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= Würschmidt =
= Würschmidt =
('''Würschmidt''' is sometimes spelled '''Wuerschmidt''')


('''Würschmidt''' is sometimes spelled '''Wuerschmidth''')
[[Comma]]: 393216/390625


Comma: 393216/390625
[[Mapping]]: [<1 7 3|, <0 -8 -1|]


[[POTE generator]]: ~5/4 = 387.799
[[POTE generator]]: ~5/4 = 387.799


Map: [<1 7 3|, <0 -8 -1|]
[[EDO|Vals]]: {{Val list| 31, 34, 65, 99, 164, 721c, 885c }}


EDOs: {{EDOs| 31, 34, 65, 99, 164, 721c, 885c }}
[[Badness]]: 0.040603


== Music ==
== Music ==
* [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


[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
* [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.
 
[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.


== Seven limit children ==
== Seven limit children ==
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>, worschmidt adds 65625/65536 = |-16 1 5 1>, whirrschmidt adds 4375/4374 = |-1 -7 4 1> and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2>.
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>, worschmidt adds 65625/65536 = |-16 1 5 1>, whirrschmidt adds 4375/4374 = |-1 -7 4 1> and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2>.


= 7-limit =
= Septimal Würschmidt =
Würschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo]] or [[127edo]] can be used as tunings. Würschmidt has <<8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version <<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.


Würschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo]] or [[127edo]] can be used as tunings. Würschmidt has <<8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version <<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.
[[Comma list]]: [[225/224]], 8748/8575


Commas: [[225/224]], 8748/8575
[[Mapping]]: [<1 7 3 15|, <0 -8 -1 -18|]


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


Map: [<1 7 3 15|, <0 -8 -1 -18|]
[[EDO|Vals]]: {{Val list| 31, 96, 127, 285bd, 412bbdd }}


EDOs: {{EDOs| 31, 96, 127, 285bd, 412bbdd }}
[[Badness]]: 0.050776
 
Badness: 0.0508


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


Commas: [[99/98]], 176/175, [[243/242]]
Mapping: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]


POTE generator: ~5/4 = 387.447
POTE generator: ~5/4 = 387.447


Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]
Vals: {{Val list| 31, 65d, 96, 127, 223d }}


EDOs: {{EDOs| 31, 65d, 96, 127, 223d }}
Badness: 0.024413


Badness: 0.0244
== 13-limit ==
 
Comma list: 99/98, 144/143, 176/175, 275/273
=== 13-limit ===


Commas: [[99/98]], [[144/143]], 176/175, 275/273
Mapping: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]


POTE generator: ~5/4 = 387.626
POTE generator: ~5/4 = 387.626


Map: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]
Vals: {{Val list| 31, 65d, 161df }}


EDOs: {{EDOs| 31, 65d, 161df }}
Badness: 0.023593


Badness: 0.0236
== Worseschmidt ==
Commas: 66/65, 99/98, 105/104, 243/242


=== Worseschmidt ===
Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]
 
Commas: 66/65, [[99/98]], 105/104, [[243/242]]


POTE generator: ~5/4 = 387.099
POTE generator: ~5/4 = 387.099


Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]
Vals: {{EDOs| 3def, 28def, 31 }}
 
EDOs: {{EDOs| 31 }}


Badness: 0.0344
Badness: 0.034382


= 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 <127 201 295 356| and not <127 201 295 357| as with würschmidt. The wedgie now is <<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.


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| and not <127 201 295 357| as with wurschmidt. The wedgie now is <<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.
[[Comma list]]: 126/125, 33075/32768


Commas: [[126/125]], 33075/32768
[[Mapping]]: [<1 7 3 -6|, <0 -8 -1 13|]


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


Map: [<1 7 3 -6|, <0 -8 -1 13|]
[[EDO|Vals]]: {{Val list| 31, 65, 96d, 127d }}


EDOs: {{EDOs| 31, 65, 96d, 127d }}
[[Badness]]: 0.064614
 
Badness: 0.0646


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


Commas: 126/125, 243/242, 385/384
Mapping: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]


POTE generator: ~5/4 = 387.407
POTE generator: ~5/4 = 387.407


Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]
Vals: {{Val list| 31, 65, 96d, 127d }}


EDOs: {{EDOs| 31, 65, 96d, 127d }}
Badness: 0.033436
 
Badness: 0.0334


= Whirrschmidt =
= Whirrschmidt =
[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with <<8 1 52 -17 60 118|| for a wedgie.


