Limit (music)

From Academic Kids

Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. This upper bound is called a limit. For example, the major and minor triads of Common practice music fall within 5-limit just intonation. By extension it may be said that Common practice music is a 5-limit genre, because those major and minor triads are the most complex harmonies considered consonant in it. Jazz and other 20th-century genres go beyond the 5-limit, but the correspondence to just intonation is less clear because of the nature of 12-tone equal temperament. 7-limit tunings are properly found in barbershop singing, and a few other relatively isolated genres.

There are two distinct types of limit in music theory literature: prime limit and odd limit. Not all authors are aware of the distinction.

In a just intonation tuning, intervals between pitches are drawn from the rational numbers. In an n-limit prime limit tuning, intervals between pitches are drawn from rational numbers that can be factored using prime numbers no greater than n, where n is prime. In an n-limit odd limit tuning, intervals between pitches are drawn from rational numbers which, after all factors of 2 are removed, have numerators and denominators no greater than n, where n is an odd whole number. Note that prime limit and odd limit do not cover the same scales even when n is an odd prime.

Harry Partch based his music on the 11-limit tonality diamond, which contains all the intervals of odd limit 11. But he also developed scales, including his famous 43-tone scale, based on a prime limit of 11.

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