Würschmidt family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 288836345 - 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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-19 00:06:34 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>303066384</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]] | ||
=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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EDOs: 31, 65d, 96, 127, 223d | EDOs: 31, 65d, 96, 127, 223d | ||
Badness: 0.0244 | Badness: 0.0244 | ||
==13-limit== | |||
Commas: 99/98, 144/143, 176/175, 275/273 | |||
POTE generator: ~5/4 = 387.626 | |||
Map: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|] | |||
EDOs: 31, 65d, 161df | |||
Badness: 0.0236 | |||
=Worschmidt= | =Worschmidt= | ||
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=Relationships to other temperaments= | =Relationships to other temperaments= | ||
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:24:&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:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Würschmidt">Würschmidt</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 2em;"><a href="#Würschmidt-Seven limit children">Seven limit children</a></div> | ||
<!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Würschmidt">Würschmidt</a></div> | |||
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 2em;"><a href="#Würschmidt-11-limit">11-limit</a></div> | |||
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 2em;"><a href="#Würschmidt-13-limit">13-limit</a></div> | |||
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Worschmidt">Worschmidt</a></div> | |||
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 2em;"><a href="#Worschmidt-11-limit">11-limit</a></div> | |||
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 1em;"><a href="#Whirrschmidt">Whirrschmidt</a></div> | |||
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Hemiwürschmidt">Hemiwürschmidt</a></div> | |||
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Hemiwürschmidt-11-limit">11-limit</a></div> | |||
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Hemiwürschmidt-Hemiwur">Hemiwur</a></div> | |||
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Relationships to other temperaments">Relationships to other temperaments</a></div> | |||
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --></div> | |||
<!-- ws:end:WikiTextTocRule:37 --><!-- 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: 0.0244<br /> | Badness: 0.0244<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id=" | <!-- 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 /> | |||
<br /> | |||
POTE generator: ~5/4 = 387.626<br /> | |||
<br /> | |||
Map: [&lt;1 7 3 15 17 1|, &lt;0 -8 -1 -18 -20 4|]<br /> | |||
EDOs: 31, 65d, 161df<br /> | |||
Badness: 0.0236<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Worschmidt"></a><!-- ws:end:WikiTextHeadingRule:10 -->Worschmidt</h1> | |||
Worschmidt tempers out 126/125 rather than 225/224, and can use <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo">34edo</a>, or <a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo">127edo</a> as a tuning. If 127 is used, note that the val is &lt;127 201 295 356| and not &lt;127 201 295 357| as with wurschmidt. The wedgie now is &lt;&lt;8 1 -13 -17 -43 -33|. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.<br /> | Worschmidt tempers out 126/125 rather than 225/224, and can use <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo">34edo</a>, or <a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo">127edo</a> as a tuning. If 127 is used, note that the val is &lt;127 201 295 356| and not &lt;127 201 295 357| as with wurschmidt. The wedgie now is &lt;&lt;8 1 -13 -17 -43 -33|. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.<br /> | ||
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Badness: 0.0646<br /> | Badness: 0.0646<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Worschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> | ||
Commas: 126/125, 243/242, 385/384<br /> | Commas: 126/125, 243/242, 385/384<br /> | ||
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Badness: 0.0334<br /> | Badness: 0.0334<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Whirrschmidt"></a><!-- ws:end:WikiTextHeadingRule:14 -->Whirrschmidt</h1> | ||
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99edo</a> is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with &lt;&lt;8 1 52 -17 60 118|| for a wedgie.<br /> | <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99edo</a> is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with &lt;&lt;8 1 52 -17 60 118|| for a wedgie.<br /> | ||
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EDOs: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo">34</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo">65</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99</a><br /> | EDOs: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo">34</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo">65</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99</a><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Hemiwürschmidt"></a><!-- ws:end:WikiTextHeadingRule:16 -->Hemiwürschmidt</h1> | ||
Hemiwürschmidt, which splits the major third in two and uses that for a generator, is the most important of these temperaments even with the rather large complexity for the fifth. It tempers out 3136/3125, 6144/6125 and 2401/2400. <a class="wiki_link" href="http://xenharmonic.wikispaces.com/68edo">68edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/130edo">130edo</a> can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, &lt;&lt;16 2 5 40 -39 -49 -48 28...<br /> | Hemiwürschmidt, which splits the major third in two and uses that for a generator, is the most important of these temperaments even with the rather large complexity for the fifth. It tempers out 3136/3125, 6144/6125 and 2401/2400. <a class="wiki_link" href="http://xenharmonic.wikispaces.com/68edo">68edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/130edo">130edo</a> can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, &lt;&lt;16 2 5 40 -39 -49 -48 28...<br /> | ||
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Badness: 0.0203<br /> | Badness: 0.0203<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Hemiwürschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> | ||
Commas: 243/242, 441/440, 3136/3125<br /> | Commas: 243/242, 441/440, 3136/3125<br /> | ||
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<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Hemiwürschmidt-Hemiwur"></a><!-- ws:end:WikiTextHeadingRule:20 -->Hemiwur</h2> | ||
Commas: 121/120, 176/175, 1375/1372<br /> | Commas: 121/120, 176/175, 1375/1372<br /> | ||
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Badness: 0.0293<br /> | Badness: 0.0293<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Relationships to other temperaments"></a><!-- ws:end:WikiTextHeadingRule:22 -->Relationships to other temperaments</h1> | ||
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> | ||