Würschmidt: Difference between revisions

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'''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 can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.

'''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 can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. 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]].

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	'''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 can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator\|generated]] by a [[5/4\|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.		'''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 can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator\|generated]] by a [[5/4\|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. 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]].		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]].