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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.  


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]].  
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]].  


[[MOS scale]]s of würschmidt are even more extreme than those of [[magic]]. [[Proper]] scales does not appear until 28, 31 or even 34 notes.
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]].  


The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Würschmidt adds {{monzo| 12 3 -6 -1 }}, worschmidt adds 65625/65536 = {{monzo| -16 1 5 1 }}, whirrschmidt adds 4375/4374 = {{monzo| -1 -7 4 1 }} and hemiwürschmidt adds 6144/6125 = {{monzo| 11 1 -3 -2 }}.
[[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 ==
== Würschmidt ==
Subgroup: 2.3.5
{{Main| Würschmidt }}
 
[[Subgroup]]: 2.3.5


[[Comma list]]: 393216/390625
[[Comma list]]: 393216/390625


[[Mapping]]: [{{Val|1 7 3}}, {{Val|0 -8 -1}}]
{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}


[[POTE generator]]: ~5/4 = 387.799
: mapping generators: ~2, ~5/4


{{Optimal ET sequence|legend=1|3, 28, 31, 34, 65, 99, 164, 721c, 885c }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.734
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.799


[[Badness]]: 0.040603
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}


; Music
[[Badness]] (Smith): 0.040603
* [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.
=== 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.  


== Septimal 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]].  
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.


Subgroup: 2.3.5.7
==== 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]]).


[[Comma list]]: [[225/224]], 8748/8575
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.


[[Mapping]]: [{{val| 1 7 3 15 }}, {{Val| 0 -8 -1 -18 }}]
=== 2.3.5.23 subgroup ===
Subgroup: 2.3.5.23


{{Multival|legend=1| 8 1 18 -17 6 39 }}
Comma list: 576/575, 12167/12150


[[POTE generator]]: ~5/4 = 387.383
Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}


{{Optimal ET sequence|legend=1| 31, 96, 127, 285bd, 412bbdd }}
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.734
* POTE: ~2 = 1\1, ~5/4 = 387.805


[[Badness]]: 0.050776
Optimal ET sequence: {{optimal ET sequence| 3, …, 28i, 31, 34, 65, 99, 164 }}


=== 11-limit ===
Badness (Smith): 0.00530
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 243/242
==== 2.3.5.11.23 subgroup ====
Subgroup: 2.3.5.11.23


Mapping: [{{val| 1 7 3 15 17 }}, {{val| 0 -8 -1 -18 -20 }}]
Comma list: 243/242, 276/275, 529/528


POTE generator: ~5/4 = 387.447
Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}


{{Optimal ET sequence|legend=1| 31, 65d, 96, 127, 223d }}
Optimal tuning:
* CTE: ~2 = 1\1, ~5/4 = 387.652
* POTE: ~2 = 1\1, ~5/4 = 387.690


Badness: 0.024413
Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }}


==== 13-limit ====
Badness (Smith): 0.00660
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 144/143, 176/175, 275/273
== 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.
Mapping: [{{val| 1 7 3 15 17 1 }}, {{val| 0 -8 -1 -18 -20 4 }}]
 
POTE generator: ~5/4 = 387.626
 
{{Optimal ET sequence|legend=1| 31, 65d, 161df }}
 
Badness: 0.023593
 
==== Worseschmidt ====
Subgroup: 2.3.5.7.11.13
 
Commas: 66/65, 99/98, 105/104, 243/242
 
Mapping: [{{val| 1 7 3 15 17 22 }}, {{val| 0 -8 -1 -18 -20 -27 }}]
 
POTE generator: ~5/4 = 387.099
 
{{Optimal ET sequence|legend=1| 3def, 28def, 31 }}
 
Badness: 0.034382
 
== 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]]: [{{val| 1 7 3 -6 }}, {{val| 0 -8 -1 13 }}]
 
{{Multival|legend=1| 8 1 -13 -17 -43 -33 }}


[[POTE generator]]: ~5/4 = 387.392
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.


