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 = | |||
Comma: 393216/390625 | |||
[[POTE generator]]: 387.799 | [[POTE generator]]: 387.799 | ||
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EDOs: {{EDOs| 31, 34, 65, 99, 164, 721c, 885c }} | EDOs: {{EDOs| 31, 34, 65, 99, 164, 721c, 885c }} | ||
== Music == | |||
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. | ||
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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 = | ||
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: 0.0508 | Badness: 0.0508 | ||
== 11-limit == | |||
Commas: [[99/98]], 176/175, [[243/242]] | Commas: [[99/98]], 176/175, [[243/242]] | ||
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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]] | ||
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Badness: 0.0344 | 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. | 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. | ||
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Badness: 0.0646 | Badness: 0.0646 | ||
== 11-limit == | |||
Commas: 126/125, 243/242, 385/384 | Commas: 126/125, 243/242, 385/384 | ||
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Badness: 0.0334 | 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. | [[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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EDOs: {{EDOs| 31, 34, 65, 99 }} | EDOs: {{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... | 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... | ||
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Badness: 0.0203 | Badness: 0.0203 | ||
== 11-limit == | |||
Commas: 243/242, 441/440, 3136/3125 | Commas: 243/242, 441/440, 3136/3125 | ||
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=== 13-limit === | === 13-limit === | ||
Commas: 243/242 351/350 441/440 3584/3575 | Commas: 243/242, 351/350, 441/440, 3584/3575 | ||
POTE generator: ~28/25 = 193.840 | POTE generator: ~28/25 = 193.840 | ||
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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 | ||
POTE generator: ~28/25 = 193.918 | POTE generator: ~28/25 = 193.918 | ||
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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 | ||
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Badness: 0.0449 | Badness: 0.0449 | ||
= 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> | ||