# 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 2^{1/72}, 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.*] andIvan 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 byJulian 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 andIannis Xenakis who used it mainly in an atonal way. Many other composers use it freely and intuitively, such as jazz musicianJoe Maneri , and classically-oriented composers such asJulia Werntz and others associated with theBoston Microtonal Society . Others, such as New York composerJoseph Pehrson are interested in it because it supports the use ofmiracle temperament , and still others simply because it approximates higher-limit just intonation, such asEzra Sims andJames 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 , andStanislav 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*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).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 onratio s 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.**References****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*]

*Wikimedia Foundation.
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### Look at other dictionaries:

**Equal temperament**— is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio. In equal temperament tunings an interval mdash; usually the octave mdash; is divided into a series of equal steps (equal… … Wikipedia**Equal temperament**— Equal E qual, a. [L. aequalis, fr. aequus even, equal; akin to Skr. ?ka, and perh. to L. unus for older oinos one, E. one.] 1. Agreeing in quantity, size, quality, degree, value, etc.; having the same magnitude, the same value, the same degree,… … The Collaborative International Dictionary of English**Equal temperament**— Temperament Tem per*a*ment, n. [L. temperamentum a mixing in due proportion, proper measure, temperament: cf. F. temp[ e]rament. See {Temper}, v. t.] 1. Internal constitution; state with respect to the relative proportion of different qualities,… … The Collaborative International Dictionary of English**equal temperament**— Music. the division of an octave into 12 equal semitones, as in the tuning of a piano. * * * ▪ music in music, a tuning system in which the octave is divided into 12 semitones of equal size. Because it enables keyboard instruments (keyboard … Universalium**equal temperament**— noun the division of the scale based on an octave that is divided into twelve exactly equal semitones equal temperament is the system commonly used in keyboard instruments • Hypernyms: ↑temperament … Useful english dictionary**53 equal temperament**— In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into fifty three equally large steps. Each step represents a frequency ratio of 21/53, or 22.6415 cents, an interval sometimes… … Wikipedia**19 equal temperament**— In music, 19 equal temperament, called 19 TET, 19 EDO, or 19 ET, is the tempered scale derived by dividing the octave into 19 equally large steps. Each step represents a frequency ratio of 21/19, or 63.16 cents. Because 19 is a prime number, one… … Wikipedia**31 equal temperament**— In music, 31 equal temperament, which can be abbreviated 31 tET, 31 EDO, 31 ET, is the tempered scale derived by dividing the octave into 31 equal sized steps. Each step represents a frequency ratio of 21/31, or 38.71 cents.Division of the octave … Wikipedia**22 equal temperament**— In music, 22 equal temperament, called 22 tet, 22 edo, or 22 et, is the tempered scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of 21/22, or 54.55 cents.The idea of dividing the octave… … Wikipedia**34 equal temperament**— In musical theory, 34 equal temperament, also referred to as 34 tet, 34 edo or 34 et, is the tempered tuning derived by dividing the octave into 34 equal sized steps. Each step represents a frequency ratio of 21/34, or 35.29 cents.Unlike… … Wikipedia