Schismatic family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 318557610 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 330108644 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-04 13:44:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>330108644</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf | EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf | ||
Badness: 0.0207 | Badness: 0.0207 | ||
==Helenus== | |||
Commas: 99/98 176/175 3125/3087 | |||
POTE generator: ~3/2 = 701.725 | |||
Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|] | |||
EDOs: 12, 53, 118d, 171de | |||
Badness: 0.0356 | |||
===13-limit Helenus=== | |||
Commas: 99/98, 176/175, 275/273, 847/845 | |||
POTE generator: ~3/2 = 701.747 | |||
Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|] | |||
EDOs: 53, 118d, 171de | |||
Badness: 0.0263 | |||
=Guiron= | =Guiron= | ||
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Badness: 0.0315</pre></div> | Badness: 0.0315</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Schismatic family</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Schismatic family</title></head><body><!-- ws:start:WikiTextTocRule:66:&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:66 --><!-- ws:start:WikiTextTocRule:67: --><a href="#Five limit">Five limit</a><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --> | <a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --> | <a href="#Garibaldi">Garibaldi</a><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --> | <a href="#Guiron">Guiron</a><!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --> | <a href="#Pogo">Pogo</a><!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --> | <a href="#Schism">Schism</a><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --> | <a href="#Pontiac">Pontiac</a><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --> | <a href="#Grackle">Grackle</a><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --> | <a href="#Bischismic">Bischismic</a><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --> | <a href="#Kleischismic">Kleischismic</a><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --> | <a href="#Term">Term</a><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --> | <a href="#Sesquiquartififths">Sesquiquartififths</a><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --> | <a href="#Sextilififths">Sextilififths</a><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --> | <a href="#Tsaharuk">Tsaharuk</a><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --> | <a href="#Quanharuk">Quanharuk</a><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:100 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Five limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Five limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit</h1> | ||
The 5-limit parent comma for the schismatic family is the <a class="wiki_link" href="/schisma">schisma</a> of 32805/32768, which is the amount by which the Pythagorean comma exceeds the <a class="wiki_link" href="/Didymus%20comma">Didymus comma</a> (<a class="wiki_link" href="/81_80">81/80</a>), or alternatively put, the difference between a just major third and a just diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1&gt;, and flipping that yields &lt;&lt;1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. <br /> | The 5-limit parent comma for the schismatic family is the <a class="wiki_link" href="/schisma">schisma</a> of 32805/32768, which is the amount by which the Pythagorean comma exceeds the <a class="wiki_link" href="/Didymus%20comma">Didymus comma</a> (<a class="wiki_link" href="/81_80">81/80</a>), or alternatively put, the difference between a just major third and a just diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1&gt;, and flipping that yields &lt;&lt;1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. <br /> | ||
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Badness: 0.0207<br /> | Badness: 0.0207<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id=" | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Garibaldi-Helenus"></a><!-- ws:end:WikiTextHeadingRule:14 -->Helenus</h2> | ||
Commas: 99/98 176/175 3125/3087<br /> | |||
<br /> | |||
POTE generator: ~3/2 = 701.725<br /> | |||
<br /> | |||
Map: [&lt;1 0 15 25 51|, &lt;0 1 -8 -14 -30|]<br /> | |||
EDOs: 12, 53, 118d, 171de<br /> | |||
Badness: 0.0356<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Garibaldi-Helenus-13-limit Helenus"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit Helenus</h3> | |||
Commas: 99/98, 176/175, 275/273, 847/845<br /> | |||
<br /> | |||
POTE generator: ~3/2 = 701.747<br /> | |||
<br /> | |||
Map: [&lt;1 0 15 25 51 56|, &lt;0 1 -8 -14 -30 -33|]<br /> | |||
EDOs: 53, 118d, 171de<br /> | |||
Badness: 0.0263<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Guiron"></a><!-- ws:end:WikiTextHeadingRule:18 -->Guiron</h1> | |||
<a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 10976/10935<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 10976/10935<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0475<br /> | Badness: 0.0475<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Guiron-11-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 385/384, 441/440, 10976/10935<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 385/384, 441/440, 10976/10935<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0266<br /> | Badness: 0.0266<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Guiron-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h2> | ||
Commas: 196/195, 352/351, 385/384, 729/728<br /> | Commas: 196/195, 352/351, 385/384, 729/728<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0284<br /> | Badness: 0.0284<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Pogo"></a><!-- ws:end:WikiTextHeadingRule:24 -->Pogo</h1> | ||
Commas: 32805/32768, 118098/117649<br /> | Commas: 32805/32768, 118098/117649<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0796<br /> | Badness: 0.0796<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Pogo-11-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->11-limit</h2> | ||
