Würschmidt family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 321827204 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 322312438 - 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>2012-04-17 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-18 17:13:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>322312438</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: 31, 99e, 130, 650ce, 811ce | EDOs: 31, 99e, 130, 650ce, 811ce | ||
Badness: 0.0211 | Badness: 0.0211 | ||
===13-limit=== | |||
Commas: 243/242 351/350 441/440 3584/3575 | |||
POTE generator: ~28/25 = 193.840 | |||
Map: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|] | |||
EDOs: 31, 99e, 130, 291, 421e, 551ce | |||
Badness: 0.0231 | |||
===Hemithir=== | |||
Commas: 121/120 176/175 196/195 275/273 | |||
POTE generator: ~28/25 = 193.918 | |||
Map: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|] | |||
EDOs: 31, 68e, 99ef | |||
Badness: 0.0312 | |||
==Hemiwur== | ==Hemiwur== | ||
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=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. | 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> | ||
</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>Würschmidt 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>Würschmidt family</title></head><body><!-- ws:start:WikiTextTocRule:30:&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:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#Würschmidt">Würschmidt</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Würschmidt-Seven limit children">Seven limit children</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Würschmidt">Würschmidt</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Würschmidt-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Würschmidt-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Worschmidt">Worschmidt</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Worschmidt-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 1em;"><a href="#Whirrschmidt">Whirrschmidt</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Hemiwürschmidt">Hemiwürschmidt</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 2em;"><a href="#Hemiwürschmidt-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 3em;"><a href="#Hemiwürschmidt-11-limit-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 3em;"><a href="#Hemiwürschmidt-11-limit-Hemithir">Hemithir</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Hemiwürschmidt-Hemiwur">Hemiwur</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 3em;"><a href="#Hemiwürschmidt-Hemiwur-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 1em;"><a href="#Relationships to other temperaments">Relationships to other temperaments</a></div> | ||
<!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --></div> | |||
<!-- ws:end:WikiTextTocRule:46 --><!-- 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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Badness: 0.0211<br /> | Badness: 0.0211<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id=" | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Hemiwürschmidt-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h3> | ||
Commas: 243/242 351/350 441/440 3584/3575<br /> | |||
<br /> | |||
POTE generator: ~28/25 = 193.840<br /> | |||
<br /> | |||
Map: [&lt;1 15 4 7 37 -29|, &lt;0 -16 -2 -5 -40 39|]<br /> | |||
EDOs: 31, 99e, 130, 291, 421e, 551ce<br /> | |||
Badness: 0.0231<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Hemiwürschmidt-11-limit-Hemithir"></a><!-- ws:end:WikiTextHeadingRule:22 -->Hemithir</h3> | |||
Commas: 121/120 176/175 196/195 275/273<br /> | |||
<br /> | |||
POTE generator: ~28/25 = 193.918<br /> | |||
<br /> | |||
Map: [&lt;1 15 4 7 37 -3|, &lt;0 -16 -2 -5 -40 8|]<br /> | |||
EDOs: 31, 68e, 99ef<br /> | |||
Badness: 0.0312<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Hemiwürschmidt-Hemiwur"></a><!-- ws:end:WikiTextHeadingRule:24 -->Hemiwur</h2> | |||
Commas: 121/120, 176/175, 1375/1372<br /> | Commas: 121/120, 176/175, 1375/1372<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0293<br /> | Badness: 0.0293<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Hemiwürschmidt-Hemiwur-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->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 /> | ||
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Badness: 0.0284<br /> | Badness: 0.0284<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Relationships to other temperaments"></a><!-- ws:end:WikiTextHeadingRule:28 -->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> | ||