Kleismic family

Revision as of 11:47, 25 March 2011 by Wikispaces>genewardsmith (**Imported revision 213985678 - Original comment: **)

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2011-03-25 11:47:49 UTC.
The original revision id was 213985678.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

[[toc|flat]]

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is |-6 -5 6>, and flipping that yields <<6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. 14/53 is about perfect as a generator, though 9/34 also makes sense and using [[19edo]] is possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].

[[POTE tuning|POTE generator]]: 317.007

Map: [<1 0 1|, <0 6 5|]

=Seven limit children=
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.

=Keemun=
[[Comma|Commas]]: 49/48, 126/125

[[POTE tuning|POTE generator]]: ~6/5 = 316.473

Map: [<1 0 1 2|, <0 6 5 3|]
Wedgie: <<6 5 3 -6 -12 -7||
EDOs: [[15edo|15]], [[19edo|19]], [[91edo|91]]
Badness: 0.0274

==11-limit==
Commas: 49/48, 56/55, 100/99

[[POTE tuning|POTE generator]]: ~6/5 = 317.576

Map: [<1 0 1 2 4|, <0 6 5 3 -2|]
EDOs: 4, 15, 19, 34
Badness: 0.0274

==13-limit==
Commas: 49/48, 56/55, 78/77, 100/99

[[POTE tuning|POTE generator]]: ~6/5 = 316.611

Map: [<1 0 1 2 4 5|, <0 6 5 3 -2 -5|]
EDOs: 4, 15, 19, 72
Badness: 0.0297

=Catakleismic=
[[Comma|Commas]]: 225/224, 4375/4374

[[POTE tuning|POTE generator]]: 316.732

Map: [<1 0 1 -3|, <0 6 5 22|]
Wedgie: <<6 5 22 -6 18 37||
EDOs: [[19edo|19]], [[53edo|15]], [[72edo|72]], [[197edo|197]], [[269edo|269]]
Badness: 0.0215

==11-limit==
[[Comma|Commas]]: 225/224, 385/384, 4375/4374

[[POTE tuning|POTE generator]]: 316.719

Map: [<1 0 1 -3 9|, <0 6 5 22 -21|]
EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[269edo|269]], [[341edo|341]]
Badness: 0.0218

==13-limit==
[[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539

[[POTE tuning|POTE generator]]: 316.738

Map: [<1 0 1 -3 9 0|, <0 6 5 22 -21 14|]
EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[197edo|197]], [[269edo|269]], [[466edo|466]]
Badness: 0.0169

=Countercata=
[[Comma|Commas]]: 15625/15552, 5120/5103

[[POTE tuning|POTE generator]]: 317.121

Map: [<1 0 1 11|, <0 6 5 -31|]
Wedgie: <<6 5 -31 -6 -66 -86||
EDOs: 15, 19, 34, 53, 87, 140, 333, 473, 806
Badness: 0.0521

==11-limit==
Commas: 385/384, 2200/2187, 3388/3375

POTE generator: ~6/5 = 317.162

Map: [<1 0 1 11 -5|, <0 6 5 -31 32|]
EDOs: 34, 53, 87, 140, 227
Badness: 0.0398

==13-limit==
Commas: 325/324, 352/351, 385/384, 625/624

POTE generator: ~6/5 = 317.162

Map: [<1 0 1 11 -5 0|, <0 6 5 -31 32 14|]
EDOs: 34, 53, 87, 140, 367
Badness: 0.0202

=Metakleismic=
Commas: 15625/15552, 179200/177147

POTE generator: ~6/5 = 317.314

Map: [<1 0 1 -12|, <0 6 5 56|]
Wedgie: <<6 5 56 -6 72 116||
EDOs: 15, 19, 34, 87, 121, 208
Badness: 0.1635

