Würschmidt family: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

__FORCETOC__

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-14 12:22:05 UTC</tt>.<br>~~

~~: The original revision id was <tt>589302500</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>~~

~~<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]

Würschmidt

The [[xenharmonic/5-limit|5-limit]] parent comma for the würschmidt 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

The [[5-limit|5-limit]] parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its [[monzo|monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[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 [[Minimax_tuning|minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[MOS|MOS]] all possibilities.

[[POTE_tuning|POTE generator]]: 387.799

Map: [<1 7 3|, <0 -8 -1|]

EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[99edo|99]], [[164edo|164]], [[721edo|721c]], [[885edo|885c]]

[http://chrisvaisvil.com/ancient-stardust-wurschmidt13/ Ancient Stardust] [http://micro.soonlabel.com/jake_freivald/tunings_by_jake_freivald/20130811_wurschmidt%5b13%5d.mp3 play] by Chris Vaisvil

~~[[http://chrisvaisvil.com/ancient-stardust-wurschmidt13/|Ancient Stardust]] [[http://micro.soonlabel.com/jake_freivald/tunings_by_jake_freivald/20130811_wurschmidt%5b13%5d.mp3|play]] by Chris Vaisvil~~

Würschmidt[13] in 5-limit minimax tuning

[[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt%5b16%5d-out.mp3|Extrospection]] by [[https://soundcloud.com/jdfreivald/extrospection|Jake Freivald]]; Würschmidt[16] tuned in 31et.

~~==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=

[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt%5b16%5d-out.mp3 Extrospection] by [https://soundcloud.com/jdfreivald/extrospection Jake Freivald]; Würschmidt[16] tuned in 31et.

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.

==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>.

=Würschmidt=

Würschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo|31edo]] or [[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. [[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

[[POTE_tuning|POTE generator]]: 387.383

Map: [<1 7 3 15|, <0 -8 -1 -18|]

EDOs: [[31edo|31]], [[96edo|96]], [[127edo|127]], [[285edo|28bd]], [[412edo|412bd]]

Badness: 0.0508

==11-limit==

Commas: 99/98, 176/175, 243/242

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Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]

EDOs: 31, 65d, 96, 127, 223d

Badness: 0.0244

==13-limit==

Commas: 99/98, 144/143, 176/175, 275/273

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Map: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]

EDOs: 31, 65d, 161df

Badness: 0.0236

==Worseschmidt==

Commas: 66/65, 99/98, 105/104, 243/242

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Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]

EDOs: 31

Badness: 0.0344

=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.

Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo|31edo]], [[34edo|34edo]], or [[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

[[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]]

EDOs: [[31edo|31]], [[65edo|65]], [[96edo|96d]], [[127edo|127d]]

Badness: 0.0646

==11-limit==

Commas: 126/125, 243/242, 385/384

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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.

[[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

[[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]]

EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[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...

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. [[68edo|68edo]], [[99edo|99edo]] and [[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

[[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]], [[37edo|37]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]], [[557edo|557c]], [[885edo|885c]]

Badness: 0.0203

==11-limit==

Commas: 243/242, 441/440, 3136/3125

[[~~xenharmonic/POTE tuning~~|POTE generator]]: ~28/25 = 193.840

[[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

===13-limit===

Commas: 243/242 351/350 441/440 3584/3575

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Map: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]

EDOs: 31, 99e, 130, 291, 421e, 551ce

Badness: 0.0231

===Hemithir===

Commas: 121/120 176/175 196/195 275/273

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Map: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]

EDOs: 31, 68e, 99ef

Badness: 0.0312

==Hemiwur==

Commas: 121/120, 176/175, 1375/1372

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Map: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]

EDOs: 6, 31, 68, 99, 130e, 229e

Badness: 0.0293

===13-limit===

Commas: 121/120, 176/175, 196/195, 275/273

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Map: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]

EDOs: 6, 31, 68, 99f, 167ef

Badness: 0.0284

===Hemiwar===

Commas: 66/65, 105/104, 121/120, 1375/1372

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Map: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]

EDOs: 31

Badness: 0.0449

=Relationships to other temperaments=

~~<~~span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"~~>~~around 775.489 which is approximately~~<~~/span~~>~~

<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span>

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.~~</pre></div>~~

~~<h4>Original HTML content:</h4>~~

2-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to [[Skwares|skwares]] as a 2.3.7.11 temperament.

