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 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]]). | '''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. | ||
== Temperaments == | == Temperaments == | ||
Revision as of 19:03, 16 December 2022
| Interval information |
didacus comma
3136/3125, the hemimean comma or didacus comma, is a 7-limit small comma measuring about 6.1 ¢. 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.
Temperaments
Didacus (2.5.7)
Tempering out this comma in its minimal prime subgroup of 2.5.7 leads to 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 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 (2.5.7.17.19)
As 28/25 is close to 19/17 and as the latter is a precise approximation of half of 5/4, it is natural to temper (28/25)/(19/17) = 476/475 and the semiparticular (5/4)/(19/17)2 = 1445/1444 which together imply tempering 3136/3125 and 2128/2125, resulting in a rank 3 temperament.
Mapping:
[⟨1 0 -3 0 -1]
⟨0 2 5 0 1]
⟨0 0 0 1 1]]
CTE generators: 2/1, ~28/25 = 193.642, ~17/16 = 104.434
See also
- Hemimean family, the rank-3 family where it is tempered out
- Hemimean clan, the rank-2 clan where it is tempered out