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{{Technical data page}}
Würschmidt
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.


The [[5-limit|5-limit]] parent comma for the würschmidt family is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt, Its [[monzo|monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[generator|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|minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[MOS|MOS]] all possibilities.
== Würschmidt ==
{{Main| Würschmidt }}


[[POTE_tuning|POTE generator]]: 387.799
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.  


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


EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[99edo|99]], [[164edo|164]], [[721edo|721c]], [[885edo|885c]]
[[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.


[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
[[Subgroup]]: 2.3.5


Würschmidt[13] in 5-limit minimax tuning
[[Comma list]]: 393216/390625


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


==Seven limit children==
[[Optimal tuning]]s:
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;.
* [[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 }}


=Würschmidt=
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}
Würschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo|31edo]] or [[127edo|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|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.


Commas: 225/224, 8748/8575
[[Badness]] (Sintel): 0.951


[[POTE_tuning|POTE generator]]: 387.383
=== 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.


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


EDOs: [[31edo|31]], [[96edo|96]], [[127edo|127]], [[285edo|28bd]], [[412edo|412bd]]
==== 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]].


Badness: 0.0508
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.  


==11-limit==
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]].
Commas: 99/98, 176/175, 243/242


POTE generator: ~5/4 = 387.447
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11


Map: [&lt;1 7 3 15 17|, &lt;0 -8 -1 -18 -20|]
Comma list: 243/242, 5632/5625


EDOs: 31, 65d, 96, 127, 223d
{{Mapping|legend=2| 1 -1 2 -3 | 0 8 1 20 }}


Badness: 0.0244
Optimal tuning:  
* WE: ~2 = 1199.7508{{c}}, ~5/4 = 387.6058{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6697{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 31, 34, 65 }}
Commas: 99/98, 144/143, 176/175, 275/273


POTE generator: ~5/4 = 387.626
Badness (Sintel): 0.477


Map: [&lt;1 7 3 15 17 1|, &lt;0 -8 -1 -18 -20 4|]
==== 2.3.5.11.23 subgroup ====
Subgroup: 2.3.5.11.23


EDOs: 31, 65d, 161df
Comma list: 243/242, 276/275, 529/528


Badness: 0.0236
{{Mapping|legend=2| 1 -1 2 -3 0 | 0 8 1 20 14 }}


==Worseschmidt==
Optimal tuning:
Commas: 66/65, 99/98, 105/104, 243/242
* WE: ~2 = 1199.8205{{c}}, ~5/4 = 387.6316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6770{{c}}
 
{{Optimal ET sequence|legend=0| 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.


POTE generator: ~5/4 = 387.099
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.


Map: [&lt;1 7 3 15 17 22|, &lt;0 -8 -1 -18 -20 -27|]
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.


EDOs: 31
[[Subgroup]]: 2.3.5.7


Badness: 0.0344
[[Comma list]]: 225/224, 8748/8575


=Worschmidt=
{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }}
Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo|31edo]], [[34edo|34edo]], or [[127edo|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.


Commas: 126/125, 33075/32768
[[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 }}


[[POTE_tuning|POTE generator]]: 387.392
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


Map: [&lt;1 7 3 -6|, &lt;0 -8 -1 13|]
[[Badness]] (Sintel): 1.28


EDOs: [[31edo|31]], [[65edo|65]], [[96edo|96d]], [[127edo|127d]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0646
Comma list: 99/98, 176/175, 243/242


==11-limit==
{{Mapping|legend=0| 1 -1 2 -3 -3 | 0 8 1 18 20 }}
Commas: 126/125, 243/242, 385/384


POTE generator: ~5/4 = 387.407
Optimal tunings:  
* WE: ~2 = 1199.9618{{c}}, ~5/4 = 387.4347{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4446{{c}}


Map: [&lt;1 7 3 -6 17|, &lt;0 -8 -1 13 -20|]
{{Optimal ET sequence|legend=0| 31, 65d, 96, 127 }}


EDOs: 31, 65, 96d, 127d
Badness (Sintel): 0.807


Badness: 0.0334
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Whirrschmidt=
Comma list: 99/98, 144/143, 176/175, 275/273
[[99edo|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: 4375/4374, 393216/390625
{{Mapping|legend=0| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


[[POTE_tuning|POTE generator]]: 387.881
Optimal tunings:  
* WE: ~2 = 1199.0325{{c}}, ~5/4 = 387.3137{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.5893{{c}}


Map: [&lt;1 7 3 38|, &lt;0 -8 -1 -52|]
{{Optimal ET sequence|legend=0| 31, 65d }}


EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[99edo|99]]
Badness (Sintel): 0.975


