11edo: Difference between revisions

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[[toc|flat]]


=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Intervals=

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==Intervals== 
|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||


===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===11edo Instant Ensemble=== 

In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&q=hi&newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&q=hi&newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&q=hi&newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott [[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar & voice.

===Videos=== 

The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

Original HTML content:

<html><head><title>11edo</title></head><body><!-- ws:start:WikiTextTocRule:18:&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:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#x11 tone equal temperament">11 tone equal temperament</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Subgroup">Subgroup</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: -->
<!-- ws:end:WikiTextTocRule:28 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x11 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->11 tone equal temperament</h1>
 <br />
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1>
<br />
Compared to 12edo, the intervals of 11edo are stretched:<br />
<ul><li>The &quot;minor second,&quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.</li><li>The &quot;major second,&quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &quot;bluesy&quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.</li><li>The &quot;minor third,&quot; at 327.27 cents, is rather sharp and encroaching upon &quot;neutral third.&quot;</li><li>The &quot;major third,&quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.</li><li>The &quot;perfect fourth,&quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Subgroup"></a><!-- ws:end:WikiTextHeadingRule:4 -->Subgroup</h1>
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Harmonic<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>JI interval from 1/1<br />
</td>
        <td>1/1 = 0 cents<br />
</td>
        <td><br />
</td>
        <td>9/8 = 204<br />
</td>
        <td><br />
</td>
        <td>11/8 = 551<br />
</td>
        <td><br />
</td>
        <td>7/4 = 969<br />
</td>
        <td><br />
</td>
        <td>2/1 = 1200<br />
</td>
    </tr>
    <tr>
        <td>nearest 11edo interval<br />
</td>
        <td>0\11edo = 0 cents<br />
</td>
        <td><br />
</td>
        <td>2\11 = 218<br />
</td>
        <td><br />
</td>
        <td>5\11 = 545<br />
</td>
        <td><br />
</td>
        <td>9\11 = 982<br />
</td>
        <td><br />
</td>
        <td>11\11 = 1200<br />
</td>
    </tr>
    <tr>
        <td>difference<br />
</td>
        <td>0<br />
</td>
        <td><br />
</td>
        <td>+14<br />
</td>
        <td><br />
</td>
        <td>-6<br />
</td>
        <td><br />
</td>
        <td>+13<br />
</td>
        <td><br />
</td>
        <td>0<br />
</td>
    </tr>
    <tr>
        <td>JI interval between<br />
</td>
        <td><br />
</td>
        <td>9:8 = 204 cents<br />
</td>
        <td><br />
</td>
        <td>11:9 = 347<br />
</td>
        <td><br />
</td>
        <td>14/11 = 418<br />
</td>
        <td><br />
</td>
        <td>8:7 = 231<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>nearest 11edo interval<br />
</td>
        <td><br />
</td>
        <td>2\11 = 218<br />
</td>
        <td><br />
</td>
        <td>3\11 = 327<br />
</td>
        <td><br />
</td>
        <td>4\11 = 436<br />
</td>
        <td><br />
</td>
        <td>2\11 = 218<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>difference<br />
</td>
        <td><br />
</td>
        <td>+14<br />
</td>
        <td><br />
</td>
        <td>-20<br />
</td>
        <td><br />
</td>
        <td>+18<br />
</td>
        <td><br />
</td>
        <td>-13<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Subgroup-Intervals"></a><!-- ws:end:WikiTextHeadingRule:6 -->Intervals</h2>
 

