Würschmidt family: Difference between revisions

Revision as of 17:18, 16 December 2011

IMPORTED REVISION FROM WIKISPACES

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This revision was by author genewardsmith and made on 2011-12-16 17:18:15 UTC.

The original revision id was 286976428.

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[[toc|flat]]
=Wuerschmidt= 
The [[5-limit]]parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma, and named after José Würschmidt, Its [[monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[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 Wuerschmift comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]]. Wuerschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[MOS]] all possibilities.

[[POTE tuning|POTE generator]]: 387.799

Map: [<1 7 3|, <0 -8 -1|]

EDOs: [[31edo|31]], [[34edo|34]], [[46edo]], [[53edo|53]], [[65edo|65]], [[99edo|99]], [[164edo|164]], [[721edo|721c]]

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

=Wurschmidt= 
Wurschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo]] or [[127edo]] can be used as tunings. Wurschmidt has <<8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version <<8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. [[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

[[POTE tuning|POTE generator]]: 387.383

Map: [<1 7 3 15|, <0 -8 -1 -18|]
EDOs: [[31edo|31]], [[43edo|43]], [[53edo|53]], [[74edo|74]], [[84edo|84]], [[96edo|96]], [[127edo|127]], [[285edo|28bd]], [[412edo|412bd]]
Badness: 0.0508

==11-limit==
Commas: 99/98, 176/175, 243/242

POTE generator: ~5/4 = 387.447

Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]
EDOs: 31, 65d, 96, 127, 223d
Badness: 0.0244

=Worschmidt= 
Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo]], [[34edo]], or [[127edo]] as a tuning. If 127 is used, note that the val is <127 201 295 356| and not <127 201 295 357| as with wurschmidt. The wedgie now is <<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.

Commas: 126/125, 33075/32768

[[POTE tuning|POTE generator]]: 387.392

Map: [<1 7 3 -6|, <0 -8 -1 13|]
EDOs: [[31edo|31]], [[77edo|77]], [[86edo|86]], [[96edo|96d]], [[127edo|127d]]
Badness: 0.0646

==11-limit==
Commas: 126/125, 243/242, 385/384

POTE generator: ~5/4 = 387.407

Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]
EDOs: 31, 65, 96d, 127d
Badness: 0.0334

=Whirrschmidt= 
[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with <<8 1 52 -17 60 118|| for a wedgie.

Commas: 4375/4374, 393216/390625

[[POTE tuning|POTE generator]]: 387.881

Map: [<1 7 3 38|, <0 -8 -1 -52|]

EDOs: [[31edo|31]], [[34edo|34]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[68edo|68]], [[87edo|87]], [[99edo|99]]

=Hemiwuerschmidt= 
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. [[68edo]], [[99edo]] and [[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, <<16 2 5 40 -39 -49 -48 28...

Commas: 2401/2400, 3136/3125

[[POTE tuning|POTE generator]]: ~28/25 = 193.898

Map: [<1 15 4 7|, <0 -16 -2 -5|]
<<16 2 5 -34 -37 6||
EDOs: [[6edo|6]], [[31edo|31]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[77edo|77]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]], [[557edo|557c]], [[885edo|885c]]
Badness: 0.0203

==11-limit== 
Commas: 243/242, 441/440, 3136/3125

[[POTE tuning|POTE generator]]: ~28/25 = 193.840

Map: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]
EDOs: 31, 41, 46, 58, 72, 84, 89, 99e, 108, 130, 650ce, 811ce
Badness: 0.0211
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span>
=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 [[skwares]] as a 2.3.7.11 temperament.

Original HTML content:

