3136/3125: Difference between revisions
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=== 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]]. | ||
==== 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 an extension of hemimean because the ~17/16 generator of orion is no longer present and instead we have a ~3/2 generator. Orion is described next. | |||
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 }}] | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647 | |||
Optimal GPV sequence: {{Val list| 12, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }} | |||
Badness: 0.456 | |||
=== 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]] 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]]. | ||
[[Subgroup]]: 2.5.7.17.19 | [[Subgroup]]: 2.5.7.17.19 | ||
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[[Badness]]: 0.0150 | [[Badness]]: 0.0150 | ||
=== Semiorion === | |||
As [[1445/1444]] = [[289/288|S17]]/[[361/360|S19]] we can extend orion 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 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]]. As a result, this equates the generators of hemimean orion and orion up to period-equivalence and is a weak extension of both and neither. | |||
As [[1445/1444]] = [[289/288|S17]]/[[361/360|S19]] we can | |||
Subgroup: 2.3.5.7.17.19 | Subgroup: 2.3.5.7.17.19 | ||
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Badness: 0.569 | Badness: 0.569 | ||
[[Category:Hemimean]] | [[Category:Hemimean]] | ||