Würschmidt family: Difference between revisions

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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.
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 {{Monzo|17 1 -8}}, and flipping that yields {{Multival|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 ==
== Würschmidt ==
('''Würschmidt''' is sometimes spelled '''Wuerschmidt''')
('''Würschmidt''' is sometimes spelled '''Wuerschmidt''')


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


[[Mapping]]: [<1 7 3|, <0 -8 -1|]
[[Mapping]]: [{{Val|1 7 3}}, {{Val|0 -8 -1}}]


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


[[EDO|Vals]]: {{Val list| 31, 34, 65, 99, 164, 721c, 885c }}
{{Val list|legend=1| 31, 34, 65, 99, 164, 721c, 885c }}


[[Badness]]: 0.040603
[[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 31EDO.


=== 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. Würschmidt adds {{Monzo|12 3 -6 -1}}, worschmidt adds 65625/65536 = {{Monzo|-16 1 5 1}}, whirrschmidt adds 4375/4374 = {{Monzo|-1 -7 4 1}} and hemiwuerschmidt adds 6144/6125 = {{Monzo|11 1 -3 -2}}.


== Septimal Würschmidt ==
== 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|31EDO]] or [[127edo|127EDO]] can be used as tunings. Würschmidt has {{Multival|8 1 18 -17 6 39}} for a wedgie. 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 three temperament tempering out 99/98 and 176/175.


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


[[Mapping]]: [<1 7 3 15|, <0 -8 -1 -18|]
[[Mapping]]: [{{val|1 7 3 15}}, {{Val|0 -8 -1 -18}}]


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


[[EDO|Vals]]: {{Val list| 31, 96, 127, 285bd, 412bbdd }}
{{Val list|legend=1| 31, 96, 127, 285bd, 412bbdd }}


[[Badness]]: 0.050776
[[Badness]]: 0.050776


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


Mapping: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]
Mapping: [{{val|1 7 3 15 17}}, {{val|0 -8 -1 -18 -20}}]


POTE generator: ~5/4 = 387.447
POTE generator: ~5/4 = 387.447
Line 46: Line 46:
Badness: 0.024413
Badness: 0.024413


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


Mapping: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]
Mapping: [{{val|1 7 3 15 17 1}}, {{val|0 -8 -1 -18 -20 4}}]


POTE generator: ~5/4 = 387.626
POTE generator: ~5/4 = 387.626
Line 57: Line 57:
Badness: 0.023593
Badness: 0.023593


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


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


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


Vals: {{EDOs| 3def, 28def, 31 }}
Vals: {{Val list| 3def, 28def, 31 }}


Badness: 0.034382
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|31EDO]], [[34edo|34EDO]], or [[127edo|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. The wedgie now is {{Multival|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
[[Comma list]]: 126/125, 33075/32768


[[Mapping]]: [<1 7 3 -6|, <0 -8 -1 13|]
[[Mapping]]: [{{val|1 7 3 -6}}, {{val|0 -8 -1 13}}]


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


[[EDO|Vals]]: {{Val list| 31, 65, 96d, 127d }}
{{Val list|legend=1| 31, 65, 96d, 127d }}


[[Badness]]: 0.064614
[[Badness]]: 0.064614


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


Mapping: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]
Mapping: [{{val|1 7 3 -6 17}}, {{val|0 -8 -1 13 -20}}]


POTE generator: ~5/4 = 387.407
POTE generator: ~5/4 = 387.407
Line 92: Line 92:
Badness: 0.033436
Badness: 0.033436


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


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


[[Mapping]]: [<1 7 3 38|, <0 -8 -1 -52|]
[[Mapping]]: [{{val|1 7 3 38}}, {{val|0 -8 -1 -52}}]


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


[[EDO|Vals]]: {{Val list| 31dd, 34d, 65, 99 }}
{{Val list|legend=1| 31dd, 34d, 65, 99 }}


[[Badness]]: 0.086334
[[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|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, {{Multival|16 2 5 40 -39 -49 -48 28...}}


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


[[Mapping]]: [<1 15 4 7|, <0 -16 -2 -5|]
[[Mapping]]: [{{val|1 15 4 7}}, {{val|0 -16 -2 -5}}]


[[Wedgie]]: <<16 2 5 -34 -37 6||
{{Multival|legend=1|16 2 5 -34 -37 6}}


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


[[EDO|Vals]]: {{Val list| 31, 68, 99, 229, 328, 557c, 885cc }}
{{Val list|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}


[[Badness]]: 0.0203
[[Badness]]: 0.020307


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


Mapping: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]
Mapping: [{{val|1 15 4 7 37}}, {{val|0 -16 -2 -5 -40}}]