<html><head><title>Würschmidt family</title></head><body><!-- ws:start:WikiTextTocRule:20:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#Wuerschmidt">Wuerschmidt</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Wurschmidt">Wurschmidt</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Worschmidt">Worschmidt</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Whirrschmidt">Whirrschmidt</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Hemiwuerschmidt">Hemiwuerschmidt</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Relationships to other temperaments">Relationships to other temperaments</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: -->
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Wuerschmidt"></a><!-- ws:end:WikiTextHeadingRule:0 -->Wuerschmidt</h1>
 The <a class="wiki_link" href="/5-limit">5-limit</a>parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma, and named after José Würschmidt, Its <a class="wiki_link" href="/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="/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 Wuerschmift comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the <a class="wiki_link" href="/minimax%20tuning">minimax tuning</a>. Wuerschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note <a class="wiki_link" href="/MOS">MOS</a> all possibilities.<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 387.799<br />
<br />
Map: [&lt;1 7 3|, &lt;0 -8 -1|]<br />
<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/46edo">46edo</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a>, <a class="wiki_link" href="/99edo">99</a>, <a class="wiki_link" href="/164edo">164</a>, <a class="wiki_link" href="/721edo">721c</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Wuerschmidt-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Seven limit children</h2>
 The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> 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;.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Wurschmidt"></a><!-- ws:end:WikiTextHeadingRule:4 -->Wurschmidt</h1>
 Wurschmidt, aside from the commas listed above, also tempers out 225/224. <a class="wiki_link" href="/31edo">31edo</a> or <a class="wiki_link" href="/127edo">127edo</a> 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. <a class="wiki_link" href="/127edo">127edo</a> 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.<br />
<br />
Commas: 225/224, 8748/8575<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 387.383<br />
<br />
Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/43edo">43</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/74edo">74</a>, <a class="wiki_link" href="/84edo">84</a>, <a class="wiki_link" href="/96edo">96</a>, <a class="wiki_link" href="/127edo">127</a>, <a class="wiki_link" href="/285edo">28bd</a>, <a class="wiki_link" href="/412edo">412bd</a><br />
Badness: 0.0508<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Wurschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2>
Commas: 99/98, 176/175, 243/242<br />
<br />
POTE generator: ~5/4 = 387.447<br />
<br />
Map: [&lt;1 7 3 15 17|, &lt;0 -8 -1 -18 -20|]<br />
EDOs: 31, 65d, 96, 127, 223d<br />
Badness: 0.0244<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Worschmidt"></a><!-- ws:end:WikiTextHeadingRule:8 -->Worschmidt</h1>
 Worschmidt tempers out 126/125 rather than 225/224, and can use <a class="wiki_link" href="/31edo">31edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, or <a class="wiki_link" href="/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 />
<br />
Commas: 126/125, 33075/32768<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 387.392<br />
<br />
Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/86edo">86</a>, <a class="wiki_link" href="/96edo">96d</a>, <a class="wiki_link" href="/127edo">127d</a><br />
Badness: 0.0646<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Worschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2>
Commas: 126/125, 243/242, 385/384<br />
<br />
POTE generator: ~5/4 = 387.407<br />
<br />
Map: [&lt;1 7 3 -6 17|, &lt;0 -8 -1 13 -20|]<br />
EDOs: 31, 65, 96d, 127d<br />
Badness: 0.0334<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Whirrschmidt"></a><!-- ws:end:WikiTextHeadingRule:12 -->Whirrschmidt</h1>
 <a class="wiki_link" href="/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 />
<br />
Commas: 4375/4374, 393216/390625<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 387.881<br />
<br />
Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]<br />
<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/99edo">99</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Hemiwuerschmidt"></a><!-- ws:end:WikiTextHeadingRule:14 -->Hemiwuerschmidt</h1>
 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. <a class="wiki_link" href="/68edo">68edo</a>, <a class="wiki_link" href="/99edo">99edo</a> and <a class="wiki_link" href="/130edo">130edo</a> 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...<br />
<br />
Commas: 2401/2400, 3136/3125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~28/25 = 193.898<br />
<br />
Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]<br />
&lt;&lt;16 2 5 -34 -37 6||<br />
EDOs: <a class="wiki_link" href="/6edo">6</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/99edo">99</a>, <a class="wiki_link" href="/229edo">229</a>, <a class="wiki_link" href="/328edo">328</a>, <a class="wiki_link" href="/557edo">557c</a>, <a class="wiki_link" href="/885edo">885c</a><br />
Badness: 0.0203<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Hemiwuerschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->11-limit</h2>
 Commas: 243/242, 441/440, 3136/3125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~28/25 = 193.840<br />
<br />
Map: [&lt;1 15 4 7 37|, &lt;0 -16 -2 -5 -40|]<br />
EDOs: 31, 41, 46, 58, 72, 84, 89, 99e, 108, 130, 650ce, 811ce<br />
Badness: 0.0211<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 />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Relationships to other temperaments"></a><!-- ws:end:WikiTextHeadingRule:18 -->Relationships to other temperaments</h1>