~~<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><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 2em;"><a href="#x-Seven limit children">Seven limit children</a></div>

[[Category:family]]

~~<div style="margin-left: 1em;"><a href="#Würschmidt">Würschmidt</a></div>~~

[[Category:hemiwuerschmidt]]

~~<div style="margin-left: 2em;"><a href="#Würschmidt-11-limit">11-limit</a></div>~~

[[Category:theory]]

~~<div style="margin-left: 2em;"><a href="#Würschmidt-13-limit">13-limit</a></div>~~

[[Category:wuerschmidt]]

~~<div style="margin-left: 2em;"><a href="#Würschmidt-Worseschmidt">Worseschmidt</a></div>~~

[[Category:wurschmidt]]

~~<div style="margin-left: 1em;"><a href="#Worschmidt">Worschmidt</a></div>~~

~~<div style="margin-left: 2em;"><a href="#Worschmidt-11-limit">11-limit</a></div>~~

~~<div style="margin-left: 1em;"><a href="#Whirrschmidt">Whirrschmidt</a></div>~~

~~<div style="margin-left: 1em;"><a href="#Hemiwürschmidt">Hemiwürschmidt</a></div>~~

~~<div style="margin-left: 2em;"><a href="#Hemiwürschmidt-11-limit">11-limit</a></div>~~

<div style="margin-left: 3em;"><a href="#Hemiwürschmidt-11-limit-13-limit">13-limit</a></div>

<div style="margin-left: 3em;"><a href="#Hemiwürschmidt-11-limit-Hemithir">Hemithir</a></div>

~~<div style="margin-left: 2em;"><a href="#Hemiwürschmidt-Hemiwur">Hemiwur</a></div>~~

~~<div style="margin-left: 3em;"><a href="#Hemiwürschmidt-Hemiwur-13-limit">13-limit</a></div>~~

~~<div style="margin-left: 3em;"><a href="#Hemiwürschmidt-Hemiwur-Hemiwar">Hemiwar</a></div>~~

<div style="margin-left: 1em;"><a href="#Relationships to other temperaments">Relationships to other temperaments</a></div>

~~</div>~~

~~Würschmidt<br />~~

The <a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit">5-limit</a> parent comma for the würschmidt 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&gt;, and flipping that yields &lt;&lt;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: [&lt;1 7 3|, &lt;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 />~~

<a class="wiki_link_ext" href="http://chrisvaisvil.com/ancient-stardust-wurschmidt13/" rel="nofollow">Ancient Stardust</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/jake_freivald/tunings_by_jake_freivald/20130811_wurschmidt%5b13%5d.mp3" rel="nofollow">play</a> by Chris Vaisvil<br />

~~Würschmidt~~[~~13] in 5-limit minimax tuning<br />~~

<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt%5b16%5d-out.mp3" rel="nofollow">Extrospection</a> by <a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/extrospection" rel="nofollow">Jake Freivald</a>; Würschmidt[~~16] tuned in 31et.<br />~~

<h2 id="toc0"><a name="x-Seven limit children"></a>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&gt;, worschmidt adds 65625/65536 = |-16 1 5 1&gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&gt; and hemiwuerschmidt ~~adds 6144/6125 = |11 1 -3 -2&gt;.<br />~~

~~<br />~~

~~<h1 id="toc1"><a name="Würschmidt"></a>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 &lt;&lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &lt;&lt;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: [&lt;1 7 3 15|, &lt;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 />~~

