Kleismic family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 262692940 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 268961418 - 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>2011-10- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-26 19:01:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>268961418</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: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[197edo|197]], [[269edo|269]], [[466edo|466]] | EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[197edo|197]], [[269edo|269]], [[466edo|466]] | ||
[[Badness]]: 0.0169 | [[Badness]]: 0.0169 | ||
=Cataclysmic= | |||
Commas: 99/98, 176/175, 2200/2187 | |||
POTE generator: ~6/5 = 317.042 | |||
Map: [<1 0 1 -3 -5|, <0 6 5 22 32|] | |||
EDOs: 53, 87d, 140d, 193de | |||
Badness: 0.0400 | |||
==13-limit== | |||
Commas: 99/98, 169/168, 176/175, 275/273 | |||
POTE generator: ~6/5 = 317.036 | |||
Map: [<1 0 1 -3 -5 0|, <0 6 5 22 32 14|] | |||
EDOs: 53, 87d, 140d, 193de | |||
Badness: 0.0226 | |||
=Countercata= | =Countercata= | ||
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</pre></div> | </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>Kleismic 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>Kleismic family</title></head><body><!-- ws:start:WikiTextTocRule:58:&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:58 --><!-- ws:start:WikiTextTocRule:59: --><a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --> | <a href="#Keemun">Keemun</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --> | <a href="#Catakleismic">Catakleismic</a><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --> | <a href="#Cataclysmic">Cataclysmic</a><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --> | <a href="#Countercata">Countercata</a><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --> | <a href="#Metakleismic">Metakleismic</a><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --> | <a href="#Hemikleismic">Hemikleismic</a><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --> | <a href="#Clyde">Clyde</a><!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --> | <a href="#Tritikleismic">Tritikleismic</a><!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --> | <a href="#Quadritikleismic">Quadritikleismic</a><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --> | <a href="#Sqrtphi">Sqrtphi</a><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:88 --><br /> | ||
<br /> | <br /> | ||
The <a class="wiki_link" href="/5-limit">5-limit</a> parent comma for the <strong>kleismic family</strong> 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. This 5-limit temperament is commonly called <strong>Hanson</strong>, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense and 5\19 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 /> | The <a class="wiki_link" href="/5-limit">5-limit</a> parent comma for the <strong>kleismic family</strong> 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. This 5-limit temperament is commonly called <strong>Hanson</strong>, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense and 5\19 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 /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0169<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0169<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Countercata"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Cataclysmic"></a><!-- ws:end:WikiTextHeadingRule:14 -->Cataclysmic</h1> | ||
Commas: 99/98, 176/175, 2200/2187<br /> | |||
<br /> | |||
POTE generator: ~6/5 = 317.042<br /> | |||
<br /> | |||
Map: [&lt;1 0 1 -3 -5|, &lt;0 6 5 22 32|]<br /> | |||
EDOs: 53, 87d, 140d, 193de<br /> | |||
Badness: 0.0400<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Cataclysmic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit</h2> | |||
Commas: 99/98, 169/168, 176/175, 275/273<br /> | |||
<br /> | |||
POTE generator: ~6/5 = 317.036<br /> | |||
<br /> | |||
Map: [&lt;1 0 1 -3 -5 0|, &lt;0 6 5 22 32 14|]<br /> | |||
EDOs: 53, 87d, 140d, 193de<br /> | |||
Badness: 0.0226<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Countercata"></a><!-- ws:end:WikiTextHeadingRule:18 -->Countercata</h1> | |||
<a class="wiki_link" href="/Comma">Commas</a>: 15625/15552, 5120/5103<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 15625/15552, 5120/5103<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0521<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0521<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Countercata-11-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 385/384, 2200/2187, 3388/3375<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 385/384, 2200/2187, 3388/3375<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0398<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0398<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Countercata-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 352/351, 385/384, 625/624<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 352/351, 385/384, 625/624<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0202<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0202<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Metakleismic"></a><!-- ws:end:WikiTextHeadingRule:24 -->Metakleismic</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 179200/177147<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 179200/177147<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.1635<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.1635<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Metakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 896/891, 2200/2187, 14700/14641<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 896/891, 2200/2187, 14700/14641<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0486<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0486<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Metakleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 352/351, 364/363, 625/624<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 352/351, 364/363, 625/624<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0244<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0244<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Hemikleismic"></a><!-- ws:end:WikiTextHeadingRule:30 -->Hemikleismic</h1> | ||
Commas: 4000/3969, 6144/6125<br /> | Commas: 4000/3969, 6144/6125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0521<br /> | Badness: 0.0521<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Hemikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> | ||
Commas: 121/120, 176/175, 4000/3969<br /> | Commas: 121/120, 176/175, 4000/3969<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0380<br /> | Badness: 0.0380<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Clyde"></a><!-- ws:end:WikiTextHeadingRule:34 -->Clyde</h1> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 245/243, 3136/3125<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 245/243, 3136/3125<br /> | ||
<br /> | <br /> | ||
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<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 /> | <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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="Tritikleismic"></a><!-- ws:end:WikiTextHeadingRule:36 -->Tritikleismic</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 1029/1024<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 1029/1024<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0563<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0563<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Tritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 385/384, 441/440, 4000/3993<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 385/384, 441/440, 4000/3993<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0193<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0193<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Tritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->13-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 364/363, 441/440, 625/624<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 364/363, 441/440, 625/624<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0157<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0157<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Quadritikleismic"></a><!-- ws:end:WikiTextHeadingRule:42 -->Quadritikleismic</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 2401/2400<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 2401/2400<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0392<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0392<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Quadritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2> | ||
Commas: 385/384, 1375/1372, 6250/6237<br /> | Commas: 385/384, 1375/1372, 6250/6237<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0234<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0234<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Quadritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2> | ||
Commas: 325/324, 385/384, 625/624, 1573/1568<br /> | Commas: 325/324, 385/384, 625/624, 1573/1568<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/Badness">Badness</a>: 0.0187<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0187<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Sqrtphi"></a><!-- ws:end:WikiTextHeadingRule:48 -->Sqrtphi</h1> | ||
Commas: 15625/15552, 16875/16807<br /> | Commas: 15625/15552, 16875/16807<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0704<br /> | Badness: 0.0704<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc25"><a name="Sqrtphi-11-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->11-limit</h2> | ||
Commas: 540/539, 1375/1372, 4375/4356<br /> | Commas: 540/539, 1375/1372, 4375/4356<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0255<br /> | Badness: 0.0255<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Sqrtphi-13-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->13-limit</h2> | ||
Commas: 325/324, 364/363, 625/624, 1375/1372<br /> | Commas: 325/324, 364/363, 625/624, 1375/1372<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0200<br /> | Badness: 0.0200<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc27"><a name="Sqrtphi-17-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->17-limit</h2> | ||
Commas: 325/324, 364/363, 375/374, 540/539, 595/594<br /> | Commas: 325/324, 364/363, 375/374, 540/539, 595/594<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0130<br /> | Badness: 0.0130<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Sqrtphi-19-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->19-limit</h2> | ||
Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594<br /> | Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594<br /> | ||
<br /> | <br /> | ||