Kleismic family: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 292546906 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 300189938 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-09 13:02:51 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>300189938</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: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]] | EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]] | ||
[[Badness]]: 0.0187 | [[Badness]]: 0.0187 | ||
=Novemkleismic= | |||
Commas: 15625/15552, 40353607/40310784 | |||
POTE generator: ~6/5 = 317.005 | |||
Map: [<9 0 9 11|, <0 6 5 6|] | |||
Wedgie: <<54 45 54 -54 -66 -1|| | |||
EDOs: 72, 261, 333, 405, 477c, 882c | |||
Badness: 0.1934 | |||
==11-limit== | |||
Commas: 1375/1372, 4000/3993, 15625/15552 | |||
POTE generator: ~6/5 = 317.010 | |||
Map: [<9 0 9 11 24|, <0 6 5 6 3|] | |||
EDOs: 72, 261, 333, 405, 882c | |||
Badness: 0.05172 | |||
==13-limit== | |||
Commas: 325/324, 625/624, 1375/1372, 4000/3993 | |||
POTE generator: ~6/5 = 317.086 | |||
Map: [<9 0 9 11 24 0|, <0 6 5 6 3 14|] | |||
EDOs: 72, 261, 333, 738cf, 1071bcf | |||
Badness: 0.0391 | |||
=Sqrtphi= | =Sqrtphi= | ||
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[[http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3|A Fight for Phi]] by Vito Sicurella</pre></div> | [[http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3|A Fight for Phi]] by Vito Sicurella</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:82:&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:82 --><!-- ws:start:WikiTextTocRule:83: --><a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --> | <a href="#Keemun">Keemun</a><!-- 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: --> | <a href="#Catakleismic">Catakleismic</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="#Cataclysmic">Cataclysmic</a><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --> | <a href="#Catalytic">Catalytic</a><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --> | <a href="#Countercata">Countercata</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="#Metakleismic">Metakleismic</a><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --> | <a href="#Hemikleismic">Hemikleismic</a><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --> | <a href="#Clyde">Clyde</a><!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --> | <a href="#Bikleismic">Bikleismic</a><!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --> | <a href="#Tritikleismic">Tritikleismic</a><!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --> | <a href="#Quadritikleismic">Quadritikleismic</a><!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --> | <a href="#Novemkleismic">Novemkleismic</a><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --> | <a href="#Sqrtphi">Sqrtphi</a><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:124 --><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>: 0.0187<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0187<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:62:&lt;h1&gt; --><h1 id="toc31"><a name="Sqrtphi"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h1&gt; --><h1 id="toc31"><a name="Novemkleismic"></a><!-- ws:end:WikiTextHeadingRule:62 -->Novemkleismic</h1> | ||
Commas: 15625/15552, 40353607/40310784<br /> | |||
<br /> | |||
POTE generator: ~6/5 = 317.005<br /> | |||
<br /> | |||
Map: [&lt;9 0 9 11|, &lt;0 6 5 6|]<br /> | |||
Wedgie: &lt;&lt;54 45 54 -54 -66 -1||<br /> | |||
EDOs: 72, 261, 333, 405, 477c, 882c<br /> | |||
Badness: 0.1934<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc32"><a name="Novemkleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:64 -->11-limit</h2> | |||
Commas: 1375/1372, 4000/3993, 15625/15552<br /> | |||
<br /> | |||
POTE generator: ~6/5 = 317.010<br /> | |||
<br /> | |||
Map: [&lt;9 0 9 11 24|, &lt;0 6 5 6 3|]<br /> | |||
EDOs: 72, 261, 333, 405, 882c<br /> | |||