~~<h2 id="toc2"><a name="Würschmidt-11-limit"></a>11-limit</h2>~~

~~Commas: 99/98, 176/175, 243/242<br />~~

~~<br />~~

~~POTE generator: ~5/4 = 387.447<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 />~~

~~<h2 id="toc3"><a name="Würschmidt-13-limit"></a>13-limit</h2>~~

~~Commas: 99/98, 144/143, 176/175, 275/273<br />~~

~~<br />~~

~~POTE generator: ~5/4 = 387.626<br />~~

~~<br />~~

~~Map:~~ [~~&lt;1 7 3 15 17 1|, &lt;0 -8 -1 -18 -20 4|]<br />~~

~~EDOs: 31, 65d, 161df<br />~~

~~Badness: 0.0236<br />~~

~~<br />~~

<h2 id="toc4"><a name="Würschmidt-Worseschmidt"></a>Worseschmidt</h2>

~~Commas: 66/65, 99/98, 105/104, 243/242<br />~~

~~<br />~~

~~POTE generator: ~5/4 = 387.099<br />~~

~~<br />~~

~~Map:~~ [~~&lt;1 7 3 15 17 22|, &lt;0 -8 -1 -18 -20 -27|]<br />~~

~~EDOs: 31<br />~~

~~Badness: 0.0344<br />~~

~~<br />~~

~~<h1 id="toc5"><a name="Worschmidt"></a>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 &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 />

~~<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: [&lt;1 7 3 -6|, &lt;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 />~~

~~<h2 id="toc6"><a name="Worschmidt-11-limit"></a>11-limit</h2>~~

~~Commas: 126/125, 243/242, 385/384<br />~~

~~<br />~~

~~POTE generator: ~5/4 = 387.407<br />~~

~~<br />~~

~~Map: [&lt;1 7 3 -6 17|, &lt;0 -8 -1 13 -20|~~]~~<br />~~

~~EDOs: 31, 65, 96d, 127d<br />~~

~~Badness: 0.0334<br />~~

~~<br />~~

~~<h1 id="toc7"><a name="Whirrschmidt"></a>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 &lt;&lt;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:~~ [~~&lt;1 7 3 38|, &lt;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 />~~

~~<h1 id="toc8"><a name="Hemiwürschmidt"></a>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, &lt;&lt;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:~~ [~~&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="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 />~~

~~<h2 id="toc9"><a name="Hemiwürschmidt-11-limit"></a>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: [&lt;1 15 4 7 37|, &lt;0 -16 -2 -5 -40|~~]~~<br />~~

