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:genewardsmith|genewardsmith]] and made on <tt>2014-02-21 09:31:14 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-21 09:35:24 UTC</tt>.<br>

: The original revision id was <tt>~~490996976~~</tt>.<br>

: The original revision id was <tt>490998086</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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== ==

=Animism=

The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In 13 edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction

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=**Compositions**=

<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]]</span> by [[http://danielthompson.blogspot.com/|Daniel Thompson]]

<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3|Prelude in 13ET]]</span> by [[Aaron Andrew Hunt]]

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[[http://www.seraph.it/dep/det/MP13.mp3|MP13]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/0039a08cabcf860c813c37912c8898a0-183.html|blog entry]])

[[@http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/|Liber Stellarum]] by [[@http://votta.wordpress.com|Alfredo Votta]]

[[http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3|For Claudi]] by [[http://soonlabel.com/xenharmonic/archives/~~category/music/tunings-temperaments/equal-divisions-octave/13-edo~~|Margo Schulter]]

[[http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3|For Claudi]] by [[http://soonlabel.com/xenharmonic/archives/1788|Margo Schulter]]

=Chuckles McGee's 13-EDO diatonic approaches=

From a temperament perspective, we can probably make the "best" use of 13-EDO as a 2.5.9.11.13.21 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 simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, 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 horagram above. This gives rise to "Sephiroth" modes, in which the generator is any flatly tempered 13th harmonic. 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 horagram above. This scale bears a superficial resemblance to the 9-note MOS of Orwell temperament, although its approximations to the 3rd, 5th, and 7th harmonics are much more distant than in more optimal tunings of the temperament (on the other hand, its approximations to the 11th and 13th harmonics are much better than in optimal tunings of the temperament).

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<h2 id="toc3"> </h2>

<h1 id="toc4"><a name="Animism"></a>Animism</h1>

~~<br />~~

The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In 13 edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction<br />

<br />

0 4 5 8 9 13 pentatonic<br />

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<br />

<h1 id="toc5"><a name="Compositions"></a><strong>Compositions</strong></h1>

~~<br />~~

<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a></span> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br />

<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><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></span> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />

<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><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></span> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />

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<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/MP13.mp3" rel="nofollow">MP13</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/0039a08cabcf860c813c37912c8898a0-183.html" rel="nofollow">blog entry</a>)<br />

<a class="wiki_link_ext" href="http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/" rel="nofollow" target="_blank">Liber Stellarum</a> by <a class="wiki_link_ext" href="http://votta.wordpress.com" rel="nofollow" target="_blank">Alfredo Votta</a><br />

<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3" rel="nofollow">For Claudi</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/~~category/music/tunings-temperaments/equal-divisions-octave/13-edo~~" rel="nofollow">Margo Schulter</a><br />

<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3" rel="nofollow">For Claudi</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1788" rel="nofollow">Margo Schulter</a><br />

<br />

<h1 id="toc6"><a name="Chuckles McGee's 13-EDO diatonic approaches"></a>Chuckles McGee's 13-EDO diatonic approaches</h1>

~~<br />~~

From a temperament perspective, we can probably make the &quot;best&quot; use of 13-EDO as a 2.5.9.11.13.21 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 simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, 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 horagram above. This gives rise to &quot;Sephiroth&quot; modes, in which the generator is any flatly tempered 13th harmonic. 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 horagram above. This scale bears a superficial resemblance to the 9-note MOS of Orwell temperament, although its approximations to the 3rd, 5th, and 7th harmonics are much more distant than in more optimal tunings of the temperament (on the other hand, its approximations to the 11th and 13th harmonics are much better than in optimal tunings of the temperament).<br />