Badness: 0.05172<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:66:&lt;h2&gt; --><h2 id="toc33"><a name="Novemkleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:66 -->13-limit</h2> | |||
Commas: 325/324, 625/624, 1375/1372, 4000/3993<br /> | |||
<br /> | |||
POTE generator: ~6/5 = 317.086<br /> | |||
<br /> | |||
Map: [&lt;9 0 9 11 24 0|, &lt;0 6 5 6 3 14|]<br /> | |||
EDOs: 72, 261, 333, 738cf, 1071bcf<br /> | |||
Badness: 0.0391<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:68:&lt;h1&gt; --><h1 id="toc34"><a name="Sqrtphi"></a><!-- ws:end:WikiTextHeadingRule:68 -->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:70:&lt;h2&gt; --><h2 id="toc35"><a name="Sqrtphi-11-limit"></a><!-- ws:end:WikiTextHeadingRule:70 -->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:72:&lt;h2&gt; --><h2 id="toc36"><a name="Sqrtphi-13-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->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:74:&lt;h2&gt; --><h2 id="toc37"><a name="Sqrtphi-17-limit"></a><!-- ws:end:WikiTextHeadingRule:74 -->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:76:&lt;h2&gt; --><h2 id="toc38"><a name="Sqrtphi-19-limit"></a><!-- ws:end:WikiTextHeadingRule:76 -->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 /> | ||
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Badness: 0.0147<br /> | Badness: 0.0147<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><h2 id="toc39"><a name="Sqrtphi-Scales"></a><!-- ws:end:WikiTextHeadingRule:78 -->Scales</h2> | ||
<a class="wiki_link" href="/sqrtphi17">sqrtphi17</a><br /> | <a class="wiki_link" href="/sqrtphi17">sqrtphi17</a><br /> | ||
<a class="wiki_link" href="/sqrtphi23">sqrtphi23</a><br /> | <a class="wiki_link" href="/sqrtphi23">sqrtphi23</a><br /> | ||
<a class="wiki_link" href="/sqrtphi49">sqrtphi49</a><br /> | <a class="wiki_link" href="/sqrtphi49">sqrtphi49</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:80:&lt;h2&gt; --><h2 id="toc40"><a name="Sqrtphi-Music"></a><!-- ws:end:WikiTextHeadingRule:80 -->Music</h2> | ||
<a class="wiki_link_ext" href="http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3" rel="nofollow">Prelude for Piano in Square root of Phi Tuning</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | <a class="wiki_link_ext" href="http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3" rel="nofollow">Prelude for Piano in Square root of Phi Tuning</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | ||
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3" rel="nofollow">A Fight for Phi</a> by Vito Sicurella</body></html></pre></div> | <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3" rel="nofollow">A Fight for Phi</a> by Vito Sicurella</body></html></pre></div> | ||
Revision as of 13:02, 9 February 2012
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 2012-02-09 13:02:51 UTC.
- The original revision id was 300189938.
- 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. This 5-limit temperament is commonly called **Hanson**, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense and 5\19 is possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]]. [[POTE tuning|POTE generator]]: 317.007 Map: [<1 0 1|, <0 6 5|] EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[458edo|458]], [[882edo|882c]] Music: [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3|Hanson Improv]] by [[Petr Parizek]] [[http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3|In Hanson11]] by [[Chris Vaisvil]] =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]], [[53edo|53d]], [[72edo|72d]], [[91edo|91d]] [[Badness]]: 0.0274 ==11-limit== [[Comma]]s: 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: [[4edo|4]], [[15edo|15]], [[19edo|19]], [[34edo|34]], [[102edo|102de]] [[Badness]]: 0.0274 ==Darjeeling== Commas: 49/48, 55/54, 77/75 POTE generator: ~6/5 = 317.656 Map: [<1 0 1 2 0|, <0 6 5 3 13|] EDOs: 15, 19e, 34e Badness: 0.0276 ==13-limit== [[Comma]]s: 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: [[4edo|4]], [[15edo|15]], [[19edo|19]], [[53def]], [[72edo|72def]] [[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|269c]] [[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: [[19edo|19]], [[53edo|53]], [[72edo|72]], [[197edo|197e]], [[269edo|269ce]], [[341edo|341ce]], [[610edo|610bce]] [[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]], [[125edo|125f]], [[197edo|197ef]], [[269edo|269cef]] [[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, 171de, 181de, 193de, 224de, 246de, 277de 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, 246de Badness: 0.0226 =Catalytic= Commas: 225/224, 441/440, 4375/4374 POTE generator: ~6/5 = 316.653 Map: [<1 0 1 -3 -10|, <0 6 5 22 51|] EDOs: 53e, 72 Badness: 0.0304 =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: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[87edo|87]], [[140edo|140]], [[333edo|333]], [[473edo|473]], [[806edo|806b]] [[Badness]]: 0.0521 ==11-limit== [[Comma]]s: 385/384, 2200/2187, 3388/3375 POTE generator: ~6/5 = 317.162 Map: [<1 0 1 11 -5|, <0 6 5 -31 32|] EDOs: [[34edo|34]], [[53edo|53]], [[87edo|87]], [[140edo|140]], [[227edo|227]] [[Badness]]: 0.0398 ==13-limit== [[Comma]]s: 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: [[34edo|34]], [[53edo|53]], [[87edo|87]], [[140edo|140]], [[227edo]],[[367edo|367e]], [[507edo|507e]] [[Badness]]: 0.0202 =Metakleismic= [[Comma]]s: 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: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[87edo|87]], [[121edo|121]], [[208edo|208]] [[Badness]]: 0.1635 ==11-limit== [[Comma]]s: 896/891, 2200/2187, 14700/14641 POTE generator: ~6/5 = 317.311 Map: [<1 0 1 -12 -5|, <0 6 5 56 32|] EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[87edo|87]], [[121edo|121]], [[208edo|208]] [[Badness]]: 0.0486 ==13-limit== [[Comma]]s: 325/324, 352/351, 364/363, 625/624 POTE generator: ~6/5 = 317.311 Map: [<1 0 1 -12 -5 0|, <0 6 5 56 32 14|] EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[87edo|87]], [[121edo|121]], [[208edo|208]] [[Badness]]: 0.0244 =Hemikleismic= Commas: 4000/3969, 6144/6125 [[POTE tuning|POTE generator]]: 158.649 Map: [<1 0 1 4|, <0 12 10 -9|] EDOs: [[53edo|53]], [[121edo|121]] Badness: 0.0521 ==11-limit== Commas: 121/120, 176/175, 4000/3969 POTE generator: ~11/10 = 158.677 Map: [<1 0 1 4 2|, <0 12 10 -9 11|] EDOs: 15, 38, 53, 68, 121e Badness: 0.0380 ==13-limit== Commas: 121/120, 176/175, 275/273, 325/324 POTE generator: ~11/10 = 158.655 Map: [<1 0 1 4 2 0|, <0 12 10 -9 11 28|] EDOs: 15, 53, 121e Badness: 0.0260 =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]]: ~9/7 = 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]] Badness: 0.0473 ==11-limit== Commas: 245/243, 3136/3125, 385/384 POTE generator: ~9/7 = 441.355 Map: [<1 6 6 12 -5|, <0 -12 -10 -25 23|] EDOs: 19, 68, 87, 329bd, 419bd, 503bd, 590bd Badness: 0.0474 ==13-limit== Commas: 196/195, 245/243, 385/384, 625/624 POTE generator: ~9/7 = 441.363 Map: [<1 6 6 12 -5 14|, <0 -12 -10 -25 23 -28|] EDOs: 19, 68, 87, 503bdf, 590bdf Badness: 0.0268 =Bikleismic= Commas: 225/224, 243/242, 4375/4356 POTE generator: ~6/5 = 316.721 Map: [<2 0 2 -6 -1|, <0 6 5 22 15|] EDOs: 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc Badness: 0.0293 ==13-limit== Commas: 169/168, 225/224, 243/242, 325/324 POTE generator: ~6/5 = 316.726 Map: [<2 0 2 -6 -1 0|, <0 6 5 22 15 14|] EDOs: 72, 106, 322cff, 394cff, 466bcff, 538bcfff Badness: 0.0218 =Tritikleismic= [[Comma]]s: 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: [[12edo|12]], [[15edo|15]], [[57edo|57]], [[72edo|72]], [[159edo|159]], [[231edo|231]] [[Badness]]: 0.0563 ==11-limit== [[Comma]]s: 385/384, 441/440, 4000/3993 [[POTE tuning|POTE generator]]: 316.881 Map: [<3 0 3 10 8|, <0 6 5 -2 3|] EDOs: [[12edo|12]], [[15edo|15]], [[57edo|57]], [[72edo|72]], [[159edo|159]], [[231edo|231]] [[Badness]]: 0.0193 ==13-limit== [[Comma]]s: 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: [[12edo|12]], [[15edo|15]], [[72edo|72]], 87, 159, 867, 1026 [[Badness]]: 