22edo
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[[toc|flat]] ---- =Theory= 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 [[cent]]s. 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-limit|7-]] and [[11-limit]]s. 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. See also: [[22edo Solfege]], [[22edo Tetrachords]], [[22edo Modes]] ==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|45/32]], which is a major third above a [[major whole tone]] representing [[9_8|9/8]], is equated to its inverted form, [[64_45|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 tone]]s ([[10_9|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|50/49]], (the [[jubilee comma]]), and [[64_63|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. ===Commas=== 22 EDO tempers out the following commas. (Note: This assumes the val < 22 35 51 62 76 81 |.) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || ||= 250/243 ||< | 1 -5 3 > ||> 49.17 ||= Maximal Diesis ||= Porcupine Comma ||= || ||= 3125/3072 ||< | -10 -1 5 > ||> 29.61 ||= Small Diesis ||= Magic Comma ||= || ||= 2048/2025 ||< | 11 -4 -2 > ||> 19.55 ||= Diaschisma ||= ||= || ||= 2109375/2097152 ||< | -21 3 7 > ||> 10.06 ||= Semicomma ||= Fokker Comma ||= || ||= 9193891/9143623 ||< | 32 -7 -9 > ||> 9.49 ||= Escapade Comma || ||= || ||= 4758837/4757272 ||< | -53 10 16 > ||> 0.57 ||= Kwazy ||= ||= || ||= 50/49 ||< | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||= || ||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma || ||= 875/864 ||< | -5 -3 3 1 > ||> 21.90 ||= Keema ||= ||= || ||= 2430/2401 ||< | 1 5 1 -4 > ||> 20.79 ||= Nuwell ||= ||= || ||= 245/243 ||< | 0 -5 1 2 > ||> 14.19 ||= Sensamagic ||= ||= || ||= 1728/1715 ||< | 6 3 -1 -3 > ||> 13.07 ||= Orwellisma ||= Orwell Comma ||= || ||= 225/224 ||< | -5 2 2 -1 > ||> 7.71 ||= Septimal Kleisma ||= Marvel Comma ||= || ||= 10976/10935 ||< | 5 -7 -1 3 > ||> 6.48 ||= Hemimage ||= ||= || ||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||= ||= || ||= 65625/65536 ||< | -16 1 5 1 > ||> 2.35 ||= Horwell ||= ||= || ||= 420175/419904 ||< | -6 -8 2 5 > ||> 1.12 ||= Wizma ||= ||= || ||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||= ||= || ||= 100/99 ||< | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||= ||= || ||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || ||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||= ||= || ||= 896/891 ||< | 7 -4 0 1 -1 > ||> 9.69 ||= Pentacircle ||= ||= || ||= 65536/65219 ||< | 16 0 0 -2 -3 > ||> 8.39 ||= Orgonisma ||= ||= || ||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||= ||= || ||= 540/539 ||< | 2 3 1 -2 -1 > ||> 3.21 ||= Swetisma ||= ||= || ||= 4000/3993 ||< | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||= ||= || ||= 9801/9800 ||< | -3 4 -2 -2 2 > ||> 0.18 ||= Kalisma ||= Gauss' Comma ||= || ||= 91/90 ||< | -1 -2 -1 1 0 1 > ||> 19.13 ||= Superleap ||= ||= || ===A Superpythagorean System=== The 22edo fifth, measuring approximately 709.1 cents, is wider than the 702-cent [[3-limit]] fifth, thus making 22edo a "super-pythagorean" system. As with any superpyth, a chain of fifths produces relatively wide major thirds and narrow minor thirds. In the case of 22edo, the thirds are stretched out to the [[7-limit]] ; the [[subminor third]] comes close to [[7_6|7/6]] and the [[supermajor third]] to [[9_7|9/7]]. Thus, the resulting diatonic scale, which no longer approximates 5-limit thirds, sounds oddly consonant. The ratio of major 2nd to minor 2nd in this diatonic scale is stretched out to 4:1, with the M2 falling between 9/8 and [[8_7|8/7]], and the m2 falling close to a quarter-tone. ===11edo=== As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo]], in the same way that 12edo can play 6edo (the whole tone scale). ==External links== [[http://lumma.org/tuning/erlich/erlich-decatonic.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.webcitation.org/5kjJcrhEx|''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 ---- =Compositions= [[http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3|Tibia]] by [[Paul Erlich]] [[http://lumma.org/music/theory/tctmo/glassic.mp3|Glassic]] by Paul Erlich and [[Ara Sarkissian]] [[http://lumma.org/tuning/erlich/decatonic-swing.mp3|Decatonic Swing]] by Paul Erlich and Ara Sarkissian (jazz) [[http://soundclick.com/share?songid=5683765|Dragged by a Storm Across the Desert Years]] by [[IgliashonJones|Igliashon Jones]] (synth with electric guitar) Numerology by Iglashion Jones (progressive metal) Revenge of the inorganic compounds by Iglashion Jones (progressive metal) [[http://chrisvaisvil.com/?p=267|My Crazy Aunt Sophie]] by [[Chris Vaisvil]]. Blatantly xenharmonic piano. [[http://soundclick.com/share?songid=8839058|where words are said to mean]] by [[Andrew Heathwaite]], a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This & the following pieces by Andrew are for 22-tone guitar & voice. [[http://soundclick.com/share?songid=9101704|I've come with a bucket of roses]] by Andrew Heathwaite (orwell-9: 3 2 3 2 3 2 3 2 2). [[http://soundclick.com/share?songid=9101705|one drop of rain]] by Andrew Heathwaite (orwell-9). [[http://soundclick.com/share?songid=8839060|being a]] by Andrew Heathwaite (porcupine-8: 3 1 3 3 3 3 3). [[http://soundclick.com/share?songid=8839071|my own house]] by Andrew Heathwaite (a pelog-flavored subset of orwell-9: 3 2 7 3 7). [[http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3|Night on Porcupine Mountain]] Mussorgsky-Smith || < 22 35 51 62 76 81 | ||
