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Würschmidt family: Difference between revisions

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**Imported revision 287009366 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 288008360 - 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:keenanpepper|keenanpepper]] and made on <tt>2011-12-16 23:09:09 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-21 15:39:41 UTC</tt>.<br>
: The original revision id was <tt>287009366</tt>.<br>
: The original revision id was <tt>288008360</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|flat]]
<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|flat]]
=Wuerschmidt=  
=Würschmidt=  
The [[xenharmonic/5-limit|5-limit]] parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma, and named after José Würschmidt, Its [[xenharmonic/monzo|monzo]] is |17 1 -8&gt;, and flipping that yields &lt;&lt;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 Wuerschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[xenharmonic/minimax tuning|minimax tuning]]. Wuerschmidt 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 wuerschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its [[xenharmonic/monzo|monzo]] is |17 1 -8&gt;, and flipping that yields &lt;&lt;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.


[[xenharmonic/POTE tuning|POTE generator]]: 387.799
[[xenharmonic/POTE tuning|POTE generator]]: 387.799
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The second comma of the [[xenharmonic/Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Wurschmidt adds |12 3 -6 -1&gt;, worschmidt adds 65625/65536 = |-16 1 5 1&gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&gt; and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2&gt;.
The second comma of the [[xenharmonic/Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Wurschmidt adds |12 3 -6 -1&gt;, worschmidt adds 65625/65536 = |-16 1 5 1&gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&gt; and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2&gt;.


=Wurschmidt=  
=Würschmidt=  
Wurschmidt, aside from the commas listed above, also tempers out 225/224. [[xenharmonic/31edo|31edo]] or [[xenharmonic/127edo|127edo]] can be used as tunings. Wurschmidt has &lt;&lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &lt;&lt;8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. [[xenharmonic/127edo|127edo]] is again an excellent tuning for 11-limit wurschmidt, as well as for minerva, the 11-limit rank three temperament tempering out 99/98 and 176/175.
Würschmidt, aside from the commas listed above, also tempers out 225/224. [[xenharmonic/31edo|31edo]] or [[xenharmonic/127edo|127edo]] can be used as tunings. Würschmidt has &lt;&lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &lt;&lt;8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. [[xenharmonic/127edo|127edo]] is again an excellent tuning for 11-limit wurschmidt, as well as for minerva, the 11-limit rank three temperament tempering out 99/98 and 176/175.


Commas: 225/224, 8748/8575
Commas: 225/224, 8748/8575
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EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/34edo|34]], [[xenharmonic/65edo|65]], [[xenharmonic/99edo|99]]
EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/34edo|34]], [[xenharmonic/65edo|65]], [[xenharmonic/99edo|99]]


=Hemiwuerschmidt=  
=Hemiwürschmidt=  
Hemiwuerschmidt, 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. [[xenharmonic/68edo|68edo]], [[xenharmonic/99edo|99edo]] and [[xenharmonic/130edo|130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwuerschmidt extends to a higher limit temperament, &lt;&lt;16 2 5 40 -39 -49 -48 28...
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. [[xenharmonic/68edo|68edo]], [[xenharmonic/99edo|99edo]] and [[xenharmonic/130edo|130edo]] 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...


