Würschmidt family: Difference between revisions

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__FORCETOC__
The [[5-limit]] parent comma for the '''würschmidt family''' is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt, Its [[monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[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]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[MOS]] all possibilities.
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.
[[POTE generator]]: 387.799
 
[[POTE_tuning|POTE generator]]: 387.799


Map: [<1 7 3|, <0 -8 -1|]
Map: [<1 7 3|, <0 -8 -1|]


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


[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
[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
Line 16: Line 13:
[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>.


=Würschmidt=
== Würschmidt ==
Würschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo|31edo]] or [[127edo|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|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
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.


[[POTE_tuning|POTE generator]]: 387.383
Commas: [[225/224]], 8748/8575
 
[[POTE generator]]: 387.383


Map: [<1 7 3 15|, <0 -8 -1 -18|]
Map: [<1 7 3 15|, <0 -8 -1 -18|]


EDOs: [[31edo|31]], [[96edo|96]], [[127edo|127]], [[285edo|28bd]], [[412edo|412bd]]
EDOs: {{EDOs| 31, 96, 127, 28bd, 412bd }}


Badness: 0.0508
Badness: 0.0508


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


POTE generator: ~5/4 = 387.447
POTE generator: ~5/4 = 387.447
Line 39: Line 39:
Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]
Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]


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


Badness: 0.0244
Badness: 0.0244


==13-limit==
=== 13-limit ===
Commas: 99/98, 144/143, 176/175, 275/273
 
Commas: [[99/98]], [[144/143]], 176/175, 275/273


POTE generator: ~5/4 = 387.626
POTE generator: ~5/4 = 387.626
Line 50: Line 51:
Map: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]
Map: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]


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


Badness: 0.0236
Badness: 0.0236


==Worseschmidt==
== Worseschmidt ==
Commas: 66/65, 99/98, 105/104, 243/242
 
Commas: 66/65, [[99/98]], 105/104, [[243/242]]


POTE generator: ~5/4 = 387.099
POTE generator: ~5/4 = 387.099
Line 61: Line 63:
Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]
Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]


EDOs: 31
EDOs: {{EDOs| 31 }}


Badness: 0.0344
Badness: 0.0344


=Worschmidt=
== Worschmidt ==
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 <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.
 
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.


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


[[POTE_tuning|POTE generator]]: 387.392
[[POTE generator]]: 387.392


Map: [<1 7 3 -6|, <0 -8 -1 13|]
Map: [<1 7 3 -6|, <0 -8 -1 13|]


EDOs: [[31edo|31]], [[65edo|65]], [[96edo|96d]], [[127edo|127d]]
EDOs: {{EDOs| 31, 65, 96d, 127d }}


Badness: 0.0646
Badness: 0.0646


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


Line 85: Line 89:
Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]
Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]


EDOs: 31, 65, 96d, 127d
EDOs: {{EDOs| 31, 65, 96d, 127d }}


Badness: 0.0334
Badness: 0.0334


=Whirrschmidt=
== Whirrschmidt ==
[[99edo|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.


Commas: 4375/4374, 393216/390625
Commas: 4375/4374, 393216/390625


[[POTE_tuning|POTE generator]]: 387.881
[[POTE generator]]: 387.881


Map: [<1 7 3 38|, <0 -8 -1 -52|]
Map: [<1 7 3 38|, <0 -8 -1 -52|]


EDOs: [[31edo|31]], [[34edo|34]], [[65edo|65]], [[99edo|99]]
EDOs: {{EDOs| 31, 34, 65, 99 }}


=Hemiwürschmidt=
== Hemiwürschmidt ==
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, <<16 2 5 40 -39 -49 -48 28...
 
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]], [[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
Commas: 2401/2400, 3136/3125


[[POTE_tuning|POTE generator]]: ~28/25 = 193.898
[[POTE generator]]: ~28/25 = 193.898


Map: [<1 15 4 7|, <0 -16 -2 -5|]
Map: [<1 15 4 7|, <0 -16 -2 -5|]
Line 111: Line 117:
<<16 2 5 -34 -37 6||
<<16 2 5 -34 -37 6||


EDOs: [[6edo|6]], [[31edo|31]], [[37edo|37]], [[68edo|68]], [[99edo|99]], [[229edo|229]], [[328edo|328]], [[557edo|557c]], [[885edo|885c]]
EDOs: {{EDOs| 6, 31, 37, 68, 99, 229, 328, 557c, 885c }}


Badness: 0.0203
Badness: 0.0203


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


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


Map: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]
Map: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]


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


Badness: 0.0211
Badness: 0.0211


===13-limit===
=== 13-limit ===
 
Commas: 243/242 351/350 441/440 3584/3575
Commas: 243/242 351/350 441/440 3584/3575


