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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

A number that is characteristic of, and is meant to quantify in some sense the complexity of, the ratio or ratios associated with the tuning (pitch-height, frequency) of some musical structure.

When the word "limit" is used without a qualifier in tuning theory, it may refer to either "prime-limit" or "odd-limit" (each defined more fully below); hopefully the context indicates which. "Odd-limit" is generally considered to be the more important when the context is a consideration of concordance, whereas "prime-limit" is generally the reference in most other cases.

A third type of limit is the intervallic limit, also described below.

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[Paul Erlich, adapted and expanded from Partch, Genesis of a Music]
1. prime limit

A pitch system in just-intonation where all ratios are of integers containing no prime factors higher than prime-number n is said to be an "n-limit" system.

A non-just system directly mappable, for example thru temperament, from n-limit just-intonation can be called an "n-limit" system as well. Examples:

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2. odd limit

The n-limit is the set all ratios of all odd numbers no greater than odd-number n, i.e., those ratios with odd factors no larger than n. Any ratio belonging to the n-limit is termed an "n-limit ratio". Examples:

Partch considered these ratios to be the sets of intervals more concordant, respectively, than some increasingly discordant cutoff -- hence each higher limit represented to him a successively more inclusive standard for intervallic concordance.

When interpreted as a set of pitches instead of as a set of intervals, the n-limit is known as the "n-limit Tonality Diamond" (after Partch).

A chord in Just Intonation where the largest odd factor in the terms making up the harmonic, or "otonal", representation of the chord (that is, its representation as an M-term frequency ratio where M is the number of notes in the chord) is an odd number no higher than n, is said to be within the "otonal limit" n. Examples:

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3. intervallic limit

A chord in Just Intonation where all interval-ratios belong to the the n-limit, is said to be within the "intervallic limit" n. See "saturated". Examples:

A composition or style where chords of the n-limit are considered consonant and chords of any higher limit are considered dissonant is said to be an "n-limit" composition or style. Sometimes the "intervallic limit" is meant here; other times it's the "otonal limit" that's meant -- it's best to say which.

All the usages of 'odd limit' may also apply to approximations, as in a temperament, of the Just Intonation intervals and chords referred to above. For example, see "consistent".

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[John Chalmers, Divisions of the Tetrachord]

A pitch system in Just Intonation whose ratios contain the prime number n and no higher primes is said to be an "n-Prime-Limit" system.

By usage, certain odd non-primes such as 9, 15, and 21 may also be said to define "n-limit" systems.

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