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.
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:
. . . 1/16, 1/8, 1/4, 1/2, 1/1, 2/1, 4/1, 8/1, 16/1, 32/1 . . .which is really only a single element, 1/1, if octave-equivalence is assumed.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . , . . , . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., 81/64, 81/32, 81/16, 81/8, 81/4, 81/2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27/32, 27/16, 27/8, 27/4, 27/2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., 9/16, 9/8, 9/4, 9/2, 9/1, 18/1, 36/1, 72/1, . . . . . . . . . . . . . . . . . . . . . . . . . . ., 3/16, 3/8, 3/4, 3/2, 3/1, 6/1, 12/1, 24/1, 48/1, . . . . . . . . . . . . . . . . . . . . . . . ., 1/16, 1/8, 1/4, 1/2, 1/1, 2/1, 4/1, 8/1, 16/1, 32/1, . . . . . . . . . . . . . . . . . . . . ., 1/48, 1/24, 1/12, 1/6, 1/3, 2/3, 4/3, 8/3, 16/3, 32/3, . . . . . . . . . . . . . . . . . . . . . . . . ., 1/72, 1/36, 1/18, 1/9, 2/9, 4/9, 8/9, 16/9, 32/9, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., 1/27, 2/27, 4/27, 8/27, 16/27, 32/27, 64/27, . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., 1/81, 2/81, 4/81, 8/81, 16/81, 32/81, 64/81, 128/81, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .If one assumes 8ve-equivalence, it is typical to represent each row of this set by a single ratio between 1/1 and 2/1, so the set of ratios becomes 1-dimensional:
. . . 128/81, 32/27, 16/9, 4/3, 1/1, 3/2, 9/8, 27/16, 81/64 . . . .
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
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".
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.
The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.