Original HTML content:
<html><head><title>22edo</title></head><body><!-- ws:start:WikiTextTocRule:16:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Theory">Theory</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: -->
<!-- ws:end:WikiTextTocRule:25 --><hr />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theory</h1>
<br />
In music, '''22 equal temperament''', called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 22 equally large steps. Each step represents a frequency ratio of twenty-second root of 2, or 54.55 <a class="wiki_link" href="/cent">cent</a>s.<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 <a class="wiki_link" href="/5-limit">5-limit</a> music. However, there is more to it than that; unlike 12 or 19 it is able to do rough justice to the <a class="wiki_link" href="/7-limit">7-</a> and <a class="wiki_link" href="/11-limit">11-limit</a>s. 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 />
See also: <a class="wiki_link" href="/22edo%20Solfege">22edo Solfege</a>, <a class="wiki_link" href="/22edo%20Tetrachords">22edo Tetrachords</a>, <a class="wiki_link" href="/22edo%20Modes">22edo Modes</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Theory-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''' "temper out" 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 <a class="wiki_link" href="/diatonic%20tritone">diatonic tritone</a> <a class="wiki_link" href="/45_32">45/32</a>, which is a major third above a <a class="wiki_link" href="/major%20whole%20tone">major whole tone</a> representing <a class="wiki_link" href="/9_8">9/8</a>, is equated to its inverted form, <a class="wiki_link" href="/64_45">64/45</a>. 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 <a class="wiki_link" href="/minor%20whole%20tone">minor whole tone</a>s (<a class="wiki_link" href="/10_9">10/9</a> 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 <a class="wiki_link" href="/50_49">50/49</a>, (the <a class="wiki_link" href="/jubilee%20comma">jubilee comma</a>), and <a class="wiki_link" href="/64_63">64/63</a>, (the <a class="wiki_link" href="/septimal%20comma">septimal comma</a>), 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 <a class="wiki_link" href="/septimal%20kleisma">septimal kleisma</a>, 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 <a class="wiki_link" href="/orwell%20tetrad">orwell tetrad</a> is also a chord of 22-et.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="Theory-Properties of 22 equal temperament-Commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->Commas</h3>
22 EDO tempers out the following commas. (Note: This assumes the val < 22 35 51 62 76 81 |.)<br />
<table class="wiki_table">
<tr>
<th>Comma<br />
</th>
<th>Monzo<br />
</th>
<th>Value (Cents)<br />
</th>
<th>Name 1<br />
</th>
<th>Name 2<br />
</th>
<th>Name 3<br />
</th>
</tr>
<tr>
<td style="text-align: center;">250/243<br />
</td>
<td style="text-align: left;">| 1 -5 3 ><br />
</td>
<td style="text-align: right;">49.17<br />
</td>
<td style="text-align: center;">Maximal Diesis<br />
</td>
<td style="text-align: center;">Porcupine Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3125/3072<br />
</td>
<td style="text-align: left;">| -10 -1 5 ><br />
</td>
<td style="text-align: right;">29.61<br />
</td>
<td style="text-align: center;">Small Diesis<br />
</td>
<td style="text-align: center;">Magic Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2048/2025<br />
</td>
<td style="text-align: left;">| 11 -4 -2 ><br />
</td>
<td style="text-align: right;">19.55<br />
</td>
<td style="text-align: center;">Diaschisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2109375/2097152<br />
</td>
<td style="text-align: left;">| -21 3 7 ><br />
</td>
<td style="text-align: right;">10.06<br />
</td>
<td style="text-align: center;">Semicomma<br />
</td>
<td style="text-align: center;">Fokker Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9193891/9143623<br />
</td>
<td style="text-align: left;">| 32 -7 -9 ><br />
</td>
<td style="text-align: right;">9.49<br />
</td>
<td style="text-align: center;">Escapade Comma<br />
</td>
<td><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4758837/4757272<br />
</td>
<td style="text-align: left;">| -53 10 16 ><br />
