Schismatic family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 241117729 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 241254089 - 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>2011-07-13 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-13 20:22:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>241254089</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, 77, 118 | EDOs: 41, 77, 118 | ||
Badness: 0.0284 | Badness: 0.0284 | ||
=Pogo= | |||
Commas: 32805/32768, 118098/117469 | |||
POTE generator: ~9/7 = 433.901 | |||
Map: [<2 1 22 2|, <0 3 -24 5|] | |||
EDOs: 36, 94, 130, 224, 354 | |||
Badness: 0.0796 | |||
==11-limit== | |||
Commas: 540/539, 4000/3993, 32805/32768 | |||
POTE generator: ~9/7 = 433.911 | |||
Map: [<2 1 22 2 25|, <0 3 -24 5 -25|] | |||
EDOs: 36, 94, 130, 224, 354, 578 | |||
Badness: 0.0319 | |||
=Pontiac= | =Pontiac= | ||
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Badness: 0.0253</pre></div> | Badness: 0.0253</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:44:&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:44 --><!-- ws:start:WikiTextTocRule:45: --><a href="#Five limit">Five limit</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | <a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --> | <a href="#Garibaldi">Garibaldi</a><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --> | <a href="#Guiron">Guiron</a><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --> | <a href="#Pogo">Pogo</a><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --> | <a href="#Pontiac">Pontiac</a><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --> | <a href="#Grackle">Grackle</a><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --> | <a href="#Bischismic">Bischismic</a><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Term">Term</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --> | <a href="#Sesquiquartififths">Sesquiquartififths</a><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --> | <a href="#Sextilififths">Sextilififths</a><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:67 --><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: 0.0284<br /> | Badness: 0.0284<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Pontiac"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Pogo"></a><!-- ws:end:WikiTextHeadingRule:20 -->Pogo</h1> | ||
Commas: 32805/32768, 118098/117469<br /> | |||
<br /> | |||
POTE generator: ~9/7 = 433.901<br /> | |||
<br /> | |||
Map: [&lt;2 1 22 2|, &lt;0 3 -24 5|]<br /> | |||
EDOs: 36, 94, 130, 224, 354<br /> | |||
Badness: 0.0796<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Pogo-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2> | |||
Commas: 540/539, 4000/3993, 32805/32768<br /> | |||
<br /> | |||
POTE generator: ~9/7 = 433.911<br /> | |||
<br /> | |||
Map: [&lt;2 1 22 2 25|, &lt;0 3 -24 5 -25|]<br /> | |||
EDOs: 36, 94, 130, 224, 354, 578<br /> | |||
Badness: 0.0319<br /> | |||
<br /> | |||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Pontiac"></a><!-- ws:end:WikiTextHeadingRule:24 -->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:26:&lt;h1&gt; --><h1 id="toc13"><a name="Grackle"></a><!-- ws:end:WikiTextHeadingRule:26 -->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:28:&lt;h1&gt; --><h1 id="toc14"><a name="Bischismic"></a><!-- ws:end:WikiTextHeadingRule:28 -->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:30:&lt;h2&gt; --><h2 id="toc15"><a name="Bischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->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:32:&lt;h2&gt; --><h2 id="toc16"><a name="Bischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->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:34:&lt;h1&gt; --><h1 id="toc17"><a name="Term"></a><!-- ws:end:WikiTextHeadingRule:34 -->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:36:&lt;h1&gt; --><h1 id="toc18"><a name="Sesquiquartififths"></a><!-- ws:end:WikiTextHeadingRule:36 -->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:38:&lt;h1&gt; --><h1 id="toc19"><a name="Sextilififths"></a><!-- ws:end:WikiTextHeadingRule:38 -->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:40:&lt;h2&gt; --><h2 id="toc20"><a name="Sextilififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->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:42:&lt;h2&gt; --><h2 id="toc21"><a name="Sextilififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->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 /> | ||