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Wikispaces>xenwolf **Imported revision 235960016 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 240555349 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-07-08 16:44:29 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>240555349</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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Commas: 2401/2400, 3136/3125 | Commas: 2401/2400, 3136/3125 | ||
[[POTE tuning|POTE generator]]: 193.898 | [[POTE tuning|POTE generator]]: ~28/25 = 193.898 | ||
Map: [<1 15 4 7|, <0 -16 -2 -5|] | Map: [<1 15 4 7|, <0 -16 -2 -5|] | ||
<<16 2 5 -34 -37 6|| | |||
EDOs: [[6edo|6]], [[31edo|31]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]] | |||
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, 130 | |||
Badness: 0.0211 | |||
</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:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Wuerschmidt">Wuerschmidt</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Wurschmidt">Wurschmidt</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Worschmidt">Worschmidt</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Whirrschmidt">Whirrschmidt</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Hemiwuerschmift">Hemiwuerschmift</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><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 <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 /> | 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 /> | ||
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Commas: 2401/2400, 3136/3125<br /> | Commas: 2401/2400, 3136/3125<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 193.898<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~28/25 = 193.898<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]<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="/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><br /> | |||
Badness: 0.0203<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Hemiwuerschmift-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->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 /> | <br /> | ||
Map: [&lt;1 15 4 7 37|, &lt;0 -16 -2 -5 -40|]<br /> | |||
EDOs: 31, 130<br /> | |||
Badness: 0.0211</body></html></pre></div> |
Revision as of 16:44, 8 July 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 genewardsmith and made on 2011-07-08 16:44:29 UTC.
- The original revision id was 240555349.
- 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]]: ~28/25 = 193.898 Map: [<1 15 4 7|, <0 -16 -2 -5|] <<16 2 5 -34 -37 6|| EDOs: [[6edo|6]], [[31edo|31]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]] 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, 130 Badness: 0.0211
Original HTML content:
<html><head><title>Würschmidt family</title></head><body><!-- ws:start:WikiTextTocRule:14:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Wuerschmidt">Wuerschmidt</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Wurschmidt">Wurschmidt</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Worschmidt">Worschmidt</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Whirrschmidt">Whirrschmidt</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Hemiwuerschmift">Hemiwuerschmift</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> <!-- ws:end:WikiTextTocRule:22 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><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>, and flipping that yields <<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: [<1 7 3|, <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:<h2> --><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>, 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>.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><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 <<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. <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: [<1 7 3 15|, <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:<h1> --><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 <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. <br /> <br /> Commas: 126/125, 33075/32768<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 387.392<br /> <br /> Map: [<1 7 3 -6|, <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:<h1> --><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 <<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: [<1 7 3 38|, <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:<h1> --><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, <<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: [<1 15 4 7|, <0 -16 -2 -5|]<br /> <<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="/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><br /> Badness: 0.0203<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Hemiwuerschmift-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->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: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]<br /> EDOs: 31, 130<br /> Badness: 0.0211</body></html>