13edo: 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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2011-06-~~30 10~~:08:39 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2011-07-18 16:38:28 UTC</tt>.<br>

: The original revision id was <tt>~~239494451~~</tt>.<br>

: The original revision id was <tt>241829697</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>

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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;white-space: pre-wrap ! important" class="old-revision-html">=13 tone equal temperament / 13edo=

13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, & 6\13, respectively.

13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo).

|| Degree || Cents || Approximate Ratios* ||

|| 0 || 0 || 1/1 ||

|| 1 || 92.3077 || ||

|| 2 || 184.6154 || 10/9, 9/8, 11/10 ||

|| 3 || 276.9231 || 13/11 ||

|| 4 || 369.2308 || 5/4, 16/13, 11/9 ||

|| 5 || 461.5385 || 13/10 ||

|| 6 || 553.84 || 11/8, 18/13 ||

|| 7 || 646.15 || 16/11, 13/9 ||

|| 8 || 738.46 || 20/13 ||

|| 9 || 830.77 || 8/5, 13/8, 18/11 ||

|| 10 || 923.08 || 22/13 ||

|| 11 || 1015.38 || 9/5, 16/9, 20/11 ||

|| 12 || 1107.69 || ||

|| 13 || 1200 || 2/1 ||

*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.

==Harmony in 13edo==

One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely.

Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings), and the most successful approaches do not always make the most sense in terms of JI.

==Scales in 13edo==

Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, & 6\13, respectively.

[[image:13edo_horograms.jpg]]

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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>13edo</title></head><body><h1 id="toc0"><a name="x13 tone equal temperament / 13edo"></a>13 tone equal temperament / 13edo</h1>

<br />

13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp; major sixth are xenharmonic (not similar to anything available in 12edo). Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br />

13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp; major sixth are xenharmonic (not similar to anything available in 12edo). <br />

<tr>

<td>Degree<br />

</td>

<td>Cents<br />

</td>

<td>Approximate Ratios*<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>

<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>

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<tr>

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</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

</table>

*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.<br />

<h2 id="toc1"><a name="x13 tone equal temperament / 13edo-Harmony in 13edo"></a>Harmony in 13edo</h2>

<br />

One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely.<br />

<br />

Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings), and the most successful approaches do not always make the most sense in terms of JI.<br />

<br />

<h2 id="toc2"><a name="x13 tone equal temperament / 13edo-Scales in 13edo"></a>Scales in 13edo</h2>

Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br />

<br />

<img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /><br />

<img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /><br />

<div class="objectEmbed"><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /></a><div><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');" class="filename" title="13edo horograms.pdf">13edo horograms.pdf</a><br /><ul><li><a href="/file/detail/13edo%20horograms.pdf">Details</a></li><li><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf">Download</a></li><li style="color: #666">242 KB</li></ul></div></div><br />

<div class="objectEmbed"><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /></a><div><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');" class="filename" title="13edo horograms.pdf">13edo horograms.pdf</a><br /><ul><li><a href="/file/detail/13edo%20horograms.pdf">Details</a></li><li><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf">Download</a></li><li style="color: #666">242 KB</li></ul></div></div><br />

~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson<br />

<br />

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<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=835265" rel="nofollow">Spikey Hair in 13tET</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br />

<br />

<h2 id="~~toc1~~"><a name="x13 tone equal temperament / 13edo-Commas"></a>Commas</h2>

<h2 id="toc3"><a name="x13 tone equal temperament / 13edo-Commas"></a>Commas</h2>

13 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; 13 21 30 36 45 48 |.)<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:xenwolf|xenwolf]] and made on <tt>2011-06-30 10:08:39 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-07-18 16:38:28 UTC</tt>.<br>
-: The original revision id was <tt>239494451</tt>.<br>
+: The original revision id was <tt>241829697</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">=13 tone equal temperament / 13edo=
-edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp; major sixth are xenharmonic (not similar to anything available in 12edo). Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.
+edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp; major sixth are xenharmonic (not similar to anything available in 12edo).
+|| Degree || Cents || Approximate Ratios* ||
+|| 0 || 0 || 1/1 ||
+|| 1 || 92.3077 ||   ||
+|| 2 || 184.6154 || 10/9, 9/8, 11/10 ||
+|| 3 || 276.9231 || 13/11 ||
+|| 4 || 369.2308 || 5/4, 16/13, 11/9 ||
+|| 5 || 461.5385 || 13/10 ||
+|| 6 || 553.84 || 11/8, 18/13 ||
+|| 7 || 646.15 || 16/11, 13/9 ||
+|| 8 || 738.46 || 20/13 ||
+|| 9 || 830.77 || 8/5, 13/8, 18/11 ||
+|| 10 || 923.08 || 22/13 ||
+|| 11 || 1015.38 || 9/5, 16/9, 20/11 ||
+|| 12 || 1107.69 ||   ||
+|| 13 || 1200 || 2/1 ||
+*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.
+==Harmony in 13edo==
+One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely.
+Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings), and the most successful approaches do not always make the most sense in terms of JI.
+==Scales in 13edo==
+Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.
 [[image:13edo_horograms.jpg]]
@@ Line 37: / Line 61: @@
 <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;13edo&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="x13 tone equal temperament / 13edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;13 tone equal temperament / 13edo&lt;/h1&gt;
   &lt;br /&gt;
-edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp;amp; major sixth are xenharmonic (not similar to anything available in 12edo). Due to the prime character of the number 13, 13edo can form several xenharmonic &lt;a class="wiki_link" href="/MOSScales"&gt;moment of symmetry scales&lt;/a&gt;. The diagram below shows five &amp;quot;families&amp;quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp;amp; 6\13, respectively.&lt;br /&gt;
+edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp;amp; major sixth are xenharmonic (not similar to anything available in 12edo). &lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Degree&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Cents&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Approximate Ratios*&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1/1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;92.3077&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;184.6154&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10/9, 9/8, 11/10&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;276.9231&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;369.2308&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5/4, 16/13, 11/9&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;461.5385&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13/10&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;553.84&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11/8, 18/13&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;646.15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;16/11, 13/9&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;738.46&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;20/13&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;830.77&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8/5, 13/8, 18/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;923.08&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;22/13&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1015.38&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9/5, 16/9, 20/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1107.69&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1200&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2/1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x13 tone equal temperament / 13edo-Harmony in 13edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harmony in 13edo&lt;/h2&gt;
+ &lt;br /&gt;
+One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely.&lt;br /&gt;
+&lt;br /&gt;
+Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings), and the most successful approaches do not always make the most sense in terms of JI.&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="x13 tone equal temperament / 13edo-Scales in 13edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Scales in 13edo&lt;/h2&gt;
+ Due to the prime character of the number 13, 13edo can form several xenharmonic &lt;a class="wiki_link" href="/MOSScales"&gt;moment of symmetry scales&lt;/a&gt;. The diagram below shows five &amp;quot;families&amp;quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp;amp; 6\13, respectively.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:146:&amp;lt;img src=&amp;quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:146 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:272:&amp;lt;img src=&amp;quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:272 --&gt;&lt;br /&gt;
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 ~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson&lt;br /&gt;
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 &lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=835265" rel="nofollow"&gt;Spikey Hair in 13tET&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;&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="x13 tone equal temperament / 13edo-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Commas&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x13 tone equal temperament / 13edo-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Commas&lt;/h2&gt;
 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 13 21 30 36 45 48 |.)&lt;br /&gt;