0.0157 =Quadritikleismic= [[Comma]]s: 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: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]] [[Badness]]: 0.0187 =Novemkleismic= Commas: 15625/15552, 40353607/40310784 POTE generator: ~6/5 = 317.005 Map: [<9 0 9 11|, <0 6 5 6|] Wedgie: <<54 45 54 -54 -66 -1|| EDOs: 72, 261, 333, 405, 477c, 882c Badness: 0.1934 ==11-limit== Commas: 1375/1372, 4000/3993, 15625/15552 POTE generator: ~6/5 = 317.010 Map: [<9 0 9 11 24|, <0 6 5 6 3|] EDOs: 72, 261, 333, 405, 882c Badness: 0.05172 ==13-limit== Commas: 325/324, 625/624, 1375/1372, 4000/3993 POTE generator: ~6/5 = 317.086 Map: [<9 0 9 11 24 0|, <0 6 5 6 3 14|] EDOs: 72, 261, 333, 738cf, 1071bcf Badness: 0.0391 =Sqrtphi= Commas: 15625/15552, 16875/16807 [[POTE tuning|POTE generator]]: ~125/98 = 416.603 cents Sqrt(phi) = 416.545 cents Map: [<1 12 11 16|, <0 -30 -25 -38|] EDOs: [[49edo|49]], 72, [[193edo|193]], [[265edo|265]] Badness: 0.0704 ==11-limit== Commas: 540/539, 1375/1372, 4375/4356 POTE generator: ~14/11 = 416.604 Map: [<1 12 11 16 17|, <0 -30 -25 -38 -39|] EDOs: 49, 72, 193, 265 Badness: 0.0255 ==13-limit== Commas: 325/324, 364/363, 625/624, 1375/1372 POTE generator: ~14/11 = 416.585 Map: [<1 12 11 16 17 28|, <0 -30 -25 -38 -39 -70|] EDOs: 72, 121, 193 Badness: 0.0200 ==17-limit== Commas: 325/324, 364/363, 375/374, 540/539, 595/594 POTE generator: ~14/11 = 416.585 Map: [<1 12 11 16 17 28 27|, <0 -30 -25 -38 -39 -70 -66|] EDOs: 72, 121, 193 Badness: 0.0130 ==19-limit== Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594 POTE generator: ~14/11 = 416.580 Map: [<1 12 11 16 17 28 27 -2|, <0 -30 -25 -38 -39 -70 -66 18|] EDOs: 72, 121, 193 Badness: 0.0147 ==Scales== [[sqrtphi17]] [[sqrtphi23]] [[sqrtphi49]] ==Music== [[http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3|Prelude for Piano in Square root of Phi Tuning]] by [[Chris Vaisvil]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3|A Fight for Phi]] by Vito Sicurella
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<html><head><title>Kleismic family</title></head><body><!-- ws:start:WikiTextTocRule:82:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --> | <a href="#Keemun">Keemun</a><!-- 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: --> | <a href="#Catakleismic">Catakleismic</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="#Cataclysmic">Cataclysmic</a><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --> | <a href="#Catalytic">Catalytic</a><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --> | <a href="#Countercata">Countercata</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="#Metakleismic">Metakleismic</a><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --> | <a href="#Hemikleismic">Hemikleismic</a><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --> | <a href="#Clyde">Clyde</a><!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --> | <a href="#Bikleismic">Bikleismic</a><!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --> | <a href="#Tritikleismic">Tritikleismic</a><!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --> | <a href="#Quadritikleismic">Quadritikleismic</a><!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --> | <a href="#Novemkleismic">Novemkleismic</a><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --> | <a href="#Sqrtphi">Sqrtphi</a><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --> <!-- ws:end:WikiTextTocRule:124 --><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>, 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. 