3136/3125: Difference between revisions
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== Temperaments == | == 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]].) | |||
=== 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 [[square superparticular|semiparticular]] ([[5/4]])/([[19/17]])<sup>2</sup> = [[1445/1444]] which together imply tempering [[3136/3125]] and [[2128/2125]], resulting in a rank 3 temperament. | |||
[[Mapping]]:<br> | |||
[{{val| 1 0 -3 0 -1 }}<br> | |||
{{val| 0 2 5 0 1 }}<br> | |||
{{val| 0 0 0 1 1 }}] | |||
[[CTE]] generators: 2/1, ~28/25 = 193.642, ~17/16 = 104.434 | |||
[[Val]]s: {{Val list| 12, 25, 31, 37, 43, 50, 56, 68, 93}} | |||
== See also == | == See also == | ||
Revision as of 18:55, 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).
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