Pythagorean family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 280729286 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 281165174 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-01 13:14:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>281165174</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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Commas: 225/224, 250047/250000 | Commas: 225/224, 250047/250000 | ||
[[POTE tuning|POTE generator]]: 16.225 | [[POTE tuning|POTE generator]]: ~5/4 = 383.775 (16.225) | ||
Map: [<12 19 0 -22|, <0 0 1 2|] | Map: [<12 19 0 -22|, <0 0 1 2|] | ||
EDOs: 12, [[60edo|60]], 72, 228, | EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc | ||
11-limit | ==11-limit== | ||
Commas: 225/224, 441/440, 4375/4356 | Commas: 225/224, 441/440, 4375/4356 | ||
[[POTE tuning|POTE generator]]: 16.734 | [[POTE tuning|POTE generator]]: ~5/4 = 383.266 (16.734) | ||
Map: [<12 19 0 -22 -42|, <0 0 1 2 3|] | Map: [<12 19 0 -22 -42|, <0 0 1 2 3|] | ||
EDOs: 12, | EDOs: 12, 60e, 72 | ||
==13-limit== | |||
Commas: 225/224, 441/440, 351/350, 364/363 | |||
POTE generator: ~5/4 = 383.963 (16.037) | |||
Map: [<12 19 0 -22 -42 -67|, <0 0 1 2 3 4|] | |||
EDOs: 72, 228f, 300cf | |||
Badness: 0.0219 | |||
==Comptone== | |||
Commas: 225/224, 441/440, 325/324, 1001/1000 | |||
POTE generator: ~5/4 = 382.612 (17.388) | |||
Map: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|] | |||
EDOs: 12, 60e, 72, 204cdef, 276cdef | |||
Badness: 0.0251 | |||
=Catler temperament= | =Catler temperament= | ||
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</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><!-- 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>Pythagorean family</title></head><body><!-- ws:start:WikiTextTocRule:12:&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:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Compton temperament">Compton temperament</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Catler temperament">Catler temperament</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Omicronbeta temperament">Omicronbeta temperament</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:19 --><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 /> | 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 /> | ||
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Commas: 225/224, 250047/250000<br /> | Commas: 225/224, 250047/250000<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 16.225<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.775 (16.225)<br /> | ||
<br /> | <br /> | ||
Map: [&lt;12 19 0 -22|, &lt;0 0 1 2|]<br /> | Map: [&lt;12 19 0 -22|, &lt;0 0 1 2|]<br /> | ||
EDOs: 12, <a class="wiki_link" href="/60edo">60</a>, 72, 228, | EDOs: 12, <a class="wiki_link" href="/60edo">60</a>, 72, 228, 300c, 372bc, 444bc<br /> | ||
<br /> | <br /> | ||
11-limit< | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Compton temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> | ||
Commas: 225/224, 441/440, 4375/4356<br /> | Commas: 225/224, 441/440, 4375/4356<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 16.734<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.266 (16.734)<br /> | ||
<br /> | <br /> | ||
Map: [&lt;12 19 0 -22 -42|, &lt;0 0 1 2 3|]<br /> | Map: [&lt;12 19 0 -22 -42|, &lt;0 0 1 2 3|]<br /> | ||
EDOs: 12, | EDOs: 12, 60e, 72<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Compton temperament-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> | |||
Commas: 225/224, 441/440, 351/350, 364/363<br /> | |||
<br /> | |||
POTE generator: ~5/4 = 383.963 (16.037)<br /> | |||
<br /> | |||
Map: [&lt;12 19 0 -22 -42 -67|, &lt;0 0 1 2 3 4|]<br /> | |||
EDOs: 72, 228f, 300cf<br /> | |||
Badness: 0.0219<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Compton temperament-Comptone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Comptone</h2> | |||
Commas: 225/224, 441/440, 325/324, 1001/1000<br /> | |||
<br /> | |||
POTE generator: ~5/4 = 382.612 (17.388)<br /> | |||
<br /> | |||
Map: [&lt;12 19 0 -22 -42 100|, &lt;0 0 1 2 3 -2|]<br /> | |||
EDOs: 12, 60e, 72, 204cdef, 276cdef<br /> | |||
Badness: 0.0251<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Catler temperament"></a><!-- ws:end:WikiTextHeadingRule:8 -->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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EDOs: 12, <a class="wiki_link" href="/36edo">36</a>, <a class="wiki_link" href="/48edo">48</a>, 132, 180<br /> | EDOs: 12, <a class="wiki_link" href="/36edo">36</a>, <a class="wiki_link" href="/48edo">48</a>, 132, 180<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Omicronbeta temperament"></a><!-- ws:end:WikiTextHeadingRule:10 -->Omicronbeta temperament</h1> | ||
Commas: 225/224, 243/242, 441/440, 4375/4356<br /> | Commas: 225/224, 243/242, 441/440, 4375/4356<br /> | ||
<br /> | <br /> | ||