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 /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.007<br /> <br /> Map: [<1 0 1|, <0 6 5|]<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/458edo">458</a>, <a class="wiki_link" href="/882edo">882c</a><br /> <br /> Music:<br /> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3" rel="nofollow">Hanson Improv</a> by <a class="wiki_link" href="/Petr%20Parizek">Petr Parizek</a><br /> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3" rel="nofollow">In Hanson11</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><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 <a class="wiki_link" href="/7-limit">7-limit</a> 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:<h1> --><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: [<1 0 1 2|, <0 6 5 3|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<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="/53edo">53d</a>, <a class="wiki_link" href="/72edo">72d</a>, <a class="wiki_link" href="/91edo">91d</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Keemun-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 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: [<1 0 1 2 4|, <0 6 5 3 -2|]<br /> EDOs: <a class="wiki_link" href="/4edo">4</a>, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/102edo">102de</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Keemun-Darjeeling"></a><!-- ws:end:WikiTextHeadingRule:6 -->Darjeeling</h2> Commas: 49/48, 55/54, 77/75<br /> <br /> POTE generator: ~6/5 = 317.656<br /> <br /> Map: [<1 0 1 2 0|, <0 6 5 3 13|]<br /> EDOs: 15, 19e, 34e<br /> Badness: 0.0276<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Keemun-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 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: [<1 0 1 2 4 5|, <0 6 5 3 -2 -5|]<br /> EDOs: <a class="wiki_link" href="/4edo">4</a>, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/53def">53def</a>, <a class="wiki_link" href="/72edo">72def</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0297<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Catakleismic"></a><!-- ws:end:WikiTextHeadingRule:10 -->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: [<1 0 1 -3|, <0 6 5 22|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<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">269c</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0215<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Catakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->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: [<1 0 1 -3 9|, <0 6 5 22 -21|]<br /> EDOs: <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">197e</a>, <a class="wiki_link" href="/269edo">269ce</a>, <a class="wiki_link" href="/341edo">341ce</a>, <a class="wiki_link" href="/610edo">610bce</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0218<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Catakleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->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: [<1 0 1 -3 9 0|, <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="/125edo">125f</a>, <a class="wiki_link" href="/197edo">197ef</a>, <a class="wiki_link" href="/269edo">269cef</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0169<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Cataclysmic"></a><!-- ws:end:WikiTextHeadingRule:16 -->Cataclysmic</h1> Commas: 99/98, 176/175, 2200/2187<br /> <br /> POTE generator: ~6/5 = 317.042<br /> <br /> Map: [<1 0 1 -3 -5|, <0 6 5 22 32|]<br /> EDOs: 53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de<br /> Badness: 0.0400<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Cataclysmic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->13-limit</h2> Commas: 99/98, 169/168, 176/175, 275/273<br /> <br /> POTE generator: ~6/5 = 317.036<br /> <br /> Map: [<1 0 1 -3 -5 0|, <0 6 5 22 32 14|]<br /> EDOs: 53, 87d, 140d, 193de, 246de<br /> Badness: 0.0226<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Catalytic"></a><!-- ws:end:WikiTextHeadingRule:20 -->Catalytic</h1> Commas: 225/224, 441/440, 4375/4374<br /> <br /> POTE generator: ~6/5 = 316.653<br /> <br /> Map: [<1 0 1 -3 -10|, <0 6 5 22 51|]<br /> EDOs: 53e, 72<br /> Badness: 0.0304<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Countercata"></a><!