3136/3125: Difference between revisions

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== 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]].

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 representing [[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]].

~~==== Hemimean orion ====~~

As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension of hemimean as we have a ~3/2 generator slicing the second generator of orion into five. (The temperament orion is described next on this page.)

~~See [[Hemimean family #Hemimean orion]].~~

=== Orion ===

As [[28/25]] is close to [[19/17]] and as the latter is the mediant 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 out [[3136/3125]] and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].

As [[28/25]] is close to [[19/17]] and as the latter is the mediant of [[5/4]], it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[square superparticular|semiparticular]] ([[5/4]])/([[19/17]])<sup>2</sup> = [[1445/1444]], which together imply tempering out 3136/3125 and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].

[[Subgroup]]: 2.5.7.17.19

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[[Badness]]: 0.0150

==== Hemimean orion ====

As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension of hemimean and weak extension of orion, as we have a ~3/2 generator slicing the second generator of orion into five.

See [[Hemimean family #Hemimean orion]].

==== Semiorion ====

@@ Line 10: / Line 10: @@
 == 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]].
+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 representing [[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]].
-==== Hemimean orion ====
-As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension of hemimean as we have a ~3/2 generator slicing the second generator of orion into five. (The temperament orion is described next on this page.)
-See [[Hemimean family #Hemimean orion]].
 === Orion ===
-As [[28/25]] is close to [[19/17]] and as the latter is the mediant 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 out [[3136/3125]] and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].
+As [[28/25]] is close to [[19/17]] and as the latter is the mediant of [[5/4]], it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[square superparticular|semiparticular]] ([[5/4]])/([[19/17]])<sup>2</sup> = [[1445/1444]], which together imply tempering out 3136/3125 and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].
 [[Subgroup]]: 2.5.7.17.19
@@ Line 36: / Line 31: @@
 [[Badness]]: 0.0150
+==== Hemimean orion ====
+As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension of hemimean and weak extension of orion, as we have a ~3/2 generator slicing the second generator of orion into five.
+See [[Hemimean family #Hemimean orion]].
 ==== Semiorion ====