~~EDOs: 31, 99e, 130, 650ce, 811ce<br />~~

~~Badness: 0.0211<br />~~

~~<br />~~

<h3 id="toc10"><a name="Hemiwürschmidt-11-limit-13-limit"></a>13-limit</h3>

~~Commas: 243/242 351/350 441/440 3584/3575<br />~~

~~<br />~~

~~POTE generator: ~28/25 = 193.840<br />~~

~~<br />~~

~~Map: [&lt;1 15 4 7 37 -29|, &lt;0 -16 -2 -5 -40 39|~~]~~<br />~~

~~EDOs: 31, 99e, 130, 291, 421e, 551ce<br />~~

~~Badness: 0.0231<br />~~

~~<br />~~

<h3 id="toc11"><a name="Hemiwürschmidt-11-limit-Hemithir"></a>Hemithir</h3>

~~Commas: 121/120 176/175 196/195 275/273<br />~~

~~<br />~~

~~POTE generator: ~28/25 = 193.918<br />~~

~~<br />~~

~~Map:~~ [~~&lt;1 15 4 7 37 -3|, &lt;0 -16 -2 -5 -40 8|]<br />~~

~~EDOs: 31, 68e, 99ef<br />~~

~~Badness: 0.0312<br />~~

~~<br />~~

~~<h2 id="toc12"><a name="Hemiwürschmidt-Hemiwur"></a>Hemiwur</h2>~~

~~Commas: 121/120, 176/175, 1375/1372<br />~~

~~<br />~~

~~POTE generator: ~28/25 = 193.884<br />~~

~~<br />~~

~~Map:~~ [~~&lt;1 15 4 7 11|, &lt;0 -16 -2 -5 -9|]<br />~~

~~EDOs~~: ~~6, 31, 68, 99, 130e, 229e<br />~~

~~Badness: 0.0293<br />~~

~~<br />~~

<h3 id="toc13"><a name="Hemiwürschmidt-Hemiwur-13-limit"></a>13-limit</h3>

~~Commas: 121/120, 176/175, 196/195, 275/273<br />~~

~~<br />~~

~~POTE generator: ~28/25 = 194.004<br />~~

~~<br />~~

~~Map: [&lt;1 15 4 7 11 -3|, &lt;0 -16 -2 -5 -9 8|~~]~~<br />~~

~~EDOs: 6, 31, 68, 99f, 167ef<br />~~

~~Badness: 0.0284<br />~~

~~<br />~~

<h3 id="toc14"><a name="Hemiwürschmidt-Hemiwur-Hemiwar"></a>Hemiwar</h3>

~~Commas: 66/65, 105/104, 121/120, 1375/1372<br />~~

~~<br />~~

~~POTE generator: ~28/25 = 193.698<br />~~

~~<br />~~

~~Map: [&lt;1 15 4 7 11 23|, &lt;0 -16 -2 -5 -9 -23|~~]~~<br />~~

~~EDOs: 31<br />~~

~~Badness: 0.0449<br />~~

~~<br />~~

<h1 id="toc15"><a name="Relationships to other temperaments"></a>Relationships to other temperaments</h1>