[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with <<8 1 52 -17 60 118|| for a wedgie.
[[Comma list]]: 4375/4374, 393216/390625


Commas: 4375/4374, 393216/390625
[[Mapping]]: [<1 7 3 38|, <0 -8 -1 -52|]


[[POTE generator]]: ~5/4 = 387.881
[[POTE generator]]: ~5/4 = 387.881


Map: [<1 7 3 38|, <0 -8 -1 -52|]
[[EDO|Vals]]: {{Val list| 31dd, 34d, 65, 99 }}


EDOs: {{EDOs| 31dd, 34d, 65, 99 }}
[[Badness]]: 0.086334


= Hemiwürschmidt =
= 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 [[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, <<16 2 5 40 -39 -49 -48 28...
'''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, <<16 2 5 40 -39 -49 -48 28...


Commas: 2401/2400, 3136/3125
[[Comma list]]: 2401/2400, 3136/3125


[[POTE generator]]: ~28/25 = 193.898
[[Mapping]]: [<1 15 4 7|, <0 -16 -2 -5|]


Map: [<1 15 4 7|, <0 -16 -2 -5|]
[[Wedgie]]: <<16 2 5 -34 -37 6||


<<16 2 5 -34 -37 6||
[[POTE generator]]: ~28/25 = 193.898


EDOs: {{EDOs| 31, 68, 99, 229, 328, 557c, 885cc }}
[[EDO|Vals]]: {{Val list| 31, 68, 99, 229, 328, 557c, 885cc }}


Badness: 0.0203
[[Badness]]: 0.0203


== 11-limit ==
== 11-limit ==
Comma list: 243/242, 441/440, 3136/3125


Commas: 243/242, 441/440, 3136/3125
Mapping: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]


[[POTE generator]]: ~28/25 = 193.840
POTE generator: ~28/25 = 193.840


Map: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]
Vals: {{Val list| 31, 99e, 130, 650ce, 811ce }}


EDOs: {{EDOs| 31, 99e, 130, 650ce, 811ce }}
Badness: 0.021069
 
Badness: 0.0211


=== 13-limit ===
=== 13-limit ===
Comma list: 243/242, 351/350, 441/440, 3584/3575


Commas: 243/242, 351/350, 441/440, 3584/3575
Mapping: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]


POTE generator: ~28/25 = 193.840
POTE generator: ~28/25 = 193.829


Map: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]
Vals: {{Val list| 31, 99e, 130, 291, 421e, 551ce }}


EDOs: {{EDOs| 31, 99e, 130, 291, 421e, 551ce }}
Badness: 0.023074
 
Badness: 0.0231


=== Hemithir ===
=== Hemithir ===
Comma list: 121/120, 176/175, 196/195, 275/273


Commas: 121/120, 176/175, 196/195, 275/273
Mapping: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]


POTE generator: ~28/25 = 193.918
POTE generator: ~28/25 = 193.918


Map: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]
Vals: {{Val list| 31, 68e, 99ef }}


EDOs: {{EDOs| 31, 68e, 99ef }}
Badness: 0.031199
 
Badness: 0.0312


== Hemiwur ==
== Hemiwur ==
Comma list: 121/120, 176/175, 1375/1372


Commas: 121/120, 176/175, 1375/1372
Mapping: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]


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


Map: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]
Vals: {{Val list| 31, 68, 99, 130e, 229e }}
 
EDOs: {{EDOs| 31, 68, 99, 130e, 229e }}


Badness: 0.0293
Badness: 0.029270


=== 13-limit ===
=== 13-limit ===
Comma list: 121/120, 176/175, 196/195, 275/273


Commas: 121/120, 176/175, 196/195, 275/273
Mapping: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]


POTE generator: ~28/25 = 194.004
POTE generator: ~28/25 = 194.004


Map: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]
Vals: {{Val list| 31, 68, 99f, 167ef }}
 
EDOs: {{EDOs| 31, 68, 99f, 167ef }}


Badness: 0.0284
Badness: 0.028432


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


Commas: 66/65, 105/104, 121/120, 1375/1372
Mapping: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]


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


Map: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]
Vals: {{Val list| 6f, 31 }}


EDOs: {{EDOs| 31 }}
Badness: 0.044886
 
Badness: 0.0449


= Relationships to other temperaments =
= 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>
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: -25px; width: 1px;">around 775.489 which is approximately</span>


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