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 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, 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.

'''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...}}