~~<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span><br />~~

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></pre></div>

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+__FORCETOC__
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-14 12:22:05 UTC</tt>.<br>
-: The original revision id was <tt>589302500</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>
-<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]
 Würschmidt
-The [[xenharmonic/5-limit|5-limit]] parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its [[xenharmonic/monzo|monzo]] is |17 1 -8&gt;, and flipping that yields &lt;&lt;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
+The [[5-limit|5-limit]] parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its [[monzo|monzo]] is |17 1 -8&gt;, and flipping that yields &lt;&lt;8 1 17|| for the wedgie. This tells us the [[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 [[Minimax_tuning|minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[MOS|MOS]] all possibilities.
+[[POTE_tuning|POTE generator]]: 387.799
 Map: [&lt;1 7 3|, &lt;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]]
+EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[99edo|99]], [[164edo|164]], [[721edo|721c]], [[885edo|885c]]
+[http://chrisvaisvil.com/ancient-stardust-wurschmidt13/ Ancient Stardust] [http://micro.soonlabel.com/jake_freivald/tunings_by_jake_freivald/20130811_wurschmidt%5b13%5d.mp3 play] by Chris Vaisvil
-[[http://chrisvaisvil.com/ancient-stardust-wurschmidt13/|Ancient Stardust]] [[http://micro.soonlabel.com/jake_freivald/tunings_by_jake_freivald/20130811_wurschmidt%5b13%5d.mp3|play]] by Chris Vaisvil
 Würschmidt[13] in 5-limit minimax tuning
-[[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt%5b16%5d-out.mp3|Extrospection]] by [[https://soundcloud.com/jdfreivald/extrospection|Jake Freivald]]; Würschmidt[16] tuned in 31et.
-==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&gt;, worschmidt adds 65625/65536 = |-16 1 5 1&gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&gt; and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2&gt;.
-=Würschmidt=
+[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt%5b16%5d-out.mp3 Extrospection] by [https://soundcloud.com/jdfreivald/extrospection Jake Freivald]; Würschmidt[16] tuned in 31et.
-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 &lt;&lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &lt;&lt;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.
+==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&gt;, worschmidt adds 65625/65536 = |-16 1 5 1&gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&gt; and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2&gt;.
+=Würschmidt=
+Würschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo|31edo]] or [[127edo|127edo]] can be used as tunings. Würschmidt has &lt;&lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &lt;&lt;8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. [[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
+[[POTE_tuning|POTE generator]]: 387.383
 Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]
-EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/96edo|96]], [[xenharmonic/127edo|127]], [[xenharmonic/285edo|28bd]], [[xenharmonic/412edo|412bd]]
+EDOs: [[31edo|31]], [[96edo|96]], [[127edo|127]], [[285edo|28bd]], [[412edo|412bd]]
 Badness: 0.0508
 ==11-limit==
 Commas: 99/98, 176/175, 243/242
@@ Line 39: / Line 38: @@
 Map: [&lt;1 7 3 15 17|, &lt;0 -8 -1 -18 -20|]
 EDOs: 31, 65d, 96, 127, 223d
 Badness: 0.0244
 ==13-limit==
 Commas: 99/98, 144/143, 176/175, 275/273
@@ Line 48: / Line 49: @@
 Map: [&lt;1 7 3 15 17 1|, &lt;0 -8 -1 -18 -20 4|]
 EDOs: 31, 65d, 161df
 Badness: 0.0236
 ==Worseschmidt==
 Commas: 66/65, 99/98, 105/104, 243/242
@@ Line 57: / Line 60: @@
 Map: [&lt;1 7 3 15 17 22|, &lt;0 -8 -1 -18 -20 -27|]
 EDOs: 31
 Badness: 0.0344
 =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 &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.
+Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo|31edo]], [[34edo|34edo]], or [[127edo|127edo]] 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.
 Commas: 126/125, 33075/32768
-[[xenharmonic/POTE tuning|POTE generator]]: 387.392
+[[POTE_tuning|POTE generator]]: 387.392
 Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]
-EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/65edo|65]], [[xenharmonic/96edo|96d]], [[xenharmonic/127edo|127d]]
+EDOs: [[31edo|31]], [[65edo|65]], [[96edo|96d]], [[127edo|127d]]
 Badness: 0.0646
 ==11-limit==
 Commas: 126/125, 243/242, 385/384
@@ Line 77: / Line 84: @@
 Map: [&lt;1 7 3 -6 17|, &lt;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 &lt;&lt;8 1 52 -17 60 118|| for a wedgie.
+[[99edo|99edo]] 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.
