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 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. It has been first theoreticized as twelth-tone by Alois Hába [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.] and Ivan Wyschnegradsky [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 Pascal Criton, edited by Franck Jedrzejewski. ISBN 2-940068-09-7; "Une philosophie dialectique de l'art musical" (Manuscript, 1936), Ed. L'Harmattan, Paris, 2005, edited by Franck Jedrzejewski. ISBN 2-7475-8578-6.] who considered it as a good approach to the "continuum" of sound. It is also cited among the divisions of tone by Julian Carrillo, who preferred 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, Julian Carrillo, Ivan Wyschnegradsky and Iannis Xenakis who used it mainly in an atonal way. 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 [ [http://smc07.uoa.gr/SMC07%20Proceedings/SMC07%20Paper%2020.pdf] G. Chryssochoidis, D. Delviniotis and G. Kouroupetroglou, "A semi-automated tagging methodology forOrthodox Ecclesiastic Chant Acoustic corpora", Proceedings SMC'07, 4th Sound and Music Computing Conference, Lefkada, Greece (11-13 July 2007).] , 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:

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.

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.

In terms of just intonation tuning theory Fact|date=April 2008, the 72 equal harmonic system equates to the unison, or "tempers out" Fact|date=April 2008, 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 Fact|date=April 2008.

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.


External links

* [http://bostonmicrotonalsociety.org/ The Boston Microtonal Society official site]
* [http://www.72note.com 72note.com]
* [http://users.bigpond.net.au/d.keenan/sagittal/gift/Wysch.gif| Wyschnegradsky notation for twelfth-tone]
* [http://www.mindspring.com/%7Etmook/micro.html Maneri-Sims notation for 72-et]

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