Pythagorean family: Difference between revisions

Revision as of 16:43, 22 February 2012

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This revision was by author guest and made on 2012-02-22 16:43:33 UTC.

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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
==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
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

==Catcall== 
Commas: 56/55, 81/80, 128/125

POTE generator: ~36/35 = 32.776

Map: [<12 19 28 0 8|, <0 0 0 1 1|]
EDOs: 12, 24, 36, 72ce
Badness: 0.0345

=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: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: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 />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 15.116<br />
<br />
Map: [&lt;12 19 0|, &lt;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:&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 />
<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: [&lt;12 19 0 -22|, &lt;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:&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 />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 383.266 (16.734)<br />
<br />
Map: [&lt;12 19 0 -22 -42|, &lt;0 0 1 2 3|]<br />
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 />
<!-- 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 />
<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 />
Badness: 0.0582<br />
<br />
<!-- 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 />
<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: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 />
<br />
POTE generator: ~36/35 = 32.776<br />
<br />
Map: [&lt;12 19 28 0 8|, &lt;0 0 0 1 1|]<br />
EDOs: 12, 24, 36, 72ce<br />
Badness: 0.0345<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 />
<br />
Generator: ~13/8 = 837.814<br />
<br />
Map: [&lt;72 114 167 202 249 266|, &lt;0 0 0 0 0 1|]<br />
EDOs: 72, 144, 216c, 288cdf, 504bcdef<br />
Badness: 0.0300<br />
<br />
<!-- 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 />
<br />
POTE generator: ~225/224 = 2.100<br />
<br />
Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]<br />
Wedgie: &lt;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: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 />
<br />
POTE generator: ~225/224 = 2.161<br />
<br />
Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]<br />
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde<br />
Badness: 0.0362<br />
<br />
<!-- 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 />
<br />
POTE generator: ~225/224 = 3.955<br />
<br />
Map: [&lt;24 38 0 123 83 33|, &lt;0 0 1 -1 0 1|]<br />
EDOs: 24, 48f, 72, 168df, 240df<br />
Badness: 0.0269</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-12-29 15:16:54 UTC</tt>.<br>
+: 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>288762942</tt>.<br>
+: The original revision id was <tt>304177420</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>
 <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]]
 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.
@@ Line 15: / Line 14: @@
 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&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.
@@ Line 29: / Line 28: @@
 EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc
 ==11-limit==
 Commas: 225/224, 441/440, 4375/4356
@@ Line 37: / Line 36: @@
 EDOs: 12, 60e, 72
 ==13-limit==
 Commas: 225/224, 441/440, 351/350, 364/363
@@ Line 46: / Line 45: @@
 Badness: 0.0219
 ==Comptone==
 Commas: 225/224, 441/440, 325/324, 1001/1000
@@ Line 53: / Line 52: @@
 Map: [&lt;12 19 0 -22 -42 100|, &lt;0 0 1 2 3 -2|]
 EDOs: 12, 60e, 72, 204cdef, 276cdef
-Badness: 0.0251
+==11-limit==
-=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&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.
-Commas: 81/80, 128/125
-[[POTE tuning|POTE generator]]: 26.790
-Map: [&lt;12 19 28 0|, &lt;0 0 0 1|]
-EDOs: 12, [[36edo|36]], [[48edo|48]], 132, 180
-==11-limit==
 Commas: 81/80, 99/98, 128/125
@@ Line 74: / Line 61: @@
 Badness: 0.0582
 ==Catlat==
 Commas: 81/80, 128/125, 540/539
@@ Line 83: / Line 70: @@
 Badness: 0.0819
 ==Catcall==
 Commas: 56/55, 81/80, 128/125
@@ Line 92: / Line 79: @@
 Badness: 0.0345
 =Omicronbeta temperament=
 Commas: 225/224, 243/242, 441/440, 4375/4356
@@ Line 101: / Line 88: @@
 Badness: 0.0300
 =Hours=
 Commas: 19683/19600, 33075/32768
@@ Line 111: / Line 98: @@
 Badness: 0.1161
 ==11-limit==
 Commas: 243/242, 385/384, 9801/9800
@@ Line 120: / Line 107: @@
 Badness: 0.0362
 ==13-limit==
 Commas: 243/242, 351/350, 364/363, 385/384
@@ Line 127: / Line 114: @@
 Map: [&lt;24 38 0 123 83 33|, &lt;0 0 1 -1 0 1|]
 EDOs: 24, 48f, 72, 168df, 240df
-Badness: 0.0269
+Badness: 0.0269</pre></div>
-</pre></div>
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-<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:24:&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:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;a href="#Compton temperament"&gt;Compton temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&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;a href="#Catler temperament"&gt;Catler temperament&lt;/a&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;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Omicronbeta temperament"&gt;Omicronbeta temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Hours"&gt;Hours&lt;/a&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;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;
+<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:22:&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:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;a href="#Compton temperament"&gt;Compton temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&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;a href="#Omicronbeta temperament"&gt;Omicronbeta temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Hours"&gt;Hours&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:37 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:34 --&gt;The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12&amp;gt;, and hence the fifths form a closed 12-note circle of fifths, identical to &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. 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.&lt;br /&gt;
