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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-08-10 15:19:09 UTC</tt>.<br>
: The original revision id was <tt>155965685</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">The 5-limit parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma. Its monzo is |17 1 -8&gt;, and flipping that yields &lt;&lt;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. 14/53 is about perfect as a generator, though 9/34 also makes sense and using 19edo is possible. Other tunings include 72edo, 87edo  and 140edo. 21/65 makes an excellent wuerschmidt generator and 53/164 is even more accurate, but deciding between [[31edo]] or [34edo]] on the basis of whether you prefer flat fifths or sharp ones does also. 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. Wuerschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28 31 and 34 note MOS all possibilities.


==Seven limit children==
== Würschmidt ==
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;.
{{Main| Würschmidt }}


===Wurschmidt===
The [[generator]] of würschmidt is a [[5/4|classical major third]], and to get to the interval class of the [[3/1|3rd harmonic]] requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.  
Wurschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo]] or [[127edo]] can be used as tunings. Wurschmidt 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]] 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.


===Worschmidt===
Similar to [[meantone]], würschmidt implies that [[3/2]] will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]].  
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 &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.  


===Whirrschmidt===
[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.  
[[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.


===Hemiwuerschmift===
[[Subgroup]]: 2.3.5
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, &lt;&lt;16 2 5 40 -39 -49 -48 28...</pre></div>
 
<h4>Original HTML content:</h4>
[[Comma list]]: 393216/390625
<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;The 5-limit parent comma for the wuerschmidt family is 393216/390625, known as Wuerschmidt's comma. Its monzo is |17 1 -8&amp;gt;, and flipping that yields &amp;lt;&amp;lt;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. 14/53 is about perfect as a generator, though 9/34 also makes sense and using 19edo is possible. Other tunings include 72edo, 87edo  and 140edo. 21/65 makes an excellent wuerschmidt generator and 53/164 is even more accurate, but deciding between &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; or [34edo]] on the basis of whether you prefer flat fifths or sharp ones does also. 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. Wuerschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28 31 and 34 note MOS all possibilities.&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}
&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;
: mapping generators: ~2, ~5/4
The second comma of the &lt;a class="wiki_link" href="/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;
[[Optimal tuning]]s:
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Seven limit children-Wurschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Wurschmidt&lt;/h3&gt;
* [[WE]]: ~2 = 1199.6942{{c}}, ~5/4 = 387.7005{{c}}
Wurschmidt, aside from the commas listed above, also tempers out 225/224. &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; or &lt;a class="wiki_link" href="/127edo"&gt;127edo&lt;/a&gt; can be used as tunings. Wurschmidt 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="/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;
: [[error map]]: {{val| -0.306 -0.045 +0.775 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.7762{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Seven limit children-Worschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Worschmidt&lt;/h3&gt;
: error map: {{val| 0.000 +0.255 +1.463 }}
Worschmidt tempers out 126/125 rather than 225/224, and can use &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, or &lt;a class="wiki_link" href="/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;
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Seven limit children-Whirrschmidt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Whirrschmidt&lt;/h3&gt;
 
&lt;a class="wiki_link" href="/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;
[[Badness]] (Sintel): 0.951
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Seven limit children-Hemiwuerschmift"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Hemiwuerschmift&lt;/h3&gt;
=== Overview to extensions ===
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. &lt;a class="wiki_link" href="/68edo"&gt;68edo&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; and &lt;a class="wiki_link" href="/130edo"&gt;130edo&lt;/a&gt; can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwuerschmidt extends to a higher limit temperament, &amp;lt;&amp;lt;16 2 5 40 -39 -49 -48 28...&lt;/body&gt;&lt;/html&gt;</pre></div>
==== 7-limit extensions ====
The 7-limit extensions can be obtained by adding another comma. Septimal würschmidt adds [[225/224]], worschmidt adds [[126/125]], whirrschmidt adds [[4375/4374]]. These all use the same generator as 5-limit würschmidt.
 
Hemiwürschmidt adds [[3136/3125]] and splits the generator in two. This temperament is the best extension available for würschmidt despite its complexity. The details can be found in [[Hemimean clan #Hemiwürschmidt|Hemimean clan]].  
 
==== Subgroup extensions ====
Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to [[11/9]], that is by tempering out [[5632/5625]] in addition to [[243/242]].
 
With this accuracy level of [[3/2]] available, extensions that add prime [[19/1|19]] exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo), but they are very complex.
 
The "free" higher prime for würschmidt outside the 5-limit is in fact [[23/1|23]], via tempering out [[576/575]] ({{S|24}}) and [[12167/12150]] ([[S-expression|S46<sup>2</sup>⋅S47]]). This is considered immediately below, and the no-11 restriction thereof is considered in [[#Other subgroup extensions]].
 
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11
 
Comma list: 243/242, 5632/5625
 
Subgroup-val mapping: {{mapping| 1 -1 2 -3 | 0 8 1 20 }}
 
Optimal tuning:  
* WE: ~2 = 1199.7508{{c}}, ~5/4 = 387.6058{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6697{{c}}
 
{{Optimal ET sequence|legend=0| 31, 34, 65 }}
 
Badness (Sintel): 0.477
 
==== 2.3.5.11.23 subgroup ====
Subgroup: 2.3.5.11.23
 
Comma list: 243/242, 276/275, 529/528
 
Subgroup-val mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}
 
Optimal tuning:
* WE: ~2 = 1199.8205{{c}}, ~5/4 = 387.6316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6770{{c}}
 
{{Optimal ET sequence|legend=0| 31, 34, 65 }}
 
Badness (Sintel): 0.300
 
== Septimal würschmidt ==
Septimal würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.
 
2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a [[2.3.7.11-subgroup|2.3.7.11 subgroup]] temperament.
 
The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 8748/8575
 
{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9741{{c}}, ~5/4 = 387.3742{{c}}
: [[error map]]: {{val| -0.026 -2.936 +1.009 +3.987 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.3809{{c}}
: error map: {{val| 0.000 -2.907 +1.067 +4.031 }}
 
{{Optimal ET sequence|legend=1| 31, 96, 127 }}
 
[[Badness]] (Sintel): 1.28
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 243/242
 
Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}
 
Optimal tunings:
* WE: ~2 = 1199.9618{{c}}, ~5/4 = 387.4347{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4446{{c}}
 
{{Optimal ET sequence|legend=0| 31, 65d, 96, 127 }}
 
Badness (Sintel): 0.807
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 144/143, 176/175, 275/273
 
Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.0325{{c}}, ~5/4 = 387.3137{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.5893{{c}}
 
{{Optimal ET sequence|legend=0| 31, 65d }}
 
Badness (Sintel): 0.975
 
==== Worseschmidt ====
Subgroup: 2.3.5.7.11.13
 
Commas: 66/65, 99/98, 105/104, 243/242
 
Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}
 
Optimal tunings:
* WE: ~2 = 1200.4712{{c}}, ~5/4 = 387.2511{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1252{{c}}
 
{{Optimal ET sequence|legend=0| 3def, 28def, 31 }}
 
Badness (Sintel): 1.42
 
== 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 {{val| 127 201 295 '''356''' }} (127d) and not {{val| 127 201 295 '''357''' }} as with würschmidt. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 33075/32768
 
{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.406{{c}}
: [[error map]]: {{val| +0.763 -1.610 +2.851 -2.786 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.3950{{c}}
: error map: {{val| 0.000 -2.795 +1.081 -4.960 }}
 
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}
 
[[Badness]] (Sintel): 1.64
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 385/384
 
Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.7911{{c}}, ~5/4 = 387.6624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4189{{c}}
 
{{Optimal ET sequence|legend=0| 31, 65, 96d, 127d }}
 
Badness (Sintel): 1.11
 
== Whirrschmidt ==
[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with 7 mapped to the 52nd generator step.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 393216/390625
 
{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5703{{c}}, ~5/4 = 387.7422{{c}}
: [[error map]]: {{val| -0.430 +0.412 +0.569 -0.216 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.8729{{c}}
: error map: {{val| 0.000 +1.029 +1.559 +0.567 }}
 
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
 
[[Badness]] (Sintel): 2.18
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 896/891, 4375/4356
 
Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}
 
Optimal tunings:
* WE: ~2 = 1199.2839{{c}}, ~5/4 = 387.6507{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.8682{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 65, 99e }}
 
Badness (Sintel): 1.93
 
== Other subgroup extensions ==
=== Würschmidt (2.3.5.23) ===
Extensions to harmonics [[47/1|47]] and [[49/1|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].
 
Subgroup: 2.3.5.23
 
Comma list: 576/575, 12167/12150
 
Subgroup-val mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}
 
Optimal tunings:
* WE: ~2 = 1199.7075{{c}}, ~5/4 = 387.7106{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.7807{{c}}
 
{{Optimal ET sequence|legend=0| 31, 34, 65, 99, 164 }}
 
Badness (Sintel): 0.216
 
[[Category:Temperament families]]
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Würschmidt_family"