36edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 213160824 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 214752008 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-03- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-28 14:12:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>214752008</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">36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents. | <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. | |||
36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo. | 36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo. | ||
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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= | |||
==3-limit (Pythagorean) approximations (same as 12edo):== | ==3-limit (Pythagorean) approximations (same as 12edo):== | ||
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72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents. | 72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents. | ||
64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents. | 64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents. | ||
63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.</pre></div> | 63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents. | ||
=Music= | |||
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3|Something]] by Herman Klein | |||
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3|Hay]] by Joe Hayseed</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>36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.<br /> | <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:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#As a harmonic temperament">As a harmonic temperament</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Approximations">Approximations</a><!-- 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 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | ||
<!-- ws:end:WikiTextTocRule:22 --><br /> | |||
<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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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:2:&lt;h2&gt; --><h2 id=" | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Approximations"></a><!-- ws:end:WikiTextHeadingRule:2 -->Approximations</h1> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Approximations-3-limit (Pythagorean) approximations (same as 12edo):"></a><!-- ws:end:WikiTextHeadingRule:4 -->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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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Approximations-7-limit approximations:"></a><!-- ws:end:WikiTextHeadingRule:6 -->7-limit approximations:</h2> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Approximations-7-limit approximations:-7 only:"></a><!-- ws:end:WikiTextHeadingRule:8 -->7 only:</h3> | ||
7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.<br /> | 7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.<br /> | ||
8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.<br /> | 8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.<br /> | ||
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64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.<br /> | 64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Approximations-7-limit approximations:-7 &amp; 3:"></a><!-- ws:end:WikiTextHeadingRule:10 -->7 &amp; 3:</h3> | ||
7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.<br /> | 7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.<br /> | ||
12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.<br /> | 12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.<br /> | ||
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72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.<br /> | 72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.<br /> | ||
64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.<br /> | 64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.<br /> | ||
63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.</body></html></pre></div> | 63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:12 -->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 Herman Klein</li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3" rel="nofollow">Hay</a> by Joe Hayseed</li></ul></body></html></pre></div> | |||