==11-limit==
Commas: 896/891, 2200/2187, 14700/14641

POTE generator: ~6/5 = 317.311

Map: [<1 0 1 -12 -5|, <0 6 5 56 32|]
EDOs: 15, 19, 34, 87, 121, 208
Badness: 0.0486

=Hemikleismic=
Commas: 4000/3969, 6144/6125

[[POTE tuning|POTE generator]]: 158.649

Map: [<1 0 1 4|, <0 12 10 -9|]
EDOs: 53, 121

=Clyde=
[[Comma|Commas]]: 245/243, 3136/3125

7 and 9 limit minimax
[|1 0 0 0>, |6/25 0 0 12/25>, |6/5 0 0 2/5>, |0 0 0 1>]
[[Eigenmonzo|Eigenmonzos]]: 2, 7

[[POTE tuning|POTE generator]]: 441.335

Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[Cent|cents]]. Associated recurrence relationship quickly converges.

Map: [<1 6 6 12|, <0 -12 -10 -25|]
[[Generator|Generators]]: 2, 9/7
[[Edo|Edos]]: [[19edo|19]], [[49edo|49]], [[68edo|68]], [[87edo|87]], [[155edo|155]]

=Tritikleismic=
Commas: 15625/15552, 1029/1024

[[POTE tuning|POTE generator]]: 316.872

Map: [<3 0 3 10|, <0 6 5 -2|]
Wedgie: <<18 15 -6 -18 -60 -56||
EDOs: 12, 15, 57, 72, 159, 231
Badness: 0.0563

==11-limit==
Commas: 385/384, 441/440, 4000/3993

[[POTE tuning|POTE generator]]: 316.881

Map: [<3 0 3 10 8|, <0 6 5 -2 3|]
EDOs: 12, 15, 57, 72, 159, 231
Badness: 0.0193

==13-limit==
Commas: 325/324, 364/363, 441/440, 625/624

[[POTE tuning|POTE generator]]: 316.959

Map: [<3 0 3 10 8 0|, <0 6 5 -2 3 14|]
EDOs: 12, 15, 72, 87, 159, 867, 1026
Badness: 0.0157

=Quadritikleismic=
Commas: 15625/15552, 2401/2400

[[POTE tuning|POTE generator]]: 316.9999

Map: [<4 0 4 7|, <0 6 5 4|]
Wedgie: <<24 20 16 -24 -42 -19||
EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[1200edo|1200]]
Badness: 0.0392

==11-limit==
Commas: 385/384, 1375/1372, 6250/6237

[[POTE tuning|POTE generator]]: 316.925

Map: [<4 0 4 7 17|, <0 6 5 4 -3|]
EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[284edo|284]], [[496edo|496]], [[780edo|780]]
Badness: 0.0234

==13-limit==
Commas: 325/324, 385/384, 625/624, 1573/1568

[[POTE tuning|POTE generator]]: 316.989

Map: [<4 0 4 7 17 0|, <0 6 5 4 -3 14|]
EDOs: 68, 72, 140, 212
Badness: 0.0187

Original HTML content:

