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The [[5-limit]] parent comma for the '''würschmidt family''' is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt, Its [[monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[generator]] is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[MOS]] all possibilities.
{{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.  


= Würschmidt =
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]].


('''Würschmidt''' is sometimes spelled '''Wuerschmidth''')
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]].


Comma: 393216/390625
[[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.


[[POTE generator]]: ~5/4 = 387.799
== Würschmidt ==
{{Main| Würschmidt }}


Map: [&lt;1 7 3|, &lt;0 -8 -1|]
[[Subgroup]]: 2.3.5


EDOs: {{EDOs| 31, 34, 65, 99, 164, 721c, 885c }}
[[Comma list]]: 393216/390625


== Music ==
{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}


[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
: mapping generators: ~2, ~5/4


[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.
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.734
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.799


== Seven limit children ==
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}


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;.
[[Badness]] (Smith): 0.040603


= 7-limit =
=== 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.


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


Commas: [[225/224]], 8748/8575
==== 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]]; 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]]).


[[POTE generator]]: ~5/4 = 387.383
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.


Map: [&lt;1 7 3 15|, &lt;0 -8 -1 -18|]
=== 2.3.5.23 subgroup ===
Subgroup: 2.3.5.23


EDOs: {{EDOs| 31, 96, 127, 285bd, 412bbdd }}
Comma list: 576/575, 12167/12150


Badness: 0.0508
Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}


== 11-limit ==
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.734
* POTE: ~2 = 1\1, ~5/4 = 387.805


Commas: [[99/98]], 176/175, [[243/242]]
Optimal ET sequence: {{optimal ET sequence| 3, …, 28i, 31, 34, 65, 99, 164 }}


POTE generator: ~5/4 = 387.447
Badness (Smith): 0.00530


Map: [&lt;1 7 3 15 17|, &lt;0 -8 -1 -18 -20|]
==== 2.3.5.11.23 subgroup ====
Subgroup: 2.3.5.11.23


EDOs: {{EDOs| 31, 65d, 96, 127, 223d }}
Comma list: 243/242, 276/275, 529/528


Badness: 0.0244
Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}


=== 13-limit ===
Optimal tuning:
* CTE: ~2 = 1\1, ~5/4 = 387.652
* POTE: ~2 = 1\1, ~5/4 = 387.690


Commas: [[99/98]], [[144/143]], 176/175, 275/273
Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }}


POTE generator: ~5/4 = 387.626
Badness (Smith): 0.00660


Map: [&lt;1 7 3 15 17 1|, &lt;0 -8 -1 -18 -20 4|]
== 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 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.


EDOs: {{EDOs| 31, 65d, 161df }}
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.


Badness: 0.0236
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.


=== Worseschmidt ===
(<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).)


Commas: 66/65, [[99/98]], 105/104, [[243/242]]
[[Subgroup]]: 2.3.5.7


POTE generator: ~5/4 = 387.099
[[Comma list]]: 225/224, 8748/8575


Map: [&lt;1 7 3 15 17 22|, &lt;0 -8 -1 -18 -20 -27|]
{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }}


EDOs: {{EDOs| 31 }}
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.379
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.383


Badness: 0.0344
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


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


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.
=== 11-limit ===
Subgroup: 2.3.5.7.11


Commas: [[126/125]], 33075/32768
Comma list: 99/98, 176/175, 243/242


[[POTE generator]]: ~5/4 = 387.392
Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}


Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.441
* POTE: ~2 = 1\1, ~5/4 = 387.447


EDOs: {{EDOs| 31, 65, 96d, 127d }}
Optimal ET sequence: {{optimal ET sequence| 31, 65d, 96, 127 }}


Badness: 0.0646
Badness (Smith): 0.024413


== 11-limit ==
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Commas: 126/125, 243/242, 385/384
Comma list: 99/98, 144/143, 176/175, 275/273


POTE generator: ~5/4 = 387.407
Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


Map: [&lt;1 7 3 -6 17|, &lt;0 -8 -1 13 -20|]
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.469
* POTE: ~2 = 1\1, ~5/4 = 387.626


EDOs: {{EDOs| 31, 65, 96d, 127d }}
Optimal ET sequence: {{optimal ET sequence| 31, 65d }}


Badness: 0.0334
Badness (Smith): 0.023593


= Whirrschmidt =
==== Worseschmidt ====
Subgroup: 2.3.5.7.11.13


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


Commas: 4375/4374, 393216/390625
Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


[[POTE generator]]: ~5/4 = 387.881
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.179
* POTE: ~2 = 1\1, ~5/4 = 387.099


Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]
Optimal ET sequence: {{optimal ET sequence| 3def, 28def, 31 }}


EDOs: {{EDOs| 31dd, 34d, 65, 99 }}
Badness (Smith): 0.034382


= Hemiwürschmidt =
== 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.


