Breedsmic temperaments: Difference between revisions

Revision as of 17:01, 6 September 2011

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[[toc|flat]]
Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4> = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.

It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.

=Hemififths= 
Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing good tunings, and [[239edo]] an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament and has wedgie <<2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.

By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. [[99edo]] is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.

Commas: 2401/2400, 5120/5103

7 and 9-limit minimax
[|1 0 0 0>, |7/5, 0, 2/25, 0>, |0 0 1 0>, |8/5 0 13/25 0>]
Eigenvalues: 2, 5

Algebraic generator: (2 + sqrt(2))/2

Map: [<1 1 -5 -1|, <0 2 25 13|]
EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]]
Badness: 0.0222

==11-limit== 
Commas: 243/242, 441/440, 896/891

POTE generator: ~11/9 = 351.521

Map: [<1 1 -5 -1 2|, <0 2 25 13 5|]
EDOs: 7, 17, 41, 58, 99
Badness: 0.0235

==13-limit== 
Commas: 144/143, 196/195, 243/242, 364/363

POTE generator: ~11/9 = 351.573

Map: [<1 1 -5 -1 2 4|, <0 2 25 13 5 -1|]
EDOs: 7, 17, 41, 58, 99
Badness: 0.0191

=Tertiaseptal= 
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.

Commas: 2401/2400, 65625/65536

POTE generator: ~256/245 = 77.191

Map: [<1 3 2 3|, <0 -22 5 -3|]
EDOs: 15, 16, 31, 109, 140, 171
Badness: 0.0130

==11-limit== 
Commas: 243/242, 441/440, 65625/65536

POTE generator: ~256/245 = 77.227

Map: [<1 3 2 3 7|, <0 -22 5 -3 -55|]
EDOs: 15, 16, 31, 171, 202
Badness: 0.0356

=Harry= 
Commas: 2401/2400, 19683/19600

Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament, with wedgie <<12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.

Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is <<12 34 20 30 ...||.

Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with <<12 34 20 30 52 ...|| as the octave wedgie. [[130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.

[[POTE tuning|POTE generator]]: ~21/20 = 83.156

Map: [<2 4 7 7|, <0 -6 -17 -10|]
Wedgie: <<12 34 20 26 -2 -49||
EDOs: 14, 58, 72, 130, 202, 534, 938
Badness: 0.0341

==11-limit== 
Commas: 243/242, 441/440, 4000/3993

[[POTE tuning|POTE generator]]: ~21/20 = 83.167

Map: [<2 4 7 7 9|, <0 -6 -17 -10 -15|]
EDOs: 14, 58, 72, 130, 202
Badness: 0.0159

==13-limit== 
Commas: 243/242, 351/350, 441/440, 676/675

[[POTE tuning|POTE generator]]: ~21/20 = 83.116

Map: [<2 4 7 7 9 11|, <0 -6 -17 -10 -15 -26|]
EDOs: 14, 58, 72, 130, 462
Badness: 0.0130

=Quasiorwell= 
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1>. It has a wedgie <<38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.

Adding 3025/3024 extends to the 11-limit and gives <<38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.

Commas: 2401/2400, 29360128/29296875

POTE generator: ~1024/875 = 271.107

Map: [<1 31 0 9|, <0 -38 3 -8|]
EDOs: [[31edo|31]], [[177edo|177]], [[208edo|208]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Badness: 0.0358

==11-limit== 
Commas: 2401/2400, 3025/3024, 5632/5625

POTE generator: ~90/77 = 271.111

Map: [<1 31 0 9 53|, <0 -38 3 -8 -64|]
EDOs: [[31edo|31]], [[208edo|208]], [[239edo|239]], [[270edo|270]]
Badness: 0.0175

==13-limit== 
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095

POTE generator: ~90/77 = 271.107

Map: [<1 31 0 9 53 -59|, <0 -38 3 -8 -64 81|]
EDOs: [[31edo|31]], [[239edo|239]], [[270edo|270]], [[571edo|571]], [[841edo|841]], [[1111edo|1111]]
Badness: 0.0179

=Decoid= 
Commas: 2401/2400, 67108864/66976875

POTE generator: ~8/7 = 231.099

Map: [<10 0 47 36|, <0 2 -3 -1|]
Wedgie: <<20 -30 -10 -94 -72 61||
EDOs: 10, 120, 130, 270
Badness: 0.0339

==11-limit== 
Commas: 2401/2400, 5832/5825, 9801/9800

POTE generator: ~8/7 = 231.070

Map: [<10 0 47 36 98|, <0 2 -3 -1 -8|]
EDOs: 130, 270, 670, 940, 1210
Badness: 0.0187

