Pythagorean family: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 281658632 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 287020500 - 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>2011-12-02 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-17 02:33:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>287020500</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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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 | ||
==11-limit== | |||
Commas: 81/80, 99/98, 128/125 | |||
POTE generator: ~36/35 = 22.723 | |||
Map: [<12 19 28 0 -26|, <0 0 0 1 2|] | |||
EDOs: 12, 48c, 108cd | |||
=Omicronbeta temperament= | =Omicronbeta 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:20:&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:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#Compton temperament">Compton temperament</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Catler temperament">Catler temperament</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Omicronbeta temperament">Omicronbeta temperament</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Hours">Hours</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:31 --><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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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:10:&lt;h1&gt; --><h1 id=" | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Catler temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2> | ||
Commas: 81/80, 99/98, 128/125<br /> | |||
<br /> | |||
POTE generator: ~36/35 = 22.723<br /> | |||
<br /> | |||
Map: [&lt;12 19 28 0 -26|, &lt;0 0 0 1 2|]<br /> | |||
EDOs: 12, 48c, 108cd<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Omicronbeta temperament"></a><!-- ws:end:WikiTextHeadingRule:12 -->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 /> | ||
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Badness: 0.0300<br /> | Badness: 0.0300<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Hours"></a><!-- ws:end:WikiTextHeadingRule:14 -->Hours</h1> | ||
Commas: 19683/19600, 33075/32768<br /> | Commas: 19683/19600, 33075/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.1161<br /> | Badness: 0.1161<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Hours-11-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->11-limit</h2> | ||
Commas: 243/242, 385/384, 9801/9800<br /> | Commas: 243/242, 385/384, 9801/9800<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0362<br /> | Badness: 0.0362<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Hours-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->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 /> | ||
Revision as of 02:33, 17 December 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-12-17 02:33:17 UTC.
- The original revision id was 287020500.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[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 Map: [<12 19 0|, <0 0 1|] 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 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. Commas: 225/224, 250047/250000 [[POTE tuning|POTE generator]]: ~5/4 = 383.775 (16.225) Map: [<12 19 0 -22|, <0 0 1 2|] EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc ==11-limit== Commas: 225/224, 441/440, 4375/4356 [[POTE tuning|POTE generator]]: ~5/4 = 383.266 (16.734) Map: [<12 19 0 -22 -42|, <0 0 1 2 3|] 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= 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. Commas: 81/80, 128/125 [[POTE tuning|POTE generator]]: 26.790 Map: [<12 19 28 0|, <0 0 0 1|] EDOs: 12, [[36edo|36]], [[48edo|48]], 132, 180 ==11-limit== Commas: 81/80, 99/98, 128/125 POTE generator: ~36/35 = 22.723 Map: [<12 19 28 0 -26|, <0 0 0 1 2|] EDOs: 12, 48c, 108cd =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 =Hours= Commas: 19683/19600, 33075/32768 POTE generator: ~225/224 = 2.100 Map: [<24 38 0 123 83|, <0 0 1 -1 0|] Wedgie: <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: [<24 38 0 123 83|, <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: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|] EDOs: 24, 48f, 72, 168df, 240df Badness: 0.0269
Original HTML content:
<html><head><title>Pythagorean family</title></head><body><!-- ws:start:WikiTextTocRule:20:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#Compton temperament">Compton temperament</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Catler temperament">Catler temperament</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Omicronbeta temperament">Omicronbeta temperament</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Hours">Hours</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> <!-- ws:end:WikiTextTocRule:31 --><br /> 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 <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 /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 15.116<br /> <br /> Map: [<12 19 0|, <0 0 1|]<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 /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><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> 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 <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 /> In the either the 5 or 7-limit, <a class="wiki_link" href="/240edo">240edo</a> 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.<br /> <br /> 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 <a class="wiki_link" href="/72edo">72edo</a> can be recommended as a tuning.<br /> <br /> Commas: 225/224, 250047/250000<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.775 (16.225)<br /> <br /> Map: [<12 19 0 -22|, <0 0 1 2|]<br /> EDOs: 12, <a class="wiki_link" href="/60edo">60</a>, 72, 228, 300c, 372bc, 444bc<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><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 /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.266 (16.734)<br /> <br /> Map: [<12 19 0 -22 -42|, <0 0 1 2 3|]<br /> EDOs: 12, 60e, 72<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><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: [<12 19 0 -22 -42 -67|, <0 0 1 2 3 4|]<br /> EDOs: 72, 228f, 300cf<br /> Badness: 0.0219<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><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: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|]<br /> EDOs: 12, 60e, 72, 204cdef, 276cdef<br /> Badness: 0.0251<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><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&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 /> Commas: 81/80, 128/125<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 26.790<br /> <br /> Map: [<12 19 28 0|, <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<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Catler temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2> Commas: 81/80, 99/98, 128/125<br /> <br /> POTE generator: ~36/35 = 22.723<br /> <br /> Map: [<12 19 28 0 -26|, <0 0 0 1 2|]<br /> EDOs: 12, 48c, 108cd<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Omicronbeta temperament"></a><!-- ws:end:WikiTextHeadingRule:12 -->Omicronbeta temperament</h1> Commas: 225/224, 243/242, 441/440, 4375/4356<br /> <br /> Generator: ~13/8 = 837.814<br /> <br /> Map: [<72 114 167 202 249 266|, <0 0 0 0 0 1|]<br /> EDOs: 72, 144, 216c, 288cdf, 504bcdef<br /> Badness: 0.0300<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Hours"></a><!-- ws:end:WikiTextHeadingRule:14 -->Hours</h1> Commas: 19683/19600, 33075/32768<br /> <br /> POTE generator: ~225/224 = 2.100<br /> <br /> Map: [<24 38 0 123 83|, <0 0 1 -1 0|]<br /> Wedgie: <0 24 -24 38 -38 -123|<br /> EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd<br /> Badness: 0.1161<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Hours-11-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->11-limit</h2> Commas: 243/242, 385/384, 9801/9800<br /> <br /> POTE generator: ~225/224 = 2.161<br /> <br /> Map: [<24 38 0 123 83|, <0 0 1 -1 0|]<br /> EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde<br /> Badness: 0.0362<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Hours-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->13-limit</h2> Commas: 243/242, 351/350, 364/363, 385/384<br /> <br /> POTE generator: ~225/224 = 3.955<br /> <br /> Map: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|]<br /> EDOs: 24, 48f, 72, 168df, 240df<br /> Badness: 0.0269</body></html>