=Hemiwürschmidt=
==== Worseschmidt ====
Hemiwürschmidt, 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|68edo]], [[99edo|99edo]] and [[130edo|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: 2.3.5.7.11.13


Commas: 2401/2400, 3136/3125
Commas: 66/65, 99/98, 105/104, 243/242


[[POTE_tuning|POTE generator]]: ~28/25 = 193.898
{{Mapping|legend=0| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


Map: [&lt;1 15 4 7|, &lt;0 -16 -2 -5|]
Optimal tunings:  
* WE: ~2 = 1200.4712{{c}}, ~5/4 = 387.2511{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1252{{c}}


&lt;&lt;16 2 5 -34 -37 6||
{{Optimal ET sequence|legend=0| 3def, 28def, 31 }}


EDOs: [[6edo|6]], [[31edo|31]], [[37edo|37]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]], [[557edo|557c]], [[885edo|885c]]
Badness (Sintel): 1.42


Badness: 0.0203
== 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.


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 243/242, 441/440, 3136/3125


[[POTE_tuning|POTE generator]]: ~28/25 = 193.840
[[Comma list]]: 126/125, 33075/32768


Map: [&lt;1 15 4 7 37|, &lt;0 -16 -2 -5 -40|]
{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }}


EDOs: 31, 99e, 130, 650ce, 811ce
[[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 }}


Badness: 0.0211
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}


===13-limit===
[[Badness]] (Sintel): 1.64
Commas: 243/242 351/350 441/440 3584/3575


POTE generator: ~28/25 = 193.840
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 15 4 7 37 -29|, &lt;0 -16 -2 -5 -40 39|]
Comma list: 126/125, 243/242, 385/384


EDOs: 31, 99e, 130, 291, 421e, 551ce
{{Mapping|legend=0| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


Badness: 0.0231
Optimal tunings:  
* WE: ~2 = 1200.7911{{c}}, ~5/4 = 387.6624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4189{{c}}


===Hemithir===
{{Optimal ET sequence|legend=0| 31, 65, 96d, 127d }}
Commas: 121/120 176/175 196/195 275/273


POTE generator: ~28/25 = 193.918
Badness (Sintel): 1.11


Map: [&lt;1 15 4 7 37 -3|, &lt;0 -16 -2 -5 -40 8|]
== 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.


EDOs: 31, 68e, 99ef
[[Subgroup]]: 2.3.5.7


Badness: 0.0312
[[Comma list]]: 4375/4374, 393216/390625


==Hemiwur==
{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }}
Commas: 121/120, 176/175, 1375/1372


POTE generator: ~28/25 = 193.884
[[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 }}


Map: [&lt;1 15 4 7 11|, &lt;0 -16 -2 -5 -9|]
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}


EDOs: 6, 31, 68, 99, 130e, 229e
[[Badness]] (Sintel): 2.18


Badness: 0.0293
=== 11-limit ===
Subgroup: 2.3.5.7.11


===13-limit===
Comma list: 243/242, 896/891, 4375/4356
Commas: 121/120, 176/175, 196/195, 275/273


POTE generator: ~28/25 = 194.004
{{Mapping|legend=0| 1 -1 2 -14 -3 | 0 8 1 52 20 }}


Map: [&lt;1 15 4 7 11 -3|, &lt;0 -16 -2 -5 -9 8|]
Optimal tunings:  
* WE: ~2 = 1199.2839{{c}}, ~5/4 = 387.6507{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.8682{{c}}


EDOs: 6, 31, 68, 99f, 167ef
{{Optimal ET sequence|legend=0| 34d, 65, 99e }}


Badness: 0.0284
Badness (Sintel): 1.93


===Hemiwar===
== Other subgroup extensions ==
Commas: 66/65, 105/104, 121/120, 1375/1372
=== 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]].


POTE generator: ~28/25 = 193.698
Subgroup: 2.3.5.23


Map: [&lt;1 15 4 7 11 23|, &lt;0 -16 -2 -5 -9 -23|]
Comma list: 576/575, 12167/12150


EDOs: 31
{{Mapping|legend=2| 1 -1 2 0 | 0 8 1 14 }}


Badness: 0.0449
Optimal tunings:  
* WE: ~2 = 1199.7075{{c}}, ~5/4 = 387.7106{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.7807{{c}}


=Relationships to other temperaments=
{{Optimal ET sequence|legend=0| 31, 34, 65, 99, 164 }}
<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|skwares]] as a 2.3.7.11 temperament.
Badness (Sintel): 0.216


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Wuerschmidt]]
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Wurschmidt]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Hemiwuerschmidt]]
Retrieved from "https://en.xen.wiki/w/Würschmidt_family"