Hemimean family: Difference between revisions

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[[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7

==== Hemimean orion ====

=== 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~~ [[~~3136~~/~~3125#Orion|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 an extension of hemimean because the~~ ~17/~~16 generator of orion is no longer present and instead we have a~~ ~3/2 ~~generator~~.

As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[5/4]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[square superparticular|semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 3136/3125

Sval mapping: [{{val| 1 0 0 -3 -5 }}, {{val| 0 1 0 0 5 }}, {{val| 0 0 2 5 1 }}]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647

Optimal GPV sequence: {{Val list| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}

Badness: 0.573

==== 2.3.5.7.17.19 subgroup ====

Subgroup: 2.3.5.7.17.19

Comma list: 476/475, 1216/1215, 1445/1444

~~Mapping~~: [{{val| 1 0 0 -3 -5 -6 }}, {{val| 0 1 0 0 5 5 }}, {{val| 0 0 2 5 1 2 }}]

Sval mapping: [{{val| 1 0 0 -3 -5 -6 }}, {{val| 0 1 0 0 5 5 }}, {{val| 0 0 2 5 1 2 }}]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647

Optimal GPV sequence: {{Val list| 12, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}

Optimal GPV sequence: {{Val list| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}

Badness: 0.456

@@ Line 31: / Line 31: @@
 [[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7
-==== Hemimean orion ====
+=== 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 [[3136/3125#Orion|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 an extension of hemimean because the ~17/16 generator of orion is no longer present and instead we have a ~3/2 generator.
+As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[5/4]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[square superparticular|semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.
+Subgroup: 2.3.5.7.17
+Comma list: 1701/1700, 3136/3125
+Sval mapping: [{{val| 1 0 0 -3 -5 }}, {{val| 0 1 0 0 5 }}, {{val| 0 0 2 5 1 }}]
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647
+Optimal GPV sequence: {{Val list| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}
+Badness: 0.573
+==== 2.3.5.7.17.19 subgroup ====
 Subgroup: 2.3.5.7.17.19
 Comma list: 476/475, 1216/1215, 1445/1444
-Mapping: [{{val| 1 0 0 -3 -5 -6 }}, {{val| 0 1 0 0 5 5 }}, {{val| 0 0 2 5 1 2 }}]
+Sval mapping: [{{val| 1 0 0 -3 -5 -6 }}, {{val| 0 1 0 0 5 5 }}, {{val| 0 0 2 5 1 2 }}]
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647
-Optimal GPV sequence: {{Val list| 12, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
+Optimal GPV sequence: {{Val list| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
 Badness: 0.456