Würschmidt family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 286973024 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 286976428 - 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>2011-12-16 | : 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> | : The original revision id was <tt>286976428</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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Map: [<1 7 3 15|, <0 -8 -1 -18|] | 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= | ||
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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.</pre></div> | 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.</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: | <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: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: | <!-- 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 /> | 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 /> | ||
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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 /> | ||
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 /> | <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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | 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 /> | ||
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Badness: 0.0646<br /> | Badness: 0.0646<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | Commas: 126/125, 243/242, 385/384<br /> | ||
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Badness: 0.0334<br /> | Badness: 0.0334<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | <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 /> | ||
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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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | 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 /> | ||
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Badness: 0.0203<br /> | Badness: 0.0203<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | Commas: 243/242, 441/440, 3136/3125<br /> | ||
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Badness: 0.0211<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 /> | <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: | <!-- 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></pre></div> | 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></pre></div> | ||