<br />

2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horagram 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 a full 2.5.9.11.13 pentad, 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 as a 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 13-EDO notation, leading to a notation very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart (in the 6L1s column). It can be thought of as a &quot;circle of major 2nds&quot; rather than a circle of 5ths.<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:genewardsmith|genewardsmith]] and made on <tt>2014-02-21 09:31:14 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-21 09:35:24 UTC</tt>.<br>
-: The original revision id was <tt>490996976</tt>.<br>
+: The original revision id was <tt>490998086</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 50: / Line 50: @@
 == ==
 =Animism=
 The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In 13 edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction
@@ Line 59: / Line 58: @@
 =**Compositions**=
 &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]]&lt;/span&gt; by [[http://danielthompson.blogspot.com/|Daniel Thompson]]
 &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3|Prelude in 13ET]]&lt;/span&gt; by [[Aaron Andrew Hunt]]
@@ Line 82: / Line 80: @@
 [[http://www.seraph.it/dep/det/MP13.mp3|MP13]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/0039a08cabcf860c813c37912c8898a0-183.html|blog entry]])
 [[@http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/|Liber Stellarum]] by [[@http://votta.wordpress.com|Alfredo Votta]]
-[[http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3|For Claudi]] by [[http://soonlabel.com/xenharmonic/archives/category/music/tunings-temperaments/equal-divisions-octave/13-edo|Margo Schulter]]
+[[http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3|For Claudi]] by [[http://soonlabel.com/xenharmonic/archives/1788|Margo Schulter]]
 =Chuckles McGee's 13-EDO diatonic approaches=
 From a temperament perspective, we can probably make the "best" use of 13-EDO as a 2.5.9.11.13.21 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 simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, 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 horagram above. This gives rise to "Sephiroth" modes, in which the generator is any flatly tempered 13th harmonic. 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 horagram above. This scale bears a superficial resemblance to the 9-note MOS of Orwell temperament, although its approximations to the 3rd, 5th, and 7th harmonics are much more distant than in more optimal tunings of the temperament (on the other hand, its approximations to the 11th and 13th harmonics are much better than in optimal tunings of the temperament).
@@ Line 371: / Line 369: @@
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt; &lt;/h2&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Animism"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Animism&lt;/h1&gt;
-  &lt;br /&gt;
+  The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In 13 edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction&lt;br /&gt;
-The animist comma, 105/104, appears whenever 3*5*7=13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (=3*7) and is also an animist temperament. In 13 edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction&lt;br /&gt;
 &lt;br /&gt;
 4 5 8 9 13 pentatonic&lt;br /&gt;
@@ Line 380: / Line 377: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;&lt;/h1&gt;
-  &lt;br /&gt;
+  &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow"&gt;Slow Dance&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow"&gt;Daniel Thompson&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow"&gt;Slow Dance&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow"&gt;Daniel Thompson&lt;/a&gt;&lt;br /&gt;
 &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow"&gt;Prelude in 13ET&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;br /&gt;
 &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3" rel="nofollow"&gt;Two-Part Invention in 13ET&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;br /&gt;
@@ Line 403: / Line 399: @@
 &lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/MP13.mp3" rel="nofollow"&gt;MP13&lt;/a&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/0039a08cabcf860c813c37912c8898a0-183.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/" rel="nofollow" target="_blank"&gt;Liber Stellarum&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://votta.wordpress.com" rel="nofollow" target="_blank"&gt;Alfredo Votta&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3" rel="nofollow"&gt;For Claudi&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/category/music/tunings-temperaments/equal-divisions-octave/13-edo" rel="nofollow"&gt;Margo Schulter&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Margo-Schulter-For_Claudi-2014-02-17-13edo.mp3" rel="nofollow"&gt;For Claudi&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1788" rel="nofollow"&gt;Margo Schulter&lt;/a&gt;&lt;br /&gt;
+&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Chuckles McGee's 13-EDO diatonic approaches"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Chuckles McGee's 13-EDO diatonic approaches&lt;/h1&gt;
-  &lt;br /&gt;
+  From a temperament perspective, we can probably make the &amp;quot;best&amp;quot; use of 13-EDO as a 2.5.9.11.13.21 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 simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, 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 &amp;lt;1 -1| (for 5 and 13), corresponding to the 3rd horagram above. This gives rise to &amp;quot;Sephiroth&amp;quot; modes, in which the generator is any flatly tempered 13th harmonic. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping &amp;lt;2 3| (for 11 and 13). This corresponds to the 2nd horagram above. This scale bears a superficial resemblance to the 9-note MOS of Orwell temperament, although its approximations to the 3rd, 5th, and 7th harmonics are much more distant than in more optimal tunings of the temperament (on the other hand, its approximations to the 11th and 13th harmonics are much better than in optimal tunings of the temperament).&lt;br /&gt;
-From a temperament perspective, we can probably make the &amp;quot;best&amp;quot; use of 13-EDO as a 2.5.9.11.13.21 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 simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, 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 &amp;lt;1 -1| (for 5 and 13), corresponding to the 3rd horagram above. This gives rise to &amp;quot;Sephiroth&amp;quot; modes, in which the generator is any flatly tempered 13th harmonic. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping &amp;lt;2 3| (for 11 and 13). This corresponds to the 2nd horagram above. This scale bears a superficial resemblance to the 9-note MOS of Orwell temperament, although its approximations to the 3rd, 5th, and 7th harmonics are much more distant than in more optimal tunings of the temperament (on the other hand, its approximations to the 11th and 13th harmonics are much better than in optimal tunings of the temperament).&lt;br /&gt;
 &lt;br /&gt;
 .5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horagram above, having the (octave-equivalent) mappings of &amp;lt;2 1| (for 5 and 9) and &amp;lt;2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating a full 2.5.9.11.13 pentad, which it achieves in the space of 5 generators via the octave-equivalent mapping &amp;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 as a 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 13-EDO notation, leading to a notation very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart (in the 6L1s column). It can be thought of as a &amp;quot;circle of major 2nds&amp;quot; rather than a circle of 5ths.&lt;br /&gt;