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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. 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 '''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.  


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 ==
{{Main| Würschmidt }}


[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales does not appear until 28, 31 or even 34 notes, depending on the specific 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.  


The 7-limit extensions can be obtained by adding another comma. Septimal würschmidt adds [[225/224]], worschmidt adds [[126/125]], whirrschmidt adds 4375/4374 = {{| -1 -7 4 1 }}. These all use the same generator as 5-limit 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]] 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]].  


Hemiwürschmidt adds 6144/6125 = {{monzo| 11 1 -3 -2 }} 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]].  
[[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.  


== Würschmidt ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


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


{{Mapping|legend=1| 1 7 3 | 0 -8 -1 }}
{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}
: mapping generators: ~2, ~5/4
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6942{{c}}, ~5/4 = 387.7005{{c}}
: [[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 }}
 
[[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 #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
 
{{Mapping|legend=2| 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


[[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 }}
{{Mapping|legend=2| 1 -1 2 -3 0 | 0 8 1 20 14 }}


[[Badness]]: 0.040603
Optimal tuning:  
* WE: ~2 = 1199.8205{{c}}, ~5/4 = 387.6316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6770{{c}}


; Music
{{Optimal ET sequence|legend=0| 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 (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 36: Line 82:
[[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]]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 tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.383
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


{{Optimal ET sequence|legend=1| 31, 96, 127, 285bd, 412bbdd }}
[[Badness]] (Sintel): 1.28
 
[[Badness]]: 0.050776


=== 11-limit ===
=== 11-limit ===
Line 51: Line 99:
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|legend=0| 1 -1 2 -3 -3 | 0 8 1 18 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.447
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=1| 31, 65d, 96, 127, 223d }}
{{Optimal ET sequence|legend=0| 31, 65d, 96, 127 }}


Badness: 0.024413
Badness (Sintel): 0.807


==== 13-limit ====
==== 13-limit ====
Line 64: 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 7 3 15 17 1 | 0 -8 -1 -18 -20 4 }}
{{Mapping|legend=0| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.626
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=1| 31, 65d, 161df }}
{{Optimal ET sequence|legend=0| 31, 65d }}


Badness: 0.023593
Badness (Sintel): 0.975


==== Worseschmidt ====
==== Worseschmidt ====
Line 77: Line 129:
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|legend=0| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.099
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=1| 3def, 28def, 31 }}
{{Optimal ET sequence|legend=0| 3def, 28def, 31 }}


Badness: 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 92: Line 146:
[[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:
* [[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 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]] (Sintel): 1.64
 
[[Badness]]: 0.064614


=== 11-limit ===
=== 11-limit ===
Line 107: Line 163:
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|legend=0| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.407
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=1| 31, 65, 96d, 127d }}
{{Optimal ET sequence|legend=0| 31, 65, 96d, 127d }}


Badness: 0.033436
Badness (Sintel): 1.11


== Whirrschmidt ==
== Whirrschmidt ==
Line 122: Line 180:
[[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:
* [[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 }}
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}


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


=== 11-limit ===
=== 11-limit ===
Line 137: Line 197:
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|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 tuning (POTE): ~2 = 1\1, ~5/4 = 387.882
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=1| 34d, 65, 99e }}
{{Optimal ET sequence|legend=0| 31, 34, 65, 99, 164 }}


Badness: 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]]
Retrieved from "https://en.xen.wiki/w/Würschmidt_family"