<table class="wiki_table">
    <tr>
        <td>degrees of 11edo<br />
</td>
        <td>cents value<br />
</td>
        <td>solfege<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.00<br />
</td>
        <td><strong>do</strong><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>109.09<br />
</td>
        <td><strong>ra</strong><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>218.18<br />
</td>
        <td><strong>re</strong><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>327.27<br />
</td>
        <td><strong>me</strong><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>436.36<br />
</td>
        <td><strong>mo</strong><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>545.45<br />
</td>
        <td><strong>fu</strong><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>654.55<br />
</td>
        <td><strong>su</strong><br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>763.64<br />
</td>
        <td><strong>lo</strong><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>872.73<br />
</td>
        <td><strong>la</strong><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>981.82<br />
</td>
        <td><strong>te</strong><br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>1090.91<br />
</td>
        <td><strong>ti</strong><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Subgroup-Intervals-MOS Scales"></a><!-- ws:end:WikiTextHeadingRule:8 -->MOS Scales</h3>
 <br />
Although 11edo has one fewer interval in the octave than 12edo, in terms of <a class="wiki_link" href="/MOSScales">moment-of-symmetry scales</a>, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.<br />
<br />
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.<br />
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.<br />
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.<br />
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.<br />
<br />
See <a class="wiki_link" href="/11edo%20Modes">11edo Modes</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Subgroup-Intervals-11edo Solfege"></a><!-- ws:end:WikiTextHeadingRule:10 -->11edo Solfege</h3>
 <br />
An 11edo solfege system can easily be applied from the <a class="wiki_link" href="/22edo%20solfege">22edo solfege</a> system.<br />
A chromatic scale would thus be sung: <strong>do ra re me mo fu su lo la te ti do</strong>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Subgroup-Intervals-11edo Instant Ensemble"></a><!-- ws:end:WikiTextHeadingRule:12 -->11edo Instant Ensemble</h3>
 <br />
In February 2011, <a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow">Oddmusic U-C</a>, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Subgroup-Intervals-Compositions"></a><!-- ws:end:WikiTextHeadingRule:14 -->Compositions</h3>
 <br />
<a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow">Cool My Head</a> by David Hamill, 2010<br />
Hyperimprovisations Nuggetwarp (<a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;q=hi&amp;newref=1%27%29;" rel="nofollow">I</a> <a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;q=hi&amp;newref=1%27%29;" rel="nofollow">II</a> <a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;q=hi&amp;newref=1%27%29;" rel="nofollow">III</a>) by Jacob Barton, 2009<br />
She Is My Lilac-Hued Obsession on City of the Asleep, <a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow">Map of an Internal Landscape</a> (2009)<br />
<a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow">Turquoise Dabo Girl</a> by <a class="wiki_link" href="/Bill%20Sethares">Bill Sethares</a> (spectrally bent synth ens.)<br />
<a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow">Prelude11ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Hunt</a> (neo-Baroque)<br />
<a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow">The Stuffed Ones</a> by <a class="wiki_link" href="/Christopher%20Bailey">Christopher Bailey</a> (keyboards concréte)<br />
<a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow">Icicle Caverns</a> by Dr. Ozan Yarman<br />
<a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow">Angkor Wat, September 1066</a> by X. J. Scott <a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow">conversation is</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a>.<br />
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp; voice.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Subgroup-Intervals-Videos"></a><!-- ws:end:WikiTextHeadingRule:16 -->Videos</h3>
 <br />
The Stuffed Ones: <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related" rel="nofollow">Goopy</a>, <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related" rel="nofollow">Ziggy</a>, <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related" rel="nofollow">Ellie</a>, <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related" rel="nofollow">Towelbear</a> by <a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow">zipzappoozoo</a></body></html>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-02-22 15:56:54 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-29 01:38:18 UTC</tt>.<br>
-: The original revision id was <tt>204106668</tt>.<br>
+: The original revision id was <tt>214940224</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>
 <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">=11 tone equal temperament=
+<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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
+=11 tone equal temperament=
 -tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.
-===Theory===
+=Intervals=
 Compared to 12edo, the intervals of 11edo are stretched:
@@ Line 19: / Line 22: @@
 * The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.
-edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.
+=Subgroup=
+edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.
 || Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
@@ Line 80: / Line 84: @@
 The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]</pre></div>
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-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;11edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x11 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;11 tone equal temperament&lt;/h1&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;11edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:18:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;a href="#x11 tone equal temperament"&gt;11 tone equal temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Subgroup"&gt;Subgroup&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;
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+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x11 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;11 tone equal temperament&lt;/h1&gt;
   &lt;br /&gt;
 -tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x11 tone equal temperament--Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Theory&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals&lt;/h1&gt;
- &lt;br /&gt;
+&lt;br /&gt;
 Compared to 12edo, the intervals of 11edo are stretched:&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;The &amp;quot;minor second,&amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The &amp;quot;major second,&amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;quot;bluesy&amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The &amp;quot;minor third,&amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;quot;neutral third.&amp;quot;&lt;/li&gt;&lt;li&gt;The &amp;quot;major third,&amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The &amp;quot;perfect fourth,&amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Subgroup"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Subgroup&lt;/h1&gt;
+edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x11 tone equal temperament-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals&lt;/h2&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x11 tone equal temperament-Intervals-MOS Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;MOS Scales&lt;/h3&gt;
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   &lt;br /&gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &lt;a class="wiki_link" href="/MOSScales"&gt;moment-of-symmetry scales&lt;/a&gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&lt;br /&gt;
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 See &lt;a class="wiki_link" href="/11edo%20Modes"&gt;11edo Modes&lt;/a&gt;&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x11 tone equal temperament-Intervals-11edo Solfege"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11edo Solfege&lt;/h3&gt;
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   &lt;br /&gt;
 An 11edo solfege system can easily be applied from the &lt;a class="wiki_link" href="/22edo%20solfege"&gt;22edo solfege&lt;/a&gt; system.&lt;br /&gt;
 A chromatic scale would thus be sung: &lt;strong&gt;do ra re me mo fu su lo la te ti do&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x11 tone equal temperament-Intervals-11edo Instant Ensemble"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11edo Instant Ensemble&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Subgroup-Intervals-11edo Instant Ensemble"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11edo Instant Ensemble&lt;/h3&gt;
   &lt;br /&gt;
 In February 2011, &lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&gt;Oddmusic U-C&lt;/a&gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x11 tone equal temperament-Intervals-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Compositions&lt;/h3&gt;
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   &lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&gt;Cool My Head&lt;/a&gt; by David Hamill, 2010&lt;br /&gt;
@@ Line 386: / Line 394: @@
 Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x11 tone equal temperament-Intervals-Videos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Videos&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Subgroup-Intervals-Videos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Videos&lt;/h3&gt;
   &lt;br /&gt;
 The Stuffed Ones: &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related" rel="nofollow"&gt;Goopy&lt;/a&gt;, &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related" rel="nofollow"&gt;Ziggy&lt;/a&gt;, &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related" rel="nofollow"&gt;Ellie&lt;/a&gt;, &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related" rel="nofollow"&gt;Towelbear&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&gt;zipzappoozoo&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

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