37edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 232649138 - Original comment: ** |
Wikispaces>hstraub **Imported revision 238143977 - 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:hstraub|hstraub]] and made on <tt>2011-06-22 07:17:42 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>238143977</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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<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">37edo is the scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a [[Porcupine family|porcupine temperament]] tuning. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, giving a temperament where three minor whole tones make up a fifth. | <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">37edo is the scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a [[Porcupine family|porcupine temperament]] tuning. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, giving a temperament where three minor whole tones make up a fifth. | ||
=Subgroups= | [[toc|flat]] | ||
---- | |||
=Subgroups= | |||
37edo offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13: | 37edo offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13: | ||
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This means 37 is quite accurate on the 2.5.7.11 subgroup, where it shares the same tuning as 111et. In fact, on the larger [[k*N subgroups|3*37 subgroup]] 2.27.5.7.11.51.57 subgroup not only shares the same tuning as 19-limit 111et, it tempers out the same commas. | This means 37 is quite accurate on the 2.5.7.11 subgroup, where it shares the same tuning as 111et. In fact, on the larger [[k*N subgroups|3*37 subgroup]] 2.27.5.7.11.51.57 subgroup not only shares the same tuning as 19-limit 111et, it tempers out the same commas. | ||
=The Two Fifths= | =The Two Fifths= | ||
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo: | The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo: | ||
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37edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. | 37edo has great potential as a xenharmonic system, which high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. | ||
=Intervals= | =Intervals= | ||
|| degrees of 37edo || cents value || | || degrees of 37edo || cents value || | ||
|| 0 || 0.00 || | || 0 || 0.00 || | ||
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|| 34 || 1102.70 || | || 34 || 1102.70 || | ||
|| 35 || 1135.14 || | || 35 || 1135.14 || | ||
|| 36 || 1167.57 ||</pre></div> | || 36 || 1167.57 || | ||
=Scales= | |||
[[roulette6]] | |||
[[roulette7]] | |||
[[roulette13]] | |||
[[roulette19]]</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>37edo</title></head><body>37edo is the scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a <a class="wiki_link" href="/Porcupine%20family">porcupine temperament</a> tuning. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, giving a temperament where three minor whole tones make up a fifth.<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>37edo</title></head><body>37edo is the scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. Using its best (and sharp) fifth, it tempers out 250/243, making it a <a class="wiki_link" href="/Porcupine%20family">porcupine temperament</a> tuning. Using its alternative flat fifth, it tempers out 16875/16384, making it a negri tuning. It also tempers out 2187/2000, giving a temperament where three minor whole tones make up a fifth.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextTocRule:8:&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:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Subgroups">Subgroups</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#The Two Fifths">The Two Fifths</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | |||
<!-- ws:end:WikiTextTocRule:13 --><br /> | |||
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<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Subgroups"></a><!-- ws:end:WikiTextHeadingRule:0 -->Subgroups</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Subgroups"></a><!-- ws:end:WikiTextHeadingRule:0 -->Subgroups</h1> | ||
37edo offers close approximations to <a class="wiki_link" href="/OverToneSeries">harmonics</a> 5, 7, 11, and 13:<br /> | 37edo offers close approximations to <a class="wiki_link" href="/OverToneSeries">harmonics</a> 5, 7, 11, and 13:<br /> | ||
<br /> | <br /> | ||
12\37 = 389.2 cents<br /> | 12\37 = 389.2 cents<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="The Two Fifths"></a><!-- ws:end:WikiTextHeadingRule:2 -->The Two Fifths</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="The Two Fifths"></a><!-- ws:end:WikiTextHeadingRule:2 -->The Two Fifths</h1> | ||
The just <a class="wiki_link" href="/perfect%20fifth">perfect fifth</a> of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:<br /> | The just <a class="wiki_link" href="/perfect%20fifth">perfect fifth</a> of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:<br /> | ||
<br /> | <br /> | ||
21\37 = 681.1 cents<br /> | 21\37 = 681.1 cents<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
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</table> | </table> | ||
</body></html></pre></div> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->Scales</h1> | |||
<br /> | |||
<a class="wiki_link" href="/roulette6">roulette6</a><br /> | |||
<a class="wiki_link" href="/roulette7">roulette7</a><br /> | |||
<a class="wiki_link" href="/roulette13">roulette13</a><br /> | |||
<a class="wiki_link" href="/roulette19">roulette19</a></body></html></pre></div> | |||