ \tfrac {1}{2}(1 + 2\rm {i}) $ 1 2 (1 + 2 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. [ABSTRACT FROM AUTHOR]

Investigations of tone systems are motivated by mathematical models in acoustics. In the present paper, tone systems are represented by the systems of fuzzy numbers. The construction idea of these systems consists of using approximations of 3⁄2. A formalization of well-tempered systems is given. Uncertainty-based-information measures are considered. The Huygens­Fokker, Opelt, Hba, Sumec, Petzval, and more other tone systems (used in the musical theory and practice and also new ones) are derived as consequences of the theory. [ABSTRACT FROM AUTHOR]