13edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-26 23:11:23 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-27 13:38:51 UTC</tt>.<br>
: The original revision id was <tt>243012613</tt>.<br>
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<h4>Original Wikitext content:</h4>
<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">=13 tone equal temperament / 13edo=
<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]]


=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 &amp; major sixth are xenharmonic (not similar to anything available in 12edo).
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).
|| Degree || Cents ||= Approximate Ratios* || Note Name** ||
|| Degree || Cents ||= Approximate Ratios* || Note Name** ||
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*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.
*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.
**based on the 6L1s heptatonic scale; see below.
**based on the 6L1s heptatonic scale; see below.
==Harmony in 13edo==  
==Harmony in 13edo==  
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). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.


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). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. The simplest MOS scale to support this pentad uses the 2nd degree (~185 cents) as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).


==Scales in 13edo==  
==Scales in 13edo==  
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||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||=  ||</pre></div>
||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||=  ||</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;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;
<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:WikiTextTocRule:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#x13 tone equal temperament / 13edo"&gt;13 tone equal temperament / 13edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
  &lt;br /&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
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;amp; major sixth are xenharmonic (not similar to anything available in 12edo).&lt;br /&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;
  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;amp; major sixth are xenharmonic (not similar to anything available in 12edo).&lt;br /&gt;




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*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.&lt;br /&gt;
*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.&lt;br /&gt;
&lt;/strong&gt;based on the 6L1s heptatonic scale; see below.&lt;br /&gt;
&lt;/strong&gt;based on the 6L1s heptatonic scale; see below.&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-Harmony in 13edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harmony in 13edo&lt;/h2&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;
  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). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &amp;quot;stack of 3rds&amp;quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;2*13 subgroup&lt;/a&gt; 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. &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). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &amp;quot;stack of 3rds&amp;quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. The simplest MOS scale to support this pentad uses the 2nd degree (~185 cents) as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).&lt;br /&gt;
&lt;br /&gt;
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).&lt;br /&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="x13 tone equal temperament / 13edo-Scales in 13edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Scales in 13edo&lt;/h2&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;
  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;br /&gt;
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&lt;!-- ws:start:WikiTextLocalImageRule:314:&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:314 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextFileRule:307:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/13edo%20horograms.pdf?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@13edo horograms.pdf&amp;quot; title=&amp;quot;File: 13edo horograms.pdf&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /&gt;&lt;/a&gt;&lt;div&gt;&lt;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"&gt;13edo horograms.pdf&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/13edo%20horograms.pdf"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;242 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:307 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextFileRule:315:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/13edo%20horograms.pdf?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@13edo horograms.pdf&amp;quot; title=&amp;quot;File: 13edo horograms.pdf&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /&gt;&lt;/a&gt;&lt;div&gt;&lt;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"&gt;13edo horograms.pdf&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/13edo%20horograms.pdf"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;242 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:315 --&gt;&lt;br /&gt;
~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson&lt;br /&gt;
~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;

Revision as of 13:38, 27 July 2011

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[[toc|flat]]

=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).
|| Degree || Cents ||= Approximate Ratios* || Note Name** ||
|| 0 || 0 ||= 1/1 || C ||
|| 1 || 92.3077 ||= 55/52, 117/110, 26/25 || C#/Db ||
|| 2 || 184.6154 ||= 10/9, 9/8, 11/10 || D ||
|| 3 || 276.9231 ||= 13/11 || D#/Eb ||
|| 4 || 369.2308 ||= 5/4, 16/13, 11/9 || E ||
|| 5 || 461.5385 ||= 13/10 || E#/Fb ||
|| 6 || 553.84 ||= 11/8, 18/13 || F ||
|| 7 || 646.15 ||= 16/11, 13/9 || F#/Gb ||
|| 8 || 738.46 ||= 20/13 || G ||
|| 9 || 830.77 ||= 8/5, 13/8, 18/11 || G#/Ab ||
|| 10 || 923.08 ||= 22/13 || A ||
|| 11 || 1015.38 ||= 9/5, 16/9, 20/11 || A#/Bb ||
|| 12 || 1107.69 ||= 25/13, 104/55 || B/Cb ||
|| 13 || 1200 ||= 2/1 || C/B# ||
*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.
**based on the 6L1s heptatonic scale; see below.

==Harmony in 13edo== 
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). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. 

The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).

==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]]
[[file:13edo horograms.pdf]]
~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson

Another neat facet of 13-EDO is the fact that any 12-EDO scale can be "turned into" a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.