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


[[Badness]]: 0.064614
(<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).)
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 385/384
 
Mapping: [{{val| 1 7 3 -6 17 }}, {{val| 0 -8 -1 13 -20 }}]
 
POTE generator: ~5/4 = 387.407
 
{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
 
Badness: 0.033436
 
== 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]]: [{{val| 1 7 3 38 }}, {{val| 0 -8 -1 -52 }}]
 
{{Multival|legend=1| 8 1 52 -17 60 118 }}
 
[[POTE generator]]: ~5/4 = 387.881
 
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
 
[[Badness]]: 0.086334
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 896/891, 4375/4356
 
Mapping: [{{val| 1 7 3 38 17 }}, {{val| 0 -8 -1 -52 -20 }}]
 
POTE generator: ~5/4 = 387.882
 
{{Optimal ET sequence|legend=1| 34d, 65, 99e }}
 
Badness: 0.058325
 
== Hemiwürschmidt ==
{{See also| Hemimean clan #Hemiwürschmidt }}
 
'''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 [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[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, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 3136/3125
[[Comma list]]: 225/224, 8748/8575
 
{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}


{{Multival|legend=1| 16 2 5 -34 -37 6 }}
{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
[[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, 68, 99, 229, 328, 557c, 885cc }}
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


[[Badness]]: 0.020307
[[Badness]] (Smith): 0.050776


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


Comma list: 243/242, 441/440, 3136/3125
Comma list: 99/98, 176/175, 243/242


Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.441
* POTE: ~2 = 1\1, ~5/4 = 387.447


{{Optimal ET sequence|legend=1| 31, 99e, 130, 650ce, 811ce }}
Optimal ET sequence: {{optimal ET sequence| 31, 65d, 96, 127 }}


Badness: 0.021069
Badness (Smith): 0.024413


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


Comma list: 243/242, 351/350, 441/440, 3584/3575
Comma list: 99/98, 144/143, 176/175, 275/273


Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.469
* POTE: ~2 = 1\1, ~5/4 = 387.626


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


Badness: 0.023074
Badness (Smith): 0.023593


==== Hemithir ====
==== Worseschmidt ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 275/273
Commas: 66/65, 99/98, 105/104, 243/242


Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.179
* POTE: ~2 = 1\1, ~5/4 = 387.099


{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
Optimal ET sequence: {{optimal ET sequence| 3def, 28def, 31 }}


Badness: 0.031199
Badness (Smith): 0.034382


=== Hemiwur ===
== Worschmidt ==
Subgroup: 2.3.5.7.11
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.


Comma list: 121/120, 176/175, 1375/1372
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
[[Comma list]]: 126/125, 33075/32768


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }}


{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.406
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.392


Badness: 0.029270
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}


==== 13-limit ====
[[Badness]] (Smith): 0.064614
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 275/273
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
Comma list: 126/125, 243/242, 385/384


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.472
* POTE: ~2 = 1\1, ~5/4 = 387.407


Badness: 0.028432
Optimal ET sequence: {{optimal ET sequence| 31, 65, 96d, 127d }}


==== Hemiwar ====
Badness (Smith): 0.033436
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 105/104, 121/120, 1375/1372
== 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.


Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
[[Comma list]]: 4375/4374, 393216/390625


{{Optimal ET sequence|legend=1| 6f, 31 }}
{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }}


Badness: 0.044886
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.853
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.881


=== Quadrawürschmidt ===
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.


Subgroup: 2.3.5.7.11
[[Badness]] (Smith): 0.086334


Comma list: 2401/2400, 3025/3024, 3136/3125
=== 11-limit ===
 
Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
 
: mapping generators: ~2, ~147/110
 
Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404
 
{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
 
Badness: 0.034814
 
=== Semihemiwür ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3136/3125, 9801/9800
Comma list: 243/242, 896/891, 4375/4356
 
Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
 
: mapping generators: ~99/70, ~495/392
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
 
Badness: 0.044848
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
 
Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
 
Badness: 0.023388
 
===== Semihemiwürat =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
 
Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
 
Badness: 0.028987
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
 
Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
 
Badness: 0.021707
 
===== Semihemiwürand =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
 
Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
 
{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
 
Badness: 0.029718
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
 
Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428
Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}


{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.829
* POTE: ~2 = 1\1, ~5/4 = 387.882


Badness: 0.029545
Optimal ET sequence: {{optimal ET sequence| 34d, 65, 99e }}


== Relationships to other temperaments ==
Badness (Smith): 0.058325
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: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:Rank 2]]
[[Category:Rank 2]]
[[Category:Würschmidt|#]] <!-- list on top of cat -->

Latest revision as of 00:34, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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)8 × 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 11\34 and especially 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.

