72 Тонды Темперация: Музыкалық Теория және Қолданылуы

In music, 72 equal temperament, called twelfth tone, 72 TET, 72 EDO, or 72 ET, is the tempered scale derived by dividing the octave into twelfth tones, or in other words 72 equal steps (equal frequency ratios). Each step represents a frequency ratio of , or 16 2/3 cents, which divides the 100 cent "halftone" into 6 equal parts (100 ÷ 16 2/3 = 6) and is thus a "twelfth tone" Since 72 is divisible by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72, 72 EDO includes all those equal temperaments. Since it contains so many temperaments, 72 EDO contains at the same time tempered semitones, third tones, quartertones and sixth tones, which makes it a very versatile temperament. This division of the octave has attracted much attention from tuning theorists, since on the one hand it subdivides the standard 12 equal temperament and on the other hand it accurately represents overtones up to the twelfth partial tone, and hence can be used for 11 limit music. It was theoreticized in the form of twelfth tones by Alois Hába and Ivan Wyschnegradsky, who considered it as a good approach to the continuum of sound. 72 EDO is also cited among the divisions of the tone by Julián Carrillo, who preferred the sixteenth tone as an approximation to continuous sound in discontinuous scales.

The 72 equal temperament is used in Byzantine music theory, dividing the octave into 72 equal moria, which itself derives from interpretations of the theories of Aristoxenos, who used something similar. Although the 72 equal temperament is based on irrational intervals (see above), as is the 12 tone equal temperament mostly commonly used in Western music (and which is contained as a subset within 72 equal temperament), 72 equal temperament, as a much finer division of the octave, is an excellent tuning for both representing the division of the octave according to the diatonic and the chromatic genera in which intervals are based on ratios between notes, and for representing with great accuracy many rational intervals as well as irrational intervals.

A number of composers have made use of it, and these represent widely different points of view and types of musical practice. These include Alois Hába, Julián Carrillo, Ivan Wyschnegradsky and Iannis Xenakis. Many other composers use it freely and intuitively, such as jazz musician Joe Maneri, and classically oriented composers such as Julia Werntz and others associated with the Boston Microtonal Society. Others, such as New York composer Joseph Pehrson are interested in it because it supports the use of miracle temperament, and still others simply because it approximates higher limit just intonation, such as Ezra Sims and James Tenney. There was also an active Soviet school of 72 equal composers, with less familiar names: Evgeny Alexandrovich Murzin, Andrei Volkonsky, Nikolai Nikolsky, Eduard Artemiev, Alexander Nemtin, Andrei Eshpai, Gennady Gladkov, Pyotr Meshchianinov, and Stanislav Kreichi. The ANS synthesizer uses 72 equal temperament.

The Maneri Sims notation system designed for 72 et uses the accidentals and for 1/12 tone down and up (1 step = 16 2/3 cents), and for 1/6 down and up (2 steps = 33 1/3 cents), and and for 1/4 up and down (3 steps = 50 cents). They may be combined with the traditional sharp and flat symbols (6 steps = 100 cents) by being placed before them, for example: or , but without the intervening space. A 1/3 tone may be one of the following , , , or (4 steps = 66 2/3) while 5 steps may be , , or (83 1/3 cents).

Below are the sizes of some intervals (common and esoteric) in this tuning. For reference, differences of less than 5 cents are melodically imperceptible to most people. Interval NameSize (steps)Size (cents)MIDIJust RatioJust (cents)MIDIErroroctave7212002:112000harmonic seventh58966.677:4968.83−2.16perfect fifth427003:2701.96−1.96septendecimal tritone3660017:12603.00−3.00septimal tritone35583.337:5582.51+0.82tridecimal tritone34566.6718:13563.38+3.2811th harmonic3355011:8551.32−1.32(15:11) augmented fourth32533.3315:11536.95−3.62perfect fourth305004:3498.04+1.96septimal narrow fourth28466.6621:16470.78−4.1117:13 narrow fourth17:13464.43+2.24tridecimal major third2745013:10454.21−4.21septendecimal supermajor third22:17446.36+3.64septimal major third26433.339:7435.08−1.75undecimal major third25416.6714:11417.51−0.84quasi tempered major third244005:4386.3113.69major third23383.335:4386.31−2.98tridecimal neutral third22366.6716:13359.47+7.19neutral third2135011:9347.41+2.59septendecimal supraminor third20333.3317:14336.13−2.80minor third19316.676:5315.64+1.03quasi tempered minor third1830025:21301.85 1.85tridecimal minor third17283.3313:11289.21−5.88septimal minor third16266.677:6266.87−0.20tridecimal 5/4 tone1525015:13247.74+2.26septimal whole tone14233.338:7231.17+2.16septendecimal whole tone13216.6717:15216.69−0.02whole tone, major tone122009:8203.91−3.91whole tone, minor tone11183.3310:9182.40+0.93greater undecimal neutral second10166.6711:10165.00+1.66lesser undecimal neutral second915012:11150.64−0.64greater tridecimal 2/3 tone8133.3313:12138.57−5.24great limma27:25133.24+0.09lesser tridecimal 2/3 tone14:13128.30+5.04septimal diatonic semitone7116.6715:14119.44−2.78diatonic semitone16:15111.73+4.94greater septendecimal semitone610017:16104.95 4.95lesser septendecimal semitone18:1798.95+1.05septimal chromatic semitone583.3321:2084.47−1.13chromatic semitone466.6725:2470.67−4.01septimal third tone28:2762.96+3.71septimal quarter tone35036:3548.77+1.23septimal diesis233.3349:4835.70−2.36undecimal comma116.67100:9917.40−0.73

Although 12 ET can be viewed as a subset of 72 ET, the closest matches to most commonly used intervals under 72 ET are distinct from the closest matches under 12 ET. For example, the major third of 12 ET, which is sharp, exists as the 24 step interval within 72 ET, but the 23 step interval is a much closer match to the 5:4 ratio of the just major third. 12 ET has a very good approximation for the perfect fifth (third harmonic), especially for such a small number of steps per octave, but compared to the equally tempered versions in 12 ET, the just major third (fifth harmonic) is off by about a sixth of a step, the seventh harmonic is off by about a third of a step, and the eleventh harmonic is off by about half of a step. This suggests that if each step of 12 ET were divided in six, the fifth, seventh, and eleventh harmonics would now be well approximated, while 12 ET's excellent approximation of the third harmonic would be retained. Indeed, all intervals involving harmonics up through the 11th are matched very closely in 72 ET; no intervals formed as the difference of any two of these intervals are tempered out by this tuning system. Thus, 72 ET can be seen as offering an almost perfect approximation to 7 , 9 , and 11 limit music. When it comes to the higher harmonics, a number of intervals are still matched quite well, but some are tempered out. For instance, the comma 169:168 is tempered out, but other intervals involving the 13 th harmonic are distinguished. Unlike tunings such as 31 ET and 41 ET, 72 ET contains many intervals which do not closely match any small number (<16) harmonics in the harmonic series.

Because 72 EDO contains 12 EDO, the scale of 12 EDO is in 72 EDO. However, the true scale can be approximated better by other intervals.