 Commas: 4375/4374, 393216/390625
-[[xenharmonic/POTE tuning|POTE generator]]: 387.881
+[[POTE_tuning|POTE generator]]: 387.881
 Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]
-EDOs: [[xenharmonic/31edo|31]], [[xenharmonic/34edo|34]], [[xenharmonic/65edo|65]], [[xenharmonic/99edo|99]]
+EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[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, &lt;&lt;16 2 5 40 -39 -49 -48 28...
+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. [[68edo|68edo]], [[99edo|99edo]] and [[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, &lt;&lt;16 2 5 40 -39 -49 -48 28...
 Commas: 2401/2400, 3136/3125
-[[xenharmonic/POTE tuning|POTE generator]]: ~28/25 = 193.898
+[[POTE_tuning|POTE generator]]: ~28/25 = 193.898
 Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]
 &lt;&lt;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]]
+EDOs: [[6edo|6]], [[31edo|31]], [[37edo|37]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]], [[557edo|557c]], [[885edo|885c]]
 Badness: 0.0203
 ==11-limit==
 Commas: 243/242, 441/440, 3136/3125
-[[xenharmonic/POTE tuning|POTE generator]]: ~28/25 = 193.840
+[[POTE_tuning|POTE generator]]: ~28/25 = 193.840
 Map: [&lt;1 15 4 7 37|, &lt;0 -16 -2 -5 -40|]
 EDOs: 31, 99e, 130, 650ce, 811ce
 Badness: 0.0211
 ===13-limit===
 Commas: 243/242 351/350 441/440 3584/3575
@@ Line 118: / Line 132: @@
 Map: [&lt;1 15 4 7 37 -29|, &lt;0 -16 -2 -5 -40 39|]
 EDOs: 31, 99e, 130, 291, 421e, 551ce
 Badness: 0.0231
 ===Hemithir===
 Commas: 121/120 176/175 196/195 275/273
@@ Line 127: / Line 143: @@
 Map: [&lt;1 15 4 7 37 -3|, &lt;0 -16 -2 -5 -40 8|]
 EDOs: 31, 68e, 99ef
 Badness: 0.0312
 ==Hemiwur==
 Commas: 121/120, 176/175, 1375/1372
@@ Line 136: / Line 154: @@
 Map: [&lt;1 15 4 7 11|, &lt;0 -16 -2 -5 -9|]
 EDOs: 6, 31, 68, 99, 130e, 229e
 Badness: 0.0293
 ===13-limit===
 Commas: 121/120, 176/175, 196/195, 275/273
@@ Line 145: / Line 165: @@
 Map: [&lt;1 15 4 7 11 -3|, &lt;0 -16 -2 -5 -9 8|]
 EDOs: 6, 31, 68, 99f, 167ef
 Badness: 0.0284
 ===Hemiwar===
 Commas: 66/65, 105/104, 121/120, 1375/1372
@@ Line 154: / Line 176: @@
 Map: [&lt;1 15 4 7 11 23|, &lt;0 -16 -2 -5 -9 -23|]
 EDOs: 31
 Badness: 0.0449
 =Relationships to other temperaments=
-&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;
+<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span>
--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.</pre></div>
-<h4>Original HTML content:</h4>
+-Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to [[Skwares|skwares]] as a 2.3.7.11 temperament.
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Würschmidt family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:32:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x-Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
+[[Category:family]]
-&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Würschmidt"&gt;Würschmidt&lt;/a&gt;&lt;/div&gt;
+[[Category:hemiwuerschmidt]]
-&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
+[[Category:theory]]
-&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
+[[Category:wuerschmidt]]
-&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Würschmidt-Worseschmidt"&gt;Worseschmidt&lt;/a&gt;&lt;/div&gt;
+[[Category:wurschmidt]]
-&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Worschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Whirrschmidt"&gt;Whirrschmidt&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hemiwürschmidt"&gt;Hemiwürschmidt&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-11-limit-Hemithir"&gt;Hemithir&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hemiwürschmidt-Hemiwur"&gt;Hemiwur&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-Hemiwur-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Hemiwürschmidt-Hemiwur-Hemiwar"&gt;Hemiwar&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Relationships to other temperaments"&gt;Relationships to other temperaments&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:49 --&gt;Würschmidt&lt;br /&gt;
-The &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit"&gt;5-limit&lt;/a&gt; parent comma for the würschmidt family is 393216/390625, known as Würschmidt's comma, and named after José Würschmidt, Its &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/monzo"&gt;monzo&lt;/a&gt; is |17 1 -8&amp;gt;, and flipping that yields &amp;lt;&amp;lt;8 1 17|| for the wedgie. This tells us the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/generator"&gt;generator&lt;/a&gt; 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/minimax%20tuning"&gt;minimax tuning&lt;/a&gt;. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS"&gt;MOS&lt;/a&gt; all possibilities.&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.799&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3|, &amp;lt;0 -8 -1|]&lt;br /&gt;
