72 equal temperament
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In music, 72 equal temperament, called 72-tet, 72-edo, or 72-et, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equally large steps. Each step represents a frequency ratio of 21/72, or 16.667 cents.
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.
A number of composers have made use of it, and these represent widely different points of view and types of musical practice. Many 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. Other composers who have used it include Alois Haba, Julian Carrillo, Ivan Wyschnegradsky and Iannis Xenakis. 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.
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[edit] Byzantine music
The 72 equal temperament is used in Byzantine music theory[1], 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.
[edit] Interval size
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 name | size (steps) | size (cents) | just ratio | just (cents) | difference |
| perfect fifth | 42 | 700.00 | 3:2 | 701.96 | 1.96 |
| tritone | 35 | 583.33 | 7:5 | 582.51 | -0.82 |
| tritone, inverted 13th harmonic | 34 | 566.67 | 18:13 | 563.38 | -3.28 |
| tritone, 11th harmonic | 33 | 550.00 | 11:8 | 551.32 | 1.32 |
| (15:11) augmented fourth | 32 | 533.33 | 15:11 | 536.95 | 3.62 |
| perfect fourth | 30 | 500.00 | 4:3 | 498.04 | -1.96 |
| tridecimal major third | 27 | 450 | 13:10 | 454.21 | 4.21 |
| septimal major third | 26 | 433.33 | 9:7 | 435.08 | 1.75 |
| undecimal major third | 25 | 416.67 | 14:11 | 417.51 | 0.84 |
| major third | 23 | 383.33 | 5:4 | 386.31 | 2.98 |
| tridecimal neutral third | 22 | 366.67 | 16:13 | 359.47 | -7.19 |
| neutral third | 21 | 350.00 | 11:9 | 347.41 | -2.59 |
| minor third | 19 | 316.67 | 6:5 | 315.64 | -1.03 |
| tridecimal minor third | 17 | 283.33 | 13:11 | 289.21 | 5.88 |
| septimal minor third | 16 | 266.67 | 7:6 | 266.87 | 0.20 |
| tridecimal 5/4 tone | 15 | 250.00 | 15:13 | 247.74 | -2.26 |
| septimal whole tone | 14 | 233.33 | 8:7 | 231.17 | -2.16 |
| whole tone, major tone | 12 | 200.00 | 9:8 | 203.91 | 3.91 |
| whole tone, minor tone | 11 | 183.33 | 10:9 | 182.40 | -0.93 |
| greater undecimal neutral second | 10 | 166.67 | 11:10 | 165.00 | -1.66 |
| lesser undecimal neutral second | 9 | 150 | 12:11 | 150.64 | 0.64 |
| greater tridecimal 2/3 tone | 8 | 133.33 | 13:12 | 138.57 | 5.24 |
| lesser tridecimal 2/3rd tone | 8 | 133.33 | 14:13 | 128.3 | -5.04 |
| diatonic semitone | 7 | 116.67 | 16:15 | 111.73 | -4.94 |
| septimal chromatic semitone | 5 | 83.33 | 21:20 | 84.47 | 1.13 |
| chromatic semitone | 4 | 66.67 | 25:24 | 70.67 | 4.01 |
| septimal quarter tone | 3 | 50 | 36:35 | 48.77 | -1.23 |
| septimal diesis | 2 | 33.33 | 49:48 | 35.7 | 2.36 |
| undecimal comma | 1 | 16.67 | 100:99 | 17.4 | 0.73 |
Although 12-ET can be viewed as a subset of 72-ET, the closest matches to many intervals of the harmonic series 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.
All intervals involving harmonics up through the 11th are matched very closely in this system; 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.
[edit] Theoretical properties
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In terms of tuning theory[citation needed], the 72 equal harmonic system equates to the unison, or "tempers out"[citation needed], the small intervals 169/168, 225/224, 243/242, 1029/1024, 385/384, 441/440, 540/539, as well as the Pythagorean comma and 15625/15552, among countless others; this gives it its own particular character in terms of functional harmony. It also means that 72 supports various temperaments which temper out some, but not all, of the above small intervals[citation needed].
It is important to notice, however, that it does not temper out the syntonic comma of 81/80, and is therefore not a meantone system. Instead, 81/80 can be treated as one step of the scale. Hence, common practice music needs to be adapted for it to be played in this harmonic system, though the option always remains to use only twelve of the 72 notes.
This tuning system also does not temper out the comma 121/120, which means that it distinguishes between the greater (11:10) and lesser (12:10) undecimal neutral second. The comma 121/120, about 14.37 cents wide, is only slightly smaller than one step of the 72-ET scale.
[edit] References
[edit] External links
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