3136/3125: Difference between revisions

Line 4:

| Comma = yes

}}

'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[small comma]] measuring about 6.1{{cent}}. It is the difference between five classic major thirds ([[5/4]]) and two subminor sevenths ([[7/4~~]]); it is also the difference between the septimal semicomma ([[126/125]]) and the septimal kleisma ([[225/224~~]]). Perhaps ~~most~~ importantly, it is ([[28/25]])2/([[5/4]]) and ~~(because~~ [[28/25]] = ([[7/5]])/([[5/4]])) ~~therefore~~ also ([[28/25]])3/([[7/5]]) which means that its square is equal to the difference between ([[28/25]])5 and [[7/4]]. ~~This~~ 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.

'''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 that 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]]).

== Temperaments ==

=== Didacus (2.5.7) ===

Tempering out this comma in its minimal prime [[subgroup]] of 2.5.7 leads to [[Hemimean clan #Didacus|didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator of [[28/25]].

=== Hemimean (2.3.5.7) ===

Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family]] of temperaments, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. (Note that if we temper both of those commas individually we get [[septimal meantone]].)

Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family]] of temperaments, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. Note that if we temper both of those commas individually we get [[septimal meantone]].

=== Orion ===

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
-'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[small comma]] measuring about 6.1{{cent}}. It is the difference between five classic major thirds ([[5/4]]) and two subminor sevenths ([[7/4]]); it is also the difference between the septimal semicomma ([[126/125]]) and the septimal kleisma ([[225/224]]). Perhaps most importantly, it is ([[28/25]])<sup>2</sup>/([[5/4]]) and (because [[28/25]] = ([[7/5]])/([[5/4]])) therefore also ([[28/25]])<sup>3</sup>/([[7/5]]) which means that its square is equal to the difference between ([[28/25]])<sup>5</sup> and [[7/4]]. This 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.
+'''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 that 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]]).
 == Temperaments ==
 === Didacus (2.5.7) ===
 Tempering out this comma in its minimal prime [[subgroup]] of 2.5.7 leads to [[Hemimean clan #Didacus|didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator of [[28/25]].
 === Hemimean (2.3.5.7) ===
-Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family]] of temperaments, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. (Note that if we temper both of those commas individually we get [[septimal meantone]].)
+Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family]] of temperaments, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. Note that if we temper both of those commas individually we get [[septimal meantone]].
 === Orion ===