Commas: 2401/2400, 3136/3125
Commas: 2401/2400, 3136/3125
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&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;
&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;
=Relationships to other temperaments=  
=Relationships to other temperaments=  
2-Wuerschmidt, the temperament with all the same commas as Wuerschmidt 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>
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Würschmidt family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Wuerschmidt"&gt;Wuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Wurschmidt"&gt;Wurschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Hemiwuerschmidt"&gt;Hemiwuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Relationships to other temperaments"&gt;Relationships to other temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Würschmidt family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Würschmidt"&gt;Würschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Würschmidt"&gt;Würschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Hemiwürschmidt"&gt;Hemiwürschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Relationships to other temperaments"&gt;Relationships to other temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;
&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Wuerschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Wuerschmidt&lt;/h1&gt;
&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Würschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Würschmidt&lt;/h1&gt;
  The &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit"&gt;5-limit&lt;/a&gt; parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma, and named after José Würschmidt, Its &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/monzo"&gt;monzo&lt;/a&gt; is |17 1 -8&amp;gt;, and flipping that yields &amp;lt;&amp;lt;8 1 17|| for the wedgie. This tells us the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/generator"&gt;generator&lt;/a&gt; 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 Wuerschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/minimax%20tuning"&gt;minimax tuning&lt;/a&gt;. Wuerschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS"&gt;MOS&lt;/a&gt; all possibilities.&lt;br /&gt;
  The &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit"&gt;5-limit&lt;/a&gt; parent comma for the wuerschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/monzo"&gt;monzo&lt;/a&gt; is |17 1 -8&amp;gt;, and flipping that yields &amp;lt;&amp;lt;8 1 17|| for the wedgie. This tells us the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/generator"&gt;generator&lt;/a&gt; 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/minimax%20tuning"&gt;minimax tuning&lt;/a&gt;. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS"&gt;MOS&lt;/a&gt; all possibilities.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.799&lt;br /&gt;
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.799&lt;br /&gt;
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EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/164edo"&gt;164&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/721edo"&gt;721c&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/885edo"&gt;885c&lt;/a&gt;&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/164edo"&gt;164&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/721edo"&gt;721c&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/885edo"&gt;885c&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Wuerschmidt-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Seven limit children&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Würschmidt-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Seven limit children&lt;/h2&gt;
  The second comma of the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Wurschmidt adds |12 3 -6 -1&amp;gt;, worschmidt adds 65625/65536 = |-16 1 5 1&amp;gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&amp;gt; and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2&amp;gt;.&lt;br /&gt;
  The second comma of the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Wurschmidt adds |12 3 -6 -1&amp;gt;, worschmidt adds 65625/65536 = |-16 1 5 1&amp;gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&amp;gt; and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Wurschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Wurschmidt&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Würschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Würschmidt&lt;/h1&gt;
  Wurschmidt, aside from the commas listed above, also tempers out 225/224. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31edo&lt;/a&gt; or &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127edo&lt;/a&gt; can be used as tunings. Wurschmidt has &amp;lt;&amp;lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &amp;lt;&amp;lt;8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127edo&lt;/a&gt; is again an excellent tuning for 11-limit wurschmidt, as well as for minerva, the 11-limit rank three temperament tempering out 99/98 and 176/175.&lt;br /&gt;
  Würschmidt, aside from the commas listed above, also tempers out 225/224. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31edo&lt;/a&gt; or &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127edo&lt;/a&gt; can be used as tunings. Würschmidt has &amp;lt;&amp;lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &amp;lt;&amp;lt;8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127edo&lt;/a&gt; is again an excellent tuning for 11-limit wurschmidt, as well as for minerva, the 11-limit rank three temperament tempering out 99/98 and 176/175.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Commas: 225/224, 8748/8575&lt;br /&gt;
Commas: 225/224, 8748/8575&lt;br /&gt;
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Badness: 0.0508&lt;br /&gt;
Badness: 0.0508&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Wurschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;11-limit&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Würschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;11-limit&lt;/h2&gt;
  Commas: 99/98, 176/175, 243/242&lt;br /&gt;
  Commas: 99/98, 176/175, 243/242&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99&lt;/a&gt;&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99&lt;/a&gt;&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Hemiwuerschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Hemiwuerschmidt&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Hemiwürschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Hemiwürschmidt&lt;/h1&gt;
  Hemiwuerschmidt, 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. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/68edo"&gt;68edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99edo&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/130edo"&gt;130edo&lt;/a&gt; can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwuerschmidt extends to a higher limit temperament, &amp;lt;&amp;lt;16 2 5 40 -39 -49 -48 28...&lt;br /&gt;
  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. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/68edo"&gt;68edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99edo&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/130edo"&gt;130edo&lt;/a&gt; can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, &amp;lt;&amp;lt;16 2 5 40 -39 -49 -48 28...&lt;br /&gt;
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Commas: 2401/2400, 3136/3125&lt;br /&gt;
Commas: 2401/2400, 3136/3125&lt;br /&gt;
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Badness: 0.0203&lt;br /&gt;
Badness: 0.0203&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Hemiwuerschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Hemiwürschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
  Commas: 243/242, 441/440, 3136/3125&lt;br /&gt;
  Commas: 243/242, 441/440, 3136/3125&lt;br /&gt;
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&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;&lt;br /&gt;
&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Relationships to other temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Relationships to other temperaments&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Relationships to other temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Relationships to other temperaments&lt;/h1&gt;
  2-Wuerschmidt, the temperament with all the same commas as Wuerschmidt but a generator of twice the size, is equivalent to &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/skwares"&gt;skwares&lt;/a&gt; as a 2.3.7.11 temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
  2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/skwares"&gt;skwares&lt;/a&gt; as a 2.3.7.11 temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
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