Würschmidt family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
Wikispaces>genewardsmith
**Imported revision 235889414 - Original comment: **
Wikispaces>xenwolf
**Imported revision 235960016 - Original comment: **
Line 1: Line 1:
<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>2011-06-10 23:17:37 UTC</tt>.<br>
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-11 15:17:52 UTC</tt>.<br>
: The original revision id was <tt>235889414</tt>.<br>
: The original revision id was <tt>235960016</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>
Line 9: Line 9:


=Wuerschmidt=
=Wuerschmidt=
The 5-limit parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma. Its monzo is |17 1 -8&gt;, and flipping that yields &lt;&lt;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.
The [[5-limit]] parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma. Its [[monzo]] is |17 1 -8&gt;, and flipping that yields &lt;&lt;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
[[POTE tuning|POTE generator]]: 387.799
Line 15: Line 15:
Map: [&lt;1 7 3|, &lt;0 -8 -1|]
Map: [&lt;1 7 3|, &lt;0 -8 -1|]


EDOs: 31, 34, 65, 164
EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[164edo|164]]


==Seven limit children==
==Seven limit children==
Line 29: Line 29:
Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]
Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]


EDOs: 31, 127
EDOs: [[31edo|31]], [[127edo|127]]


=Worschmidt=
=Worschmidt=
Line 40: Line 40:
Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]
Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]


EDOs: 31, 127
EDOs: [[31edo|31]], [[127edo|127]]


=Whirrschmidt=
=Whirrschmidt=
Line 51: Line 51:
Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]
Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]


EDOs: 31, 34, 99
EDOs: [[31edo|31]], [[34edo|34]], [[99edo|99]]


=Hemiwuerschmift=
=Hemiwuerschmift=
Line 62: Line 62:
Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]
Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]


EDOs: 31, 99, 229</pre></div>
EDOs: [[31edo|31]], [[99edo|99]], [[229edo|229]]</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:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Wuerschmidt"&gt;Wuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Wurschmidt"&gt;Wurschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Hemiwuerschmift"&gt;Hemiwuerschmift&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&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:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Wuerschmidt"&gt;Wuerschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Wurschmidt"&gt;Wurschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Hemiwuerschmift"&gt;Hemiwuerschmift&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&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: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 5-limit parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma. Its monzo is |17 1 -8&amp;gt;, and flipping that yields &amp;lt;&amp;lt;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.&lt;br /&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. 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;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.799&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.799&lt;br /&gt;
Line 73: Line 73:
Map: [&amp;lt;1 7 3|, &amp;lt;0 -8 -1|]&lt;br /&gt;
Map: [&amp;lt;1 7 3|, &amp;lt;0 -8 -1|]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
EDOs: 31, 34, 65, 164&lt;br /&gt;
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="/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="/164edo"&gt;164&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="Wuerschmidt-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Seven limit children&lt;/h2&gt;
Line 87: Line 87:
Map: [&amp;lt;1 7 3 15|, &amp;lt;0 -8 -1 -18|]&lt;br /&gt;
Map: [&amp;lt;1 7 3 15|, &amp;lt;0 -8 -1 -18|]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
EDOs: 31, 127&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/127edo"&gt;127&lt;/a&gt;&lt;br /&gt;
&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: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;
Line 98: Line 98:
Map: [&amp;lt;1 7 3 -6|, &amp;lt;0 -8 -1 13|]&lt;br /&gt;
Map: [&amp;lt;1 7 3 -6|, &amp;lt;0 -8 -1 13|]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
EDOs: 31, 127&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/127edo"&gt;127&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Whirrschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Whirrschmidt&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Whirrschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Whirrschmidt&lt;/h1&gt;
Line 109: Line 109:
Map: [&amp;lt;1 7 3 38|, &amp;lt;0 -8 -1 -52|]&lt;br /&gt;
Map: [&amp;lt;1 7 3 38|, &amp;lt;0 -8 -1 -52|]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
EDOs: 31, 34, 99&lt;br /&gt;
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="/99edo"&gt;99&lt;/a&gt;&lt;br /&gt;
&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="Hemiwuerschmift"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Hemiwuerschmift&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Hemiwuerschmift"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Hemiwuerschmift&lt;/h1&gt;
Line 120: Line 120:
Map: [&amp;lt;1 15 4 7|, &amp;lt;0 -16 -2 -5|]&lt;br /&gt;
Map: [&amp;lt;1 15 4 7|, &amp;lt;0 -16 -2 -5|]&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
EDOs: 31, 99, 229&lt;/body&gt;&lt;/html&gt;</pre></div>
EDOs: &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/229edo"&gt;229&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 15:17, 11 June 2011

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author xenwolf and made on 2011-06-11 15:17:52 UTC.
The original revision id was 235960016.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

[[toc|flat]]

=Wuerschmidt=
The [[5-limit]] parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma. 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]], [[65edo|65]], [[164edo|164]]

==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]], [[127edo|127]]

=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]], [[127edo|127]]

=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]], [[99edo|99]]

=Hemiwuerschmift=
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]]: 193.898

Map: [<1 15 4 7|, <0 -16 -2 -5|]

EDOs: [[31edo|31]], [[99edo|99]], [[229edo|229]]

Original HTML content:

<html><head><title>Würschmidt family</title></head><body><!-- ws:start:WikiTextTocRule:12:&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:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Wuerschmidt">Wuerschmidt</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Wurschmidt">Wurschmidt</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Worschmidt">Worschmidt</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Whirrschmidt">Whirrschmidt</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Hemiwuerschmift">Hemiwuerschmift</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: -->
<!-- ws:end:WikiTextTocRule:19 --><br />
<!-- 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. 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="/65edo">65</a>, <a class="wiki_link" href="/164edo">164</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 />
<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/127edo">127</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Worschmidt"></a><!-- ws:end:WikiTextHeadingRule:6 -->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 />
<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/127edo">127</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Whirrschmidt"></a><!-- ws:end:WikiTextHeadingRule:8 -->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="/99edo">99</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Hemiwuerschmift"></a><!-- ws:end:WikiTextHeadingRule:10 -->Hemiwuerschmift</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>: 193.898<br />
<br />
Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]<br />
<br />
EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/99edo">99</a>, <a class="wiki_link" href="/229edo">229</a></body></html>