3136/3125: Difference between revisions
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=== Orion === | === Orion === | ||
As [[28/25]] is close to [[19/17]] and as the latter is | 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]]. | ||
==== 2.5.7.17.19 ==== | ==== 2.5.7.17.19 ==== | ||
[[Subgroup]]: 2.5.7.17.19 | |||
[[ | [[Comma list]]: 476/475, 1445/1444 | ||
[[ | [[Mapping]]: [{{val| 1 0 -3 0 -1 }}, {{val| 0 2 5 0 1 }}, {{val| 0 0 0 1 1 }}] | ||
Mapping generators: ~2, ~56/25, ~17 | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434 | |||
{{Val list|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }} | |||
[[Badness]]: 0.0150 | |||
<!-- names await consolidation | |||
==== 2.3.5.7.17.19 ==== | ==== 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. | 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. | ||
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 | |||
==== Semiorion ==== | ==== Semiorion ==== | ||
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Subgroup: 2.3.5.7.17.19 | Subgroup: 2.3.5.7.17.19 | ||
Comma list: | Comma list: 289/288, 361/360, 476/475 | ||
Mapping: [{{val| 2 0 0 -6 5 3 }}, {{val| 0 1 0 0 1 1 }}, {{val| 0 0 2 5 0 1 }}] | |||
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669 | |||
Optimal GPV sequence: {{Val list| 12, …, 50, 68, 106d, 118, 248g, 316g }} | |||
Badness: 0.569 | |||
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[[Category:Hemimean]] | [[Category:Hemimean]] | ||
Revision as of 10:43, 17 December 2022
| Interval information |
didacus comma
3136/3125, the hemimean comma or didacus comma, is a 7-limit small comma measuring about 6.1 ¢. It is the difference between a stack of five classic major thirds (5/4) and a stack of two subminor sevenths (7/4). Perhaps more importantly, it is (28/25)2/(5/4), and in light of the fact that 28/25 = (7/5)/(5/4)), it is also (28/25)3/(7/5), which means that its square is equal to the difference between (28/25)5 and 7/4. The associated temperament has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of 28/25's, itself a 2.5.7 subgroup interval.
In terms of commas, it is the difference between the septimal semicomma (126/125) and the septimal kleisma (225/224), or between the augmented comma (128/125) and the jubilisma (50/49).
Temperaments
Didacus (2.5.7)
Tempering out this comma in its minimal prime subgroup of 2.5.7 leads to 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 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
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 semiparticular (5/4)/(19/17)2 = 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 The Milky Way realm.
2.5.7.17.19
Subgroup: 2.5.7.17.19
Comma list: 476/475, 1445/1444
Mapping: [⟨1 0 -3 0 -1], ⟨0 2 5 0 1], ⟨0 0 0 1 1]]
Mapping generators: ~2, ~56/25, ~17
Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434
Badness: 0.0150