Würschmidt family: Difference between revisions

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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. 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.  
{{Technical data page}}
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.  


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 ==
{{Main| Würschmidt }}


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 [[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.  


[[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.  
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]].  


== Würschmidt ==
[[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.
{{Main| Würschmidt }}


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 15: Line 16:


{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}
{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}
: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.734
* [[WE]]: ~2 = 1199.6942{{c}}, ~5/4 = 387.7005{{c}}
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.799
: [[error map]]: {{val| -0.306 -0.045 +0.775 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.7762{{c}}
: error map: {{val| 0.000 +0.255 +1.463 }}


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


[[Badness]] (Smith): 0.040603
[[Badness]] (Sintel): 0.951


=== Overview to extensions ===
=== Overview to extensions ===
Line 33: Line 35:


==== Subgroup extensions ====
==== 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]]).
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.  


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.
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]].


=== 2.3.5.23 subgroup ===
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.23
Subgroup: 2.3.5.11


Comma list: 576/575, 12167/12150
Comma list: 243/242, 5632/5625


Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}
{{Mapping|legend=2| 1 -1 2 -3 | 0 8 1 20 }}


Optimal tunings:  
Optimal tuning:  
* CTE: ~2 = 1\1, ~5/4 = 387.734
* WE: ~2 = 1199.7508{{c}}, ~5/4 = 387.6058{{c}}
* POTE: ~2 = 1\1, ~5/4 = 387.805
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6697{{c}}


Optimal ET sequence: {{optimal ET sequence| 3, …, 28i, 31, 34, 65, 99, 164 }}
{{Optimal ET sequence|legend=0| 31, 34, 65 }}


Badness (Smith): 0.00530
Badness (Sintel): 0.477


==== 2.3.5.11.23 subgroup ====
==== 2.3.5.11.23 subgroup ====
Line 57: Line 61:
Comma list: 243/242, 276/275, 529/528
Comma list: 243/242, 276/275, 529/528


Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}
{{Mapping|legend=2| 1 -1 2 -3 0 | 0 8 1 20 14 }}


Optimal tuning:  
Optimal tuning:  
* CTE: ~2 = 1\1, ~5/4 = 387.652
* WE: ~2 = 1199.8205{{c}}, ~5/4 = 387.6316{{c}}
* POTE: ~2 = 1\1, ~5/4 = 387.690
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6770{{c}}


Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }}
{{Optimal ET sequence|legend=0| 31, 34, 65 }}


Badness (Smith): 0.00660
Badness (Sintel): 0.300


== 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.
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 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|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
[[Subgroup]]: 2.3.5.7
Line 77: Line 83:


{{Mapping|legend=1| 1 -1 2 -3 | 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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.379
* [[WE]]: ~2 = 1199.9741{{c}}, ~5/4 = 387.3742{{c}}
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.383
: [[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 }}
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


[[Badness]] (Smith): 0.050776
[[Badness]] (Sintel): 1.28


=== 11-limit ===
=== 11-limit ===
Line 93: Line 99:
Comma list: 99/98, 176/175, 243/242
Comma list: 99/98, 176/175, 243/242


Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}
{{Mapping|legend=0| 1 -1 2 -3 -3 | 0 8 1 18 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.441
* WE: ~2 = 1199.9618{{c}}, ~5/4 = 387.4347{{c}}
* POTE: ~2 = 1\1, ~5/4 = 387.447
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4446{{c}}


Optimal ET sequence: {{optimal ET sequence| 31, 65d, 96, 127 }}
{{Optimal ET sequence|legend=0| 31, 65d, 96, 127 }}


Badness (Smith): 0.024413
Badness (Sintel): 0.807


==== 13-limit ====
==== 13-limit ====
Line 108: Line 114:
Comma list: 99/98, 144/143, 176/175, 275/273
Comma list: 99/98, 144/143, 176/175, 275/273


Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}
{{Mapping|legend=0| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.469
* WE: ~2 = 1199.0325{{c}}, ~5/4 = 387.3137{{c}}
* POTE: ~2 = 1\1, ~5/4 = 387.626
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.5893{{c}}


Optimal ET sequence: {{optimal ET sequence| 31, 65d }}
{{Optimal ET sequence|legend=0| 31, 65d }}


Badness (Smith): 0.023593
Badness (Sintel): 0.975


==== Worseschmidt ====
==== Worseschmidt ====
Line 123: Line 129:
Commas: 66/65, 99/98, 105/104, 243/242
Commas: 66/65, 99/98, 105/104, 243/242


Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}
{{Mapping|legend=0| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.179
* WE: ~2 = 1200.4712{{c}}, ~5/4 = 387.2511{{c}}
* POTE: ~2 = 1\1, ~5/4 = 387.099
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1252{{c}}


Optimal ET sequence: {{optimal ET sequence| 3def, 28def, 31 }}
{{Optimal ET sequence|legend=0| 3def, 28def, 31 }}


Badness (Smith): 0.034382
Badness (Sintel): 1.42


== Worschmidt ==
== 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.
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
[[Subgroup]]: 2.3.5.7
Line 141: Line 147:


{{Mapping|legend=1| 1 -1 2 7 | 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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.406
* [[WE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.406{{c}}
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.392
: [[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 }}
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}


[[Badness]] (Smith): 0.064614
[[Badness]] (Sintel): 1.64


=== 11-limit ===
=== 11-limit ===
Line 157: Line 163:
Comma list: 126/125, 243/242, 385/384
Comma list: 126/125, 243/242, 385/384


Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}
{{Mapping|legend=0| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.472
* WE: ~2 = 1200.7911{{c}}, ~5/4 = 387.6624{{c}}
* POTE: ~2 = 1\1, ~5/4 = 387.407
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4189{{c}}