==13-limit== 
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224

POTE generator: ~8/7 = 231.083

Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]
EDOs: 130, 270, 940, 1480
Badness: 0.0135

Original HTML content:

<html><head><title>Breedsmic temperaments</title></head><body><!-- ws:start:WikiTextTocRule:28:&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:28 --><!-- ws:start:WikiTextTocRule:29: --><a href="#Hemififths">Hemififths</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Tertiaseptal">Tertiaseptal</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Harry">Harry</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#Quasiorwell">Quasiorwell</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#Decoid">Decoid</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: -->
<!-- ws:end:WikiTextTocRule:43 -->Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4&gt; = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.<br />
<br />
It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Hemififths"></a><!-- ws:end:WikiTextHeadingRule:0 -->Hemififths</h1>
 Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with <a class="wiki_link" href="/99edo">99edo</a> and <a class="wiki_link" href="/140edo">140edo</a> providing good tunings, and <a class="wiki_link" href="/239edo">239edo</a> an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;58 temperament and has wedgie &lt;&lt;2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.<br />
<br />
By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. <a class="wiki_link" href="/99edo">99edo</a> is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.<br />
<br />
Commas: 2401/2400, 5120/5103<br />
<br />
7 and 9-limit minimax<br />
[|1 0 0 0&gt;, |7/5, 0, 2/25, 0&gt;, |0 0 1 0&gt;, |8/5 0 13/25 0&gt;]<br />
Eigenvalues: 2, 5<br />
<br />
Algebraic generator: (2 + sqrt(2))/2<br />
<br />
Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|]<br />
EDOs: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/99edo">99</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/338edo">338</a><br />
Badness: 0.0222<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Hemififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2>
 Commas: 243/242, 441/440, 896/891<br />
<br />
POTE generator: ~11/9 = 351.521<br />
<br />
Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]<br />
EDOs: 7, 17, 41, 58, 99<br />
Badness: 0.0235<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Hemififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2>
 Commas: 144/143, 196/195, 243/242, 364/363<br />
<br />
POTE generator: ~11/9 = 351.573<br />
<br />
Map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]<br />
EDOs: 7, 17, 41, 58, 99<br />
Badness: 0.0191<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Tertiaseptal"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tertiaseptal</h1>
 Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. <a class="wiki_link" href="/171edo">171edo</a> makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.<br />
<br />
Commas: 2401/2400, 65625/65536<br />
<br />
POTE generator: ~256/245 = 77.191<br />
<br />
Map: [&lt;1 3 2 3|, &lt;0 -22 5 -3|]<br />
EDOs: 15, 16, 31, 109, 140, 171<br />
Badness: 0.0130<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Tertiaseptal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h2>
 Commas: 243/242, 441/440, 65625/65536<br />
<br />
POTE generator: ~256/245 = 77.227<br />
<br />
Map: [&lt;1 3 2 3 7|, &lt;0 -22 5 -3 -55|]<br />
EDOs: 15, 16, 31, 171, 202<br />
Badness: 0.0356<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Harry"></a><!-- ws:end:WikiTextHeadingRule:10 -->Harry</h1>
 Commas: 2401/2400, 19683/19600<br />
<br />
Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&amp;72 temperament, with wedgie &lt;&lt;12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.<br />
<br />
Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is &lt;&lt;12 34 20 30 ...||.<br />
<br />
Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. <a class="wiki_link" href="/130edo">130edo</a> is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~21/20 = 83.156<br />
<br />
Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]<br />
Wedgie: &lt;&lt;12 34 20 26 -2 -49||<br />
EDOs: 14, 58, 72, 130, 202, 534, 938<br />
Badness: 0.0341<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Harry-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2>
 Commas: 243/242, 441/440, 4000/3993<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~21/20 = 83.167<br />
<br />
Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]<br />
EDOs: 14, 58, 72, 130, 202<br />
Badness: 0.0159<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Harry-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2>
 Commas: 243/242, 351/350, 441/440, 676/675<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~21/20 = 83.116<br />
<br />
Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]<br />
EDOs: 14, 58, 72, 130, 462<br />
Badness: 0.0130<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Quasiorwell"></a><!-- ws:end:WikiTextHeadingRule:16 -->Quasiorwell</h1>
 In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.<br />
<br />
Adding 3025/3024 extends to the 11-limit and gives &lt;&lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.<br />
<br />
Commas: 2401/2400, 29360128/29296875<br />
<br />
POTE generator: ~1024/875 = 271.107<br />
<br />
Map: [&lt;1 31 0 9|, &lt;0 -38 3 -8|]<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/177edo">177</a>, <a class="wiki_link" href="/208edo">208</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/270edo">270</a>, <a class="wiki_link" href="/571edo">571</a>, <a class="wiki_link" href="/841edo">841</a>, <a class="wiki_link" href="/1111edo">1111</a><br />
Badness: 0.0358<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Quasiorwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2>
 Commas: 2401/2400, 3025/3024, 5632/5625<br />
<br />
POTE generator: ~90/77 = 271.111<br />
<br />
Map: [&lt;1 31 0 9 53|, &lt;0 -38 3 -8 -64|]<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/208edo">208</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/270edo">270</a><br />
Badness: 0.0175<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Quasiorwell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2>
 Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095<br />
<br />
POTE generator: ~90/77 = 271.107<br />
<br />
Map: [&lt;1 31 0 9 53 -59|, &lt;0 -38 3 -8 -64 81|]<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/270edo">270</a>, <a class="wiki_link" href="/571edo">571</a>, <a class="wiki_link" href="/841edo">841</a>, <a class="wiki_link" href="/1111edo">1111</a><br />
Badness: 0.0179<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Decoid"></a><!-- ws:end:WikiTextHeadingRule:22 -->Decoid</h1>
 Commas: 2401/2400, 67108864/66976875<br />
<br />
POTE generator: ~8/7 = 231.099<br />
<br />
Map: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]<br />
Wedgie: &lt;&lt;20 -30 -10 -94 -72 61||<br />
EDOs: 10, 120, 130, 270<br />
Badness: 0.0339<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Decoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->11-limit</h2>
 Commas: 2401/2400, 5832/5825, 9801/9800<br />
<br />
POTE generator: ~8/7 = 231.070<br />
<br />
Map: [&lt;10 0 47 36 98|, &lt;0 2 -3 -1 -8|]<br />
EDOs: 130, 270, 670, 940, 1210<br />
Badness: 0.0187<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Decoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2>
 Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br />
<br />
POTE generator: ~8/7 = 231.083<br />
<br />
Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]<br />
EDOs: 130, 270, 940, 1480<br />
Badness: 0.0135</body></html>

