Schismatic family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 457066354 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 463342692 - 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>2013-10- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-10-26 14:20:09 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>463342692</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: 41, 89, 130 | EDOs: 41, 89, 130 | ||
Badness: 0.0293 | Badness: 0.0293 | ||
==Bisesqui== | |||
Commas: 2401/2400, 9801/9800, 32805/32768 | |||
[[POTE tuning|POTE generator]]: ~448/405 = 175.435 | |||
Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|] | |||
EDOs: 130, 212, 342, 1156, 1498 | |||
Badness: 0.0170 | |||
=Sextilififths= | =Sextilififths= | ||
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Badness: 0.0304</pre></div> | Badness: 0.0304</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:80:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 1em;"><a href="#Five limit">Five limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 1em;"><a href="#Garibaldi">Garibaldi</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Garibaldi-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 3em;"><a href="#Garibaldi-11-limit-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 2em;"><a href="#Garibaldi-Cassandra">Cassandra</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 3em;"><a href="#Garibaldi-Cassandra-13-limit Cassandra">13-limit Cassandra</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 2em;"><a href="#Garibaldi-Helenus">Helenus</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 3em;"><a href="#Garibaldi-Helenus-13-limit Helenus">13-limit Helenus</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 1em;"><a href="#Hemigari">Hemigari</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 2em;"><a href="#Hemigari-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 1em;"><a href="#Guiron">Guiron</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 2em;"><a href="#Guiron-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 2em;"><a href="#Guiron-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Pogo">Pogo</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Pogo-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 2em;"><a href="#Pogo-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 1em;"><a href="#Schism">Schism</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Schism-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 1em;"><a href="#Pontiac">Pontiac</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 1em;"><a href="#Grackle">Grackle</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 1em;"><a href="#Bischismic">Bischismic</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 2em;"><a href="#Bischismic-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 2em;"><a href="#Bischismic-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 1em;"><a href="#Kleischismic">Kleischismic</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 2em;"><a href="#Kleischismic-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Kleischismic-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 1em;"><a href="#Term">Term</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 1em;"><a href="#Sesquiquartififths">Sesquiquartififths</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Sesquiquartififths-Sesquart">Sesquart</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Sesquiquartififths-Bisesqui">Bisesqui</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 1em;"><a href="#Sextilififths">Sextilififths</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><div style="margin-left: 2em;"><a href="#Sextilififths-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><div style="margin-left: 2em;"><a href="#Sextilififths-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><div style="margin-left: 1em;"><a href="#Tsaharuk">Tsaharuk</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><div style="margin-left: 1em;"><a href="#Quanharuk">Quanharuk</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 2em;"><a href="#Quanharuk-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 1em;"><a href="#Octant">Octant</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#Octant-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 2em;"><a href="#Octant-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --></div> | ||
<!-- ws:end:WikiTextTocRule:121 --><!-- 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 Pythagorean 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 Pythagorean 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: 0.0293<br /> | Badness: 0.0293<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:60:&lt;h1&gt; --><h1 id=" | <!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc30"><a name="Sesquiquartififths-Bisesqui"></a><!-- ws:end:WikiTextHeadingRule:60 -->Bisesqui</h2> | ||
Commas: 2401/2400, 9801/9800, 32805/32768<br /> | |||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~448/405 = 175.435<br /> | |||
<br /> | |||
Map: [&lt;2 2 14 10 23|, &lt;0 4 -32 -15 -55|]<br /> | |||
EDOs: 130, 212, 342, 1156, 1498<br /> | |||
Badness: 0.0170<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:62:&lt;h1&gt; --><h1 id="toc31"><a name="Sextilififths"></a><!-- ws:end:WikiTextHeadingRule:62 -->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:64:&lt;h2&gt; --><h2 id="toc32"><a name="Sextilififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:64 -->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:66:&lt;h2&gt; --><h2 id="toc33"><a name="Sextilififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:66 -->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 /> | ||
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Badness: 0.0253<br /> | Badness: 0.0253<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:68:&lt;h1&gt; --><h1 id="toc34"><a name="Tsaharuk"></a><!-- ws:end:WikiTextHeadingRule:68 -->Tsaharuk</h1> | ||
Commas: 32805/32768, 420175/419904<br /> | Commas: 32805/32768, 420175/419904<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0307<br /> | Badness: 0.0307<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:70:&lt;h1&gt; --><h1 id="toc35"><a name="Quanharuk"></a><!-- ws:end:WikiTextHeadingRule:70 -->Quanharuk</h1> | ||
Commas: 16875/16807, 32805/32768<br /> | Commas: 16875/16807, 32805/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0720<br /> | Badness: 0.0720<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc36"><a name="Quanharuk-11-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->11-limit</h2> | ||
Commas: 540/539, 1375/1372, 32805/32768<br /> | Commas: 540/539, 1375/1372, 32805/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0315<br /> | Badness: 0.0315<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h1&gt; --><h1 id="toc37"><a name="Octant"></a><!-- ws:end:WikiTextHeadingRule:74 -->Octant</h1> | ||
Commas: 32805/32768, 2259436291848/2251875390625<br /> | Commas: 32805/32768, 2259436291848/2251875390625<br /> | ||
<br /> | <br /> | ||
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Badness: 0.1572<br /> | Badness: 0.1572<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:76:&lt;h2&gt; --><h2 id="toc38"><a name="Octant-11-limit"></a><!-- ws:end:WikiTextHeadingRule:76 -->11-limit</h2> | ||
Commas: 9801/9800, 32805/32768, 46656/46585<br /> | Commas: 9801/9800, 32805/32768, 46656/46585<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0448<br /> | Badness: 0.0448<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><h2 id="toc39"><a name="Octant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:78 -->13-limit</h2> | ||
Commas: 729/728, 1575/1573, 2200/2197, 6656/6655<br /> | Commas: 729/728, 1575/1573, 2200/2197, 6656/6655<br /> | ||
<br /> | <br /> | ||