Commas: 540/539, 4000/3993, 32805/32768<br /> | Commas: 540/539, 4000/3993, 32805/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0319<br /> | Badness: 0.0319<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Schism"></a><!-- ws:end:WikiTextHeadingRule:28 -->Schism</h1> | ||
Commas: 64/63, 360/343<br /> | Commas: 64/63, 360/343<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0566<br /> | Badness: 0.0566<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Schism-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> | ||
Commas: 45/44, 64/63, 99/98<br /> | Commas: 45/44, 64/63, 99/98<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0375<br /> | Badness: 0.0375<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Pontiac"></a><!-- ws:end:WikiTextHeadingRule:32 -->Pontiac</h1> | ||
Commas: 32805/32768, 4375/4374<br /> | Commas: 32805/32768, 4375/4374<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0141<br /> | Badness: 0.0141<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Grackle"></a><!-- ws:end:WikiTextHeadingRule:34 -->Grackle</h1> | ||
Commas: 126/125, 32805/32768<br /> | Commas: 126/125, 32805/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.070<br /> | Badness: 0.070<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="Bischismic"></a><!-- ws:end:WikiTextHeadingRule:36 -->Bischismic</h1> | ||
Commas: 3136/3125, 32805/32768<br /> | Commas: 3136/3125, 32805/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0547<br /> | Badness: 0.0547<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Bischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->11-limit</h2> | ||
Commas: 441/440, 3136/3125, 8019/8000<br /> | Commas: 441/440, 3136/3125, 8019/8000<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0282<br /> | Badness: 0.0282<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Bischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->13-limit</h2> | ||
Commas: 441/440, 729/728, 1001/1000, 3136/3125<br /> | Commas: 441/440, 729/728, 1001/1000, 3136/3125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0287<br /> | Badness: 0.0287<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Kleischismic"></a><!-- ws:end:WikiTextHeadingRule:42 -->Kleischismic</h1> | ||
Commas: 32805/32768, 1500625/1492992<br /> | Commas: 32805/32768, 1500625/1492992<br /> | ||
<br /> | <br /> | ||
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Badness: 0.1106<br /> | Badness: 0.1106<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Kleischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2> | ||
Commas: 385/384, 9801/9800, 14641/14580<br /> | Commas: 385/384, 9801/9800, 14641/14580<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0367<br /> | Badness: 0.0367<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Kleischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2> | ||
Commas: 352/351, 385/384, 729/728, 1575/1573<br /> | Commas: 352/351, 385/384, 729/728, 1575/1573<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0376<br /> | Badness: 0.0376<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Term"></a><!-- ws:end:WikiTextHeadingRule:48 -->Term</h1> | ||
Commas: 32805/32768, 250047/250000<br /> | Commas: 32805/32768, 250047/250000<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0200<br /> | Badness: 0.0200<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc25"><a name="Sesquiquartififths"></a><!-- ws:end:WikiTextHeadingRule:50 -->Sesquiquartififths</h1> | ||
Commas: 2401/2400, 32805/32768<br /> | Commas: 2401/2400, 32805/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0112<br /> | Badness: 0.0112<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Sesquiquartififths-Sesquart"></a><!-- ws:end:WikiTextHeadingRule:52 -->Sesquart</h2> | ||
Commas: 243/242, 441/440, 16384/16335<br /> | Commas: 243/242, 441/440, 16384/16335<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0293<br /> | Badness: 0.0293<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc27"><a name="Sextilififths"></a><!-- ws:end:WikiTextHeadingRule:54 -->Sextilififths</h1> | ||
Commas: 32768/32805, 235298/234375<br /> | Commas: 32768/32805, 235298/234375<br /> | ||
<br /> | <br /> | ||
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Badness: 0.1088<br /> | Badness: 0.1088<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Sextilififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->11-limit</h2> | ||
Commas: 441/440, 4000/3993, 235298/234375<br /> | Commas: 441/440, 4000/3993, 235298/234375<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0455<br /> | Badness: 0.0455<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:58:&lt;h2&gt; --><h2 id="toc29"><a name="Sextilififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:58 -->13-limit</h2> | ||
Commas: 363/363, 441/440, 676/675, 10985/10976<br /> | Commas: 363/363, 441/440, 676/675, 10985/10976<br /> | ||
<br /> | <br /> | ||
| Line 669: | Line 705: | ||
Badness: 0.0253<br /> | Badness: 0.0253<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:60:&lt;h1&gt; --><h1 id="toc30"><a name="Tsaharuk"></a><!-- ws:end:WikiTextHeadingRule:60 -->Tsaharuk</h1> | ||
Commas: 32805/32768, 420175/419904<br /> | Commas: 32805/32768, 420175/419904<br /> | ||
<br /> | <br /> | ||
| Line 679: | Line 715: | ||
Badness: 0.0307<br /> | Badness: 0.0307<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h1&gt; --><h1 id="toc31"><a name="Quanharuk"></a><!-- ws:end:WikiTextHeadingRule:62 -->Quanharuk</h1> | ||
Commas: 16875/16807, 32805/32768<br /> | Commas: 16875/16807, 32805/32768<br /> | ||
<br /> | <br /> | ||
| Line 689: | Line 725: | ||
Badness: 0.0720<br /> | Badness: 0.0720<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc32"><a name="Quanharuk-11-limit"></a><!-- ws:end:WikiTextHeadingRule:64 -->11-limit</h2> | ||
Commas: 540/539, 1375/1372, 32805/32768<br /> | Commas: 540/539, 1375/1372, 32805/32768<br /> | ||
<br /> | <br /> | ||