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 <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>2011-03-21 17:10:04 UTC</tt>.<br>
+: This revision was by author [[User:Natebedell|Natebedell]] and made on <tt>2011-09-06 17:01:53 UTC</tt>.<br>
-: The original revision id was <tt>212568090</tt>.<br>
+: The original revision id was <tt>251319616</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]]
+Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4&gt; = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.
-Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.
 It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
 =Hemififths=
 Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing good tunings, and [[239edo]] an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;58 temperament and has wedgie &lt;&lt;2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.
@@ Line 29: / Line 28: @@
 Badness: 0.0222
 ==11-limit==
 Commas: 243/242, 441/440, 896/891
@@ Line 38: / Line 37: @@
 Badness: 0.0235
 ==13-limit==
 Commas: 144/143, 196/195, 243/242, 364/363
@@ Line 47: / Line 46: @@
 Badness: 0.0191
 =Tertiaseptal=
 Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
@@ Line 58: / Line 57: @@
 Badness: 0.0130
 ==11-limit==
 Commas: 243/242, 441/440, 65625/65536
@@ Line 67: / Line 66: @@
 Badness: 0.0356
 =Harry=
 Commas: 2401/2400, 19683/19600
@@ Line 83: / Line 82: @@
 Badness: 0.0341
 ==11-limit==
 Commas: 243/242, 441/440, 4000/3993
@@ Line 92: / Line 91: @@
 Badness: 0.0159
 ==13-limit==
 Commas: 243/242, 351/350, 441/440, 676/675
@@ Line 101: / Line 100: @@
 Badness: 0.0130
 =Quasiorwell=
 In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.
@@ Line 114: / Line 113: @@
 Badness: 0.0358
 ==11-limit==
 Commas: 2401/2400, 3025/3024, 5632/5625
@@ Line 123: / Line 122: @@
 Badness: 0.0175
 ==13-limit==
 Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095
@@ Line 132: / Line 131: @@
 Badness: 0.0179
 =Decoid=
 Commas: 2401/2400, 67108864/66976875
@@ Line 142: / Line 141: @@
 Badness: 0.0339
 ==11-limit==
 Commas: 2401/2400, 5832/5825, 9801/9800
@@ Line 151: / Line 150: @@
 Badness: 0.0187
 ==13-limit==
 Commas: 676/675, 1001/1000, 1716/1715, 4225/4224
@@ Line 161: / Line 160: @@
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Breedsmic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:28:&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:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;a href="#Hemififths"&gt;Hemififths&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;a href="#Tertiaseptal"&gt;Tertiaseptal&lt;/a&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;a href="#Harry"&gt;Harry&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; | &lt;a href="#Quasiorwell"&gt;Quasiorwell&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; | &lt;a href="#Decoid"&gt;Decoid&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:43 --&gt;Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4&amp;gt; = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.&lt;br /&gt;
-Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.&lt;br /&gt;
 &lt;br /&gt;
 It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.&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="Hemififths"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Hemififths&lt;/h1&gt;
-Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; and &lt;a class="wiki_link" href="/140edo"&gt;140edo&lt;/a&gt; providing good tunings, and &lt;a class="wiki_link" href="/239edo"&gt;239edo&lt;/a&gt; an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;amp;58 temperament and has wedgie &amp;lt;&amp;lt;2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.&lt;br /&gt;
+ Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; and &lt;a class="wiki_link" href="/140edo"&gt;140edo&lt;/a&gt; providing good tunings, and &lt;a class="wiki_link" href="/239edo"&gt;239edo&lt;/a&gt; an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;amp;58 temperament and has wedgie &amp;lt;&amp;lt;2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.&lt;br /&gt;
 &lt;br /&gt;
 By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.&lt;br /&gt;
@@ Line 184: / Line 182: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Hemififths-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
-Commas: 243/242, 441/440, 896/891&lt;br /&gt;
+ Commas: 243/242, 441/440, 896/891&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~11/9 = 351.521&lt;br /&gt;
