3136/3125: Difference between revisions
m made explanation terser and removed unpaired bracket |
|||
| Line 14: | Line 14: | ||
=== 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 | ||
| Line 33: | Line 47: | ||
[[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 | ||
| Line 63: | Line 61: | ||
Badness: 0.569 | Badness: 0.569 | ||
[[Category:Hemimean]] | [[Category:Hemimean]] | ||
Revision as of 18:00, 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 because 28/25 = (7/5)/(5/4), it is also (28/25)3/(7/5), which means 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.
Hemimean orion
As tempering either S16/S18 = 1216/1215 or S18/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: [⟨1 0 0 -3 -5 -6], ⟨0 1 0 0 5 5], ⟨0 0 2 5 1 2]]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647
Optimal GPV sequence: Template:Val list
Badness: 0.456
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.
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
Semiorion
As 1445/1444 = S17/S19 we can extend orion to include prime 3 in its subgroup by tempering both S17 and S19. However, note that (because of tempering 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.
Subgroup: 2.3.5.7.17.19
Comma list: 289/288, 361/360, 476/475
Mapping: [⟨2 0 0 -6 5 3], ⟨0 1 0 0 1 1], ⟨0 0 2 5 0 1]]
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669
Optimal GPV sequence: Template:Val list
Badness: 0.569