Limit (music)
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In music, a limit is a number measuring the harmony of an interval. The lower the number, the more consonant the interval is considered to be. There are two different kinds of limits: prime limits and odd limits.
The concept of limit only makes sense when applied to intervals appearing in the harmonic series which can be represented as the ratios of whole numbers; its use in describing intervals in equal temperament rests on the fact that these intervals closely approximate intervals found in the harmonic series. In just intonation, any given interval can be expressed as the ratio between two frequencies, such as 4:3 for the perfect fourth or 9:8 for the major second. The limits for such intervals are defined as follows:
The odd limit only regards pitch classes. (That is, it treats pitches the same when they differ only in the octave.) Mathematically, this is achieved by dividing any even numbers in the fraction repeatedly by 2 until both numerator and denominator are odd. The limit is then defined as the bigger number of the two. Thus the odd limit of the perfect fourth is 3, while the minor tone has an odd limit of 9.
The prime limit can be seen as a generalization that does not favor the number 2. It is defined as the largest prime number in the factorization of both numerator and denominator. That is, in number theoretic terms, it measures the smoothness of the numerator and denominator. The prime limit of the perfect fourth is 3 (the same as the odd limit), but the minor tone has a prime limit of 5, because 9 can be factorized into 3×3, and 10 into 2×5.
[edit] Prime limits, scales and microtonal music
Prime limits lend themselves for the investigation of scales. This is because in a scale in which all notes form an interval from the base note that remains within a certain prime limit, all other intervals between these notes remain within the same limit. This can be shown using the following diatonic scale:
| Note | C | D | E | F | G | A | B | C | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ratio to base note | 1/1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2/1 | ||||||||
| Limit | 1 | 3 | 5 | 3 | 3 | 5 | 5 | 2 | ||||||||
| Step | 9/8 | 10/9 | 16/15 | 9/8 | 10/9 | 9/8 | 16/15 | |||||||||
| Limit | 3 | 5 | 5 | 3 | 5 | 3 | 5 | |||||||||
This scale is defined such that all pitches remain within a 5-limit (relative to the base note). As can be seen, that same condition holds for the steps between neighboring pitches. All resulting intervals between any two pitches include all of the intervals necessary for major and minor triads, which are the building-blocks of tonal music from the common practice period. Thus, almost all music composed is in five-limit — it uses relationships based only on the fifth partial or below, and all intervals can be described as ratios of regular numbers.
In the harmonic series, every even-numbered partial is the octave duplication of another lower one. Every prime-numbered partial introduces a new relationship; just as the five-limit primes (1, 2, 3 and 5) introduce new types of intervals (unisons, octaves, fifths, and thirds, respectively), higher primes (such as 7, 11, 13 and beyond) introduce intervals that are foreign to most music. Septimal meantone temperaments such as 31 equal temperament provide approximations to 7-limit intervals. Some believe that blue notes are derived from 7-limit intervals.
In the twentieth century, Harry Partch developed a system of just intonation microtonal music that included intervals up to the 11-limit. Ben Johnston extended Partch's system, composing music based on a flexible tuning system that derives pitches from as high as the 31-limit. Others, including La Monte Young, have based music on higher primes than 31.
[edit] Tonality diamond and lattice
All the intervals of a given odd limit make up a tonality diamond, and all the intervals of a given prime limit make up an infinite n-dimensional lattice of pitches, where n is the number of primes not exceeding the limit.