@@ Line 193: / Line 191: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Hemififths-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
-Commas: 144/143, 196/195, 243/242, 364/363&lt;br /&gt;
+ Commas: 144/143, 196/195, 243/242, 364/363&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~11/9 = 351.573&lt;br /&gt;
@@ Line 202: / Line 200: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Tertiaseptal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Tertiaseptal&lt;/h1&gt;
-Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.&lt;br /&gt;
+ Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.&lt;br /&gt;
 &lt;br /&gt;
 Commas: 2401/2400, 65625/65536&lt;br /&gt;
@@ Line 213: / Line 211: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Tertiaseptal-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
-Commas: 243/242, 441/440, 65625/65536&lt;br /&gt;
+ Commas: 243/242, 441/440, 65625/65536&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~256/245 = 77.227&lt;br /&gt;
@@ Line 222: / Line 220: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Harry"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Harry&lt;/h1&gt;
-Commas: 2401/2400, 19683/19600&lt;br /&gt;
+ Commas: 2401/2400, 19683/19600&lt;br /&gt;
 &lt;br /&gt;
 Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&amp;amp;72 temperament, with wedgie &amp;lt;&amp;lt;12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.&lt;br /&gt;
@@ Line 238: / Line 236: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Harry-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
-Commas: 243/242, 441/440, 4000/3993&lt;br /&gt;
+ Commas: 243/242, 441/440, 4000/3993&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~21/20 = 83.167&lt;br /&gt;
@@ Line 247: / Line 245: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Harry-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
-Commas: 243/242, 351/350, 441/440, 676/675&lt;br /&gt;
+ Commas: 243/242, 351/350, 441/440, 676/675&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~21/20 = 83.116&lt;br /&gt;
@@ Line 256: / Line 254: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Quasiorwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Quasiorwell&lt;/h1&gt;
-In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&amp;gt;. It has a wedgie &amp;lt;&amp;lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.&lt;br /&gt;
+ In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&amp;gt;. It has a wedgie &amp;lt;&amp;lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.&lt;br /&gt;
 &lt;br /&gt;
 Adding 3025/3024 extends to the 11-limit and gives &amp;lt;&amp;lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.&lt;br /&gt;
@@ Line 269: / Line 267: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Quasiorwell-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
-Commas: 2401/2400, 3025/3024, 5632/5625&lt;br /&gt;
+ Commas: 2401/2400, 3025/3024, 5632/5625&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~90/77 = 271.111&lt;br /&gt;
@@ Line 278: / Line 276: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Quasiorwell-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
-Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095&lt;br /&gt;
+ Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~90/77 = 271.107&lt;br /&gt;
@@ Line 287: / Line 285: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Decoid"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Decoid&lt;/h1&gt;
-Commas: 2401/2400, 67108864/66976875&lt;br /&gt;
+ Commas: 2401/2400, 67108864/66976875&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~8/7 = 231.099&lt;br /&gt;
@@ Line 297: / Line 295: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Decoid-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
-Commas: 2401/2400, 5832/5825, 9801/9800&lt;br /&gt;
+ Commas: 2401/2400, 5832/5825, 9801/9800&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~8/7 = 231.070&lt;br /&gt;
@@ Line 306: / Line 304: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Decoid-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
-Commas: 676/675, 1001/1000, 1716/1715, 4225/4224&lt;br /&gt;
+ Commas: 676/675, 1001/1000, 1716/1715, 4225/4224&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~8/7 = 231.083&lt;br /&gt;

Breedsmic temperaments: Difference between revisions

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