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.
In just intonation, any given interval can be expressed as the ratio between two frequencies, such as 4:3 for the perfect fourth or 10:9 for the minor tone. 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 a prime limit of 5, because 9 can be factorized into 3×3, and 10 into 2×5.
[edit] Prime limits
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. 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 this 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. Some believe that "blue" notes are derived from 7-limit intervals.
In the twentieth century, the composer Harry Partch developed a system of just intonation microtonal music that included intervals up to the 11-limit. Composer Ben Johnston extended Partch's system, composing music based on a flexible tuning system that derives pitches from as high as the 31-limit. Other composers, including La Monte Young, have based music on higher primes than 31.
[edit] Odd limits
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.
Since the harmonic series generates overtones from each of its overtones, non-prime odd-numbered partials are compound-intervals. For instance, 9:8 (as described above), is the ninth partial; it is the "third overtone of the third overtone" (3x3=9), or, in musical terms, two perfect fifths above the fundamental.
In practice this distinction is often glossed over, and some authors use the unqualified word limit to refer to prime limits, while others use it to refer to odd limits.
