Schismatic family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 504312528 - Original comment: ** |
Wikispaces>hstraub **Imported revision 518397720 - 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: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2014-08-13 05:57:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>518397720</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc]] | <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;white-space: pre-wrap ! important" class="old-revision-html">[[toc]] | ||
<span style="display: block; text-align: right;">Other languages: [[xenharmonie/Schismatische Temperaturen|Deutsch]] | |||
</span> | |||
=Five limit= | =Five limit= | ||
The 5-limit parent comma for the schismatic family is the [[schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus comma]] ([[81_80|81/80]]), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. 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. | The 5-limit parent comma for the schismatic family is the [[schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus comma]] ([[81_80|81/80]]), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. 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. | ||
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Badness: 0.0319 | Badness: 0.0319 | ||
==13-limit== | ==13-limit== | ||
Commas: 540/539, 729/728, 4000/3993, 4225/4224 | Commas: 540/539, 729/728, 4000/3993, 4225/4224 | ||
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Badness: 0.0175 | Badness: 0.0175 | ||
=Sanjaab= | =Sanjaab= | ||
Commas: 225/224, 3125/3087, 1331/1323 | Commas: 225/224, 3125/3087, 1331/1323 | ||
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Badness: 0.0580 | Badness: 0.0580 | ||
==13-limit== | ==13-limit== | ||
Commas: 225/224, 275/273, 847/845, 1331/1323 | Commas: 225/224, 275/273, 847/845, 1331/1323 | ||
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Badness: 0.0338 | Badness: 0.0338 | ||
=Squirrel= | =Squirrel= | ||
Commmas: 686/675, 32805/32768 | Commmas: 686/675, 32805/32768 | ||
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Badness: 0.1747 | Badness: 0.1747 | ||
==11-limit== | ==11-limit== | ||
Commas: 245/242, 686/675, 896/891 | Commas: 245/242, 686/675, 896/891 | ||
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Badness: 0.0683 | Badness: 0.0683 | ||
==13-limit== | ==13-limit== | ||
Commas: 91/90, 169/168, 245/242, 896/891 | Commas: 91/90, 169/168, 245/242, 896/891 | ||
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Badness: 0.0293 | Badness: 0.0293 | ||
===13-limit=== | ===13-limit=== | ||
Commas: 243/242, 364/363, 441/440, 105644/105625 | Commas: 243/242, 364/363, 441/440, 105644/105625 | ||
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Badness: 0.0224 | Badness: 0.0224 | ||
==Bisesqui== | ==Bisesqui== | ||
Commas: 2401/2400, 9801/9800, 32805/32768 | Commas: 2401/2400, 9801/9800, 32805/32768 | ||
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Badness: 0.0315 | Badness: 0.0315 | ||
==13-limit== | ==13-limit== | ||
Commas: 540/539, 1375/1372, 4096/4095, 6656/6655 | Commas: 540/539, 1375/1372, 4096/4095, 6656/6655 | ||
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<!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Octant-13-limit">13-limit</a></div> | <!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Octant-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --></div> | <!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --></div> | ||
<!-- ws:end:WikiTextTocRule:142 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Five limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit</h1> | <!-- ws:end:WikiTextTocRule:142 --><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Schismatische%20Temperaturen">Deutsch</a><br /> | ||
</span><br /> | |||
<!-- 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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<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Pogo-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Pogo-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2> | ||
Commas: 540/539, 729/728, 4000/3993, 4225/4224<br /> | Commas: 540/539, 729/728, 4000/3993, 4225/4224<br /> | ||
<br /> | <br /> | ||
POTE generator: ~9/7 = 433.911<br /> | POTE generator: ~9/7 = 433.911<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Sanjaab"></a><!-- ws:end:WikiTextHeadingRule:34 -->Sanjaab</h1> | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Sanjaab"></a><!-- ws:end:WikiTextHeadingRule:34 -->Sanjaab</h1> | ||
Commas: 225/224, 3125/3087, 1331/1323<br /> | Commas: 225/224, 3125/3087, 1331/1323<br /> | ||
<br /> | <br /> | ||
POTE generator: ~11/10 = 165.974<br /> | POTE generator: ~11/10 = 165.974<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Sanjaab-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Sanjaab-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2> | ||
Commas: 225/224, 275/273, 847/845, 1331/1323<br /> | Commas: 225/224, 275/273, 847/845, 1331/1323<br /> | ||
<br /> | <br /> | ||
POTE generator: ~11/10 = 165.963<br /> | POTE generator: ~11/10 = 165.963<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Squirrel"></a><!-- ws:end:WikiTextHeadingRule:38 -->Squirrel</h1> | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Squirrel"></a><!-- ws:end:WikiTextHeadingRule:38 -->Squirrel</h1> | ||
Commmas: 686/675, 32805/32768<br /> | Commmas: 686/675, 32805/32768<br /> | ||
<br /> | <br /> | ||
POTE generator: ~160/147 = 166.140<br /> | POTE generator: ~160/147 = 166.140<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Squirrel-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Squirrel-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> | ||
Commas: 245/242, 686/675, 896/891<br /> | Commas: 245/242, 686/675, 896/891<br /> | ||
<br /> | <br /> | ||
POTE generator: ~11/10 = 166.097<br /> | POTE generator: ~11/10 = 166.097<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Squirrel-13-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Squirrel-13-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->13-limit</h2> | ||
Commas: 91/90, 169/168, 245/242, 896/891<br /> | Commas: 91/90, 169/168, 245/242, 896/891<br /> | ||
<br /> | <br /> | ||
POTE generator: ~11/10 = 166.054<br /> | POTE generator: ~11/10 = 166.054<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:70:&lt;h3&gt; --><h3 id="toc35"><a name="Sesquiquartififths-Sesquart-13-limit"></a><!-- ws:end:WikiTextHeadingRule:70 -->13-limit</h3> | <!-- ws:start:WikiTextHeadingRule:70:&lt;h3&gt; --><h3 id="toc35"><a name="Sesquiquartififths-Sesquart-13-limit"></a><!-- ws:end:WikiTextHeadingRule:70 -->13-limit</h3> | ||
Commas: 243/242, 364/363, 441/440, 105644/105625<br /> | Commas: 243/242, 364/363, 441/440, 105644/105625<br /> | ||
<br /> | <br /> | ||
POTE generator: ~256/231 = 175.409<br /> | POTE generator: ~256/231 = 175.409<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc36"><a name="Sesquiquartififths-Bisesqui"></a><!-- ws:end:WikiTextHeadingRule:72 -->Bisesqui</h2> | <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc36"><a name="Sesquiquartififths-Bisesqui"></a><!-- ws:end:WikiTextHeadingRule:72 -->Bisesqui</h2> | ||
Commas: 2401/2400, 9801/9800, 32805/32768<br /> | Commas: 2401/2400, 9801/9800, 32805/32768<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~448/405 = 175.435<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~448/405 = 175.435<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc43"><a name="Quanharuk-13-limit"></a><!-- ws:end:WikiTextHeadingRule:86 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc43"><a name="Quanharuk-13-limit"></a><!-- ws:end:WikiTextHeadingRule:86 -->13-limit</h2> | ||
Commas: 540/539, 1375/1372, 4096/4095, 6656/6655<br /> | Commas: 540/539, 1375/1372, 4096/4095, 6656/6655<br /> | ||
<br /> | <br /> | ||
POTE generator: ~56/45 = 380.351<br /> | POTE generator: ~56/45 = 380.351<br /> | ||