Würschmidt: Difference between revisions

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== Extensions ==

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 {{nowrap|([[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 {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|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]].

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 {{nowrap|([[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 {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|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]] = {{nowrap|[[2401/2400|S49]] / ([[25921/25920|S161]]2)}}.

Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).

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 == Extensions ==
-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 {{nowrap|([[24/23]]) &times; ([[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 {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S46<sup>2</sup> &times; 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 {{nowrap|([[24/23]]) &times; ([[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 {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S46<sup>2</sup> &times; 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]] = {{nowrap|[[2401/2400|S49]] / ([[25921/25920|S161]]<sup>2</sup>)}}.
 Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).