Würschmidt family
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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]], [[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]] =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]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]] Badness: 0.0203 =Doppelwuerschmidt= Doppelwuerschmidt uses a generator an approximate 25th harmonic for the genarator. To arrive exactly at 6/1, cents (775.489)^4 is best, and yields a spanking 14-note sLsssLssssLssss scale. In addition to 25/16 and 6/1, it also arrives approximates many other consonanzen, like 7/4, 7/6, 11/9, etc. and especially 11/8. Commas: 390625/279963 etc. [[POTE tuning|POTE generator]]: ~25/16 = 775 c Map: EDOs: (if it works for wuerschmidt it works for doppelwuerschmidt) Badness: ==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 <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately </span>
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<html><head><title>Würschmidt family</title></head><body><!-- ws:start:WikiTextTocRule:16:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Wuerschmidt">Wuerschmidt</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Wurschmidt">Wurschmidt</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Worschmidt">Worschmidt</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Whirrschmidt">Whirrschmidt</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Hemiwuerschmidt">Hemiwuerschmidt</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Doppelwuerschmidt">Doppelwuerschmidt</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> <!-- ws:end:WikiTextTocRule:25 --><!-- 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, and named after José Würschmidt, 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="Hemiwuerschmidt"></a><!-- ws:end:WikiTextHeadingRule:10 -->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, <<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:<h1> --><h1 id="toc6"><a name="Doppelwuerschmidt"></a><!-- ws:end:WikiTextHeadingRule:12 -->Doppelwuerschmidt</h1> Doppelwuerschmidt uses a generator an approximate 25th harmonic for the genarator. To arrive exactly at 6/1, cents (775.489)^4 is best, and yields a spanking 14-note sLsssLssssLssss scale. In addition to 25/16 and 6/1, it also arrives approximates many other consonanzen, like 7/4, 7/6, 11/9, etc. and especially 11/8.<br /> <br /> Commas: 390625/279963 etc.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~25/16 = 775 c<br /> <br /> Map:<br /> EDOs: (if it works for wuerschmidt it works for doppelwuerschmidt)<br /> Badness:<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Doppelwuerschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->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<br /> <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately </span></body></html>