22edo: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 83606885 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 83608783 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-08-07 | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-08-07 21:11:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>83608783</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 25: | Line 25: | ||
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. | 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. | ||
===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 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. | |||
===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== | ==External links== | ||
| Line 47: | Line 55: | ||
Revenge of the inorganic compounds by Iglashion Jones (progressive metal)</pre></div> | Revenge of the inorganic compounds by Iglashion Jones (progressive metal)</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>22edo</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>22edo</title></head><body><!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Theory">Theory</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><!-- 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: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><br /> | ||
<br /> | <br /> | ||
<hr /> | <hr /> | ||
| Line 67: | Line 75: | ||
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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id=" | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Theory-Properties of 22 equal temperament-A Superpythagorean System"></a><!-- ws:end:WikiTextHeadingRule:4 -->A Superpythagorean System</h3> | ||
<br /> | |||
The 22edo fifth, measuring approximately 709.1 cents, is wider than the 702-cent 3-limit fifth, thus making 22edo a &quot;super-pythagorean&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.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Theory-Properties of 22 equal temperament-11edo"></a><!-- ws:end:WikiTextHeadingRule:6 -->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:8:&lt;h2&gt; --><h2 id="toc4"><a name="Theory-External links"></a><!-- ws:end:WikiTextHeadingRule:8 -->External links</h2> | |||
<br /> | <br /> | ||
Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''<br /> | Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''<br /> | ||
*[<!-- ws:start:WikiTextUrlRule: | *[<!-- ws:start:WikiTextUrlRule:79:http://66.98.148.43/~xenharmo/text/tuning22.pdf --><a class="wiki_link_ext" href="http://66.98.148.43/~xenharmo/text/tuning22.pdf" rel="nofollow">http://66.98.148.43/~xenharmo/text/tuning22.pdf</a><!-- ws:end:WikiTextUrlRule:79 -->] or <!-- ws:start:WikiTextUrlRule:80: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:80 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Theory-References"></a><!-- ws:end:WikiTextHeadingRule:10 -->References</h2> | ||
<br /> | <br /> | ||
Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br /> | Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br /> | ||
<br /> | <br /> | ||
Bosanquet, R.H.M. [<!-- ws:start:WikiTextUrlRule: | Bosanquet, R.H.M. [<!-- ws:start:WikiTextUrlRule:81: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:81 --> ''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<br /> | ||
<br /> | <br /> | ||
<hr /> | <hr /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:12 -->Compositions</h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3" rel="nofollow">Tibia</a> by Paul Erlich<br /> | <a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3" rel="nofollow">Tibia</a> by Paul Erlich<br /> | ||