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 |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 == | ||
('''Würschmidt''' is sometimes spelled '''Wuerschmidt''') | ('''Würschmidt''' is sometimes spelled '''Wuerschmidt''') | ||
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[[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 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>. | ||
= 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]] 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. | ||
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[[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 | ||
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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 | ||
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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 | ||
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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]], [[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. | ||
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[[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 | ||
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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]] 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. | ||
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[[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]], [[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... | ||
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[[Badness]]: 0.0203 | [[Badness]]: 0.0203 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 243/242, 441/440, 3136/3125 | Comma list: 243/242, 441/440, 3136/3125 | ||
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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 | ||
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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 | ||
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Badness: 0.031199 | Badness: 0.031199 | ||
== Hemiwur == | === Hemiwur === | ||
Comma list: 121/120, 176/175, 1375/1372 | Comma list: 121/120, 176/175, 1375/1372 | ||
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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 | ||
| 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 | ||
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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 15:34, 31 May 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
Badness: 0.040603
Music
- 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>.
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.
Comma list: 225/224, 8748/8575
Mapping: [<1 7 3 15|, <0 -8 -1 -18|]
POTE generator: ~5/4 = 387.383
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
Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]
POTE generator: ~5/4 = 387.099
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| 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
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
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
Badness: 0.0203
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
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.