Würschmidt family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 235889414 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 235960016 - 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: | : 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> | : 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> | ||
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=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>, 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. | 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 | [[POTE tuning|POTE generator]]: 387.799 | ||
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Map: [<1 7 3|, <0 -8 -1|] | Map: [<1 7 3|, <0 -8 -1|] | ||
EDOs: 31, 34, 65, 164 | EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[164edo|164]] | ||
==Seven limit children== | ==Seven limit children== | ||
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Map: [<1 7 3 15|, <0 -8 -1 -18|] | Map: [<1 7 3 15|, <0 -8 -1 -18|] | ||
EDOs: 31, 127 | EDOs: [[31edo|31]], [[127edo|127]] | ||
=Worschmidt= | =Worschmidt= | ||
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Map: [<1 7 3 -6|, <0 -8 -1 13|] | Map: [<1 7 3 -6|, <0 -8 -1 13|] | ||
EDOs: 31, 127 | EDOs: [[31edo|31]], [[127edo|127]] | ||
=Whirrschmidt= | =Whirrschmidt= | ||
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Map: [<1 7 3 38|, <0 -8 -1 -52|] | Map: [<1 7 3 38|, <0 -8 -1 -52|] | ||
EDOs: 31, 34, 99 | EDOs: [[31edo|31]], [[34edo|34]], [[99edo|99]] | ||
=Hemiwuerschmift= | =Hemiwuerschmift= | ||
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Map: [<1 15 4 7|, <0 -16 -2 -5|] | Map: [<1 15 4 7|, <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"><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: --> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Würschmidt family</title></head><body><!-- ws:start:WikiTextTocRule: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:end:WikiTextTocRule:19 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Wuerschmidt"></a><!-- ws:end:WikiTextHeadingRule:0 -->Wuerschmidt</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Wuerschmidt"></a><!-- ws:end:WikiTextHeadingRule:0 -->Wuerschmidt</h1> | ||
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.<br /> | 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 /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 387.799<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 387.799<br /> | ||
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Map: [&lt;1 7 3|, &lt;0 -8 -1|]<br /> | Map: [&lt;1 7 3|, &lt;0 -8 -1|]<br /> | ||
<br /> | <br /> | ||
EDOs: 31, 34, 65, 164<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 /> | <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> | <!-- 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> | ||
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Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]<br /> | Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]<br /> | ||
<br /> | <br /> | ||
EDOs: 31, 127<br /> | EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/127edo">127</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Worschmidt"></a><!-- ws:end:WikiTextHeadingRule:6 -->Worschmidt</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Worschmidt"></a><!-- ws:end:WikiTextHeadingRule:6 -->Worschmidt</h1> | ||
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Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]<br /> | Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]<br /> | ||
<br /> | <br /> | ||
EDOs: 31, 127<br /> | EDOs: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/127edo">127</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Whirrschmidt"></a><!-- ws:end:WikiTextHeadingRule:8 -->Whirrschmidt</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Whirrschmidt"></a><!-- ws:end:WikiTextHeadingRule:8 -->Whirrschmidt</h1> | ||
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Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]<br /> | Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]<br /> | ||
<br /> | <br /> | ||
EDOs: 31, 34, 99<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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Hemiwuerschmift"></a><!-- ws:end:WikiTextHeadingRule:10 -->Hemiwuerschmift</h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Hemiwuerschmift"></a><!-- ws:end:WikiTextHeadingRule:10 -->Hemiwuerschmift</h1> | ||
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Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]<br /> | Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]<br /> | ||
<br /> | <br /> | ||
EDOs: 31, 99, 229</body></html></pre></div> | 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></pre></div> | ||