Würschmidt family: Difference between revisions

Revision as of 15:33, 13 October 2021

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 [17 1 -8⟩, and flipping that yields ⟨⟨ 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)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 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 hemiwürschmidt adds 6144/6125 = [11 1 -3 -2⟩.

Würschmidt

Subgroup: 2.3.5

Comma list: 393216/390625

Mapping: [⟨1 7 3], ⟨0 -8 -1]]

POTE generator: ~5/4 = 387.799

Template:Val list

Badness: 0.040603

Music

Ancient Stardust, play by Chris Vaisvil; Würschmidt[13] in 5-odd-limit minimax tuning

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

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 ⟨⟨ 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.

Subgroup: 2.3.5.7

Comma list: 225/224, 8748/8575

Mapping: [⟨1 7 3 15], ⟨0 -8 -1 -18]]

Wedgie: ⟨⟨ 8 1 18 -17 6 39 ]]

POTE generator: ~5/4 = 387.383

Template:Val list

Badness: 0.050776

11-limit

Subgroup: 2.3.5.7.11

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

@@ Line 1: / Line 1: @@
-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 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. 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''' (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. 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 second comma of the [[Normal lists #Normal interval list|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 hemiwürschmidt adds 6144/6125 = {{monzo| 11 1 -3 -2 }}.
 == Würschmidt ==
-('''Würschmidt''' is sometimes spelled '''Wuerschmidt''')
 Subgroup: 2.3.5
-[[Comma]]: 393216/390625
+[[Comma list]]: 393216/390625
 [[Mapping]]: [{{Val|1 7 3}}, {{Val|0 -8 -1}}]
@@ Line 17: / Line 17: @@
 ; 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-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.
-=== Extensions ===
+* [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.
-The second comma of the [[Normal lists #Normal interval list|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|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.
+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.
 Subgroup: 2.3.5.7
@@ Line 32: / Line 29: @@
 [[Mapping]]: [{{val| 1 7 3 15 }}, {{Val| 0 -8 -1 -18 }}]
+{{Multival|legend=1| 8 1 18 -17 6 39 }}
 [[POTE generator]]: ~5/4 = 387.383
@@ Line 52: / Line 51: @@
 Badness: 0.024413
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 65: / Line 64: @@
 Badness: 0.023593
-=== Worseschmidt ===
+==== Worseschmidt ====
 Subgroup: 2.3.5.7.11.13
@@ Line 79: / Line 78: @@
 == 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 {{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.
+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
@@ Line 86: / Line 85: @@
 [[Mapping]]: [{{val| 1 7 3 -6 }}, {{val| 0 -8 -1 13 }}]
+{{Multival|legend=1| 8 1 -13 -17 -43 -33 }}
 [[POTE generator]]: ~5/4 = 387.392
@@ Line 107: / Line 108: @@
 == 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 {{Multival|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 7 mapped to the 52nd generator step.
 Subgroup: 2.3.5.7
@@ Line 114: / Line 115: @@
 [[Mapping]]: [{{val| 1 7 3 38 }}, {{val| 0 -8 -1 -52 }}]
+{{Multival|legend=1| 8 1 52 -17 60 118 }}
 [[POTE generator]]: ~5/4 = 387.881
@@ Line 122: / Line 125: @@
 == 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|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 …}}.
+{{See also| Hemimean clan }}
+'''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 [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[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, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.
 Subgroup: 2.3.5.7
@@ Line 316: / Line 321: @@
 == Relationships to other temperaments ==
--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.
+-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:Regular temperament theory]]

Würschmidt family: Difference between revisions

Revision as of 15:33, 13 October 2021

Würschmidt

Septimal würschmidt

11-limit

13-limit

Worseschmidt

Worschmidt

11-limit

Whirrschmidt

Hemiwürschmidt

11-limit

13-limit

Hemithir

Hemiwur

13-limit

Hemiwar

Quadrawürschmidt

Semihemiwür

13-limit

Semihemiwürat

19-limit

Semihemiwürand

19-limit

Relationships to other temperaments

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