Pythagorean family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 287021814 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 287022060 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2011-12-17 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-17 03:01:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>287022060</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> | ||
| Line 73: | Line 73: | ||
EDOs: 12, 48c, 108cd | EDOs: 12, 48c, 108cd | ||
Badness: 0.0582 | Badness: 0.0582 | ||
==Catlat== | |||
Commas: 81/80, 128/125, 540/539 | |||
POTE generator: ~36/35 = 27.864 | |||
Map: [<12 19 28 0 109|, <0 0 0 1 -2|] | |||
EDOs: 36, 48c, 84c | |||
Badness: 0.0819 | |||
=Omicronbeta temperament= | =Omicronbeta temperament= | ||
| Line 112: | Line 121: | ||
</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:22:&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:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#Compton temperament">Compton temperament</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Catler temperament">Catler temperament</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Omicronbeta temperament">Omicronbeta temperament</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Hours">Hours</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:34 --><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 /> | ||
| Line 180: | Line 189: | ||
Badness: 0.0582<br /> | Badness: 0.0582<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id=" | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Catler temperament-Catlat"></a><!-- ws:end:WikiTextHeadingRule:12 -->Catlat</h2> | ||
Commas: 81/80, 128/125, 540/539<br /> | |||
<br /> | |||
POTE generator: ~36/35 = 27.864<br /> | |||
<br /> | |||
Map: [&lt;12 19 28 0 109|, &lt;0 0 0 1 -2|]<br /> | |||
EDOs: 36, 48c, 84c<br /> | |||
Badness: 0.0819<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Omicronbeta temperament"></a><!-- ws:end:WikiTextHeadingRule:14 -->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 /> | ||
| Line 189: | Line 207: | ||
Badness: 0.0300<br /> | Badness: 0.0300<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Hours"></a><!-- ws:end:WikiTextHeadingRule:16 -->Hours</h1> | ||
Commas: 19683/19600, 33075/32768<br /> | Commas: 19683/19600, 33075/32768<br /> | ||
<br /> | <br /> | ||
| Line 199: | Line 217: | ||
Badness: 0.1161<br /> | Badness: 0.1161<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Hours-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> | ||
Commas: 243/242, 385/384, 9801/9800<br /> | Commas: 243/242, 385/384, 9801/9800<br /> | ||
<br /> | <br /> | ||
| Line 208: | Line 226: | ||
Badness: 0.0362<br /> | Badness: 0.0362<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Hours-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> | ||
Commas: 243/242, 351/350, 364/363, 385/384<br /> | Commas: 243/242, 351/350, 364/363, 385/384<br /> | ||
<br /> | <br /> | ||