22edo: Difference between revisions

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-05-05 16:34:26 UTC</tt>.<br>

: The original revision id was <tt>139763515</tt>.<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 [[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.

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 and the supermajor third to 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, and the m2 falling close to a quarter-tone.

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 &lt;a class="wiki_link" href="/31edo"&gt;31 equal temperament&lt;/a&gt; does much better, 22-et at least allows the use of these higher-limit harmonies. Moreover, 22-et, unlike 12 and &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, is not a &lt;a class="wiki_link" href="/Regular%20Temperaments#meantone"&gt;meantone&lt;/a&gt; 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.&lt;br /&gt;

The 22edo fifth, measuring approximately 709.1 cents, is wider than the 702-cent 3-limit fifth, thus making 22edo a &amp;quot;super-pythagorean&amp;quot; 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 and the supermajor third to 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, and the m2 falling close to a quarter-tone.&lt;br /&gt;

*[&lt;!-- ws:start:WikiTextUrlRule:89:http://66.98.148.43/~xenharmo/text/tuning22.pdf --&gt;&lt;a class="wiki_link_ext" href="http://66.98.148.43/~xenharmo/text/tuning22.pdf" rel="nofollow"&gt;http://66.98.148.43/~xenharmo/text/tuning22.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:89 --&gt;] or &lt;!-- ws:start:WikiTextUrlRule:90:http://lumma.org/tuning/erlich/erlich-decatonic.pdf --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/erlich-decatonic.pdf" rel="nofollow"&gt;http://lumma.org/tuning/erlich/erlich-decatonic.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:90 --&gt;&lt;br /&gt;

Bosanquet, R.H.M. [&lt;!-- ws:start:WikiTextUrlRule:91:http://www.geocities.com/threesixesinarow/hindoo.htm --&gt;&lt;a class="wiki_link_ext" href="http://www.geocities.com/threesixesinarow/hindoo.htm" rel="nofollow"&gt;http://www.geocities.com/threesixesinarow/hindoo.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:91 --&gt; ''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&lt;br /&gt;