-The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12&amp;gt;, and hence the fifths form a closed 12-note circle of fifths, identical to &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. 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.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 15.116&lt;br /&gt;
@@ Line 140: / Line 125: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Compton temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Compton temperament&lt;/h1&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;
+ 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;
 &lt;br /&gt;
 In the either the 5 or 7-limit, &lt;a class="wiki_link" href="/240edo"&gt;240edo&lt;/a&gt; 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.&lt;br /&gt;
@@ Line 154: / Line 139: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Compton temperament-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
-Commas: 225/224, 441/440, 4375/4356&lt;br /&gt;
+ Commas: 225/224, 441/440, 4375/4356&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 383.266 (16.734)&lt;br /&gt;
@@ Line 162: / Line 147: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Compton temperament-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
-Commas: 225/224, 441/440, 351/350, 364/363&lt;br /&gt;
+ Commas: 225/224, 441/440, 351/350, 364/363&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~5/4 = 383.963 (16.037)&lt;br /&gt;
@@ Line 171: / Line 156: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Compton temperament-Comptone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Comptone&lt;/h2&gt;
-Commas: 225/224, 441/440, 325/324, 1001/1000&lt;br /&gt;
+ Commas: 225/224, 441/440, 325/324, 1001/1000&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~5/4 = 382.612 (17.388)&lt;br /&gt;
@@ Line 177: / Line 162: @@
 Map: [&amp;lt;12 19 0 -22 -42 100|, &amp;lt;0 0 1 2 3 -2|]&lt;br /&gt;
 EDOs: 12, 60e, 72, 204cdef, 276cdef&lt;br /&gt;
-Badness: 0.0251&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Compton temperament-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
-&lt;br /&gt;
+ Commas: 81/80, 99/98, 128/125&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Catler temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Catler temperament&lt;/h1&gt;
-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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. Catler can also be characterized as the 12&amp;amp;24 temperament. &lt;a class="wiki_link" href="/36edo"&gt;36edo&lt;/a&gt; or &lt;a class="wiki_link" href="/48edo"&gt;48edo&lt;/a&gt; are possible tunings, and 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7 or 7/5 are possible generators.  &lt;br /&gt;
-&lt;br /&gt;
-Commas: 81/80, 128/125&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 26.790&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;12 19 28 0|, &amp;lt;0 0 0 1|]&lt;br /&gt;
-EDOs: 12, &lt;a class="wiki_link" href="/36edo"&gt;36&lt;/a&gt;, &lt;a class="wiki_link" href="/48edo"&gt;48&lt;/a&gt;, 132, 180&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Catler temperament-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
-Commas: 81/80, 99/98, 128/125&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~36/35 = 22.723&lt;br /&gt;
@@ Line 198: / Line 171: @@
 Badness: 0.0582&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Catler temperament-Catlat"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Catlat&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Compton temperament-Catlat"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Catlat&lt;/h2&gt;
-Commas: 81/80, 128/125, 540/539&lt;br /&gt;
+ Commas: 81/80, 128/125, 540/539&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~36/35 = 27.864&lt;br /&gt;
@@ Line 207: / Line 180: @@
 Badness: 0.0819&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Catler temperament-Catcall"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Catcall&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Compton temperament-Catcall"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Catcall&lt;/h2&gt;
-Commas: 56/55, 81/80, 128/125&lt;br /&gt;
+ Commas: 56/55, 81/80, 128/125&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~36/35 = 32.776&lt;br /&gt;
@@ Line 216: / Line 189: @@
 Badness: 0.0345&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Omicronbeta temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Omicronbeta temperament&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Omicronbeta temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Omicronbeta temperament&lt;/h1&gt;
-Commas: 225/224, 243/242, 441/440, 4375/4356&lt;br /&gt;
+ Commas: 225/224, 243/242, 441/440, 4375/4356&lt;br /&gt;
 &lt;br /&gt;
 Generator: ~13/8 = 837.814&lt;br /&gt;
@@ Line 225: / Line 198: @@
 Badness: 0.0300&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Hours"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Hours&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Hours"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Hours&lt;/h1&gt;
-Commas: 19683/19600, 33075/32768&lt;br /&gt;
+ Commas: 19683/19600, 33075/32768&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~225/224 = 2.100&lt;br /&gt;
@@ Line 235: / Line 208: @@
 Badness: 0.1161&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Hours-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Hours-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
-Commas: 243/242, 385/384, 9801/9800&lt;br /&gt;
+ Commas: 243/242, 385/384, 9801/9800&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~225/224 = 2.161&lt;br /&gt;
@@ Line 244: / Line 217: @@
 Badness: 0.0362&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Hours-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;13-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Hours-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
-Commas: 243/242, 351/350, 364/363, 385/384&lt;br /&gt;
+ Commas: 243/242, 351/350, 364/363, 385/384&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~225/224 = 3.955&lt;br /&gt;

Pythagorean family: Difference between revisions

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