72 equal temperament

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). Play  Each step represents a frequency ratio of 72√2, 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" (Play ). 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.

"Third tone" redirects here. For the Chinese third tone, see Standard Chinese phonology § Tones.

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[1] and Ivan Wyschnegradsky,[2][3][4] 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.

History and use

Byzantine music

The 72 equal temperament is used in Byzantine music theory,[5] 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.

Other history and use

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.

Notation

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).

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) MIDI Just Ratio Just (cents) MIDI Error
octave 72 1200 2:1 1200 0
perfect fifth 42 700 play  3:2 701.96 play  −1.96
septendecimal tritone 36 600 play  17:12 603.00 −3.00
septimal tritone 35 583.33 play  7:5 582.51 play  +0.82
tridecimal tritone 34 566.67 play  18:13 563.38 +3.28
11th harmonic 33 550 play  11:8 551.32 play  −1.32
(15:11) augmented fourth 32 533.33 play  15:11 536.95 play  −3.62
perfect fourth 30 500 play  4:3 498.04 play  +1.96
septimal narrow fourth 28 466.66 play  21:16 470.78 play  −4.11
17:13 narrow fourth 17:13 464.43 +2.24
tridecimal major third 27 450 play  13:10 454.21 play  −4.21
septendecimal supermajor third 22:17 446.36 +3.64
septimal major third 26 433.33 play  9:7 435.08 play  −1.75
undecimal major third 25 416.67 play  14:11 417.51 play  −0.84
major third 23 383.33 play  5:4 386.31 play  −2.98
tridecimal neutral third 22 366.67 play  16:13 359.47 +7.19
neutral third 21 350 play  11:9 347.41 play  +2.59
septendecimal supraminor third 20 333.33 play  17:14 336.13 −2.80
minor third 19 316.67 play  6:5 315.64 play  +1.03
quasi-tempered minor third 18 300 play  25:21 301.85 -1.85
tridecimal minor third 17 283.33 play  13:11 289.21 play  −5.88
septimal minor third 16 266.67 play  7:6 266.87 play  −0.20
tridecimal ​5⁄4 tone 15 250 play  15:13 247.74 +2.26
septimal whole tone 14 233.33 play  8:7 231.17 play  +2.16
septendecimal whole tone 13 216.67 play  17:15 216.69 −0.02
whole tone, major tone 12 200 play  9:8 203.91 play  −3.91
whole tone, minor tone 11 183.33 play  10:9 182.40 play  +0.93
greater undecimal neutral second 10 166.67 play  11:10 165.00 play  +1.66
lesser undecimal neutral second 9 150 play  12:11 150.64 play  −0.64
greater tridecimal ​2⁄3 tone 8 133.33 play  13:12 138.57 play  −5.24
great limma 27:25 133.24 play  +0.09
lesser tridecimal ​2⁄3 tone 14:13 128.30 play  +5.04
septimal diatonic semitone 7 116.67 play  15:14 119.44 play  −2.78
diatonic semitone 16:15 111.73 play  +4.94
greater septendecimal semitone 6 100 play  17:16 104.95 play  -4.95
lesser septendecimal semitone 18:17 98.95 play  +1.05
septimal chromatic semitone 5 83.33 play  21:20 84.47 play  −1.13
chromatic semitone 4 66.67 play  25:24 70.67 play  −4.01
septimal third-tone 28:27 62.96 play  +3.71
septimal quarter tone 3 50 play  36:35 48.77 play  +1.23
septimal diesis 2 33.33 play  49:48 35.70 play  −2.36
undecimal comma 1 16.67 play  100:99 17.40 −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.

Scale diagram
12-tone Play  and 72-tone Play  regular diatonic scales notated with the Maneri-Sims system

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.

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See also

References
  1. A. Hába: "Harmonické základy ctvrttónové soustavy". German translation: "Neue Harmonielehre des diatonischen, chromatischen Viertel-, Drittel-, Sechstel- und Zwölftel-tonsystems" by the author. Fr. Kistner & C.F.W. Siegel, Leipzig, 1927. Universal, Wien, 1978. Revised by Erich Steinhard, "Grundfragen der mikrotonalen Musik"; Bd. 3, Musikedition Nymphenburg 2001, Filmkunst-Musikverlag, München, 251 pages.
  2. I. Wyschnegradsky: "L'ultrachromatisme et les espaces non octaviants", La Revue Musicale no. 290–291, pp. 71–141, Ed. Richard-Masse, Paris, 1972
  3. La Loi de la Pansonorité (Manuscript, 1953), Ed. Contrechamps, Geneva, 1996. Preface by Pascale Criton, edited by Franck Jedrzejewski. ISBN 978-2-940068-09-8
  4. Une philosophie dialectique de l'art musical (Manuscript, 1936), Ed. L'Harmattan, Paris, 2005, edited by Franck Jedrzejewski. ISBN 978-2-7475-8578-1.
  5. G. Chryssochoidis, D. Delviniotis and G. Kouroupetroglou, "A semi-automated tagging methodology for Orthodox Ecclesiastic Chant Acoustic corpora", Proceedings SMC'07, 4th Sound and Music Computing Conference, Lefkada, Greece (11–13 July 2007).
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