===**Compositions**=== 

[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]] by [[http://danielthompson.blogspot.com/|Daniel Thompson]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3|Prelude in 13ET]] by [[Aaron Andrew Hunt]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3|Two-Part Invention in 13ET]] by [[Aaron Andrew Hunt]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3|Triskaidekaphobia]] by [[http://www.io.com/%7Ehmiller/music/|Herman Miller]]
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=835265|Spikey Hair in 13tET]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3|play]] by [[Andrew Heathwaite]]
[[@http://cityoftheasleep.bandcamp.com/track/broken-dream-jar|Broken Dream Jar]] by [[IgliashonJones|City of the Asleep]]
[[@http://www.last.fm/music/City+of+the+Asleep/Map+of+an+Internal+Landscape/Blinding+White+Darkness|Blinding White Darkness]] by [[IgliashonJones|City of the Asleep]]

==Igliashon's 13-EDO diatonic approaches== 

From a temperament perspective, we can probably make the best use of 13-EDO as a 2.5.9.11.13 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, and 2.11.13, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping <1 -1| (for 5 and 13), corresponding to the 3rd horogram above. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping <2 3| (for 11 and 13). This corresponds to the 2nd horogram above.

2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of <2 1| (for 5 and 9) and <2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating the entire 2.5.9.11.13 subgroup, which it achieves in the space of 5 generators via the octave-equivalent mapping <2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it "the" tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for naming the notes of 13-EDO, leading to a lettering very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart.


==Commas== 
13 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val < 13 21 30 36 45 48 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 2109375/2097152 ||< | -21 3 7 > ||> 10.06 ||= Semicomma ||= Fokker Comma ||=   ||
||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||=   ||=   ||
||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||=   ||
||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||=   ||=   ||
||= 3125/3087 ||< | 0 -2 5 -3 > ||> 21.18 ||= Gariboh ||=   ||=   ||
||= 3136/3125 ||< | 6 0 -5 2 > ||> 6.08 ||= Hemimean ||=   ||=   ||
||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||=   ||=   ||
||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||=   ||=   ||

Original HTML content:

<html><head><title>13edo</title></head><body><!-- ws:start:WikiTextTocRule:12:&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:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#x13 tone equal temperament / 13edo">13 tone equal temperament / 13edo</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- 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 --><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x13 tone equal temperament / 13edo"></a><!-- ws:end:WikiTextHeadingRule:0 -->13 tone equal temperament / 13edo</h1>
 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 />


<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents<br />
</td>
        <td style="text-align: center;">Approximate Ratios*<br />
</td>
        <td>Note Name<strong><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
        <td>C<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>92.3077<br />
</td>
        <td style="text-align: center;">55/52, 117/110, 26/25<br />
</td>
        <td>C#/Db<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>184.6154<br />
</td>
        <td style="text-align: center;">10/9, 9/8, 11/10<br />
</td>
        <td>D<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>276.9231<br />
</td>
        <td style="text-align: center;">13/11<br />
</td>
        <td>D#/Eb<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>369.2308<br />
</td>
        <td style="text-align: center;">5/4, 16/13, 11/9<br />
</td>
        <td>E<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>461.5385<br />
</td>
        <td style="text-align: center;">13/10<br />
</td>
        <td>E#/Fb<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>553.84<br />
</td>
        <td style="text-align: center;">11/8, 18/13<br />
</td>
        <td>F<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>646.15<br />
</td>
        <td style="text-align: center;">16/11, 13/9<br />
</td>
        <td>F#/Gb<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>738.46<br />
</td>
        <td style="text-align: center;">20/13<br />
</td>
        <td>G<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>830.77<br />
</td>
        <td style="text-align: center;">8/5, 13/8, 18/11<br />
</td>
        <td>G#/Ab<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>923.08<br />
</td>
        <td style="text-align: center;">22/13<br />
</td>
        <td>A<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>1015.38<br />
</td>
        <td style="text-align: center;">9/5, 16/9, 20/11<br />
</td>
        <td>A#/Bb<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>1107.69<br />
</td>
        <td style="text-align: center;">25/13, 104/55<br />
</td>
        <td>B/Cb<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>1200<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
        <td>C/B#<br />
</td>
    </tr>
</table>