</td>
<td style="text-align: right;">0.57<br />
</td>
<td style="text-align: center;">Kwazy<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">50/49<br />
</td>
<td style="text-align: left;">| 1 0 2 -2 ><br />
</td>
<td style="text-align: right;">34.98<br />
</td>
<td style="text-align: center;">Tritonic Diesis<br />
</td>
<td style="text-align: center;">Jubilisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">64/63<br />
</td>
<td style="text-align: left;">| 6 -2 0 -1 ><br />
</td>
<td style="text-align: right;">27.26<br />
</td>
<td style="text-align: center;">Septimal Comma<br />
</td>
<td style="text-align: center;">Archytas' Comma<br />
</td>
<td style="text-align: center;">Leipziger Komma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">875/864<br />
</td>
<td style="text-align: left;">| -5 -3 3 1 ><br />
</td>
<td style="text-align: right;">21.90<br />
</td>
<td style="text-align: center;">Keema<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2430/2401<br />
</td>
<td style="text-align: left;">| 1 5 1 -4 ><br />
</td>
<td style="text-align: right;">20.79<br />
</td>
<td style="text-align: center;">Nuwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">245/243<br />
</td>
<td style="text-align: left;">| 0 -5 1 2 ><br />
</td>
<td style="text-align: right;">14.19<br />
</td>
<td style="text-align: center;">Sensamagic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1728/1715<br />
</td>
<td style="text-align: left;">| 6 3 -1 -3 ><br />
</td>
<td style="text-align: right;">13.07<br />
</td>
<td style="text-align: center;">Orwellisma<br />
</td>
<td style="text-align: center;">Orwell Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">225/224<br />
</td>
<td style="text-align: left;">| -5 2 2 -1 ><br />
</td>
<td style="text-align: right;">7.71<br />
</td>
<td style="text-align: center;">Septimal Kleisma<br />
</td>
<td style="text-align: center;">Marvel Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">10976/10935<br />
</td>
<td style="text-align: left;">| 5 -7 -1 3 ><br />
</td>
<td style="text-align: right;">6.48<br />
</td>
<td style="text-align: center;">Hemimage<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">6144/6125<br />
</td>
<td style="text-align: left;">| 11 1 -3 -2 ><br />
</td>
<td style="text-align: right;">5.36<br />
</td>
<td style="text-align: center;">Porwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">65625/65536<br />
</td>
<td style="text-align: left;">| -16 1 5 1 ><br />
</td>
<td style="text-align: right;">2.35<br />
</td>
<td style="text-align: center;">Horwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">420175/419904<br />
</td>
<td style="text-align: left;">| -6 -8 2 5 ><br />
</td>
<td style="text-align: right;">1.12<br />
</td>
<td style="text-align: center;">Wizma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">99/98<br />
</td>
<td style="text-align: left;">| -1 2 0 -2 1 ><br />
</td>
<td style="text-align: right;">17.58<br />
</td>
<td style="text-align: center;">Mothwellsma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">100/99<br />
</td>
<td style="text-align: left;">| 2 -2 2 0 -1 ><br />
</td>
<td style="text-align: right;">17.40<br />
</td>
<td style="text-align: center;">Ptolemisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">121/120<br />
</td>
<td style="text-align: left;">| -3 -1 -1 0 2 ><br />
</td>
<td style="text-align: right;">14.37<br />
</td>
<td style="text-align: center;">Biyatisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">176/175<br />
</td>
<td style="text-align: left;">| 4 0 -2 -1 1 ><br />
</td>
<td style="text-align: right;">9.86<br />
</td>
<td style="text-align: center;">Valinorsma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">896/891<br />
</td>
<td style="text-align: left;">| 7 -4 0 1 -1 ><br />
</td>
<td style="text-align: right;">9.69<br />
</td>
<td style="text-align: center;">Pentacircle<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">65536/65219<br />
</td>
<td style="text-align: left;">| 16 0 0 -2 -3 ><br />
</td>
<td style="text-align: right;">8.39<br />
</td>
<td style="text-align: center;">Orgonisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">385/384<br />
</td>
<td style="text-align: left;">| -7 -1 1 1 1 ><br />
</td>
<td style="text-align: right;">4.50<br />
</td>
<td style="text-align: center;">Keenanisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">540/539<br />
</td>
<td style="text-align: left;">| 2 3 1 -2 -1 ><br />
</td>
<td style="text-align: right;">3.21<br />
</td>
<td style="text-align: center;">Swetisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4000/3993<br />
</td>
<td style="text-align: left;">| 5 -1 3 0 -3 ><br />
</td>
<td style="text-align: right;">3.03<br />
</td>
<td style="text-align: center;">Wizardharry<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9801/9800<br />
</td>
