Würschmidt family
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[[toc|flat]] =Würschmidt= The [[xenharmonic/5-limit|5-limit]] parent comma for the wuerschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its [[xenharmonic/monzo|monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[xenharmonic/generator|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 Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[xenharmonic/minimax tuning|minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[xenharmonic/MOS|MOS]] all possibilities. [[xenharmonic/POTE tuning|POTE generator]]: 387.799 Map: [<1 7 3|, <0 -8 -1|] EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/34edo|34]], [[xenharmonic/65edo|65]], [[xenharmonic/99edo|99]], [[xenharmonic/164edo|164]], [[xenharmonic/721edo|721c]], [[xenharmonic/885edo|885c]] ==Seven limit children== The second comma of the [[xenharmonic/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>. =Würschmidt= Würschmidt, aside from the commas listed above, also tempers out 225/224. [[xenharmonic/31edo|31edo]] or [[xenharmonic/127edo|127edo]] can be used as tunings. Würschmidt 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. [[xenharmonic/127edo|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 [[xenharmonic/POTE tuning|POTE generator]]: 387.383 Map: [<1 7 3 15|, <0 -8 -1 -18|] EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/96edo|96]], [[xenharmonic/127edo|127]], [[xenharmonic/285edo|28bd]], [[xenharmonic/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 tempers out 126/125 rather than 225/224, and can use [[xenharmonic/31edo|31edo]], [[xenharmonic/34edo|34edo]], or [[xenharmonic/127edo|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 [[xenharmonic/POTE tuning|POTE generator]]: 387.392 Map: [<1 7 3 -6|, <0 -8 -1 13|] EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/65edo|65]], [[xenharmonic/96edo|96d]], [[xenharmonic/127edo|127d]] Badness: 0.0646 ==11-limit== Commas: 126/125, 243/242, 385/384 POTE generator: ~5/4 = 387.407 Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|] EDOs: 31, 65, 96d, 127d Badness: 0.0334 =Whirrschmidt= [[xenharmonic/99edo|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 [[xenharmonic/POTE tuning|POTE generator]]: 387.881 Map: [<1 7 3 38|, <0 -8 -1 -52|] EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/34edo|34]], [[xenharmonic/65edo|65]], [[xenharmonic/99edo|99]] =Hemiwürschmidt= Hemiwürschmidt, 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. [[xenharmonic/68edo|68edo]], [[xenharmonic/99edo|99edo]] and [[xenharmonic/130edo|130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, <<16 2 5 40 -39 -49 -48 28... Commas: 2401/2400, 3136/3125 [[xenharmonic/POTE tuning|POTE generator]]: ~28/25 = 193.898 Map: [<1 15 4 7|, <0 -16 -2 -5|] <<16 2 5 -34 -37 6|| EDOs: [[xenharmonic/6edo|6]], [[xenharmonic/31edo|31]], [[xenharmonic/37edo|37]], [[xenharmonic/68edo|68]], [[xenharmonic/99edo|99]], [[xenharmonic/229edo|229]], [[xenharmonic/328edo|328]], [[xenharmonic/557edo|557c]], [[xenharmonic/885edo|885c]] Badness: 0.0203 ==11-limit== Commas: 243/242, 441/440, 3136/3125 [[xenharmonic/POTE tuning|POTE generator]]: ~28/25 = 193.840 Map: [<1 15 4 7 37|, <0 -16 -2 -5 -40|] EDOs: 31, 99e, 130, 650ce, 811ce 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> =Relationships to other temperaments= 2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to [[xenharmonic/skwares|skwares]] as a 2.3.7.11 temperament.
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<html><head><title>Würschmidt family</title></head><body><!-- ws:start:WikiTextTocRule:20:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#Würschmidt">Würschmidt</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Würschmidt">Würschmidt</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="#Hemiwürschmidt">Hemiwürschmidt</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:31 --><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Würschmidt"></a><!-- ws:end:WikiTextHeadingRule:0 -->Würschmidt</h1> The <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit">5-limit</a> parent comma for the wuerschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/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 Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/minimax%20tuning">minimax tuning</a>. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS">MOS</a> all possibilities.<br /> <br /> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning">POTE generator</a>: 387.799<br /> <br /> Map: [<1 7 3|, <0 -8 -1|]<br /> <br /> EDOs: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo">34</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo">65</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/164edo">164</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/721edo">721c</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/885edo">885c</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Würschmidt-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Seven limit children</h2> The second comma of the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="Würschmidt"></a><!-- ws:end:WikiTextHeadingRule:4 -->Würschmidt</h1> Würschmidt, aside from the commas listed above, also tempers out 225/224. <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31edo</a> or <a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo">127edo</a> can be used as tunings. Würschmidt 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="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/POTE%20tuning">POTE generator</a>: 387.383<br /> <br /> Map: [<1 7 3 15|, <0 -8 -1 -18|]<br /> EDOs: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/96edo">96</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo">127</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/285edo">28bd</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/412edo">412bd</a><br /> Badness: 0.0508<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Würschmidt-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 /> Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]<br /> EDOs: 31, 65d, 96, 127, 223d<br /> Badness: 0.0244<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><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="http://xenharmonic.wikispaces.com/31edo">31edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo">34edo</a>, or <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/POTE%20tuning">POTE generator</a>: 387.392<br /> <br /> Map: [<1 7 3 -6|, <0 -8 -1 13|]<br /> EDOs: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo">65</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/96edo">96d</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo">127d</a><br /> Badness: 0.0646<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Worschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2> Commas: 126/125, 243/242, 385/384<br /> <br /> POTE generator: ~5/4 = 387.407<br /> <br /> Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]<br /> EDOs: 31, 65, 96d, 127d<br /> Badness: 0.0334<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Whirrschmidt"></a><!-- ws:end:WikiTextHeadingRule:12 -->Whirrschmidt</h1> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/31edo">31</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo">34</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo">65</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Hemiwürschmidt"></a><!-- ws:end:WikiTextHeadingRule:14 -->Hemiwürschmidt</h1> Hemiwürschmidt, 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="http://xenharmonic.wikispaces.com/68edo">68edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/130edo">130edo</a> can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt 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="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/6edo">6</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/37edo">37</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/68edo">68</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo">99</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/229edo">229</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/328edo">328</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/557edo">557c</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/885edo">885c</a><br /> Badness: 0.0203<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Hemiwürschmidt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->11-limit</h2> Commas: 243/242, 441/440, 3136/3125<br /> <br /> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/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, 99e, 130, 650ce, 811ce<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 /> <!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Relationships to other temperaments"></a><!-- ws:end:WikiTextHeadingRule:18 -->Relationships to other temperaments</h1> 2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/skwares">skwares</a> as a 2.3.7.11 temperament.</body></html>