21edo: Difference between revisions

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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:igliashon|igliashon]] and made on <tt>2014-06-28 16:10~~:37~~ UTC</tt>.<br>

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2014-06-28 22:34:10 UTC</tt>.<br>

: The original revision id was <tt>~~515174902~~</tt>.<br>

: The original revision id was <tt>515186756</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>

Line 8:

<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">=21 equal divisions of the octave=

Twenty one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads~~. Some other cool things about 21-edo: it has an 11-limit minor third/wide sixth, 7-limit neutral third and sixth, a 7/4 harmonic seventh and grave seventh~~. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents.

Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other ET <26.

==~~Harmony in~~ 21-EDO:==

==21-EDO as a temperament:==

In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

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[[augment9]]

[[augment12]]

==Triadic Harmony in 21-EDO:==

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:

||= **Steps** ||= **Cents** ||= **Ratio** ||

||= 0-5-10 ||= 0-286-571 ||= 23:27:32 ||

||= 0-4-11 ||= 0-229-629 ||= 7:8:10 ||

||= 0-6-11 ||= 0-343-629 ||= 9:11:13 ||

||= 0-5-13 ||= 0-286-743 ||= 11:13:17 ||

||= 0-8-13 ||= 0-457-743 ||= 13:17:20 ||

||= 0-5-15 ||= 0-286-857 ||= 11:13:18 ||

==Moment-of-Symmetry Scales in 21-EDO:==

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For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.

==Tetrachordal Scales in 21-EDO==

While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:

||= Step Pattern ||= Cents ||= Name* ||

||= 3, 3, 3 ||= 0-171-343-514 ||= Equal diatonic ||

||= 4, 3, 2 ||= 0-229-400-514 ||= Soft diatonic ||

||= 4, 4, 1 ||= 0-229-457-514 ||= Hard diatonic ||

||= 5, 3, 1 ||= 0-286-457-514 ||= Hard chromatic ||

||= 5, 2, 2 ||= 0-286-400-514 ||= Soft chromatic ||

||= 6, 2, 1 ||= 0-343-457-514 ||= Soft enharmonic ||

||= 7, 1, 1 ||= 0-400-457-514 ||= Hard enharmonic ||

*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!

The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.

==Rank two temperaments==

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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"><html><head><title>21edo</title></head><body><h1 id="toc0"><a name="x21 equal divisions of the octave"></a>21 equal divisions of the octave</h1>

<br />

Twenty one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or &quot;equi-heptatonic&quot; scales, or as seven 3-edo ''augmented'' triads~~. Some other cool things about 21-edo: it has an 11-limit minor third/wide sixth, 7-limit neutral third and sixth, a 7/4 harmonic seventh and grave seventh~~. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents.<br />

Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or &quot;equi-heptatonic&quot; scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other ET &lt;26. <br />

<br />

<h2 id="toc1"><a name="x21 equal divisions of the octave-~~Harmony in~~ 21-EDO:"></a>~~Harmony in~~ 21-EDO:</h2>

<h2 id="toc1"><a name="x21 equal divisions of the octave-21-EDO as a temperament:"></a>21-EDO as a temperament:</h2>

In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a &quot;third-fourth&quot; (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.<br />

<br />

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<a class="wiki_link" href="/augment12">augment12</a><br />

<br />

<h2 id="toc2"><a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"></a>Moment-of-Symmetry Scales in 21-EDO:</h2>

<h2 id="toc2"><a name="x21 equal divisions of the octave-Triadic Harmony in 21-EDO:"></a>Triadic Harmony in 21-EDO:</h2>

<br />

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as &quot;3rds&quot; for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy &quot;altered&quot; triads that stand out as representations to parts of the overtone series:<br />

<br />

<tr>

<td style="text-align: center;"><strong>Steps</strong><br />

</td>

<td style="text-align: center;"><strong>Cents</strong><br />

</td>

<td style="text-align: center;"><strong>Ratio</strong><br />

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

</table>

<br />

<h2 id="toc3"><a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"></a>Moment-of-Symmetry Scales in 21-EDO:</h2>

<br />

Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.<br />

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For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.<br />

<br />

<h2 id="~~toc3~~"><a name="x21 equal divisions of the octave-Rank two temperaments"></a>Rank two temperaments</h2>

<h2 id="toc4"><a name="x21 equal divisions of the octave-Tetrachordal Scales in 21-EDO"></a>Tetrachordal Scales in 21-EDO</h2>

