Würschmidt family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 328082566 - Original comment: ** |
Wikispaces>hstraub **Imported revision 445082380 - 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>2013-08-16 06:04:46 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>445082380</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/Würschmidt|Deutsch]] | |||
</span> | |||
=Würschmidt= | =Würschmidt= | ||
The [[xenharmonic/5-limit|5-limit]] parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its [[xenharmonic/monzo|monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[xenharmonic/generator|generator]] is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[xenharmonic/minimax tuning|minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[xenharmonic/MOS|MOS]] all possibilities. | The [[xenharmonic/5-limit|5-limit]] parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its [[xenharmonic/monzo|monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[xenharmonic/generator|generator]] is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[xenharmonic/minimax tuning|minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[xenharmonic/MOS|MOS]] all possibilities. | ||
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Badness: 0.0244 | Badness: 0.0244 | ||
==13-limit== | ==13-limit== | ||
Commas: 99/98, 144/143, 176/175, 275/273 | Commas: 99/98, 144/143, 176/175, 275/273 | ||
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Badness: 0.0236 | Badness: 0.0236 | ||
==Worseschmidt== | ==Worseschmidt== | ||
Commas: 66/65, 99/98, 105/104, 243/242 | Commas: 66/65, 99/98, 105/104, 243/242 | ||
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Badness: 0.0211 | Badness: 0.0211 | ||
===13-limit=== | ===13-limit=== | ||
Commas: 243/242 351/350 441/440 3584/3575 | Commas: 243/242 351/350 441/440 3584/3575 | ||
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Badness: 0.0231 | Badness: 0.0231 | ||
===Hemithir=== | ===Hemithir=== | ||
Commas: 121/120 176/175 196/195 275/273 | Commas: 121/120 176/175 196/195 275/273 | ||
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Badness: 0.0312 | Badness: 0.0312 | ||
==Hemiwur== | ==Hemiwur== | ||
Commas: 121/120, 176/175, 1375/1372 | Commas: 121/120, 176/175, 1375/1372 | ||
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Badness: 0.0293 | Badness: 0.0293 | ||
===13-limit=== | ===13-limit=== | ||
Commas: 121/120, 176/175, 196/195, 275/273 | Commas: 121/120, 176/175, 196/195, 275/273 | ||
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Badness: 0.0284 | Badness: 0.0284 | ||
===Hemiwar=== | ===Hemiwar=== | ||
Commas: 66/65, 105/104, 121/120, 1375/1372 | Commas: 66/65, 105/104, 121/120, 1375/1372 | ||
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Badness: 0.0449 | Badness: 0.0449 | ||
=Relationships to other temperaments= | =Relationships to other temperaments= | ||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span> | ||
2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to [[xenharmonic/skwares|skwares]] as a 2.3.7.11 temperament.</pre></div> | 2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to [[xenharmonic/skwares|skwares]] as a 2.3.7.11 temperament.</pre></div> | ||
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<!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#Relationships to other temperaments">Relationships to other temperaments</a></div> | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#Relationships to other temperaments">Relationships to other temperaments</a></div> | ||
<!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --></div> | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --></div> | ||
<!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Würschmidt"></a><!-- ws:end:WikiTextHeadingRule:0 -->Würschmidt</h1> | <!-- ws:end:WikiTextTocRule:52 --><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/W%C3%BCrschmidt">Deutsch</a><br /> | ||
</span><br /> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Würschmidt"></a><!-- ws:end:WikiTextHeadingRule:0 -->Würschmidt</h1> | |||