Optimal ET sequence: {{optimal ET sequence| 31, 65, 96d, 127d }}
{{Optimal ET sequence|legend=0| 31, 65, 96d, 127d }}


Badness (Smith): 0.033436
Badness (Sintel): 1.11


== Whirrschmidt ==
== Whirrschmidt ==
Line 175: Line 181:


{{Mapping|legend=1| 1 -1 2 -14 | 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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.853
* [[WE]]: ~2 = 1199.5703{{c}}, ~5/4 = 387.7422{{c}}
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.881
: [[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 }}
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}


[[Badness]] (Smith): 0.086334
[[Badness]] (Sintel): 2.18


=== 11-limit ===
=== 11-limit ===
Line 191: Line 197:
Comma list: 243/242, 896/891, 4375/4356
Comma list: 243/242, 896/891, 4375/4356


Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}
{{Mapping|legend=0| 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 for 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
 
{{Mapping|legend=2| 1 -1 2 0 | 0 8 1 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 387.829
* WE: ~2 = 1199.7075{{c}}, ~5/4 = 387.7106{{c}}
* POTE: ~2 = 1\1, ~5/4 = 387.882
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.7807{{c}}


Optimal ET sequence: {{optimal ET sequence| 34d, 65, 99e }}
{{Optimal ET sequence|legend=0| 31, 34, 65, 99, 164 }}


Badness (Smith): 0.058325
Badness (Sintel): 0.216


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 17:44, 30 September 2026

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 würschmidt family (würschmidt is sometimes spelled wuerschmidt) of temperaments tempers out 393216/390625, known as Würschmidt's comma, and named after José Würschmidt.

Würschmidt

Main article: Würschmidt

The generator of würschmidt is a classical major third, and to get to the interval class of the 3rd harmonic 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 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.

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

Subgroup: 2.3.5

Comma list: 393216/390625

Mapping: [⟨1 -1 2], ⟨0 8 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1199.6942 ¢, ~5/4 = 387.7005 ¢
error map: ⟨-0.306 -0.045 +0.775]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.7762 ¢
error map: ⟨0.000 +0.255 +1.463]

Optimal ET sequence: 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc

Badness (Sintel): 0.951

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.

With this accuracy level of 3/2 available, extensions that add prime 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, via tempering out 576/575 (S24) and 12167/12150 (S462⋅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: [⟨1 -1 2 -3], ⟨0 8 1 20]]

Optimal tuning:

  • WE: ~2 = 1199.7508 ¢, ~5/4 = 387.6058 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.6697 ¢

Optimal ET sequence: 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: [⟨1 -1 2 -3 0], ⟨0 8 1 20 14]]

Optimal tuning:

  • WE: ~2 = 1199.8205 ¢, ~5/4 = 387.6316 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.6770 ¢

Optimal ET sequence: 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 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 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 superparticulars 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: [⟨1 -1 2 -3], ⟨0 8 1 18]]

Optimal tunings:

  • WE: ~2 = 1199.9741 ¢, ~5/4 = 387.3742 ¢
error map: ⟨-0.026 -2.936 +1.009 +3.987]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.3809 ¢
error map: ⟨0.000 -2.907 +1.067 +4.031]

Optimal ET sequence: 31, 96, 127

Badness (Sintel): 1.28

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:

  • WE: ~2 = 1199.9618 ¢, ~5/4 = 387.4347 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.4446 ¢

Optimal ET sequence: 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: [⟨1 -1 2 -3 -3 5], ⟨0 8 1 18 20 -4]]

Optimal tunings:

  • WE: ~2 = 1199.0325 ¢, ~5/4 = 387.3137 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.5893 ¢

Optimal ET sequence: 31, 65d

Badness (Sintel): 0.975

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:

  • WE: ~2 = 1200.4712 ¢, ~5/4 = 387.2511 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.1252 ¢

Optimal ET sequence: 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 ⟨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:

  • WE: ~2 = 1200.000 ¢, ~5/4 = 387.406 ¢
error map: ⟨+0.763 -1.610 +2.851 -2.786]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.3950 ¢
error map: ⟨0.000 -2.795 +1.081 -4.960]

Optimal ET sequence: 31, 96d, 127d

Badness (Sintel): 1.64

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:

  • WE: ~2 = 1200.7911 ¢, ~5/4 = 387.6624 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.4189 ¢

Optimal ET sequence: 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: [⟨1 -1 2 -14], ⟨0 8 1 52]]

Optimal tunings:

  • WE: ~2 = 1199.5703 ¢, ~5/4 = 387.7422 ¢
error map: ⟨-0.430 +0.412 +0.569 -0.216]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.8729 ¢
error map: ⟨0.000 +1.029 +1.559 +0.567]

Optimal ET sequence: 34d, 65, 99

Badness (Sintel): 2.18

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:

  • WE: ~2 = 1199.2839 ¢, ~5/4 = 387.6507 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.8682 ¢

Optimal ET sequence: 34d, 65, 99e

Badness (Sintel): 1.93

Other subgroup extensions

Würschmidt (2.3.5.23)

Extensions for harmonics 47 and 49 are also available at +11 and +5 generator steps respectively, equalizing the 45::50 segment of the harmonic series. If we derive a mapping of prime 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: [⟨1 -1 2 0], ⟨0 8 1 14]]

Optimal tunings:

  • WE: ~2 = 1199.7075 ¢, ~5/4 = 387.7106 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 387.7807 ¢

Optimal ET sequence: 31, 34, 65, 99, 164

Badness (Sintel): 0.216

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