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Pythagorean family: Difference between revisions

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Wikispaces>genewardsmith
**Imported revision 281165174 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 281658632 - 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>2011-12-01 13:14:04 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-02 13:54:32 UTC</tt>.<br>
: The original revision id was <tt>281165174</tt>.<br>
: The original revision id was <tt>281658632</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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EDOs: 72, 144, 216c, 288cdf, 504bcdef
EDOs: 72, 144, 216c, 288cdf, 504bcdef
Badness: 0.0300
Badness: 0.0300
=Hours=
Commas: 19683/19600, 33075/32768
POTE generator: ~225/224 = 2.100
Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]
Wedgie: &lt;0 24 -24 38 -38 -123|
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd
Badness: 0.1161
==11-limit==
Commas: 243/242, 385/384, 9801/9800
POTE generator: ~225/224 = 2.161
Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde
Badness: 0.0362
==13-limit==
Commas: 243/242, 351/350, 364/363, 385/384
POTE generator: ~225/224 = 3.955
Map: [&lt;24 38 0 123 83 33|, &lt;0 0 1 -1 0 1|]
EDOs: 24, 48f, 72, 168df, 240df
Badness: 0.0269
</pre></div>
</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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Pythagorean family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Compton temperament"&gt;Compton temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Catler temperament"&gt;Catler temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Omicronbeta temperament"&gt;Omicronbeta temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Pythagorean family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:18:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;a href="#Compton temperament"&gt;Compton temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Catler temperament"&gt;Catler temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Omicronbeta temperament"&gt;Omicronbeta temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Hours"&gt;Hours&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;br /&gt;
The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12&amp;gt;, and hence the fifths form a closed 12-note circle of fifths, identical to &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.&lt;br /&gt;
The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12&amp;gt;, and hence the fifths form a closed 12-note circle of fifths, identical to &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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Map: [&amp;lt;72 114 167 202 249 266|, &amp;lt;0 0 0 0 0 1|]&lt;br /&gt;
Map: [&amp;lt;72 114 167 202 249 266|, &amp;lt;0 0 0 0 0 1|]&lt;br /&gt;
EDOs: 72, 144, 216c, 288cdf, 504bcdef&lt;br /&gt;
EDOs: 72, 144, 216c, 288cdf, 504bcdef&lt;br /&gt;
Badness: 0.0300&lt;/body&gt;&lt;/html&gt;</pre></div>
Badness: 0.0300&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Hours"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Hours&lt;/h1&gt;
Commas: 19683/19600, 33075/32768&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~225/224 = 2.100&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;24 38 0 123 83|, &amp;lt;0 0 1 -1 0|]&lt;br /&gt;
Wedgie: &amp;lt;0 24 -24 38 -38 -123|&lt;br /&gt;
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd&lt;br /&gt;
Badness: 0.1161&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Hours-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;11-limit&lt;/h2&gt;
Commas: 243/242, 385/384, 9801/9800&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~225/224 = 2.161&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;24 38 0 123 83|, &amp;lt;0 0 1 -1 0|]&lt;br /&gt;
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde&lt;br /&gt;
Badness: 0.0362&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Hours-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit&lt;/h2&gt;
Commas: 243/242, 351/350, 364/363, 385/384&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~225/224 = 3.955&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;24 38 0 123 83 33|, &amp;lt;0 0 1 -1 0 1|]&lt;br /&gt;
EDOs: 24, 48f, 72, 168df, 240df&lt;br /&gt;
Badness: 0.0269&lt;/body&gt;&lt;/html&gt;</pre></div>
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