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The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. Its [[monzo]] is {{monzo| 17 1 -8 }}, and flipping that yields {{multival| 8 1 17 }} for the wedgie. This tells us the [[generator]] is a classic major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.  
{{Technical data page}}
The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. The [[generator]] is a classic major third, and to get to the interval class of fifths requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.  


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]]. [[65edo]] is the point where it is combined with [[schismic]] (especially the extension to include prime 19 called [[nestoria]]) and [[gravity]], so is a very accurate 5-limit tuning that extends naturally to prime 11 through [[243/242]], [[8019/8000]] and [[4000/3993]] (which is natural because [[243/242|S9/S11]] = [[8019/8000|S9/S10]] * [[4000/3993|S10/S11]]; note that tempering all three without tempering the würschmidt comma yields [[gravity]]) and prime 19 through nestoria, among others. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]].  
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]]/[[nestoria]] and [[gravity]]/[[larry]]. 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 fifths; this is the [[minimax tuning]].  
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 fifths; this is the [[minimax tuning]].  
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[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.  
[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.  


== Würschmidt ==
{{Main| Würschmidt }}
[[Subgroup]]: 2.3.5
[[Comma list]]: 393216/390625
{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}
: mapping generators: ~2, ~5/4
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.734
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.799
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}
[[Badness]] (Smith): 0.040603
=== Overview to extensions ===
==== 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.  
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]].  
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]].  


== Würschmidt ==
==== Subgroup extensions ====
[[Subgroup]]: 2.3.5
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]]; furthermore, like practically any 5-limit temperament with this accuracy level of [[3/2]] available, extensions to prime 19 exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo and [[nestoria]]).
 
However, as discussed in the main article, the "free" higher prime for würschmidt outside the 5-limit is in fact 23, via tempering out S24 = [[576/575]] and S46<sup>2</sup> × S47 = [[12167/12150]]. Therefore, the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions.
 
=== 2.3.5.23 subgroup ===
Subgroup: 2.3.5.23
 
Comma list: 576/575, 12167/12150
 
Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.734
* POTE: ~2 = 1\1, ~5/4 = 387.805
 
Optimal ET sequence: {{optimal ET sequence| 3, …, 28i, 31, 34, 65, 99, 164 }}


[[Comma list]]: 393216/390625
Badness (Smith): 0.00530


{{Mapping|legend=1| 1 7 3 | 0 -8 -1 }}
==== 2.3.5.11.23 subgroup ====
Subgroup: 2.3.5.11.23


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.799
Comma list: 243/242, 276/275, 529/528


{{Optimal ET sequence|legend=1| 3, 28, 31, 34, 65, 99, 164, 721c, 885c }}
Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}


[[Badness]]: 0.040603
Optimal tuning:  
* CTE: ~2 = 1\1, ~5/4 = 387.652
* POTE: ~2 = 1\1, ~5/4 = 387.690


; Music
Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }}
* [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-odd-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 31edo.
Badness (Smith): 0.00660


== Septimal würschmidt ==
== Septimal würschmidt ==
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 {{Multival| 8 1 18 20 … }} 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.
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 temperament.
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 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]], but specifically, the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap| {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap| {S8/9/10/11} }}*. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7 no matter what mapping you use (with the sharp 7 being used for this temperament); even so, it's 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, 65d + 31 = [[96edo]] is also a reasonable choice, as it has the advantage of being [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.
(<nowiki>*</nowiki> The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).)


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 225/224, 8748/8575
[[Comma list]]: 225/224, 8748/8575


{{Mapping|legend=1| 1 7 3 15 | 0 -8 -1 -18 }}
{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }}
 
{{Multival|legend=1| 8 1 18 -17 6 39 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.383
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.379
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.383


{{Optimal ET sequence|legend=1| 31, 96, 127, 285bd, 412bbdd }}
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


[[Badness]]: 0.050776
[[Badness]] (Smith): 0.050776


=== 11-limit ===
=== 11-limit ===
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Comma list: 99/98, 176/175, 243/242
Comma list: 99/98, 176/175, 243/242


Mapping: {{mapping| 1 7 3 15 17 | 0 -8 -1 -18 -20 }}
Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.447
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.441
* POTE: ~2 = 1\1, ~5/4 = 387.447


{{Optimal ET sequence|legend=1| 31, 65d, 96, 127, 223d }}
Optimal ET sequence: {{optimal ET sequence| 31, 65d, 96, 127 }}


Badness: 0.024413
Badness (Smith): 0.024413


==== 13-limit ====
==== 13-limit ====
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Comma list: 99/98, 144/143, 176/175, 275/273
Comma list: 99/98, 144/143, 176/175, 275/273


Mapping: {{mapping| 1 7 3 15 17 1 | 0 -8 -1 -18 -20 4 }}
Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.626
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.469
* POTE: ~2 = 1\1, ~5/4 = 387.626


{{Optimal ET sequence|legend=1| 31, 65d, 161df }}
Optimal ET sequence: {{optimal ET sequence| 31, 65d }}


Badness: 0.023593
Badness (Smith): 0.023593


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


Mapping: {{mapping| 1 7 3 15 17 22 | 0 -8 -1 -18 -20 -27 }}
Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.099
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.179
* POTE: ~2 = 1\1, ~5/4 = 387.099


{{Optimal ET sequence|legend=1| 3def, 28def, 31 }}
Optimal ET sequence: {{optimal ET sequence| 3def, 28def, 31 }}


Badness: 0.034382
Badness (Smith): 0.034382


== Worschmidt ==
== Worschmidt ==
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[[Comma list]]: 126/125, 33075/32768
[[Comma list]]: 126/125, 33075/32768


{{Mapping|legend=1| 1 7 3 -6 | 0 -8 -1 13 }}
{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }}


{{Multival|legend=1| 8 1 -13 -17 -43 -33 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.406
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.392


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.392
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}


{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
[[Badness]] (Smith): 0.064614
 
[[Badness]]: 0.064614


=== 11-limit ===
=== 11-limit ===
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Comma list: 126/125, 243/242, 385/384
Comma list: 126/125, 243/242, 385/384


Mapping: {{mapping| 1 7 3 -6 17 | 0 -8 -1 13 -20 }}
Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.407
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.472
* POTE: ~2 = 1\1, ~5/4 = 387.407


{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
Optimal ET sequence: {{optimal ET sequence| 31, 65, 96d, 127d }}


Badness: 0.033436
Badness (Smith): 0.033436


== Whirrschmidt ==
== Whirrschmidt ==
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[[Comma list]]: 4375/4374, 393216/390625
[[Comma list]]: 4375/4374, 393216/390625


{{Mapping|legend=1| 1 7 3 38 | 0 -8 -1 -52 }}
{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }}
 
{{Multival|legend=1| 8 1 52 -17 60 118 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.881
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.853
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.881


{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}


[[Badness]]: 0.086334
[[Badness]] (Smith): 0.086334


=== 11-limit ===
=== 11-limit ===
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Comma list: 243/242, 896/891, 4375/4356
Comma list: 243/242, 896/891, 4375/4356


Mapping: {{mapping| 1 7 3 38 17 | 0 -8 -1 -52 -20 }}
Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.882
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.829
* POTE: ~2 = 1\1, ~5/4 = 387.882


{{Optimal ET sequence|legend=1| 34d, 65, 99e }}
Optimal ET sequence: {{optimal ET sequence| 34d, 65, 99e }}


Badness: 0.058325
Badness (Smith): 0.058325


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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