Limit (music)
From Wikipedia, the free encyclopedia
In music, a limit is a degree of harmonic complexity based on pitches’ relationships in “just intonation,” a system of tuning intervals with minimal harmonic “beating.” Such pitches vibrate in relation to each other in ways that parallel those of the harmonic series. In the harmonic series, a fundamental pitch is accompanied by overtones, which form a composite sound that results in a sound’s timbre.
In a “pure” harmonic series (one that does not have inharmonic, or irregularly spaced, overtones), the overtones relate to each other in simple ratios (see harmonic series for a diagram):
- The second overtone partial relates to the fundamental (the first partial) in a 2:1 ratio—the higher pitch vibrates twice as fast as the lower pitch. This sounds as an octave.
- The third overtone partial relates to the fundamental in a 3:1 ratio—the higher pitch vibrates three times for every vibration of the lower pitch. When reduced to the same octave as the fundamental, this sounds as a perfect fifth.
- The fourth overtone partial relates to the fundamental in a 4:1 ratio, which is the double-octave; when reduced by an octave, it is a duplication of an interval already expressed in the overtone series.
- The fifth overtone partial relates to the fundamental in a 5:1 ratio, which, when reduced to the same octave as the fundamental, sounds as a major third.
These intervals, along with the intervals that result from relating one to another (the fifth partial to the fourth, or the fifth partial to the third, etc.) 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.
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] Prime versus odd
Strictly speaking, there are two different kinds of limits: prime limits and odd limits. The prime limit of an interval or chord in just intonation is the largest prime number in its factorization. The odd limit, on the other hand, is the largest odd number in its factorization, which can be found by dividing by two repeatedly until the quotient is odd. Thus the 9:8 major tone has a prime limit of 3, but an odd limit of 9. 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.