-- ws:end:WikiTextHeadingRule:22 -->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: [<1 0 1 11|, <0 6 5 -31|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<6 5 -31 -6 -66 -86||<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/333edo">333</a>, <a class="wiki_link" href="/473edo">473</a>, <a class="wiki_link" href="/806edo">806b</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0521<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="Countercata-11-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 385/384, 2200/2187, 3388/3375<br /> <br /> POTE generator: ~6/5 = 317.162<br /> <br /> Map: [<1 0 1 11 -5|, <0 6 5 -31 32|]<br /> EDOs: <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/227edo">227</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0398<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="Countercata-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 352/351, 385/384, 625/624<br /> <br /> POTE generator: ~6/5 = 317.162<br /> <br /> Map: [<1 0 1 11 -5 0|, <0 6 5 -31 32 14|]<br /> EDOs: <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/227edo">227edo</a>,<a class="wiki_link" href="/367edo">367e</a>, <a class="wiki_link" href="/507edo">507e</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0202<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h1> --><h1 id="toc14"><a name="Metakleismic"></a><!-- ws:end:WikiTextHeadingRule:28 -->Metakleismic</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 179200/177147<br /> <br /> POTE generator: ~6/5 = 317.314<br /> <br /> Map: [<1 0 1 -12|, <0 6 5 56|]<br /> Wedgie: <<6 5 56 -6 72 116||<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/121edo">121</a>, <a class="wiki_link" href="/208edo">208</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.1635<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="Metakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 896/891, 2200/2187, 14700/14641<br /> <br /> POTE generator: ~6/5 = 317.311<br /> <br /> Map: [<1 0 1 -12 -5|, <0 6 5 56 32|]<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/121edo">121</a>, <a class="wiki_link" href="/208edo">208</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0486<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Metakleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 325/324, 352/351, 364/363, 625/624<br /> <br /> POTE generator: ~6/5 = 317.311<br /> <br /> Map: [<1 0 1 -12 -5 0|, <0 6 5 56 32 14|]<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/121edo">121</a>, <a class="wiki_link" href="/208edo">208</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0244<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h1> --><h1 id="toc17"><a name="Hemikleismic"></a><!-- ws:end:WikiTextHeadingRule:34 -->Hemikleismic</h1> Commas: 4000/3969, 6144/6125<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 158.649<br /> <br /> Map: [<1 0 1 4|, <0 12 10 -9|]<br /> EDOs: <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/121edo">121</a><br /> Badness: 0.0521<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Hemikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> Commas: 121/120, 176/175, 4000/3969<br /> <br /> POTE generator: ~11/10 = 158.677<br /> <br /> Map: [<1 0 1 4 2|, <0 12 10 -9 11|]<br /> EDOs: 15, 38, 53, 68, 121e<br /> Badness: 0.0380<br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h2> --><h2 id="toc19"><a name="Hemikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->13-limit</h2> Commas: 121/120, 176/175, 275/273, 325/324<br /> <br /> POTE generator: ~11/10 = 158.655<br /> <br /> Map: [<1 0 1 4 2 0|, <0 12 10 -9 11 28|]<br /> EDOs: 15, 53, 121e<br /> Badness: 0.0260<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h1> --><h1 id="toc20"><a name="Clyde"></a><!