3136/3125: Difference between revisions

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'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[small comma]] measuring about 6.1{{cent}}. It is the difference between a stack of five classic major thirds ([[5/4]]) and a stack of two subminor sevenths ([[7/4]]). Perhaps more importantly, it is ([[28/25]])2/([[5/4]]), and in light of the fact that [[28/25]] = ([[7/5]])/([[5/4]]), it is also ([[28/25]])3/([[7/5]]), which means its square is equal to the difference between ([[28/25]])5 and [[7/4]]. The associated temperament has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of [[28/25]]'s, itself a 2.5.7 subgroup interval.

In terms of commas, it is the difference between the septimal semicomma ([[126/125]]) and the septimal kleisma ([[225/224]]), or between the augmented comma ([[128/125]]) and the jubilisma ([[50/49]]).

Examining the latter expression we can observe that this gives us a relatively simple [[S-expression]] of ([[128/125|S4/S5]])/([[50/49|S5/S7]]) which can be rearranged to [[16/15|S4]]*[[49/48|S7]]/[[25/24|S5]]2.

Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = [[36/35|S6]]*[[49/48|S7]]*[[64/63|S8]]; substituting this in and simplifying yields:

S6*S7*S8*S7/S52 from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])2, meaning we split [[49/45]] into two [[25/24]]'s in the resulting temperament.

== Temperaments ==

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 '''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[small comma]] measuring about 6.1{{cent}}. It is the difference between a stack of five classic major thirds ([[5/4]]) and a stack of two subminor sevenths ([[7/4]]). Perhaps more importantly, it is ([[28/25]])<sup>2</sup>/([[5/4]]), and in light of the fact that [[28/25]] = ([[7/5]])/([[5/4]]), it is also ([[28/25]])<sup>3</sup>/([[7/5]]), which means its square is equal to the difference between ([[28/25]])<sup>5</sup> and [[7/4]]. The associated temperament has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of [[28/25]]'s, itself a 2.5.7 subgroup interval.
 In terms of commas, it is the difference between the septimal semicomma ([[126/125]]) and the septimal kleisma ([[225/224]]), or between the augmented comma ([[128/125]]) and the jubilisma ([[50/49]]).
+Examining the latter expression we can observe that this gives us a relatively simple [[S-expression]] of ([[128/125|S4/S5]])/([[50/49|S5/S7]]) which can be rearranged to [[16/15|S4]]*[[49/48|S7]]/[[25/24|S5]]<sup>2</sup>.
+Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = [[36/35|S6]]*[[49/48|S7]]*[[64/63|S8]]; substituting this in and simplifying yields:
+S6*S7*S8*S7/S5<sup>2</sup> from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])<sup>2</sup>, meaning we split [[49/45]] into two [[25/24]]'s in the resulting temperament.
 == Temperaments ==