Würschmidt

Würschmidt, is a rank-2 temperament and parent of the würschmidt family, characterized by tempering out the würschmidt comma, (393216/390625). It is generated by a classical major third (5/4), eight of which make the sixth harmonic (6/1). Four generators, therefore, reach the interval 625/512, which is equated to 768/625 and functions as a neutral third.

Another useful interpretation of the würschmidt comma is that it makes the interval of 25/24 equal to two-thirds the size of 16/15. This can be exploited, as 16/15 factorizes into near-2:1 parts as (24/23)×(46/45), and therefore it is illogical not to set 25/24 equal to 24/23 (and 128/125 equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering S24 = 576/575 and S46² × S47 = 12167/12150 in the 2.3.5.23 subgroup. 14 generators turn out to stack to 23/1, and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma 279936/279841.

Strong extensions to the 7-limit include septimal würschmidt, worschmidt, and whirrschmidt, but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be hemiwürschmidt, which splits the ~5/4 generator into two ~28/25's. Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with 45/44 and therefore 625/512 with 11/9 (by tempering out 243/242 and 5632/5625), finding the 11th harmonic at 20 generators up, is highly natural, in addition to the aforementioned extension to prime 23.

For technical data, see Würschmidt family #Würschmidt.

Other tunings

• DKW (2.3.5): ~2 = 1\1, ~5/4 = 387.8015

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