*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.<br />
</strong>based on the 6L1s heptatonic scale; see below.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x13 tone equal temperament / 13edo-Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h2>
 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). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. <br />
<br />
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x13 tone equal temperament / 13edo-Scales in 13edo"></a><!-- ws:end:WikiTextHeadingRule:4 -->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 />
<!-- ws:start:WikiTextLocalImageRule:314:&lt;img src=&quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /><!-- ws:end:WikiTextLocalImageRule:314 --><br />
<!-- ws:start:WikiTextFileRule:315:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/13edo%20horograms.pdf?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@13edo horograms.pdf&quot; title=&quot;File: 13edo horograms.pdf&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><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><!-- ws:end:WikiTextFileRule:315 --><br />
~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson<br />
<br />
Another neat facet of 13-EDO is the fact that any 12-EDO scale can be &quot;turned into&quot; a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x13 tone equal temperament / 13edo-Scales in 13edo-Compositions"></a><!-- ws:end:WikiTextHeadingRule:6 --><strong>Compositions</strong></h3>
 <br />
<a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow">Prelude in 13ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3" rel="nofollow">Two-Part Invention in 13ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow">Triskaidekaphobia</a> by <a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/music/" rel="nofollow">Herman Miller</a><br />
<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 />
<a class="wiki_link_ext" href="http://cityoftheasleep.bandcamp.com/track/broken-dream-jar" rel="nofollow" target="_blank">Broken Dream Jar</a> by <a class="wiki_link" href="/IgliashonJones">City of the Asleep</a><br />
<a class="wiki_link_ext" href="http://www.last.fm/music/City+of+the+Asleep/Map+of+an+Internal+Landscape/Blinding+White+Darkness" rel="nofollow" target="_blank">Blinding White Darkness</a> by <a class="wiki_link" href="/IgliashonJones">City of the Asleep</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x13 tone equal temperament / 13edo-Igliashon's 13-EDO diatonic approaches"></a><!-- ws:end:WikiTextHeadingRule:8 -->Igliashon's 13-EDO diatonic approaches</h2>
 <br />
From a temperament perspective, we can probably make the best use of 13-EDO as a 2.5.9.11.13 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, and 2.11.13, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping &lt;1 -1| (for 5 and 13), corresponding to the 3rd horogram above. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping &lt;2 3| (for 11 and 13). This corresponds to the 2nd horogram above.<br />
<br />
2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of &lt;2 1| (for 5 and 9) and &lt;2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating the entire 2.5.9.11.13 subgroup, which it achieves in the space of 5 generators via the octave-equivalent mapping &lt;2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it &quot;the&quot; tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for naming the notes of 13-EDO, leading to a lettering very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x13 tone equal temperament / 13edo-Commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->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 />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
        <th>Name 3<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">2109375/2097152<br />
</td>
        <td style="text-align: left;">| -21 3 7 &gt;<br />
</td>
        <td style="text-align: right;">10.06<br />
</td>
        <td style="text-align: center;">Semicomma<br />
</td>
        <td style="text-align: center;">Fokker Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1000<br />
</td>
        <td style="text-align: left;">| -3 1 -3 3 &gt;<br />
</td>
        <td style="text-align: right;">49.49<br />
</td>
        <td style="text-align: center;">Keega<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">525/512<br />
</td>
        <td style="text-align: left;">| -9 1 2 1 &gt;<br />
</td>
        <td style="text-align: right;">43.41<br />
</td>
        <td style="text-align: center;">Avicennma<br />
</td>
        <td style="text-align: center;">Avicenna's Enharmonic Diesis<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64/63<br />
</td>
        <td style="text-align: left;">| 6 -2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">27.26<br />
</td>
        <td style="text-align: center;">Septimal Comma<br />
</td>
        <td style="text-align: center;">Archytas' Comma<br />
</td>
        <td style="text-align: center;">Leipziger Komma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64827/64000<br />
</td>
        <td style="text-align: left;">| -9 3 -3 4 &gt;<br />
</td>
        <td style="text-align: right;">22.23<br />
</td>
        <td style="text-align: center;">Squalentine<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3125/3087<br />
</td>
        <td style="text-align: left;">| 0 -2 5 -3 &gt;<br />
</td>
        <td style="text-align: right;">21.18<br />
</td>
        <td style="text-align: center;">Gariboh<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3136/3125<br />
</td>
        <td style="text-align: left;">| 6 0 -5 2 &gt;<br />
</td>
        <td style="text-align: right;">6.08<br />
</td>
        <td style="text-align: center;">Hemimean<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">121/120<br />
</td>
        <td style="text-align: left;">| -3 -1 -1 0 2 &gt;<br />
</td>
        <td style="text-align: right;">14.37<br />
</td>
        <td style="text-align: center;">Biyatisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">441/440<br />
</td>
        <td style="text-align: left;">| -3 2 -1 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.93<br />
</td>
        <td style="text-align: center;">Werckisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

</body></html>