A principle of least complexity for musical scales.
The following properties of equally tempered scales are demonstrated in this Letter: For N semitones and M notes, each distinct scale (interval) structure can be represented by an associated multiplet of N scales. These scales allow themselves to be labeled by a set of integers ¿c¿. Each label c is the difference between the number of sharps and flats in a given scale. The equivalence classes [c] modulo N form a commutative ring with unity. When the ratio N/M cannot be simplified further, then each member of a given multiplet will have a unique label (modulo N), different from the other members of the same multiplet. Because this labeling depends not on the interval structure of the multiplet but only on N and M, different multiplets with the same N and M values will have members carrying the same respective labels. Each equally tempered scale (interval) structure possesses a property which will be referred to as complexity. This Letter proposes a quantitative measure for complexity which distinguishes between different scale (interval) structures. For the particular case where N= 12 and M=7, out of 462 possible different equally tempered scale structures, those with minimum complexity are the major scale and the modes, which suggests a minimum principle in music based on equally tempered scales. This simplicity of structure allows the practical use of key signatures in music.
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D008962:Models, Theoretical; D009146:Music