Mos scales 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 article: Würschmidt

Subgroup: 2.3.5

Comma list: 393216/390625

Mapping: [⟨1 -1 2], ⟨0 8 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.734
  • POTE: ~2 = 1\1, ~5/4 = 387.799

Optimal ET sequence: 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.

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.

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).

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 S462 × 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: [⟨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: 3, …, 28i, 31, 34, 65, 99, 164

Badness (Smith): 0.00530

2.3.5.11.23 subgroup

Subgroup: 2.3.5.11.23

Comma list: 243/242, 276/275, 529/528

Sval mapping: [⟨1 -1 2 -3 0], ⟨0 8 1 20 14]]

Optimal tuning:

  • CTE: ~2 = 1\1, ~5/4 = 387.652
  • POTE: ~2 = 1\1, ~5/4 = 387.690

Optimal ET sequence: 31, 34, 65

Badness (Smith): 0.00660

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.

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 {S8/S10, S9/S11, S15}. Tempering out S9 or S11 results in 31edo, and in complementary fashion, tempering out S8 or 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 superparticulars by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {S8/S9, S9/S10, S10/S11}, which may be expressed in shortened form as {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.

(* 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

Comma list: 225/224, 8748/8575

Mapping: [⟨1 -1 2 -3], ⟨0 8 1 18]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.379
  • POTE: ~2 = 1\1, ~5/4 = 387.383

Optimal ET sequence: 31, 96, 127

Badness (Smith): 0.050776

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 243/242

Mapping: [⟨1 -1 2 -3 -3], ⟨0 8 1 18 20]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.441
  • POTE: ~2 = 1\1, ~5/4 = 387.447

Optimal ET sequence: 31, 65d, 96, 127

Badness (Smith): 0.024413

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 176/175, 275/273

Mapping: [⟨1 -1 2 -3 -3 5], ⟨0 8 1 18 20 -4]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.469
  • POTE: ~2 = 1\1, ~5/4 = 387.626

Optimal ET sequence: 31, 65d

Badness (Smith): 0.023593

Worseschmidt

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 2 -3 -3 -5], ⟨0 8 1 18 20 27]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.179
  • POTE: ~2 = 1\1, ~5/4 = 387.099

Optimal ET sequence: 3def, 28def, 31

Badness (Smith): 0.034382

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 ⟨127 201 295 356] (127d) and not ⟨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: [⟨1 -1 2 7], ⟨0 8 1 -13]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.406
  • POTE: ~2 = 1\1, ~5/4 = 387.392

Optimal ET sequence: 31, 96d, 127d

Badness (Smith): 0.064614

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 385/384

Mapping: [⟨1 -1 2 7 -3], ⟨0 8 1 -13 20]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.472
  • POTE: ~2 = 1\1, ~5/4 = 387.407

Optimal ET sequence: 31, 65, 96d, 127d

Badness (Smith): 0.033436

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: [⟨1 -1 2 -14], ⟨0 8 1 52]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.853
  • POTE: ~2 = 1\1, ~5/4 = 387.881

Optimal ET sequence: 34d, 65, 99

Badness (Smith): 0.086334

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891, 4375/4356

Mapping: [⟨1 -1 2 -14 -3], ⟨0 8 1 52 20]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~5/4 = 387.829
  • POTE: ~2 = 1\1, ~5/4 = 387.882

Optimal ET sequence: 34d, 65, 99e

Badness (Smith): 0.058325

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