-- ws:end:WikiTextHeadingRule:40 -->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>, |6/25 0 0 12/25>, |6/5 0 0 2/5>, |0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~9/7 = 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: [<1 6 6 12|, <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 /> Badness: 0.0473<br /> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h2> --><h2 id="toc21"><a name="Clyde-11-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->11-limit</h2> Commas: 245/243, 3136/3125, 385/384<br /> <br /> POTE generator: ~9/7 = 441.355<br /> <br /> Map: [<1 6 6 12 -5|, <0 -12 -10 -25 23|]<br /> EDOs: 19, 68, 87, 329bd, 419bd, 503bd, 590bd<br /> Badness: 0.0474<br /> <br /> <!-- ws:start:WikiTextHeadingRule:44:<h2> --><h2 id="toc22"><a name="Clyde-13-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->13-limit</h2> Commas: 196/195, 245/243, 385/384, 625/624<br /> <br /> POTE generator: ~9/7 = 441.363<br /> <br /> Map: [<1 6 6 12 -5 14|, <0 -12 -10 -25 23 -28|]<br /> EDOs: 19, 68, 87, 503bdf, 590bdf<br /> Badness: 0.0268<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h1> --><h1 id="toc23"><a name="Bikleismic"></a><!-- ws:end:WikiTextHeadingRule:46 -->Bikleismic</h1> Commas: 225/224, 243/242, 4375/4356<br /> <br /> POTE generator: ~6/5 = 316.721<br /> <br /> Map: [<2 0 2 -6 -1|, <0 6 5 22 15|]<br /> EDOs: 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc<br /> Badness: 0.0293<br /> <br /> <!-- ws:start:WikiTextHeadingRule:48:<h2> --><h2 id="toc24"><a name="Bikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->13-limit</h2> Commas: 169/168, 225/224, 243/242, 325/324<br /> <br /> POTE generator: ~6/5 = 316.726<br /> <br /> Map: [<2 0 2 -6 -1 0|, <0 6 5 22 15 14|]<br /> EDOs: 72, 106, 322cff, 394cff, 466bcff, 538bcfff<br /> Badness: 0.0218<br /> <br /> <!-- ws:start:WikiTextHeadingRule:50:<h1> --><h1 id="toc25"><a name="Tritikleismic"></a><!-- ws:end:WikiTextHeadingRule:50 -->Tritikleismic</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 1029/1024<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.872<br /> <br /> Map: [<3 0 3 10|, <0 6 5 -2|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<18 15 -6 -18 -60 -56||<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/57edo">57</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/231edo">231</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0563<br /> <br /> <!-- ws:start:WikiTextHeadingRule:52:<h2> --><h2 id="toc26"><a name="Tritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 385/384, 441/440, 4000/3993<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.881<br /> <br /> Map: [<3 0 3 10 8|, <0 6 5 -2 3|]<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/57edo">57</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/231edo">231</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0193<br /> <br /> <!-- ws:start:WikiTextHeadingRule:54:<h2> --><h2 id="toc27"><a name="Tritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->13-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 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: [<3 0 3 10 8 0|, <0 6 5 -2 3 14|]<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/72edo">72</a>, 87, 159, 867, 1026<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0157<br /> <br /> <!-- ws:start:WikiTextHeadingRule:56:<h1> --><h1 id="toc28"><a name="Quadritikleismic"></a><!-- ws:end:WikiTextHeadingRule:56 -->Quadritikleismic</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 15625/15552, 2401/2400<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.9999<br /> <br /> Map: [<4 0 4 7|, <0 6 5 4|]<br /> Wedgie: <<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 /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0392<br /> <br /> <!-- ws:start:WikiTextHeadingRule:58:<h2> --><h2 id="toc29"><a name="Quadritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:58 -->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: [<4 0 4 7 17|, <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 /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0234<br /> <br /> <!-- ws:start:WikiTextHeadingRule:60:<h2> --><h2 id="toc30"><a name="Quadritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->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: [<4 0 4 7 17 0|, <0 6 5 4 -3 14|]<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><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0187<br /> <br /> <!