<html><head><title>Kleismic family</title></head><body><!-- ws:start:WikiTextTocRule:40:&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:40 --><!-- ws:start:WikiTextTocRule:41: --><a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --> | <a href="#Keemun">Keemun</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#Catakleismic">Catakleismic</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#Countercata">Countercata</a><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --> | <a href="#Metakleismic">Metakleismic</a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --> | <a href="#Hemikleismic">Hemikleismic</a><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --> | <a href="#Clyde">Clyde</a><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --> | <a href="#Tritikleismic">Tritikleismic</a><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --> | <a href="#Quadritikleismic">Quadritikleismic</a><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: -->
<!-- ws:end:WikiTextTocRule:61 --><br />
The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is |-6 -5 6&gt;, and flipping that yields &lt;&lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. 14/53 is about perfect as a generator, though 9/34 also makes sense and using <a class="wiki_link" href="/19edo">19edo</a> is possible. Other tunings include <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/87edo">87edo</a> and <a class="wiki_link" href="/140edo">140edo</a>.<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.007<br />
<br />
Map: [&lt;1 0 1|, &lt;0 6 5|]<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h1>
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Keemun"></a><!-- ws:end:WikiTextHeadingRule:2 -->Keemun</h1>
<a class="wiki_link" href="/Comma">Commas</a>: 49/48, 126/125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~6/5 = 316.473<br />
<br />
Map: [&lt;1 0 1 2|, &lt;0 6 5 3|]<br />
Wedgie: &lt;&lt;6 5 3 -6 -12 -7||<br />
EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/91edo">91</a><br />
Badness: 0.0274<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Keemun-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2>
Commas: 49/48, 56/55, 100/99<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~6/5 = 317.576<br />
<br />
Map: [&lt;1 0 1 2 4|, &lt;0 6 5 3 -2|]<br />
EDOs: 4, 15, 19, 34<br />
Badness: 0.0274<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Keemun-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2>
Commas: 49/48, 56/55, 78/77, 100/99<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~6/5 = 316.611<br />
<br />
Map: [&lt;1 0 1 2 4 5|, &lt;0 6 5 3 -2 -5|]<br />
EDOs: 4, 15, 19, 72<br />
Badness: 0.0297<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Catakleismic"></a><!-- ws:end:WikiTextHeadingRule:8 -->Catakleismic</h1>
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 4375/4374<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.732<br />
<br />
Map: [&lt;1 0 1 -3|, &lt;0 6 5 22|]<br />
Wedgie: &lt;&lt;6 5 22 -6 18 37||<br />
EDOs: <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/53edo">15</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/197edo">197</a>, <a class="wiki_link" href="/269edo">269</a><br />
Badness: 0.0215<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Catakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2>
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 385/384, 4375/4374<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.719<br />
<br />
Map: [&lt;1 0 1 -3 9|, &lt;0 6 5 22 -21|]<br />
EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/269edo">269</a>, <a class="wiki_link" href="/341edo">341</a><br />
Badness: 0.0218<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Catakleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit</h2>
<a class="wiki_link" href="/Comma">Commas</a>: 169/168, 225/224, 325/324, 540/539<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.738<br />
<br />
Map: [&lt;1 0 1 -3 9 0|, &lt;0 6 5 22 -21 14|]<br />
EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/197edo">197</a>, <a class="wiki_link" href="/269edo">269</a>, <a class="wiki_link" href="/466edo">466</a><br />
Badness: 0.0169<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Countercata"></a><!-- ws:end:WikiTextHeadingRule:14 -->Countercata</h1>
<a class="wiki_link" href="/Comma">Commas</a>: 15625/15552, 5120/5103<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.121<br />
<br />
Map: [&lt;1 0 1 11|, &lt;0 6 5 -31|]<br />
Wedgie: &lt;&lt;6 5 -31 -6 -66 -86||<br />
EDOs: 15, 19, 34, 53, 87, 140, 333, 473, 806<br />
Badness: 0.0521<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Countercata-11-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->11-limit</h2>