<td style="text-align: left;">| -3 4 -2 -2 2 ><br />
</td>
<td style="text-align: right;">0.18<br />
</td>
<td style="text-align: center;">Kalisma<br />
</td>
<td style="text-align: center;">Gauss' Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">91/90<br />
</td>
<td style="text-align: left;">| -1 -2 -1 1 0 1 ><br />
</td>
<td style="text-align: right;">19.13<br />
</td>
<td style="text-align: center;">Superleap<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="Theory-Properties of 22 equal temperament-A Superpythagorean System"></a><!-- ws:end:WikiTextHeadingRule:6 -->A Superpythagorean System</h3>
<br />
The 22edo fifth, measuring approximately 709.1 cents, is wider than the 702-cent <a class="wiki_link" href="/3-limit">3-limit</a> fifth, thus making 22edo a "super-pythagorean" system. As with any superpyth, a chain of fifths produces relatively wide major thirds and narrow minor thirds. In the case of 22edo, the thirds are stretched out to the <a class="wiki_link" href="/7-limit">7-limit</a> ; the <a class="wiki_link" href="/subminor%20third">subminor third</a> comes close to <a class="wiki_link" href="/7_6">7/6</a> and the <a class="wiki_link" href="/supermajor%20third">supermajor third</a> to <a class="wiki_link" href="/9_7">9/7</a>. Thus, the resulting diatonic scale, which no longer approximates 5-limit thirds, sounds oddly consonant. The ratio of major 2nd to minor 2nd in this diatonic scale is stretched out to 4:1, with the M2 falling between 9/8 and <a class="wiki_link" href="/8_7">8/7</a>, and the m2 falling close to a quarter-tone.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="Theory-Properties of 22 equal temperament-11edo"></a><!-- ws:end:WikiTextHeadingRule:8 -->11edo</h3>
<br />
As 22 is divisible by 11, a 22edo instrument can play any music in <a class="wiki_link" href="/11edo">11edo</a>, in the same way that 12edo can play 6edo (the whole tone scale).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Theory-External links"></a><!-- ws:end:WikiTextHeadingRule:10 -->External links</h2>
<br />
<a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/erlich-decatonic.pdf" rel="nofollow">Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Theory-References"></a><!-- ws:end:WikiTextHeadingRule:12 -->References</h2>
<br />
Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br />
Bosanquet, R.H.M. <a class="wiki_link_ext" href="http://www.webcitation.org/5kjJcrhEx" rel="nofollow">''On the Hindoo division of the octave, with additions to the theory of higher orders''</a>, 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<br />
<br />
<hr />
<!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:14 -->Compositions</h1>
<br />
<a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3" rel="nofollow">Tibia</a> by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a><br />
<a class="wiki_link_ext" href="http://lumma.org/music/theory/tctmo/glassic.mp3" rel="nofollow">Glassic</a> by Paul Erlich and <a class="wiki_link" href="/Ara%20Sarkissian">Ara Sarkissian</a><br />
<a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/decatonic-swing.mp3" rel="nofollow">Decatonic Swing</a> by Paul Erlich and Ara Sarkissian (jazz)<br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683765" rel="nofollow">Dragged by a Storm Across the Desert Years</a> by <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a> (synth with electric guitar)<br />
Numerology by Iglashion Jones (progressive metal)<br />
Revenge of the inorganic compounds by Iglashion Jones (progressive metal)<br />
<a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=267" rel="nofollow">My Crazy Aunt Sophie</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a>. Blatantly xenharmonic piano.<br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839058" rel="nofollow">where words are said to mean</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a>, a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This & the following pieces by Andrew are for 22-tone guitar & voice.<br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow">I've come with a bucket of roses</a> by Andrew Heathwaite (orwell-9: 3 2 3 2 3 2 3 2 2).<br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow">one drop of rain</a> by Andrew Heathwaite (orwell-9).<br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839060" rel="nofollow">being a</a> by Andrew Heathwaite (porcupine-8: 3 1 3 3 3 3 3).<br />
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow">my own house</a> by Andrew Heathwaite (a pelog-flavored subset of orwell-9: 3 2 7 3 7).<br />
<a class="wiki_link_ext" href="http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3" rel="nofollow">Night on Porcupine Mountain</a> Mussorgsky-Smith<br />
<br />
<table class="wiki_table">
<tr>
<td>< 22 35 51 62 76 81 |<br />
</td>
</tr>
</table>
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