POTE generator: ~28/25 = 193.840
POTE generator: ~28/25 = 193.840
Line 131: Line 131:
Badness: 0.021069
Badness: 0.021069


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


Mapping: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]
Mapping: [{{val|1 15 4 7 37 -29}}, {{val|0 -16 -2 -5 -40 39}}]


POTE generator: ~28/25 = 193.829
POTE generator: ~28/25 = 193.829
Line 142: Line 142:
Badness: 0.023074
Badness: 0.023074


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


Mapping: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]
Mapping: [{{val|1 15 4 7 37 -3}}, {{val|0 -16 -2 -5 -40 8}}]


POTE generator: ~28/25 = 193.918
POTE generator: ~28/25 = 193.918
Line 153: Line 153:
Badness: 0.031199
Badness: 0.031199


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


Mapping: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]
Mapping: [{{val|1 15 4 7 11}}, {{val|0 -16 -2 -5 -9}}]


POTE generator: ~28/25 = 193.884
POTE generator: ~28/25 = 193.884
Line 164: Line 164:
Badness: 0.029270
Badness: 0.029270


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


Mapping: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]
Mapping: [{{val|1 15 4 7 11 -3}}, {{val|0 -16 -2 -5 -9 8}}]


POTE generator: ~28/25 = 194.004
POTE generator: ~28/25 = 194.004
Line 175: Line 175:
Badness: 0.028432
Badness: 0.028432


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


Mapping: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]
Mapping: [{{val|1 15 4 7 11 23}}, {{val|0 -16 -2 -5 -9 -23}}]


POTE generator: ~28/25 = 193.698
POTE generator: ~28/25 = 193.698
Line 186: Line 186:
Badness: 0.044886
Badness: 0.044886


== 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>


Revision as of 03:36, 10 June 2021

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

(Würschmidt is sometimes spelled Wuerschmidt)

Comma: 393216/390625

Mapping: [⟨1 7 3], ⟨0 -8 -1]]

POTE generator: ~5/4 = 387.799

Template:Val list

Badness: 0.040603

Music

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Würschmidt 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⟩.

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 würschmidt, as well as for minerva, the 11-limit rank three temperament tempering out 99/98 and 176/175.

Comma list: 225/224, 8748/8575

Mapping: [⟨1 7 3 15], ⟨0 -8 -1 -18]]

POTE generator: ~5/4 = 387.383

Template:Val list

Badness: 0.050776

11-limit

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

Mapping: [⟨1 7 3 15 17], ⟨0 -8 -1 -18 -20]]

POTE generator: ~5/4 = 387.447

Vals: Template:Val list

Badness: 0.024413

13-limit

Comma list: 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

Vals: Template:Val list

Badness: 0.023593

Worseschmidt

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

Mapping: [⟨1 7 3 15 17 22], ⟨0 -8 -1 -18 -20 -27]]

POTE generator: ~5/4 = 387.099

Vals: Template:Val list

Badness: 0.034382

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. 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

Mapping: [⟨1 7 3 -6], ⟨0 -8 -1 13]]

POTE generator: ~5/4 = 387.392

Template:Val list

Badness: 0.064614

11-limit

Comma list: 126/125, 243/242, 385/384

Mapping: [⟨1 7 3 -6 17], ⟨0 -8 -1 13 -20]]

POTE generator: ~5/4 = 387.407

Vals: Template:Val list

Badness: 0.033436

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.

Comma list: 4375/4374, 393216/390625

Mapping: [⟨1 7 3 38], ⟨0 -8 -1 -52]]

POTE generator: ~5/4 = 387.881

Template:Val list

Badness: 0.086334

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... ]]

Comma list: 2401/2400, 3136/3125

Mapping: [⟨1 15 4 7], ⟨0 -16 -2 -5]]

Wedgie: ⟨⟨ 16 2 5 -34 -37 6 ]]

POTE generator: ~28/25 = 193.898

Template:Val list

Badness: 0.020307

11-limit

Comma list: 243/242, 441/440, 3136/3125

Mapping: [⟨1 15 4 7 37], ⟨0 -16 -2 -5 -40]]

POTE generator: ~28/25 = 193.840

Vals: Template:Val list

Badness: 0.021069

13-limit

Comma list: 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 = 193.829

Vals: Template:Val list

Badness: 0.023074

Hemithir

Comma list: 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

Vals: Template:Val list

Badness: 0.031199

Hemiwur

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

Mapping: [⟨1 15 4 7 11], ⟨0 -16 -2 -5 -9]]

POTE generator: ~28/25 = 193.884

Vals: Template:Val list

Badness: 0.029270

13-limit

Comma list: 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

Vals: Template:Val list

Badness: 0.028432

Hemiwar

Comma list: 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

Vals: Template:Val list

Badness: 0.044886

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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