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 (help·info) Each step represents a frequency ratio of 21/72, or 16.67 cents, which divides the 100 cent "halftone" into 6 equal parts (100/16.6 = 6) and is thus a "twelfth-tone" ( Play (help·info)).
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-et 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.
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.
Byzantine music 
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.
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|
|perfect fifth||42||700||play (help·info)||3:2||701.96||play (help·info)||−1.96|
|septimal tritone||35||583.33||play (help·info)||7:5||582.51||play (help·info)||+0.82|
|tridecimal tritone||34||566.67||play (help·info)||18:13||563.38||+3.28|
|11th harmonic||33||550||play (help·info)||11:8||551.32||play (help·info)||−1.32|
|(15:11) augmented fourth||32||533.33||play (help·info)||15:11||536.95||−3.62|
|perfect fourth||30||500||play (help·info)||4:3||498.04||play (help·info)||+1.96|
|septimal narrow fourth||28||466.66||play (help·info)||21:16||470.78||play (help·info)||−4.11|
|17:13 narrow fourth||28||466.66||play (help·info)||17:13||464.43||+2.24|
|tridecimal major third||27||450||play (help·info)||13:10||454.21||play (help·info)||−4.21|
|septendecimal supermajor third||27||450||play (help·info)||22:17||446.36||+3.64|
|septimal major third||26||433.33||play (help·info)||9:7||435.08||play (help·info)||−1.75|
|undecimal major third||25||416.67||play (help·info)||14:11||417.51||play (help·info)||−0.84|
|major third||23||383.33||play (help·info)||5:4||386.31||play (help·info)||−2.98|
|tridecimal neutral third||22||366.67||play (help·info)||16:13||359.47||+7.19|
|neutral third||21||350||play (help·info)||11:9||347.41||play (help·info)||+2.59|
|septendecimal supraminor third||20||333.33||17:14||336.13||−2.80|
|minor third||19||316.67||play (help·info)||6:5||315.64||play (help·info)||+1.03|
|tridecimal minor third||17||283.33||play (help·info)||13:11||289.21||play (help·info)||−5.88|
|septimal minor third||16||266.67||play (help·info)||7:6||266.87||play (help·info)||−0.20|
|tridecimal 5/4 tone||15||250||play (help·info)||15:13||247.74||+2.26|
|septimal whole tone||14||233.33||play (help·info)||8:7||231.17||play (help·info)||+2.16|
|septendecimal whole tone||13||216.67||17:15||216.69||−0.02|
|whole tone, major tone||12||200||play (help·info)||9:8||203.91||play (help·info)||−3.91|
|whole tone, minor tone||11||183.33||play (help·info)||10:9||182.40||play (help·info)||+0.93|
|greater undecimal neutral second||10||166.67||play (help·info)||11:10||165.00||play (help·info)||+1.66|
|lesser undecimal neutral second||9||150||play (help·info)||12:11||150.64||play (help·info)||−0.64|
|greater tridecimal 2/3 tone||8||133.33||13:12||138.57||−5.24|
|lesser tridecimal 2/3rd tone||8||133.33||play (help·info)||14:13||128.30||+5.04|
|septimal diatonic semitone||7||116.67||play (help·info)||15:14||119.44||play (help·info)||−2.78|
|diatonic semitone||7||116.67||play (help·info)||16:15||111.73||play (help·info)||+4.94|
|Arabic lute index finger||6||100||18:17||98.95||+1.05|
|septimal chromatic semitone||5||83.33||play (help·info)||21:20||84.47||play (help·info)||−1.13|
|chromatic semitone||4||66.67||play (help·info)||25:24||70.67||play (help·info)||−4.01|
|septimal third-tone||4||66.67||play (help·info)||28:27||62.96||+3.71|
|septimal quarter tone||3||50||play (help·info)||36:35||48.77||play (help·info)||+1.23|
|septimal diesis||2||33.33||play (help·info)||49:48||35.70||play (help·info)||−2.36|
|undecimal comma||1||16.67||play (help·info)||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.
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.
Theoretical properties 
72 equal temperament contains at the same time tempered semitones, third-tones, quartertones and sixth-tones, which makes it a very versatile temperament.
- 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.
- I. Wyschnegradsky: "L'ultrachromatisme et les espaces non octaviants", La Revue Musicale # 290-291, pp. 71-141, Ed. Richard-Masse, Paris, 1972; 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; Une philosophie dialectique de l'art musical (Manuscript, 1936), Ed. L'Harmattan, Paris, 2005, edited by Franck Jedrzejewski. ISBN 978-2-7475-8578-1.
-  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).
- The Boston Microtonal Society official site
- Wyschnegradsky notation for twelfth-tone[dead link]
- Maneri-Sims notation for 72-et at the Wayback Machine (archived May 12, 2008)