# limit

[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.

. . . . . . . . .
[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:

• A "2-limit" system would be a single note with any number of octave-transpositions of that same note. The pitch ratios and interval ratios will belong to the 1-dimensional infinite set
```. . . 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.
• A "3-limit" system would be some manifestation of pythagorean tuning. The pitch ratios and interval ratios will belong to the 2-dimensional infinite set
```. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . , . . , . . , . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . ., 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 . . . .
```
• A 5-limit system is often described as 'Classic Just Intonation', and is 3-dimensional in raw form. Assuming 8ve-equivalence as above, one obtains a 2-dimensional lattice, as shown in the first figure of Erlich, A Gentle Introduction to Fokker Periodicity Blocks, Part 2
• A 7-limit system, 4-dimensional in raw form, becomes 3-dimensional when 8ve-equivalence is assumed, and is then exemplified by the first diagram of Erlich, A Gentle Introduction to Fokker Periodicity Blocks, Part 3
. . . . . . . . .
###### 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:

• The 3-limit consists of the following ratios, and all their octave-equivalents: 1/1, 4/3, 3/2.
• The 5-limit consists of the following ratios, and all their octave-equivalents: 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3.
• The 7-limit consists of the following ratios, and all their octave-equivalents: 1/1, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4.
• The 9-limit consists of the following ratios, and all their octave-equivalents: 1/1, 10/9, 9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 16/9, 9/5.
• The 11-limit consists of the following ratios, and all their octave-equivalents: 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6.

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:

• The Just Major Triad, 4:5:6, is within "otonal limit" 5 (or any higher odd number).
• The Pythagorean Suspended Fourth chord, 6:8:9, is within "otonal limit" 9 (or any higher odd number).
• The Just Major Seventh chord, 8:10:12:15, is within "otonal limit" 15 (or any higher odd number).
• The Just Minor Seventh chord, 10:12:15:18, is within "otonal limit" 15 (or any higher odd number).
• The Just Minor Triad, 10:12:15, is within "otonal limit" 15 (or any higher odd number).
. . . . . . . . .
###### 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:

• The Just Major Triad, 4:5:6, contains one occurrence of the interval 3:2, one occurrence of the interval 5:4, and one occurrence of the interval 6:5. Hence the chord is within "intervallic limit" 5 (or any higher odd number).
• The Pythagorean Suspended Fourth chord, 6:8:9, contains one occurrence of the interval 3:2, one occurrence of the interval 4:3, and one occurrence of the interval 9:8. Hence the chord is within "intervallic limit" 9 (or any higher odd number).
• The Just Major Seventh chord, 8:10:12:15, contains two occurrences of the interval 3:2, two occurrences of the interval 5:4, one occurrence of the interval 6:5, and one occurrence of the interval 15:8. Hence the chord is within "intervallic limit" 15 (or any higher odd number).
• The Just Minor Seventh chord, 10:12:15:18, contains two occurrences of the interval 3:2, two occurrences of the interval 6:5, one occurrence of the interval 5:4, and one occurrence of the interval 9:5. Hence the chord is within "intervallic limit" 9 (or any higher odd number).
• The Just Minor Triad, 10:12:15, containes one occurrence of the interval 3:2, one occurrence of the interval 5:4, and one occurrence of the interval 6:5. Hence the chord is within "intervallic limit" 5 (or any higher odd number).

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".

. . . . . . . . .
[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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