Pythagorean family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 288762942 - Original comment: ** |
Wikispaces>guest **Imported revision 304177420 - 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:guest|guest]] and made on <tt>2012-02-22 16:43:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>304177420</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">[[toc|flat]] | <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. | 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. | ||
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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= | =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. | ||
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 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. | ||
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EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc | 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 | ||
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EDOs: 12, 60e, 72 | EDOs: 12, 60e, 72 | ||
==13-limit== | ==13-limit== | ||
Commas: 225/224, 441/440, 351/350, 364/363 | Commas: 225/224, 441/440, 351/350, 364/363 | ||
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Badness: 0.0219 | Badness: 0.0219 | ||
==Comptone== | ==Comptone== | ||
Commas: 225/224, 441/440, 325/324, 1001/1000 | Commas: 225/224, 441/440, 325/324, 1001/1000 | ||
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Map: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|] | Map: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|] | ||
EDOs: 12, 60e, 72, 204cdef, 276cdef | EDOs: 12, 60e, 72, 204cdef, 276cdef | ||
==11-limit== | |||
==11-limit== | |||
Commas: 81/80, 99/98, 128/125 | Commas: 81/80, 99/98, 128/125 | ||
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Badness: 0.0582 | Badness: 0.0582 | ||
==Catlat== | ==Catlat== | ||
Commas: 81/80, 128/125, 540/539 | Commas: 81/80, 128/125, 540/539 | ||
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Badness: 0.0819 | Badness: 0.0819 | ||
==Catcall== | ==Catcall== | ||
Commas: 56/55, 81/80, 128/125 | Commas: 56/55, 81/80, 128/125 | ||
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Badness: 0.0345 | Badness: 0.0345 | ||
=Omicronbeta temperament= | =Omicronbeta temperament= | ||
Commas: 225/224, 243/242, 441/440, 4375/4356 | Commas: 225/224, 243/242, 441/440, 4375/4356 | ||
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Badness: 0.0300 | Badness: 0.0300 | ||
=Hours= | =Hours= | ||
Commas: 19683/19600, 33075/32768 | Commas: 19683/19600, 33075/32768 | ||
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Badness: 0.1161 | Badness: 0.1161 | ||
==11-limit== | ==11-limit== | ||
Commas: 243/242, 385/384, 9801/9800 | Commas: 243/242, 385/384, 9801/9800 | ||
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Badness: 0.0362 | Badness: 0.0362 | ||
==13-limit== | ==13-limit== | ||
Commas: 243/242, 351/350, 364/363, 385/384 | Commas: 243/242, 351/350, 364/363, 385/384 | ||
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Map: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|] | Map: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|] | ||
EDOs: 24, 48f, 72, 168df, 240df | EDOs: 24, 48f, 72, 168df, 240df | ||
Badness: 0.0269 | Badness: 0.0269</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: --><!-- 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 -->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 /> | ||
<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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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Compton temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->Compton temperament</h1> | <!-- 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 /> | ||
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 /> | 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 /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Compton temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> | <!-- 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>: ~5/4 = 383.266 (16.734)<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.266 (16.734)<br /> | ||
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<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> | <!-- 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 /> | Commas: 225/224, 441/440, 351/350, 364/363<br /> | ||
<br /> | <br /> | ||
POTE generator: ~5/4 = 383.963 (16.037)<br /> | POTE generator: ~5/4 = 383.963 (16.037)<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Compton temperament-Comptone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Comptone</h2> | <!-- 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 /> | Commas: 225/224, 441/440, 325/324, 1001/1000<br /> | ||
<br /> | <br /> | ||
POTE generator: ~5/4 = 382.612 (17.388)<br /> | POTE generator: ~5/4 = 382.612 (17.388)<br /> | ||
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Map: [&lt;12 19 0 -22 -42 100|, &lt;0 0 1 2 3 -2|]<br /> | Map: [&lt;12 19 0 -22 -42 100|, &lt;0 0 1 2 3 -2|]<br /> | ||
EDOs: 12, 60e, 72, 204cdef, 276cdef<br /> | EDOs: 12, 60e, 72, 204cdef, 276cdef<br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Compton temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h2> | |||
Commas: 81/80, 99/98, 128/125<br /> | |||
<!-- ws:start:WikiTextHeadingRule:8 | |||
Commas: 81/80, 99/98, 128/125<br /> | |||
<br /> | <br /> | ||
POTE generator: ~36/35 = 22.723<br /> | POTE generator: ~36/35 = 22.723<br /> | ||
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Badness: 0.0582<br /> | Badness: 0.0582<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Compton temperament-Catlat"></a><!-- ws:end:WikiTextHeadingRule:10 -->Catlat</h2> | ||
Commas: 81/80, 128/125, 540/539<br /> | Commas: 81/80, 128/125, 540/539<br /> | ||
<br /> | <br /> | ||
POTE generator: ~36/35 = 27.864<br /> | POTE generator: ~36/35 = 27.864<br /> | ||
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Badness: 0.0819<br /> | Badness: 0.0819<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Compton temperament-Catcall"></a><!-- ws:end:WikiTextHeadingRule:12 -->Catcall</h2> | ||
Commas: 56/55, 81/80, 128/125<br /> | Commas: 56/55, 81/80, 128/125<br /> | ||
<br /> | <br /> | ||
POTE generator: ~36/35 = 32.776<br /> | POTE generator: ~36/35 = 32.776<br /> | ||
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Badness: 0.0345<br /> | Badness: 0.0345<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | ||
Generator: ~13/8 = 837.814<br /> | Generator: ~13/8 = 837.814<br /> | ||
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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 /> | ||
POTE generator: ~225/224 = 2.100<br /> | POTE generator: ~225/224 = 2.100<br /> | ||
| Line 235: | Line 208: | ||
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 /> | ||
POTE generator: ~225/224 = 2.161<br /> | POTE generator: ~225/224 = 2.161<br /> | ||
| Line 244: | Line 217: | ||
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 /> | ||
POTE generator: ~225/224 = 3.955<br /> | POTE generator: ~225/224 = 3.955<br /> | ||