While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:<br />

<br />

<tr>

<td style="text-align: center;">Step Pattern<br />

</td>

<td style="text-align: center;">Cents<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">Equal diatonic<br />

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">Soft diatonic<br />

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">Hard diatonic<br />

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">Hard chromatic<br />

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">Soft chromatic<br />

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">Soft enharmonic<br />

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">Hard enharmonic<br />

</td>

</tr>

</table>

*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!<br />

<br />

The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.<br />

<br />

<h2 id="toc5"><a name="x21 equal divisions of the octave-Rank two temperaments"></a>Rank two temperaments</h2>

<a class="wiki_link" href="/List%20of%2021edo%20rank%20two%20temperaments%20by%20badness">List of 21edo rank two temperaments by badness</a><br />

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<br />

<h2 id="~~toc4~~"><a name="x21 equal divisions of the octave-13-limit Commas"></a>13-limit Commas</h2>

<h2 id="toc6"><a name="x21 equal divisions of the octave-13-limit Commas"></a>13-limit Commas</h2>

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)<br />

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<br />

<h1 id="~~toc5~~"><a name="Books / Literature:"></a><strong>Books / Literature:</strong></h1>

<h1 id="toc7"><a name="Books / Literature:"></a><strong>Books / Literature:</strong></h1>

Sword, Ron. &quot;Icosihenaphonic Scales for Guitar&quot;. IAAA Press. 1st ed: July 2009.<br />

<img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /><br />

<img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /><br />

<strong><em>21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)</em></strong><br />

<br />

<h1 id="~~toc6~~"><a name="Compositions/Listening:"></a><strong>Compositions/Listening:</strong></h1>

<h1 id="toc8"><a name="Compositions/Listening:"></a><strong>Compositions/Listening:</strong></h1>

<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank">Short Clip of 21-edo Acoustic</a> by <a class="wiki_link" href="/Ron%20Sword">Ron Sword</a><br />

<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank">Open tuning Drone Improvisation in 21-edo</a> by Ron Sword<br />

