3136/3125: Difference between revisions

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== Temperaments ==

~~In the~~ 2.5.7 ~~[[subgroup]], tempering~~ out ~~the~~ comma ~~leads to the rank-2~~ 2.5.7 ~~subgroup temperament~~ [[Hemimean clan #Didacus|didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator of [[28/25]]. ~~In the full [[7-limit]]~~ (2.3.5.7)~~, tempering it~~ out 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. ~~It also splits~~ 5/4 ~~into two equal parts~~, ~~each representing~~ [[28/25]]. ~~Typical edos tempering out the comma include~~ {{~~EDOs~~| ~~68, 80, 87, 99, 111, 118 and 130~~ }}, ~~and all of them tune both 126~~/~~125 and 225~~/~~224 to a single step~~. ~~Smaller edos that temper out the comma are~~ {{~~EDOs~~| 19, 25, 31, 37 }}, ~~which temper out both 126/125 and 225/224~~, ~~thus also 81/80~~, ~~on the 2.9.5.7 subgroup.~~

=== 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]])2 = [[1445/1444]] which together imply tempering [[3136/3125]] and [[2128/2125]], resulting in a rank 3 temperament.

[[Mapping]]:

[{{val| 1 0 -3 0 -1 }}

{{val| 0 2 5 0 1 }}

{{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 ==

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 == Temperaments ==
-In the 2.5.7 [[subgroup]], tempering out the comma leads to the rank-2 2.5.7 subgroup temperament [[Hemimean clan #Didacus|didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator of [[28/25]]. In the full [[7-limit]] (2.3.5.7), tempering it out 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. It also splits 5/4 into two equal parts, each representing [[28/25]]. Typical edos tempering out the comma include {{EDOs| 68, 80, 87, 99, 111, 118 and 130 }}, and all of them tune both 126/125 and 225/224 to a single step. Smaller edos that temper out the comma are {{EDOs| 19, 25, 31, 37 }}, which temper out both 126/125 and 225/224, thus also 81/80, on the 2.9.5.7 subgroup.
+=== 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 ==