36edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 239659363 - Original comment: ** |
Wikispaces>vaisvil **Imported revision 323664996 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:vaisvil|vaisvil]] and made on <tt>2012-04-21 13:28:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>323664996</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <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">[[toc|flat]] | <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]] | ||
36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents. | 36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents. | ||
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36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95. | 36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95. | ||
As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. | As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. | ||
Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale. | Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale. | ||
=Approximations= | =Approximations= | ||
==3-limit (Pythagorean) approximations (same as 12edo):== | ==3-limit (Pythagorean) approximations (same as 12edo):== | ||
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63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents. | 63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents. | ||
=Music= | =Music= | ||
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3|Something]] by [[Herman Klein]] | * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3|Something]]</span> by [[Herman Klein]] | ||
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3|Hay]] by [[Joe Hayseed]] | * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3|Hay]]</span> by [[Joe Hayseed]] | ||
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3|Boomers]] by [[Ivan Bratt]][[media type="custom" key="9486498"]]</pre></div> | * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3|Boomers]]</span> by [[Ivan Bratt]][[media type="custom" key="9486498"]] | ||
* [[http://micro.soonlabel.com/36edo/20120418-36edo.mp3|Thoughts in Legolas Tuning]] by [[Chris Vaisvil]]</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>36edo</title></head><body><!-- ws:start:WikiTextTocRule:15:&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:15 --><!-- ws:start:WikiTextTocRule:16: --><a href="#As a harmonic temperament">As a harmonic temperament</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Approximations">Approximations</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <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>36edo</title></head><body><!-- ws:start:WikiTextTocRule:15:&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:15 --><!-- ws:start:WikiTextTocRule:16: --><a href="#As a harmonic temperament">As a harmonic temperament</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Approximations">Approximations</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | ||
<!-- ws:end:WikiTextTocRule:23 --> | <!-- ws:end:WikiTextTocRule:23 -->36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.<br /> | ||
36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.<br /> | |||
<br /> | <br /> | ||
36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar <a class="wiki_link" href="/12edo">12edo</a> as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called &quot;sixth tones.&quot; 36edo also contains <a class="wiki_link" href="/18edo">18edo</a> (&quot;third tones&quot;) and <a class="wiki_link" href="/9edo">9edo</a> (&quot;two-thirds tones&quot;) as subsets, not to mention the <a class="wiki_link" href="/6edo">6edo</a> whole tone scale, <a class="wiki_link" href="/4edo">4edo</a> full-diminished seventh chord, and the <a class="wiki_link" href="/3edo">3edo</a> augmented triad, all of which are present in 12edo.<br /> | 36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar <a class="wiki_link" href="/12edo">12edo</a> as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called &quot;sixth tones.&quot; 36edo also contains <a class="wiki_link" href="/18edo">18edo</a> (&quot;third tones&quot;) and <a class="wiki_link" href="/9edo">9edo</a> (&quot;two-thirds tones&quot;) as subsets, not to mention the <a class="wiki_link" href="/6edo">6edo</a> whole tone scale, <a class="wiki_link" href="/4edo">4edo</a> full-diminished seventh chord, and the <a class="wiki_link" href="/3edo">3edo</a> augmented triad, all of which are present in 12edo.<br /> | ||
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36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank four temperament tempering out 56/55, as well as the rank three temperament <a class="wiki_link" href="/Didymus%20rank%20three%20family">melpomene</a> tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.<br /> | 36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank four temperament tempering out 56/55, as well as the rank three temperament <a class="wiki_link" href="/Didymus%20rank%20three%20family">melpomene</a> tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.<br /> | ||
<br /> | <br /> | ||
As a 5-limit temperament, the patent val for 36edo is <a class="wiki_link" href="/Wedgies%20and%20Multivals">contorted</a>, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the &quot;comma&quot; |29 0 -9&gt; is also tempered out, and the &quot;fifth&quot;, 29\36, is actually approximately 7/4, whereas the &quot;major third&quot;, 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a <a class="wiki_link" href="/transversal">transversal</a> for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. <br /> | As a 5-limit temperament, the patent val for 36edo is <a class="wiki_link" href="/Wedgies%20and%20Multivals">contorted</a>, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the &quot;comma&quot; |29 0 -9&gt; is also tempered out, and the &quot;fifth&quot;, 29\36, is actually approximately 7/4, whereas the &quot;major third&quot;, 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a <a class="wiki_link" href="/transversal">transversal</a> for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.<br /> | ||
<br /> | <br /> | ||
Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.<br /> | Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Approximations"></a><!-- ws:end:WikiTextHeadingRule:3 -->Approximations</h1> | <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Approximations"></a><!-- ws:end:WikiTextHeadingRule:3 -->Approximations</h1> | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="Approximations-3-limit (Pythagorean) approximations (same as 12edo):"></a><!-- ws:end:WikiTextHeadingRule:5 -->3-limit (Pythagorean) approximations (same as 12edo):</h2> | <!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="Approximations-3-limit (Pythagorean) approximations (same as 12edo):"></a><!-- ws:end:WikiTextHeadingRule:5 -->3-limit (Pythagorean) approximations (same as 12edo):</h2> | ||
<br /> | <br /> | ||
3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.<br /> | 3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:13 -->Music</h1> | <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:13 -->Music</h1> | ||
<ul><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3" rel="nofollow">Something</a> by <a class="wiki_link" href="/Herman%20Klein">Herman Klein</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3" rel="nofollow">Hay</a> by <a class="wiki_link" href="/Joe%20Hayseed">Joe Hayseed</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3" rel="nofollow">Boomers</a> by <a class="wiki_link" href="/Ivan%20Bratt">Ivan Bratt</a><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/9486498?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;9486498&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"> | <ul><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3" rel="nofollow">Something</a></span> by <a class="wiki_link" href="/Herman%20Klein">Herman Klein</a></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3" rel="nofollow">Hay</a></span> by <a class="wiki_link" href="/Joe%20Hayseed">Joe Hayseed</a></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3" rel="nofollow">Boomers</a></span> by <a class="wiki_link" href="/Ivan%20Bratt">Ivan Bratt</a><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/9486498?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;9486498&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"> | ||
</script><!-- ws:end:WikiTextMediaRule:0 --></li></ul></body></html></pre></div> | </script><!-- ws:end:WikiTextMediaRule:0 --></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/36edo/20120418-36edo.mp3" rel="nofollow">Thoughts in Legolas Tuning</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></li></ul></body></html></pre></div> | ||