Würschmidt: Difference between revisions

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#~~redirect~~ [[Würschmidt family #Würschmidt]]

'''Würschmidt''', is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, ([[393216/390625]]). It is [[generator|generated]] by a [[5/4|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 S46<sup>2</sup> × 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 [[würschmidt family#septimal würschmidt|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 theory|DKW]] (2.3.5): ~2 = 1\1, ~5/4 = 387.8015

[[Category:Temperaments]]

[[Category:Würschmidt family]]

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-#redirect [[Würschmidt family #Würschmidt]]
+'''Würschmidt''', is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, ([[393216/390625]]). It is [[generator|generated]] by a [[5/4|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 S46<sup>2</sup> × 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 [[würschmidt family#septimal würschmidt|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 theory|DKW]] (2.3.5): ~2 = 1\1, ~5/4 = 387.8015
+{{stub}}
 [[Category:Temperaments]]
 [[Category:Würschmidt family]]