Würschmidt family: Difference between revisions

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)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.

[[Comma]]: 393216/390625

* [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

 

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 ==

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.

=== 13-limit ===

=== Worseschmidt ===

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.

[[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.

'''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 …}}.

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.