-&lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/164edo"&gt;164&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/721edo"&gt;721c&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/885edo"&gt;885c&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/ancient-stardust-wurschmidt13/" rel="nofollow"&gt;Ancient Stardust&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jake_freivald/tunings_by_jake_freivald/20130811_wurschmidt%5b13%5d.mp3" rel="nofollow"&gt;play&lt;/a&gt; by Chris Vaisvil&lt;br /&gt;
-Würschmidt[13] in 5-limit minimax tuning&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt%5b16%5d-out.mp3" rel="nofollow"&gt;Extrospection&lt;/a&gt; by &lt;a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/extrospection" rel="nofollow"&gt;Jake Freivald&lt;/a&gt;; Würschmidt[16] tuned in 31et.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
- The second comma of the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Wurschmidt adds |12 3 -6 -1&amp;gt;, worschmidt adds 65625/65536 = |-16 1 5 1&amp;gt;, whirrschmidt adds 4375/4374 = |-1 -7 4 1&amp;gt; and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2&amp;gt;.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Würschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Würschmidt&lt;/h1&gt;
- Würschmidt, aside from the commas listed above, also tempers out 225/224. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31edo&lt;/a&gt; or &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127edo&lt;/a&gt; can be used as tunings. Würschmidt has &amp;lt;&amp;lt;8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version &amp;lt;&amp;lt;8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127edo&lt;/a&gt; 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.&lt;br /&gt;
-&lt;br /&gt;
-Commas: 225/224, 8748/8575&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.383&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3 15|, &amp;lt;0 -8 -1 -18|]&lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/96edo"&gt;96&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/285edo"&gt;28bd&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/412edo"&gt;412bd&lt;/a&gt;&lt;br /&gt;
-Badness: 0.0508&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Würschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
- Commas: 99/98, 176/175, 243/242&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~5/4 = 387.447&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3 15 17|, &amp;lt;0 -8 -1 -18 -20|]&lt;br /&gt;
-EDOs: 31, 65d, 96, 127, 223d&lt;br /&gt;
-Badness: 0.0244&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Würschmidt-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
- Commas: 99/98, 144/143, 176/175, 275/273&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~5/4 = 387.626&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3 15 17 1|, &amp;lt;0 -8 -1 -18 -20 4|]&lt;br /&gt;
-EDOs: 31, 65d, 161df&lt;br /&gt;
-Badness: 0.0236&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Würschmidt-Worseschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Worseschmidt&lt;/h2&gt;
- Commas: 66/65, 99/98, 105/104, 243/242&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~5/4 = 387.099&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3 15 17 22|, &amp;lt;0 -8 -1 -18 -20 -27|]&lt;br /&gt;
-EDOs: 31&lt;br /&gt;
-Badness: 0.0344&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Worschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Worschmidt&lt;/h1&gt;
- Worschmidt tempers out 126/125 rather than 225/224, and can use &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo"&gt;34edo&lt;/a&gt;, or &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127edo&lt;/a&gt; as a tuning. If 127 is used, note that the val is &amp;lt;127 201 295 356| and not &amp;lt;127 201 295 357| as with wurschmidt. The wedgie now is &amp;lt;&amp;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.&lt;br /&gt;
-&lt;br /&gt;
-Commas: 126/125, 33075/32768&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.392&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3 -6|, &amp;lt;0 -8 -1 13|]&lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/96edo"&gt;96d&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/127edo"&gt;127d&lt;/a&gt;&lt;br /&gt;
-Badness: 0.0646&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Worschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
- Commas: 126/125, 243/242, 385/384&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~5/4 = 387.407&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3 -6 17|, &amp;lt;0 -8 -1 13 -20|]&lt;br /&gt;
-EDOs: 31, 65, 96d, 127d&lt;br /&gt;