Commas: 385/384, 2200/2187, 3388/3375<br />
<br />
POTE generator: ~6/5 = 317.162<br />
<br />
Map: [&lt;1 0 1 11 -5|, &lt;0 6 5 -31 32|]<br />
EDOs: 34, 53, 87, 140, 227<br />
Badness: 0.0398<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Countercata-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->13-limit</h2>
Commas: 325/324, 352/351, 385/384, 625/624<br />
<br />
POTE generator: ~6/5 = 317.162<br />
<br />
Map: [&lt;1 0 1 11 -5 0|, &lt;0 6 5 -31 32 14|]<br />
EDOs: 34, 53, 87, 140, 367<br />
Badness: 0.0202<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Metakleismic"></a><!-- ws:end:WikiTextHeadingRule:20 -->Metakleismic</h1>
Commas: 15625/15552, 179200/177147<br />
<br />
POTE generator: ~6/5 = 317.314<br />
<br />
Map: [&lt;1 0 1 -12|, &lt;0 6 5 56|]<br />
Wedgie: &lt;&lt;6 5 56 -6 72 116||<br />
EDOs: 15, 19, 34, 87, 121, 208<br />
Badness: 0.1635<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Metakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2>
Commas: 896/891, 2200/2187, 14700/14641<br />
<br />
POTE generator: ~6/5 = 317.311<br />
<br />
Map: [&lt;1 0 1 -12 -5|, &lt;0 6 5 56 32|]<br />
EDOs: 15, 19, 34, 87, 121, 208<br />
Badness: 0.0486<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Hemikleismic"></a><!-- ws:end:WikiTextHeadingRule:24 -->Hemikleismic</h1>
Commas: 4000/3969, 6144/6125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 158.649<br />
<br />
Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]<br />
EDOs: 53, 121<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Clyde"></a><!-- ws:end:WikiTextHeadingRule:26 -->Clyde</h1>
<a class="wiki_link" href="/Comma">Commas</a>: 245/243, 3136/3125<br />
<br />
7 and 9 limit minimax<br />
[|1 0 0 0&gt;, |6/25 0 0 12/25&gt;, |6/5 0 0 2/5&gt;, |0 0 0 1&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 441.335<br />
<br />
Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 <a class="wiki_link" href="/Cent">cents</a>. Associated recurrence relationship quickly converges.<br />
<br />
Map: [&lt;1 6 6 12|, &lt;0 -12 -10 -25|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 9/7<br />
<a class="wiki_link" href="/Edo">Edos</a>: <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/49edo">49</a>, <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/155edo">155</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Tritikleismic"></a><!-- ws:end:WikiTextHeadingRule:28 -->Tritikleismic</h1>
Commas: 15625/15552, 1029/1024<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.872<br />
<br />
Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]<br />
Wedgie: &lt;&lt;18 15 -6 -18 -60 -56||<br />
EDOs: 12, 15, 57, 72, 159, 231<br />
Badness: 0.0563<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Tritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2>
Commas: 385/384, 441/440, 4000/3993<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.881<br />
<br />
Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]<br />
EDOs: 12, 15, 57, 72, 159, 231<br />
Badness: 0.0193<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Tritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2>
Commas: 325/324, 364/363, 441/440, 625/624<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.959<br />
<br />
Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]<br />
EDOs: 12, 15, 72, 87, 159, 867, 1026<br />
Badness: 0.0157<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Quadritikleismic"></a><!-- ws:end:WikiTextHeadingRule:34 -->Quadritikleismic</h1>
Commas: 15625/15552, 2401/2400<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.9999<br />
<br />
Map: [&lt;4 0 4 7|, &lt;0 6 5 4|]<br />
Wedgie: &lt;&lt;24 20 16 -24 -42 -19||<br />
EDOs: <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/212edo">212</a>, <a class="wiki_link" href="/1200edo">1200</a><br />
Badness: 0.0392<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Quadritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2>
Commas: 385/384, 1375/1372, 6250/6237<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.925<br />
<br />
Map: [&lt;4 0 4 7 17|, &lt;0 6 5 4 -3|]<br />
EDOs: <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/212edo">212</a>, <a class="wiki_link" href="/284edo">284</a>, <a class="wiki_link" href="/496edo">496</a>, <a class="wiki_link" href="/780edo">780</a><br />
Badness: 0.0234<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Quadritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->13-limit</h2>
Commas: 325/324, 385/384, 625/624, 1573/1568<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.989<br />
<br />
Map: [&lt;4 0 4 7 17 0|, &lt;0 6 5 4 -3 14|]<br />
EDOs: 68, 72, 140, 212<br />
Badness: 0.0187</body></html>