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Würschmidt family: Difference between revisions

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**Imported revision 327990308 - Original comment: **
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**Imported revision 328082566 - 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>2012-05-01 11:49:01 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-01 14:15:48 UTC</tt>.<br>
: The original revision id was <tt>327990308</tt>.<br>
: The original revision id was <tt>328082566</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: 6, 31, 68, 99f, 167ef
EDOs: 6, 31, 68, 99f, 167ef
Badness: 0.0284
Badness: 0.0284
===Hemiwar===
Commas: 66/65, 105/104, 121/120, 1375/1372
POTE generator: ~28/25 = 193.698
Map: [&lt;1 15 4 7 11 23|, &lt;0 -16 -2 -5 -9 -23|]
EDOs: 31
Badness: 0.0449


=Relationships to other temperaments=
=Relationships to other temperaments=
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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>
<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:32:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Würschmidt"&gt;Würschmidt&lt;/a&gt;&lt;/div&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:34:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Würschmidt"&gt;Würschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Würschmidt"&gt;Würschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Würschmidt"&gt;Würschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-Worseschmidt"&gt;Worseschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-Worseschmidt"&gt;Worseschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Worschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Worschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hemiwürschmidt"&gt;Hemiwürschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hemiwürschmidt"&gt;Hemiwürschmidt&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit-Hemithir"&gt;Hemithir&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit-Hemithir"&gt;Hemithir&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemiwürschmidt-Hemiwur"&gt;Hemiwur&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemiwürschmidt-Hemiwur"&gt;Hemiwur&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-Hemiwur-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Relationships to other temperaments"&gt;Relationships to other temperaments&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-Hemiwur-Hemiwar"&gt;Hemiwar&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Relationships to other temperaments"&gt;Relationships to other temperaments&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:49 --&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;
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&lt;!-- ws:end:WikiTextTocRule:52 --&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 würschmidt 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;
  The &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit"&gt;5-limit&lt;/a&gt; parent comma for the würschmidt 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;
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Badness: 0.0284&lt;br /&gt;
Badness: 0.0284&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Hemiwürschmidt-Hemiwur-Hemiwar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Hemiwar&lt;/h3&gt;
Commas: 66/65, 105/104, 121/120, 1375/1372&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~28/25 = 193.698&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 15 4 7 11 23|, &amp;lt;0 -16 -2 -5 -9 -23|]&lt;br /&gt;
EDOs: 31&lt;br /&gt;
Badness: 0.0449&lt;br /&gt;
&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;
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>
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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