-- ws:start:WikiTextHeadingRule:62:<h1> --><h1 id="toc31"><a name="Novemkleismic"></a><!-- ws:end:WikiTextHeadingRule:62 -->Novemkleismic</h1> Commas: 15625/15552, 40353607/40310784<br /> <br /> POTE generator: ~6/5 = 317.005<br /> <br /> Map: [<9 0 9 11|, <0 6 5 6|]<br /> Wedgie: <<54 45 54 -54 -66 -1||<br /> EDOs: 72, 261, 333, 405, 477c, 882c<br /> Badness: 0.1934<br /> <br /> <!-- ws:start:WikiTextHeadingRule:64:<h2> --><h2 id="toc32"><a name="Novemkleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:64 -->11-limit</h2> Commas: 1375/1372, 4000/3993, 15625/15552<br /> <br /> POTE generator: ~6/5 = 317.010<br /> <br /> Map: [<9 0 9 11 24|, <0 6 5 6 3|]<br /> EDOs: 72, 261, 333, 405, 882c<br /> Badness: 0.05172<br /> <br /> <!-- ws:start:WikiTextHeadingRule:66:<h2> --><h2 id="toc33"><a name="Novemkleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:66 -->13-limit</h2> Commas: 325/324, 625/624, 1375/1372, 4000/3993<br /> <br /> POTE generator: ~6/5 = 317.086<br /> <br /> Map: [<9 0 9 11 24 0|, <0 6 5 6 3 14|]<br /> EDOs: 72, 261, 333, 738cf, 1071bcf<br /> Badness: 0.0391<br /> <br /> <!-- ws:start:WikiTextHeadingRule:68:<h1> --><h1 id="toc34"><a name="Sqrtphi"></a><!-- ws:end:WikiTextHeadingRule:68 -->Sqrtphi</h1> Commas: 15625/15552, 16875/16807<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~125/98 = 416.603 cents<br /> <br /> Sqrt(phi) = 416.545 cents<br /> <br /> Map: [<1 12 11 16|, <0 -30 -25 -38|]<br /> EDOs: <a class="wiki_link" href="/49edo">49</a>, 72, <a class="wiki_link" href="/193edo">193</a>, <a class="wiki_link" href="/265edo">265</a><br /> Badness: 0.0704<br /> <br /> <!-- ws:start:WikiTextHeadingRule:70:<h2> --><h2 id="toc35"><a name="Sqrtphi-11-limit"></a><!-- ws:end:WikiTextHeadingRule:70 -->11-limit</h2> Commas: 540/539, 1375/1372, 4375/4356<br /> <br /> POTE generator: ~14/11 = 416.604<br /> <br /> Map: [<1 12 11 16 17|, <0 -30 -25 -38 -39|]<br /> EDOs: 49, 72, 193, 265<br /> Badness: 0.0255<br /> <br /> <!-- ws:start:WikiTextHeadingRule:72:<h2> --><h2 id="toc36"><a name="Sqrtphi-13-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->13-limit</h2> Commas: 325/324, 364/363, 625/624, 1375/1372<br /> <br /> POTE generator: ~14/11 = 416.585<br /> <br /> Map: [<1 12 11 16 17 28|, <0 -30 -25 -38 -39 -70|]<br /> EDOs: 72, 121, 193<br /> Badness: 0.0200<br /> <br /> <!-- ws:start:WikiTextHeadingRule:74:<h2> --><h2 id="toc37"><a name="Sqrtphi-17-limit"></a><!-- ws:end:WikiTextHeadingRule:74 -->17-limit</h2> Commas: 325/324, 364/363, 375/374, 540/539, 595/594<br /> <br /> POTE generator: ~14/11 = 416.585<br /> <br /> Map: [<1 12 11 16 17 28 27|, <0 -30 -25 -38 -39 -70 -66|]<br /> EDOs: 72, 121, 193<br /> Badness: 0.0130<br /> <br /> <!-- ws:start:WikiTextHeadingRule:76:<h2> --><h2 id="toc38"><a name="Sqrtphi-19-limit"></a><!-- ws:end:WikiTextHeadingRule:76 -->19-limit</h2> Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594<br /> <br /> POTE generator: ~14/11 = 416.580<br /> <br /> Map: [<1 12 11 16 17 28 27 -2|, <0 -30 -25 -38 -39 -70 -66 18|]<br /> EDOs: 72, 121, 193<br /> Badness: 0.0147<br /> <br /> <!-- ws:start:WikiTextHeadingRule:78:<h2> --><h2 id="toc39"><a name="Sqrtphi-Scales"></a><!-- ws:end:WikiTextHeadingRule:78 -->Scales</h2> <a class="wiki_link" href="/sqrtphi17">sqrtphi17</a><br /> <a class="wiki_link" href="/sqrtphi23">sqrtphi23</a><br /> <a class="wiki_link" href="/sqrtphi49">sqrtphi49</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:80:<h2> --><h2 id="toc40"><a name="Sqrtphi-Music"></a><!-- ws:end:WikiTextHeadingRule:80 -->Music</h2> <a class="wiki_link_ext" href="http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3" rel="nofollow">Prelude for Piano in Square root of Phi Tuning</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3" rel="nofollow">A Fight for Phi</a> by Vito Sicurella</body></html>