Line 133: Line 141:
Map: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]
Map: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]


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


Badness: 0.0231
Badness: 0.0231


===Hemithir===
== Hemithir ==
 
Commas: 121/120 176/175 196/195 275/273
Commas: 121/120 176/175 196/195 275/273


Line 144: Line 153:
Map: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]
Map: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]


EDOs: 31, 68e, 99ef
EDOs: {{EDOs| 31, 68e, 99ef }}


Badness: 0.0312
Badness: 0.0312


==Hemiwur==
== Hemiwur ==
 
Commas: 121/120, 176/175, 1375/1372
Commas: 121/120, 176/175, 1375/1372


Line 155: Line 165:
Map: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]
Map: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]


EDOs: 6, 31, 68, 99, 130e, 229e
EDOs: {{EDOs| 6, 31, 68, 99, 130e, 229e }}


Badness: 0.0293
Badness: 0.0293


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


Line 166: Line 177:
Map: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]
Map: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]


EDOs: 6, 31, 68, 99f, 167ef
EDOs: {{EDOs| 6, 31, 68, 99f, 167ef }}


Badness: 0.0284
Badness: 0.0284


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


Line 177: Line 189:
Map: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]
Map: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]


EDOs: 31
EDOs: {{EDOs| 31 }}


Badness: 0.0449
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>


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.
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:Theory]]
[[Category:Theory]]

Revision as of 20:01, 13 October 2020

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

POTE generator: 387.799

Map: [<1 7 3|, <0 -8 -1|]

EDOs: 31, 34, 65, 99, 164, 721c, 885c

Ancient Stardust play by Chris Vaisvil

Würschmidt[13] in 5-limit minimax tuning

Extrospection by Jake Freivald; Würschmidt[16] tuned in 31et.

Seven limit children

The second comma of the 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>.

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.

Commas: 225/224, 8748/8575

POTE generator: 387.383

Map: [<1 7 3 15|, <0 -8 -1 -18|]

EDOs: 31, 96, 127, 28bd, 412bd

Badness: 0.0508

11-limit

Commas: 99/98, 176/175, 243/242

POTE generator: ~5/4 = 387.447

Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]

EDOs: 31, 65d, 96, 127, 223d

Badness: 0.0244

13-limit

Commas: 99/98, 144/143, 176/175, 275/273

POTE generator: ~5/4 = 387.626

Map: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]

EDOs: 31, 65d, 161df

Badness: 0.0236

Worseschmidt

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

POTE generator: ~5/4 = 387.099

Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]

EDOs: 31

Badness: 0.0344

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 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.

Commas: 126/125, 33075/32768

POTE generator: 387.392

Map: [<1 7 3 -6|, <0 -8 -1 13|]

EDOs: 31, 65, 96d, 127d

Badness: 0.0646

11-limit

Commas: 126/125, 243/242, 385/384

POTE generator: ~5/4 = 387.407

Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]

EDOs: 31, 65, 96d, 127d

Badness: 0.0334

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.

Commas: 4375/4374, 393216/390625

POTE generator: 387.881

Map: [<1 7 3 38|, <0 -8 -1 -52|]

EDOs: 31, 34, 65, 99

Hemiwürschmidt

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, 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

POTE generator: ~28/25 = 193.898

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

<<16 2 5 -34 -37 6||

EDOs: 6, 31, 37, 68, 99, 229, 328, 557c, 885c

Badness: 0.0203

11-limit

Commas: 243/242, 441/440, 3136/3125

POTE generator: ~28/25 = 193.840

Map: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]

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

Badness: 0.0211

13-limit

Commas: 243/242 351/350 441/440 3584/3575

POTE generator: ~28/25 = 193.840

Map: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]

EDOs: 31, 99e, 130, 291, 421e, 551ce

Badness: 0.0231

Hemithir

Commas: 121/120 176/175 196/195 275/273

POTE generator: ~28/25 = 193.918

Map: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]

EDOs: 31, 68e, 99ef

Badness: 0.0312

Hemiwur

Commas: 121/120, 176/175, 1375/1372

POTE generator: ~28/25 = 193.884

Map: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]

EDOs: 6, 31, 68, 99, 130e, 229e

Badness: 0.0293

13-limit

Commas: 121/120, 176/175, 196/195, 275/273

POTE generator: ~28/25 = 194.004

Map: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]

EDOs: 6, 31, 68, 99f, 167ef

Badness: 0.0284

Hemiwar

Commas: 66/65, 105/104, 121/120, 1375/1372

POTE generator: ~28/25 = 193.698

Map: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]

EDOs: 31

Badness: 0.0449

Relationships to other temperaments

around 775.489 which is approximately

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.

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