22edo

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=22 tone equal temperament= 

In music, '''22 equal temperament''', called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of twenty-second root of 2, 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 [[Indian|music theory of India]], Bosenquet 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 theorist José Würschmidt, who noted it as a possible next step after [[19edo|19 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 [[31edo|31 equal temperament]] does much better, 22-et at least allows the use of these higher-limit harmonies. Moreover, 22-et, unlike 12 and [[19edo|19]], is not a [[Regular Temperaments#meantone|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.

==Properties of 22 equal temperament== 

Possibly the most striking characteristic of 22-et to those not used to it is that it does '''not''' "temper out" the syntonic comma of 81/80, and therefore is not a system of [[Regular Temperaments#meantone|meantone]] temperament. It does, however, temper out the diaschisma, 2048/2025, the magic comma or small diesis, 3125/3072, and the porcupine 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 a [[Regular Temperaments#magic|magic]] system, where five major thirds make up a perfect fifth. That the porcupine comma is tempered out means that 22-et is a [[Regular Temperaments#porcupine|porcupine]] system, 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, (the septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritons of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and a utonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the septimal kleisma, so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell comma]]; and the orwell tetrad is also a chord of 22-et.

==External links== 

Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''
*[[[http://66.98.148.43/%7Exenharmo/text/tuning22.pdf|http://66.98.148.43/~xenharmo/text/tuning22.pdf]]] or http://lumma.org/tuning/erlich/erlich-decatonic.pdf

==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, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965

Original HTML content:

<html><head><title>22edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x22 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->22 tone equal temperament</h1>
 <br />
In music, '''22 equal temperament''', called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of twenty-second root of 2, or 54.55 cents.<br />
<br />
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 <a class="wiki_link" href="/Indian">music theory of India</a>, Bosenquet 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 theorist José Würschmidt, who noted it as a possible next step after <a class="wiki_link" href="/19edo">19 equal temperament</a>, and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.<br />
<br />
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 <a class="wiki_link" href="/31edo">31 equal temperament</a> does much better, 22-et at least allows the use of these higher-limit harmonies. Moreover, 22-et, unlike 12 and <a class="wiki_link" href="/19edo">19</a>, is not a <a class="wiki_link" href="/Regular%20Temperaments#meantone">meantone</a> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x22 tone equal temperament-Properties of 22 equal temperament"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties of 22 equal temperament</h2>
 <br />
Possibly the most striking characteristic of 22-et to those not used to it is that it does '''not''' &quot;temper out&quot; the syntonic comma of 81/80, and therefore is not a system of <a class="wiki_link" href="/Regular%20Temperaments#meantone">meantone</a> temperament. It does, however, temper out the diaschisma, 2048/2025, the magic comma or small diesis, 3125/3072, and the porcupine 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 a <a class="wiki_link" href="/Regular%20Temperaments#magic">magic</a> system, where five major thirds make up a perfect fifth. That the porcupine comma is tempered out means that 22-et is a <a class="wiki_link" href="/Regular%20Temperaments#porcupine">porcupine</a> system, where three minor whole tones (10/9 tones) give a fourth, and five give a minor sixth.<br />
<br />
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, (the septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritons of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and a utonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the septimal kleisma, so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the <a class="wiki_link" href="/orwell%20comma">orwell comma</a>; and the orwell tetrad is also a chord of 22-et.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x22 tone equal temperament-External links"></a><!-- ws:end:WikiTextHeadingRule:4 -->External links</h2>
 <br />
Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''<br />
*[<a class="wiki_link_ext" href="http://66.98.148.43/%7Exenharmo/text/tuning22.pdf" rel="nofollow">http://66.98.148.43/~xenharmo/text/tuning22.pdf</a>] or <!-- ws:start:WikiTextUrlRule:37:http://lumma.org/tuning/erlich/erlich-decatonic.pdf --><a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/erlich-decatonic.pdf" rel="nofollow">http://lumma.org/tuning/erlich/erlich-decatonic.pdf</a><!-- ws:end:WikiTextUrlRule:37 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x22 tone equal temperament-References"></a><!-- ws:end:WikiTextHeadingRule:6 -->References</h2>
 <br />
Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br />
<br />
Bosanquet, R.H.M. [<!-- ws:start:WikiTextUrlRule:38:http://www.geocities.com/threesixesinarow/hindoo.htm --><a class="wiki_link_ext" href="http://www.geocities.com/threesixesinarow/hindoo.htm" rel="nofollow">http://www.geocities.com/threesixesinarow/hindoo.htm</a><!-- ws:end:WikiTextUrlRule:38 --> ''On the Hindoo division of the octave, with additions to the theory of higher orders''], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965</body></html>