'''Hemiwürschmidt''' (sometimes spelled '''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 hemiwürschmidt extends to a higher limit temperament, &lt;&lt;16 2 5 40 -39 -49 -48 28...
[[Subgroup]]: 2.3.5.7


Commas: 2401/2400, 3136/3125
[[Comma list]]: 126/125, 33075/32768


[[POTE generator]]: ~28/25 = 193.898
{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }}


Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.406
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.392


&lt;&lt;16 2 5 -34 -37 6||
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}


EDOs: {{EDOs| 31, 68, 99, 229, 328, 557c, 885cc }}
[[Badness]] (Smith): 0.064614


Badness: 0.0203
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


Commas: 243/242, 441/440, 3136/3125
Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


[[POTE generator]]: ~28/25 = 193.840
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.472
* POTE: ~2 = 1\1, ~5/4 = 387.407


Map: [&lt;1 15 4 7 37|, &lt;0 -16 -2 -5 -40|]
Optimal ET sequence: {{optimal ET sequence| 31, 65, 96d, 127d }}


EDOs: {{EDOs| 31, 99e, 130, 650ce, 811ce }}
Badness (Smith): 0.033436


Badness: 0.0211
== 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.  


=== 13-limit ===
[[Subgroup]]: 2.3.5.7


Commas: 243/242, 351/350, 441/440, 3584/3575
[[Comma list]]: 4375/4374, 393216/390625


POTE generator: ~28/25 = 193.840
{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }}


Map: [&lt;1 15 4 7 37 -29|, &lt;0 -16 -2 -5 -40 39|]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.853
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.881


EDOs: {{EDOs| 31, 99e, 130, 291, 421e, 551ce }}
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}


Badness: 0.0231
[[Badness]] (Smith): 0.086334


=== Hemithir ===
=== 11-limit ===
Subgroup: 2.3.5.7.11


Commas: 121/120, 176/175, 196/195, 275/273
Comma list: 243/242, 896/891, 4375/4356


POTE generator: ~28/25 = 193.918
Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}


Map: [&lt;1 15 4 7 37 -3|, &lt;0 -16 -2 -5 -40 8|]
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.829
* POTE: ~2 = 1\1, ~5/4 = 387.882


EDOs: {{EDOs| 31, 68e, 99ef }}
Optimal ET sequence: {{optimal ET sequence| 34d, 65, 99e }}


Badness: 0.0312
Badness (Smith): 0.058325


== Hemiwur ==
[[Category:Temperament families]]
 
[[Category:Pages with mostly numerical content]]
Commas: 121/120, 176/175, 1375/1372
 
POTE generator: ~28/25 = 193.884
 
Map: [&lt;1 15 4 7 11|, &lt;0 -16 -2 -5 -9|]
 
EDOs: {{EDOs| 31, 68, 99, 130e, 229e }}
 
Badness: 0.0293
 
=== 13-limit ===
 
Commas: 121/120, 176/175, 196/195, 275/273
 
POTE generator: ~28/25 = 194.004
 
Map: [&lt;1 15 4 7 11 -3|, &lt;0 -16 -2 -5 -9 8|]
 
EDOs: {{EDOs| 31, 68, 99f, 167ef }}
 
Badness: 0.0284
 
=== Hemiwar ===
 
Commas: 66/65, 105/104, 121/120, 1375/1372
 
POTE generator: ~28/25 = 193.698
 
Map: [&lt;1 15 4 7 11 23|, &lt;0 -16 -2 -5 -9 -23|]
 
EDOs: {{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>
 
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 temperament.
 
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Hemiwürschmidt]]
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Rank 2]]
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