# 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 2^{1/22}, or 54.55 cents.The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist

RHM Bosanquet . Inspired by the division of the octave into 22 unequal parts in the music theory of India, Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theoristJosé Würschmidt , who noted it as a possible next step after19 equal temperament , and J. Murray Barbour in his classic survey of tuning history, "Tuning and Temperament".The 22-et system is in fact the third equal division, after 12 and 19, which is capable of tolerably dealing with 5-limit music. However, there is more to it than that; unlike 12 or 19 it is able to do rough justice to the 7- and 11-limits. While

31 equal temperament does much better, 22-et at least allows the use of these higher-limit harmonies. Moreover, 22-et, unlike 12 and 19, is not a meantone system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like.**Interval size**Here are the sizes of some common intervals in this system:

As can be seen by the table above, 22-et does not have a very good match for the smaller intervals such as the 8:7, 9:8 and 10:9; these intervals lie roughly near the middle of the intervals defined. It does offer a very close match to the greater undecimal

neutral second and thediatonic semitone .**Comparison to other equal temperaments**The matches to the

septimal major third andseptimal minor third in 22-et are excellent, significantly better than in 19-et, and the major and minor thirds are still significantly better than in 12-et. Although the 31-et improves the matches of theperfect fifth , and major, minor, and septimal minor thirds, it does not fit the septimal major third as well, nor does it fit the greater undecimal neutral second.**Properties of 22 equal temperament**22-ET does

**not**"temper out" thesyntonic comma of 81:80, which is connected to the fact that it is not ameantone temperament . It does, however, temper out thediaschisma , 2048:2025, themagic comma or smalldiesis , 3125:3072, and theporcupine comma , or maximal diesis, 250:243. In a diaschismic system, such as 12-et or 22-et, the diatonic tritone 45:32, which is a major third above a major whole tone representing 9:8, is equated to its inverted form, 64:45. That the magic comma is tempered out means that 22-et is amagic temperament , where five major thirds make up a perfect twelfth (a perfect fifth plus an octave). That the porcupine comma is tempered out means that 22-et is aporcupine temperament , where three minor whole tones (10:9 tones) give a fourth, and five give a minor sixth.In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both 50:49, the

jubilee comma , and 64:63, theseptimal comma , are tempered out in both systems. Hence, because 50:49 is equated to a unison, they both equate the twoseptimal tritone s of 7:5 and 10:7; because 64:63 is equated to a unison, neither of them distinguish between adominant seventh chord and anotonal tetrad. As 22-et equates both of these commas to a unison, they are also made equivalent to each other: theseptimal kleisma 225:224, which is the difference between the two $left(\; frac\{50\}\{49\}\; div\; frac\{64\}\{63\}\; =\; frac\{225\}\{224\}\; ight)$, is also tempered out, so the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. But not all septimal commas tempered out by 12-et are also tempered out by 22-et—1728:1715, theorwell comma , is tempered out by 12-et but not 22-et; and therefore the orwell tetrad is also a chord of 22-et.**External links*** [

*http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-22.pdf Erlich, Paul, "Tuning, Tonality, and Twenty-Two Tone Temperament"*]**References***Barbour, James Murray, "Tuning and temperament, a historical survey", East Lansing, Michigan State College Press, 1953 [c1951]

*Bosanquet, R.H.M. [*http://www.geocities.com/threesixesinarow/hindoo.htm "On the Hindoo division of the octave, with additions to the theory of higher orders"*] , "Proceedings of the Royal Society of London" vol. 26 (March 1, 1877 to December 20, 1877) Taylor & Francis, London 1878, pp. 372-384. (Reproduced in Tagore, Sourindro Mohun, "Hindu Music from Various Authors", Chowkhamba Sanskrit Series, Varanasi, India, 1965)

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**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**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… … 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