Equal temperament: Encyclopedia II - Equal temperament - Non-12 TET

Equal temperament - Non-12 TET

Five and seven tone equal temperament, with 240 and 171 cent steps relatively, seem the most common outside of 12-tET. A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-tET. A Ugandan Chop xylophone measured by Haddon (1952) also tuned to 171 cent steps. Gamelans are tuned to 5-tET according to Kunst (1949), but according to Hood (1966) and McPhee (1966) their tuning varies widely and according to Tenzer (2000) contain stretched octaves. It is now well-accepted that of the two primary tuning systems in Gamelan music, Slendro and Pelog, only Slendro somewhat resembles 5-tone equal temperament while Pelog is highly unequal. However, Wachsmann (1950) used a Stroboconn to measure a Ugandan harp and women singing unaccompanied, finding variations of 15 and 5 cents respectively. ( ← check accuracy of fragment repair) A South American Indian scale from a preinstrumental culture measured by Boiles (1969) featured 175 cent equal temperament which stretches the octave slightly as with instrumental gamelan music.

The quarter tone scale or 24-tET is, similarly, based on steps of 50 cents or powers of . Other equal divisions of the octave, though, can be better considered temperaments; 24 is usually best considered simply as an equal division (e.g., a bisection of 12-tET). 19-tET and especially 31-tET are extended varieties of Meantone temperament and approximate most just intonation intervals considerably better than 12-tET. They have been used sporadically since the 16th century, with 31-tET particularly popular in Holland, there advocated by Christiaan Huygens and Adriaan Fokker. 53-tET is much better still at approximating the traditional just intonation consonances, but has had very little use. It doesn't fit the Meantone mold that shaped the development of Western harmony and tonality since the Rennaissance, though it does fit schismatic temperament and the Pythagorean tuning of medieval music, and is sometimes used in Turkish music theory. In 53-tET, most traditional compositions would necessitate subtle microtonal pitch shifts or a drifting pitch level in order to make use of the tuning's excellent just intonation triads. Another tuning which has seen some use in practice and is not a meantone system is 22-tET. 55-tET, not as close to just intonation, was a bit closer to common practice. As an excellent representative of the variety of meantone temperament popular in the 18th century, 55-tET it was considered ideal by Georg Philipp Telemann and other prominent musicians. Wolfgang Amadeus Mozart's surviving violin lessons conform closely to such a model.

In the 20th century, standardized Western pitch and notation practices having been placed on a 12-tET foundation made the quarter tone scale a much more popular microtonal tuning. A further extension of 12-tET is 72-tET, which though not a meantone tuning, approximates most just intonation intervals, including non-traditional ones like 7/4, 9/7, 11/5, 11/6 and 11/7, much better. 72-tET has been taught, written and performed in practice, for example by Joe Maneri and his students -- whose atonal inclinations typically avoid any reference to just intonation intervals whatsoever. Still other equal temperaments occupying more than a few musicians include 5-tET, 7-tET, 15-tET, 22-tET, and 48-tET. Theoretically interesting temperaments which have found occasional use include division of the octave into 34, 41, 46, 99 or 171 parts.

More generally, every step in n tone equal temperament is 1200/n cents. However, if one wishes to create an equal tempered scale that does not repeat at the octave, a scale with n equal steps in a pseudo-octave p is based on the ratio r

This still may be easier to calculate in cents, for instance the pseudo-octave of ratio 2.1:1 is an interval of 1284 cents. Equal tempered scales can also be generated simply by picking the number of cents that each step will consist of.

Wendy Carlos created two equal tempered scales for the title track of her album Beauty In The Beast, the Alpha and Beta scales. Beta splits a perfect fourth into two equal parts, which creates a scale where each step is almost 64 cents. Alpha does the same to a minor third to create a scale of 78 cent steps.

The equal tempered version of the Bohlen-Pierce scale consists of the ratio 3:1, 1902 cents, conventionally an octave and a just fifth, used as a tritave, and split into a thirteen tone equal temperament where each step is

or 146.3 cents. This provides a very close match to justly tuned ratios consisting only of odd numbers.

Australian aboriginal music extensively measured by Ellis (1965) was based on arithmetic scales (the harmonic series is an arithmetic scale, though presumably the Australian scales began an interval smaller than an octave) with an equal separation in hertz.

Adapted from the Wikipedia article "Non-12 TET", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki