Pythagorean family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<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 [[12edo|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.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-09 01:49:34 UTC</tt>.<br>
: The original revision id was <tt>332273242</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.
[[POTE_tuning|POTE generator]]: 15.116


[[POTE tuning|POTE generator]]: 15.116
Map: [&lt;12 19 0|, &lt;0 0 1|]


Map: [&lt;12 19 0|, &lt;0 0 1|]
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&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 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|72edo]], [[84edo|84edo]] or [[240edo|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|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.
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|72edo]] can be recommended as a tuning.


Commas: 225/224, 250047/250000
Commas: 225/224, 250047/250000


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


Map: [&lt;12 19 0 -22|, &lt;0 0 1 2|]
Map: [&lt;12 19 0 -22|, &lt;0 0 1 2|]
EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc
EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc


Line 32: Line 26:
Commas: 225/224, 441/440, 4375/4356
Commas: 225/224, 441/440, 4375/4356


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


Map: [&lt;12 19 0 -22 -42|, &lt;0 0 1 2 3|]
Map: [&lt;12 19 0 -22 -42|, &lt;0 0 1 2 3|]
EDOs: 12, 60e, 72
EDOs: 12, 60e, 72


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Map: [&lt;12 19 0 -22 -42 -67|, &lt;0 0 1 2 3 4|]
Map: [&lt;12 19 0 -22 -42 -67|, &lt;0 0 1 2 3 4|]
EDOs: 72, 228f, 300cf
EDOs: 72, 228f, 300cf
Badness: 0.0219
Badness: 0.0219


Line 52: Line 49:


Map: [&lt;12 19 0 -22 -42 100|, &lt;0 0 1 2 3 -2|]
Map: [&lt;12 19 0 -22 -42 100|, &lt;0 0 1 2 3 -2|]
EDOs: 12, 60e, 72, 204cdef, 276cdef
EDOs: 12, 60e, 72, 204cdef, 276cdef
Badness: 0.0251
Badness: 0.0251


=Catler temperament=
=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.   
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|12edo]]. Catler can also be characterized as the 12&amp;24 temperament. [[36edo|36edo]] or [[48edo|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
Commas: 81/80, 128/125


[[POTE tuning|POTE generator]]: 26.790
[[POTE_tuning|POTE generator]]: 26.790


Map: [&lt;12 19 28 0|, &lt;0 0 0 1|]
Map: [&lt;12 19 28 0|, &lt;0 0 0 1|]
EDOs: 12, [[36edo|36]], [[48edo|48]], 132, 180
EDOs: 12, [[36edo|36]], [[48edo|48]], 132, 180


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Map: [&lt;12 19 28 0 -26|, &lt;0 0 0 1 2|]
Map: [&lt;12 19 28 0 -26|, &lt;0 0 0 1 2|]
EDOs: 12, 48c, 108cd
EDOs: 12, 48c, 108cd
Badness: 0.0582
Badness: 0.0582


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Map: [&lt;12 19 28 0 109|, &lt;0 0 0 1 -2|]
Map: [&lt;12 19 28 0 109|, &lt;0 0 0 1 -2|]
EDOs: 36, 48c, 84c
EDOs: 36, 48c, 84c
Badness: 0.0819
Badness: 0.0819


Line 89: Line 93:


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


Line 98: Line 104:


Map: [&lt;12 19 28 0 8 11|, &lt;0 0 0 1 1 1|]
Map: [&lt;12 19 28 0 8 11|, &lt;0 0 0 1 1 1|]
EDOs: 12f, 24, 36f, 60cf
EDOs: 12f, 24, 36f, 60cf
Badness: 0.0284
Badness: 0.0284


Line 107: Line 115:


Map: [&lt;72 114 167 202 249 266|, &lt;0 0 0 0 0 1|]
Map: [&lt;72 114 167 202 249 266|, &lt;0 0 0 0 0 1|]
EDOs: 72, 144, 216c, 288cdf, 504bcdef
EDOs: 72, 144, 216c, 288cdf, 504bcdef
Badness: 0.0300
Badness: 0.0300


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Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]
Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]
Wedgie: &lt;0 24 -24 38 -38 -123|
Wedgie: &lt;0 24 -24 38 -38 -123|
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd
Badness: 0.1161
Badness: 0.1161


Line 126: Line 139:


Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]
Map: [&lt;24 38 0 123 83|, &lt;0 0 1 -1 0|]
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde
Badness: 0.0362
Badness: 0.0362


Line 135: Line 150:


Map: [&lt;24 38 0 123 83 33|, &lt;0 0 1 -1 0 1|]
Map: [&lt;24 38 0 123 83 33|, &lt;0 0 1 -1 0 1|]
EDOs: 24, 48f, 72, 168df, 240df
EDOs: 24, 48f, 72, 168df, 240df
Badness: 0.0269</pre></div>
 
<h4>Original HTML content:</h4>
Badness: 0.0269
<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;
[[Category:family]]
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;br /&gt;
[[Category:list]]
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;
[[Category:overview]]
&lt;br /&gt;
[[Category:pythagorean]]
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 15.116&lt;br /&gt;
[[Category:rank_2]]
&lt;br /&gt;
[[Category:theory]]
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;
&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;
&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;
&lt;br /&gt;
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 &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt; can be recommended as a tuning.&lt;br /&gt;
&lt;br /&gt;
Commas: 225/224, 250047/250000&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 383.775 (16.225)&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;12 19 0 -22|, &amp;lt;0 0 1 2|]&lt;br /&gt;
EDOs: 12, &lt;a class="wiki_link" href="/60edo"&gt;60&lt;/a&gt;, 72, 228, 300c, 372bc, 444bc&lt;br /&gt;
&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;
&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;
&lt;br /&gt;
Map: [&amp;lt;12 19 0 -22 -42|, &amp;lt;0 0 1 2 3|]&lt;br /&gt;
EDOs: 12, 60e, 72&lt;br /&gt;
&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;
&lt;br /&gt;
POTE generator: ~5/4 = 383.963 (16.037)&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;12 19 0 -22 -42 -67|, &amp;lt;0 0 1 2 3 4|]&lt;br /&gt;
EDOs: 72, 228f, 300cf&lt;br /&gt;
Badness: 0.0219&lt;br /&gt;
&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;
&lt;br /&gt;
POTE generator: ~5/4 = 382.612 (17.388)&lt;br /&gt;
&lt;br /&gt;
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;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;
&lt;br /&gt;
Map: [&amp;lt;12 19 28 0 -26|, &amp;lt;0 0 0 1 2|]&lt;br /&gt;
EDOs: 12, 48c, 108cd&lt;br /&gt;
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;
Commas: 81/80, 128/125, 540/539&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~36/35 = 27.864&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;12 19 28 0 109|, &amp;lt;0 0 0 1 -2|]&lt;br /&gt;
EDOs: 36, 48c, 84c&lt;br /&gt;
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;
Commas: 56/55, 81/80, 128/125&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~36/35 = 32.776&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;12 19 28 0 8|, &amp;lt;0 0 0 1 1|]&lt;br /&gt;
EDOs: 12, 24, 36, 72ce&lt;br /&gt;
Badness: 0.0345&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Catler temperament-Catcall-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit&lt;/h3&gt;
Commas: 56/55, 66/65, 81/80, 105/104&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~36/35 = 37.232&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;12 19 28 0 8 11|, &amp;lt;0 0 0 1 1 1|]&lt;br /&gt;
EDOs: 12f, 24, 36f, 60cf&lt;br /&gt;
Badness: 0.0284&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="Omicronbeta temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Omicronbeta temperament&lt;/h1&gt;
Commas: 225/224, 243/242, 441/440, 4375/4356&lt;br /&gt;
&lt;br /&gt;
Generator: ~13/8 = 837.814&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;72 114 167 202 249 266|, &amp;lt;0 0 0 0 0 1|]&lt;br /&gt;
EDOs: 72, 144, 216c, 288cdf, 504bcdef&lt;br /&gt;
Badness: 0.0300&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Hours"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Hours&lt;/h1&gt;
Commas: 19683/19600, 33075/32768&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~225/224 = 2.100&lt;br /&gt;
&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;
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd&lt;br /&gt;
Badness: 0.1161&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-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
Commas: 243/242, 385/384, 9801/9800&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~225/224 = 2.161&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;24 38 0 123 83|, &amp;lt;0 0 1 -1 0|]&lt;br /&gt;
EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde&lt;br /&gt;
Badness: 0.0362&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Hours-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h2&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;
&lt;br /&gt;
Map: [&amp;lt;24 38 0 123 83 33|, &amp;lt;0 0 1 -1 0 1|]&lt;br /&gt;
EDOs: 24, 48f, 72, 168df, 240df&lt;br /&gt;
Badness: 0.0269&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

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 generator: 15.116

Map: [<12 19 0|, <0 0 1|]

EDOs: 12, 72, 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 generator: ~5/4 = 383.775 (16.225)

Map: [<12 19 0 -22|, <0 0 1 2|]

EDOs: 12, 60, 72, 228, 300c, 372bc, 444bc

11-limit

Commas: 225/224, 441/440, 4375/4356

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 generator: 26.790

Map: [<12 19 28 0|, <0 0 0 1|]

EDOs: 12, 36, 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

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

13-limit

Commas: 56/55, 66/65, 81/80, 105/104

POTE generator: ~36/35 = 37.232

Map: [<12 19 28 0 8 11|, <0 0 0 1 1 1|]

EDOs: 12f, 24, 36f, 60cf

Badness: 0.0284

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

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