36edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 230064374 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 230097516 - 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>2011-05-19 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-19 15:15:50 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>230097516</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> | ||
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* [[http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3|Something]] by Herman Klein | * [[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 | * [[http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3|Hay]] by Joe Hayseed | ||
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3|Boomers]] by Ivan Bratt</pre></div> | * [[http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3|Boomers]] by Ivan Bratt[[media type="custom" key="9486498"]]</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: | <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: | <!-- ws:end:WikiTextTocRule:23 --><br /> | ||
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 /> | ||
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That 36edo contains 12edo as a subset makes in compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, <a class="wiki_link_ext" href="http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/" rel="nofollow">De-quinin'</a>). Three 12edo instruments could play the entire gamut.<br /> | That 36edo contains 12edo as a subset makes in compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, <a class="wiki_link_ext" href="http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/" rel="nofollow">De-quinin'</a>). Three 12edo instruments could play the entire gamut.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="As a harmonic temperament"></a><!-- ws:end:WikiTextHeadingRule:1 -->As a harmonic temperament</h1> | ||
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For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow">Slendro diesis</a> of around 36 cents, and as 64:63, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">septimal comma</a> of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow">Septimal third-tone</a> (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as <a class="wiki_link" href="/72edo">72edo</a> does in the full <a class="wiki_link" href="/17-limit">17-limit</a>.<br /> | For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow">Slendro diesis</a> of around 36 cents, and as 64:63, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">septimal comma</a> of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow">Septimal third-tone</a> (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as <a class="wiki_link" href="/72edo">72edo</a> does in the full <a class="wiki_link" href="/17-limit">17-limit</a>.<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Approximations"></a><!-- ws:end:WikiTextHeadingRule:3 -->Approximations</h1> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- 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> | ||
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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: | <!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc3"><a name="Approximations-7-limit approximations:"></a><!-- ws:end:WikiTextHeadingRule:7 -->7-limit approximations:</h2> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:9:&lt;h3&gt; --><h3 id="toc4"><a name="Approximations-7-limit approximations:-7 only:"></a><!-- ws:end:WikiTextHeadingRule:9 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:11:&lt;h3&gt; --><h3 id="toc5"><a name="Approximations-7-limit approximations:-3 and 7:"></a><!-- ws:end:WikiTextHeadingRule:11 -->3 and 7:</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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63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.<br /> | 63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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 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><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3" rel="nofollow">Boomers</a> by Ivan Bratt</li></ul></body></html></pre></div> | <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><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3" rel="nofollow">Boomers</a> by Ivan Bratt<!-- 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> | |||