Pythagorean family: Difference between revisions

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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 [[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

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 [[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 the either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this [[72edo]] can be recommended as a tuning.

[[POTE tuning|POTE generator]]: ~5/4 = 383.775 (16.225)

[[POTE tuning|POTE generator]]: ~5/4 = 383.266 (16.734)

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&amp;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.   

[[POTE tuning|POTE generator]]: 26.790

Badness: 0.0269</pre></div>

<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:26:&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:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#Compton temperament"&gt;Compton temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Catler temperament"&gt;Catler temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt; | &lt;a href="#Omicronbeta temperament"&gt;Omicronbeta temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#Hours"&gt;Hours&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;

Map: [&amp;lt;12 19 0|, &amp;lt;0 0 1|]&lt;br /&gt;

EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/84edo"&gt;84&lt;/a&gt;, 156, 240, 396&lt;br /&gt;

In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1&amp;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;amp;72 temperament, and &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt;, &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt; or &lt;a class="wiki_link" href="/240edo"&gt;240edo&lt;/a&gt; make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80. &lt;br /&gt;

Commas: 225/224, 250047/250000&lt;br /&gt;

Map: [&amp;lt;12 19 0 -22|, &amp;lt;0 0 1 2|]&lt;br /&gt;

Map: [&amp;lt;12 19 0 -22 -42|, &amp;lt;0 0 1 2 3|]&lt;br /&gt;

Map: [&amp;lt;12 19 0 -22 -42 -67|, &amp;lt;0 0 1 2 3 4|]&lt;br /&gt;

Map: [&amp;lt;12 19 0 -22 -42 100|, &amp;lt;0 0 1 2 3 -2|]&lt;br /&gt;

Map: [&amp;lt;12 19 28 0|, &amp;lt;0 0 0 1|]&lt;br /&gt;

Map: [&amp;lt;12 19 28 0 -26|, &amp;lt;0 0 0 1 2|]&lt;br /&gt;

Map: [&amp;lt;12 19 28 0 109|, &amp;lt;0 0 0 1 -2|]&lt;br /&gt;

POTE generator: ~36/35 = 32.776&lt;br /&gt;

Map: [&amp;lt;12 19 28 0 8|, &amp;lt;0 0 0 1 1|]&lt;br /&gt;

Map: [&amp;lt;12 19 28 0 8 11|, &amp;lt;0 0 0 1 1 1|]&lt;br /&gt;

Map: [&amp;lt;72 114 167 202 249 266|, &amp;lt;0 0 0 0 0 1|]&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;

Map: [&amp;lt;24 38 0 123 83|, &amp;lt;0 0 1 -1 0|]&lt;br /&gt;

Map: [&amp;lt;24 38 0 123 83 33|, &amp;lt;0 0 1 -1 0 1|]&lt;br /&gt;