-Badness: 0.0334&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Whirrschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Whirrschmidt&lt;/h1&gt;
- &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99edo&lt;/a&gt; is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with &amp;lt;&amp;lt;8 1 52 -17 60 118|| for a wedgie.&lt;br /&gt;
-&lt;br /&gt;
-Commas: 4375/4374, 393216/390625&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 387.881&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 7 3 38|, &amp;lt;0 -8 -1 -52|]&lt;br /&gt;
-&lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Hemiwürschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Hemiwürschmidt&lt;/h1&gt;
- 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. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/68edo"&gt;68edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99edo&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/130edo"&gt;130edo&lt;/a&gt; can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, &amp;lt;&amp;lt;16 2 5 40 -39 -49 -48 28...&lt;br /&gt;
-&lt;br /&gt;
-Commas: 2401/2400, 3136/3125&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~28/25 = 193.898&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 15 4 7|, &amp;lt;0 -16 -2 -5|]&lt;br /&gt;
-&amp;lt;&amp;lt;16 2 5 -34 -37 6||&lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/6edo"&gt;6&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/68edo"&gt;68&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/229edo"&gt;229&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/328edo"&gt;328&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/557edo"&gt;557c&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/885edo"&gt;885c&lt;/a&gt;&lt;br /&gt;
-Badness: 0.0203&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Hemiwürschmidt-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
- Commas: 243/242, 441/440, 3136/3125&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~28/25 = 193.840&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 15 4 7 37|, &amp;lt;0 -16 -2 -5 -40|]&lt;br /&gt;
-EDOs: 31, 99e, 130, 650ce, 811ce&lt;br /&gt;
-Badness: 0.0211&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="Hemiwürschmidt-11-limit-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h3&gt;
- Commas: 243/242 351/350 441/440 3584/3575&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~28/25 = 193.840&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 15 4 7 37 -29|, &amp;lt;0 -16 -2 -5 -40 39|]&lt;br /&gt;
-EDOs: 31, 99e, 130, 291, 421e, 551ce&lt;br /&gt;
-Badness: 0.0231&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Hemiwürschmidt-11-limit-Hemithir"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Hemithir&lt;/h3&gt;
- Commas: 121/120 176/175 196/195 275/273&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~28/25 = 193.918&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 15 4 7 37 -3|, &amp;lt;0 -16 -2 -5 -40 8|]&lt;br /&gt;
-EDOs: 31, 68e, 99ef&lt;br /&gt;
-Badness: 0.0312&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Hemiwürschmidt-Hemiwur"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Hemiwur&lt;/h2&gt;
- Commas: 121/120, 176/175, 1375/1372&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~28/25 = 193.884&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 15 4 7 11|, &amp;lt;0 -16 -2 -5 -9|]&lt;br /&gt;
-EDOs: 6, 31, 68, 99, 130e, 229e&lt;br /&gt;
-Badness: 0.0293&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Hemiwürschmidt-Hemiwur-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h3&gt;
- Commas: 121/120, 176/175, 196/195, 275/273&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~28/25 = 194.004&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 15 4 7 11 -3|, &amp;lt;0 -16 -2 -5 -9 8|]&lt;br /&gt;
-EDOs: 6, 31, 68, 99f, 167ef&lt;br /&gt;
-Badness: 0.0284&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Hemiwürschmidt-Hemiwur-Hemiwar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Hemiwar&lt;/h3&gt;
- Commas: 66/65, 105/104, 121/120, 1375/1372&lt;br /&gt;
-&lt;br /&gt;
-POTE generator: ~28/25 = 193.698&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 15 4 7 11 23|, &amp;lt;0 -16 -2 -5 -9 -23|]&lt;br /&gt;
-EDOs: 31&lt;br /&gt;
-Badness: 0.0449&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Relationships to other temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Relationships to other temperaments&lt;/h1&gt;
- &lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;"&gt;around 775.489 which is approximately&lt;/span&gt;&lt;br /&gt;
--Würschmidt, the temperament with all the same commas as Würschmidt but a generator of twice the size, is equivalent to &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/skwares"&gt;skwares&lt;/a&gt; as a 2.3.7.11 temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>