Pythagorean family: Difference between revisions
Wikispaces>xenwolf **Imported revision 237585033 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 280729286 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-11-30 14:42:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>280729286</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">The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12>, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. 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. | <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]] | ||
The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12>, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. 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. | |||
[[POTE tuning|POTE generator]]: 15.116 | [[POTE tuning|POTE generator]]: 15.116 | ||
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EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, 396 | EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, 396 | ||
=Compton temperament= | |||
In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1> to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80. | In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1> to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80. | ||
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EDOs: 12, 60, 72, 2940 | EDOs: 12, 60, 72, 2940 | ||
=Catler temperament= | |||
In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of [[12edo]]. Catler can also be characterized as the 12&24 temperament. [[36edo]] or [[48edo]] are possible tunings, and 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7 or 7/5 are possible generators. | In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of [[12edo]]. Catler can also be characterized as the 12&24 temperament. [[36edo]] or [[48edo]] are possible tunings, and 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7 or 7/5 are possible generators. | ||
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Map: [<12 19 28 0|, <0 0 0 1|] | Map: [<12 19 28 0|, <0 0 0 1|] | ||
EDOs: 12, [[36edo|36]], [[48edo|48]], 132, 180 | EDOs: 12, [[36edo|36]], [[48edo|48]], 132, 180 | ||
=Omicronbeta temperament= | |||
Commas: 225/224, 243/242, 441/440, 4375/4356 | |||
Generator: ~13/8 = 837.814 | |||
Map: [<72 114 167 202 249 266|, <0 0 0 0 0 1|] | |||
EDOs: 72, 144, 216c, 288cdf, 504bcdef | |||
Badness: 0.0300 | |||
</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"><html><head><title>Pythagorean family</title></head><body>The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12&gt;, and hence the fifths form a closed 12-note circle of fifths, identical to <a class="wiki_link" href="/12edo">12edo</a>. 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.<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>Pythagorean family</title></head><body><!-- ws:start:WikiTextTocRule:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Compton temperament">Compton temperament</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Catler temperament">Catler temperament</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Omicronbeta temperament">Omicronbeta temperament</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | ||
<!-- ws:end:WikiTextTocRule:10 --><br /> | |||
The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12&gt;, and hence the fifths form a closed 12-note circle of fifths, identical to <a class="wiki_link" href="/12edo">12edo</a>. 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.<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 15.116<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 15.116<br /> | ||
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EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/84edo">84</a>, 156, 240, 396<br /> | EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/84edo">84</a>, 156, 240, 396<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Compton temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->Compton temperament</h1> | ||
In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1&gt; to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&amp;72 temperament, and <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/84edo">84edo</a> or <a class="wiki_link" href="/240edo">240edo</a> make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80. <br /> | In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1&gt; to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&amp;72 temperament, and <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/84edo">84edo</a> or <a class="wiki_link" href="/240edo">240edo</a> make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80. <br /> | ||
<br /> | <br /> | ||
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EDOs: 12, 60, 72, 2940<br /> | EDOs: 12, 60, 72, 2940<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt; | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Catler temperament"></a><!-- ws:end:WikiTextHeadingRule:2 -->Catler temperament</h1> | ||
In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of <a class="wiki_link" href="/12edo">12edo</a>. Catler can also be characterized as the 12&amp;24 temperament. <a class="wiki_link" href="/36edo">36edo</a> or <a class="wiki_link" href="/48edo">48edo</a> are possible tunings, and 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7 or 7/5 are possible generators. <br /> | In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of <a class="wiki_link" href="/12edo">12edo</a>. Catler can also be characterized as the 12&amp;24 temperament. <a class="wiki_link" href="/36edo">36edo</a> or <a class="wiki_link" href="/48edo">48edo</a> are possible tunings, and 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7 or 7/5 are possible generators. <br /> | ||
<br /> | <br /> | ||
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Map: [&lt;12 19 28 0|, &lt;0 0 0 1|]<br /> | Map: [&lt;12 19 28 0|, &lt;0 0 0 1|]<br /> | ||
EDOs: 12, <a class="wiki_link" href="/36edo">36</a>, <a class="wiki_link" href="/48edo">48</a>, 132, 180</body></html></pre></div> | EDOs: 12, <a class="wiki_link" href="/36edo">36</a>, <a class="wiki_link" href="/48edo">48</a>, 132, 180<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Omicronbeta temperament"></a><!-- ws:end:WikiTextHeadingRule:4 -->Omicronbeta temperament</h1> | |||
Commas: 225/224, 243/242, 441/440, 4375/4356<br /> | |||
<br /> | |||
Generator: ~13/8 = 837.814<br /> | |||
<br /> | |||
Map: [&lt;72 114 167 202 249 266|, &lt;0 0 0 0 0 1|]<br /> | |||
EDOs: 72, 144, 216c, 288cdf, 504bcdef<br /> | |||
Badness: 0.0300</body></html></pre></div> | |||