The <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit">5-limit</a> parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its <a class="wiki_link" href="http://xenharmonic.wikispaces.com/monzo">monzo</a> is |17 1 -8&gt;, and flipping that yields &lt;&lt;8 1 17|| for the wedgie. This tells us the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/generator">generator</a> is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/minimax%20tuning">minimax tuning</a>. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS">MOS</a> all possibilities.<br /> | The <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit">5-limit</a> parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its <a class="wiki_link" href="http://xenharmonic.wikispaces.com/monzo">monzo</a> is |17 1 -8&gt;, and flipping that yields &lt;&lt;8 1 17|| for the wedgie. This tells us the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/generator">generator</a> is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/minimax%20tuning">minimax tuning</a>. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS">MOS</a> all possibilities.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Würschmidt-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Würschmidt-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> | ||
Commas: 99/98, 144/143, 176/175, 275/273<br /> | Commas: 99/98, 144/143, 176/175, 275/273<br /> | ||
<br /> | <br /> | ||
POTE generator: ~5/4 = 387.626<br /> | POTE generator: ~5/4 = 387.626<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Würschmidt-Worseschmidt"></a><!-- ws:end:WikiTextHeadingRule:10 -->Worseschmidt</h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Würschmidt-Worseschmidt"></a><!-- ws:end:WikiTextHeadingRule:10 -->Worseschmidt</h2> | ||
Commas: 66/65, 99/98, 105/104, 243/242<br /> | Commas: 66/65, 99/98, 105/104, 243/242<br /> | ||
<br /> | <br /> | ||
POTE generator: ~5/4 = 387.099<br /> | POTE generator: ~5/4 = 387.099<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Hemiwürschmidt-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h3> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Hemiwürschmidt-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h3> | ||
Commas: 243/242 351/350 441/440 3584/3575<br /> | Commas: 243/242 351/350 441/440 3584/3575<br /> | ||
<br /> | <br /> | ||
POTE generator: ~28/25 = 193.840<br /> | POTE generator: ~28/25 = 193.840<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Hemiwürschmidt-11-limit-Hemithir"></a><!-- ws:end:WikiTextHeadingRule:24 -->Hemithir</h3> | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Hemiwürschmidt-11-limit-Hemithir"></a><!-- ws:end:WikiTextHeadingRule:24 -->Hemithir</h3> | ||
Commas: 121/120 176/175 196/195 275/273<br /> | Commas: 121/120 176/175 196/195 275/273<br /> | ||
<br /> | <br /> | ||
POTE generator: ~28/25 = 193.918<br /> | POTE generator: ~28/25 = 193.918<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Hemiwürschmidt-Hemiwur"></a><!-- ws:end:WikiTextHeadingRule:26 -->Hemiwur</h2> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Hemiwürschmidt-Hemiwur"></a><!-- ws:end:WikiTextHeadingRule:26 -->Hemiwur</h2> | ||
Commas: 121/120, 176/175, 1375/1372<br /> | Commas: 121/120, 176/175, 1375/1372<br /> | ||
<br /> | <br /> | ||
POTE generator: ~28/25 = 193.884<br /> | POTE generator: ~28/25 = 193.884<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Hemiwürschmidt-Hemiwur-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h3> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Hemiwürschmidt-Hemiwur-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h3> | ||
Commas: 121/120, 176/175, 196/195, 275/273<br /> | Commas: 121/120, 176/175, 196/195, 275/273<br /> | ||
<br /> | <br /> | ||
POTE generator: ~28/25 = 194.004<br /> | POTE generator: ~28/25 = 194.004<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Hemiwürschmidt-Hemiwur-Hemiwar"></a><!-- ws:end:WikiTextHeadingRule:30 -->Hemiwar</h3> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Hemiwürschmidt-Hemiwur-Hemiwar"></a><!-- ws:end:WikiTextHeadingRule:30 -->Hemiwar</h3> | ||
Commas: 66/65, 105/104, 121/120, 1375/1372<br /> | Commas: 66/65, 105/104, 121/120, 1375/1372<br /> | ||
<br /> | <br /> | ||
POTE generator: ~28/25 = 193.698<br /> | POTE generator: ~28/25 = 193.698<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Relationships to other temperaments"></a><!-- ws:end:WikiTextHeadingRule:32 -->Relationships to other temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Relationships to other temperaments"></a><!-- ws:end:WikiTextHeadingRule:32 -->Relationships to other temperaments</h1> | ||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span><br /> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span><br /> | ||
2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/skwares">skwares</a> as a 2.3.7.11 temperament.</body></html></pre></div> | 2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/skwares">skwares</a> as a 2.3.7.11 temperament.</body></html></pre></div> | ||