3136/3125: Difference between revisions
→Orion (2.5.7.17.19): corrected mapping to fit with the CTE generators listed |
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=== Didacus (2.5.7) === | === 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 of [[28/25]]. | ||
=== Hemimean (2.3.5.7) === | === 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]].) | 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 | === Orion === | ||
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. | 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. | ||
==== 2.5.7.17.19 ==== | |||
Comma list: 3136/3125, 476/475, 1445/1444 = S17/S19, 2128/2125 | |||
[[Mapping]]:<br> | [[Mapping]]:<br> | ||
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[[Val]]s: {{Val list| 12, 25, 31, 37, 43, 50, 56, 68, 93}} | [[Val]]s: {{Val list| 12, 25, 31, 37, 43, 50, 56, 68, 93}} | ||
==== 2.3.5.7.17.19 ==== | |||
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 a natural and precise extension, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. | |||
Comma list: 3136/3125, 1445/1444 = S17/S19, 1216/1215 = S18/S20 | |||
[[Mapping]]:<br> | |||
[{{val| 1 1 2 2 1 1 }}<br> | |||
{{val| 0 1 0 0 5 5 }}<br> | |||
{{val| 0 0 2 5 1 2 }}] | |||
==== Semiorion ==== | |||
As [[1445/1444]] = [[289/288|S17]]/[[361/360|S19]] we can alternatively extend this temperament to include prime 3 in its subgroup by tempering both [[289/288|S17]] and [[361/360|S19]]. However, note that (because of tempering [[289/288|S17]]) this splits the period in half, representing a [[17/12]]~[[24/17]] half-octave. This is also of course a higher damage route. This has the consequence that the [[17/16]] generator can be described as a [[3/2]] because [[17/16]] up from [[24/17]] is [[3/2]]. | |||
Subgroup: 2.3.5.7.17.19 | |||
Comma list: 3136/3125, 289/288 = S17, 361/360 = S19 | |||
[[Mapping]]:<br> | |||
[{{val| 2 2 4 4 7 7 }}<br> | |||
{{val| 0 1 0 0 1 1 }}<br> | |||
{{val| 0 0 2 5 0 1 }}] | |||
[[CTE]] generators: ~17/12 = 600.0, ~3/2 = 702.509, ~28/25 = 193.669 | |||
[[Val]]s: {{Val list|12, 50, 56, 62, 68, 80, 118, 130}} | |||
[[Category:Hemimean]] | [[Category:Hemimean]] | ||