 2-Wuerschmidt, the temperament with all the same commas as Wuerschmidt but a generator of twice the size, is equivalent to <a class="wiki_link" href="/skwares">skwares</a> as a 2.3.7.11 temperament.</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-12-16 16:56:26 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-16 17:18:15 UTC</tt>.<br>
-: The original revision id was <tt>286973024</tt>.<br>
+: The original revision id was <tt>286976428</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>
@@ Line 27: / Line 27: @@
 Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]
+EDOs: [[31edo|31]], [[43edo|43]], [[53edo|53]], [[74edo|74]], [[84edo|84]], [[96edo|96]], [[127edo|127]], [[285edo|28bd]], [[412edo|412bd]]
+Badness: 0.0508
-EDOs: [[31edo|31]], [[43edo|43]], [[53edo|53]], [[74edo|74]], [[84edo|84]], [[96edo|96]], [[127edo|127]], [[285edo|28bd]], [[412edo|412bd]]
+==11-limit==
+Commas: 99/98, 176/175, 243/242
+POTE generator: ~5/4 = 387.447
+Map: [&lt;1 7 3 15 17|, &lt;0 -8 -1 -18 -20|]
+EDOs: 31, 65d, 96, 127, 223d
+Badness: 0.0244
 =Worschmidt=
@@ Line 85: / Line 94: @@
 -Wuerschmidt, the temperament with all the same commas as Wuerschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 temperament.</pre></div>
 <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:18:&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:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;a href="#Wuerschmidt"&gt;Wuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Wurschmidt"&gt;Wurschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Hemiwuerschmidt"&gt;Hemiwuerschmidt&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="#Relationships to other temperaments"&gt;Relationships to other temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&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="#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;
-&lt;!-- ws:end:WikiTextTocRule:28 --&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="Wuerschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Wuerschmidt&lt;/h1&gt;
   The &lt;a class="wiki_link" href="/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="/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="/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 Wuerschmift comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the &lt;a class="wiki_link" href="/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="/MOS"&gt;MOS&lt;/a&gt; all possibilities.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 106: / Line 115: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 7 3 15|, &amp;lt;0 -8 -1 -18|]&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/74edo"&gt;74&lt;/a&gt;, &lt;a class="wiki_link" href="/84edo"&gt;84&lt;/a&gt;, &lt;a class="wiki_link" href="/96edo"&gt;96&lt;/a&gt;, &lt;a class="wiki_link" href="/127edo"&gt;127&lt;/a&gt;, &lt;a class="wiki_link" href="/285edo"&gt;28bd&lt;/a&gt;, &lt;a class="wiki_link" href="/412edo"&gt;412bd&lt;/a&gt;&lt;br /&gt;
+Badness: 0.0508&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;
+Commas: 99/98, 176/175, 243/242&lt;br /&gt;
+&lt;br /&gt;
+POTE generator: ~5/4 = 387.447&lt;br /&gt;
 &lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/74edo"&gt;74&lt;/a&gt;, &lt;a class="wiki_link" href="/84edo"&gt;84&lt;/a&gt;, &lt;a class="wiki_link" href="/96edo"&gt;96&lt;/a&gt;, &lt;a class="wiki_link" href="/127edo"&gt;127&lt;/a&gt;, &lt;a class="wiki_link" href="/285edo"&gt;28bd&lt;/a&gt;, &lt;a class="wiki_link" href="/412edo"&gt;412bd&lt;/a&gt;&lt;br /&gt;
+Map: [&amp;lt;1 7 3 15 17|, &amp;lt;0 -8 -1 -18 -20|]&lt;br /&gt;
+EDOs: 31, 65d, 96, 127, 223d&lt;br /&gt;
+Badness: 0.0244&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Worschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Worschmidt&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Worschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Worschmidt&lt;/h1&gt;
   Worschmidt tempers out 126/125 rather than 225/224, and can use &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, or &lt;a class="wiki_link" href="/127edo"&gt;127edo&lt;/a&gt; as a tuning. If 127 is used, note that the val is &amp;lt;127 201 295 356| and not &amp;lt;127 201 295 357| as with wurschmidt. The wedgie now is &amp;lt;&amp;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.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 120: / Line 138: @@
 Badness: 0.0646&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Worschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Worschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
 Commas: 126/125, 243/242, 385/384&lt;br /&gt;
 &lt;br /&gt;
@@ Line 129: / Line 147: @@
 Badness: 0.0334&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Whirrschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Whirrschmidt&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Whirrschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Whirrschmidt&lt;/h1&gt;
   &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with &amp;lt;&amp;lt;8 1 52 -17 60 118|| for a wedgie.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 140: / Line 158: @@
 EDOs: &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/68edo"&gt;68&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Hemiwuerschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Hemiwuerschmidt&lt;/h1&gt;
+&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;
   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="/68edo"&gt;68edo&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; and &lt;a class="wiki_link" href="/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;
 &lt;br /&gt;
@@ Line 152: / Line 170: @@
 Badness: 0.0203&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Hemiwuerschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&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="Hemiwuerschmidt-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;
 &lt;br /&gt;
@@ Line 161: / Line 179: @@
 Badness: 0.0211&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:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Relationships to other temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&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;
 -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="/skwares"&gt;skwares&lt;/a&gt; as a 2.3.7.11 temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>

Würschmidt family: Difference between revisions

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