@@ Line 1: / Line 1: @@
 <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:igliashon|igliashon]] and made on <tt>2014-06-28 16:10:37 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2014-06-28 22:34:10 UTC</tt>.<br>
-: The original revision id was <tt>515174902</tt>.<br>
+: The original revision id was <tt>515186756</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>
@@ Line 8: / Line 8: @@
 <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">=21 equal divisions of the octave=
-Twenty one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. Some other cool things about 21-edo: it has an 11-limit minor third/wide sixth, 7-limit neutral third and sixth, a 7/4 harmonic seventh and grave seventh. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents.
+Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other ET &lt;26.
-==Harmony in 21-EDO:==
+==21-EDO as a temperament:==
 In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.
@@ Line 54: / Line 54: @@
 [[augment9]]
 [[augment12]]
+==Triadic Harmony in 21-EDO:==
+One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:
+||= **Steps** ||= **Cents** ||= **Ratio** ||
+||= 0-5-10 ||= 0-286-571 ||= 23:27:32 ||
+||= 0-4-11 ||= 0-229-629 ||= 7:8:10 ||
+||= 0-6-11 ||= 0-343-629 ||= 9:11:13 ||
+||= 0-5-13 ||= 0-286-743 ||= 11:13:17 ||
+||= 0-8-13 ||= 0-457-743 ||= 13:17:20 ||
+||= 0-5-15 ||= 0-286-857 ||= 11:13:18 ||
 ==Moment-of-Symmetry Scales in 21-EDO:==
@@ Line 61: / Line 73: @@
 For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.
+==Tetrachordal Scales in 21-EDO==
+While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:
+||= Step Pattern ||= Cents ||= Name* ||
+||= 3, 3, 3 ||= 0-171-343-514 ||= Equal diatonic ||
+||= 4, 3, 2 ||= 0-229-400-514 ||= Soft diatonic ||
+||= 4, 4, 1 ||= 0-229-457-514 ||= Hard diatonic ||
+||= 5, 3, 1 ||= 0-286-457-514 ||= Hard chromatic ||
+||= 5, 2, 2 ||= 0-286-400-514 ||= Soft chromatic ||
+||= 6, 2, 1 ||= 0-343-457-514 ||= Soft enharmonic ||
+||= 7, 1, 1 ||= 0-400-457-514 ||= Hard enharmonic ||
+*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!
+The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.
 ==Rank two temperaments==
@@ Line 114: / Line 141: @@
 <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;21edo&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="x21 equal divisions of the octave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;21 equal divisions of the octave&lt;/h1&gt;
   &lt;br /&gt;
-Twenty one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or &amp;quot;equi-heptatonic&amp;quot; scales, or as seven 3-edo ''augmented'' triads. Some other cool things about 21-edo: it has an 11-limit minor third/wide sixth, 7-limit neutral third and sixth, a 7/4 harmonic seventh and grave seventh. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents.&lt;br /&gt;
+Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or &amp;quot;equi-heptatonic&amp;quot; scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other ET &amp;lt;26. &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x21 equal divisions of the octave-Harmony in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harmony in 21-EDO:&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x21 equal divisions of the octave-21-EDO as a temperament:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;21-EDO as a temperament:&lt;/h2&gt;
   In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a &amp;quot;third-fourth&amp;quot; (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 557: / Line 584: @@
 &lt;a class="wiki_link" href="/augment12"&gt;augment12&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Moment-of-Symmetry Scales in 21-EDO:&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x21 equal divisions of the octave-Triadic Harmony in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Triadic Harmony in 21-EDO:&lt;/h2&gt;
+ &lt;br /&gt;
+One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as &amp;quot;3rds&amp;quot; for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy &amp;quot;altered&amp;quot; triads that stand out as representations to parts of the overtone series:&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Steps&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Cents&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;Ratio&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-286-571&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;23:27:32&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-4-11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-229-629&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;7:8:10&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-6-11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-343-629&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;9:11:13&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-286-743&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;11:13:17&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-8-13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-457-743&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;13:17:20&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;0-5-15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-286-857&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;11:13:18&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Moment-of-Symmetry Scales in 21-EDO:&lt;/h2&gt;
   &lt;br /&gt;
 Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.&lt;br /&gt;
@@ Line 564: / Line 657: @@
 For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x21 equal divisions of the octave-Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Rank two temperaments&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x21 equal divisions of the octave-Tetrachordal Scales in 21-EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Tetrachordal Scales in 21-EDO&lt;/h2&gt;
+ While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;Step Pattern&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Cents&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Name*&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;3, 3, 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-171-343-514&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Equal diatonic&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;4, 3, 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-229-400-514&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Soft diatonic&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;4, 4, 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-229-457-514&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Hard diatonic&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;5, 3, 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-286-457-514&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Hard chromatic&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;5, 2, 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-286-400-514&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Soft chromatic&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;6, 2, 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-343-457-514&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Soft enharmonic&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;7, 1, 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0-400-457-514&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Hard enharmonic&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!&lt;br /&gt;
+&lt;br /&gt;
+The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x21 equal divisions of the octave-Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Rank two temperaments&lt;/h2&gt;
   &lt;a class="wiki_link" href="/List%20of%2021edo%20rank%20two%20temperaments%20by%20badness"&gt;List of 21edo rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
@@ Line 662: / Line 831: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x21 equal divisions of the octave-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit Commas&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x21 equal divisions of the octave-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit Commas&lt;/h2&gt;
 EDO tempers out the following 13-limit commas. (Note: This assumes the val &amp;lt; 21 33 49 59 73 78 |.)&lt;br /&gt;
@@ Line 864: / Line 1,033: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;strong&gt;Books / Literature:&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;strong&gt;Books / Literature:&lt;/strong&gt;&lt;/h1&gt;
   Sword, Ron. &amp;quot;Icosihenaphonic Scales for Guitar&amp;quot;. IAAA Press. 1st ed: July 2009.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:714:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/ron1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 188px; width: 254px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:714 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:715:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/21tetsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 191px; width: 363px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:715 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:842:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/ron1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 188px; width: 254px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:842 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:843:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/21tetsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 191px; width: 363px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:843 --&gt;&lt;br /&gt;
 &lt;strong&gt;&lt;em&gt;21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)&lt;/em&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Compositions/Listening:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;strong&gt;Compositions/Listening:&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Compositions/Listening:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;strong&gt;Compositions/Listening:&lt;/strong&gt;&lt;/h1&gt;
   &lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank"&gt;Short Clip of 21-edo Acoustic&lt;/a&gt; by &lt;a class="wiki_link" href="/Ron%20Sword"&gt;Ron Sword&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank"&gt;Open tuning Drone Improvisation in 21-edo&lt;/a&gt; by Ron Sword&lt;br /&gt;