3136/3125: Difference between revisions
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'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[small comma]] measuring about 6.1{{cent}}. 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]])<sup>2</sup>/([[5/4]]), and | '''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[small comma]] measuring about 6.1{{cent}}. 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]])<sup>2</sup>/([[5/4]]), and because [[28/25]] = ([[7/5]])/([[5/4]]), it is also ([[28/25]])<sup>3</sup>/([[7/5]]), which means its square is equal to the difference between ([[28/25]])<sup>5</sup> 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]]). | 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]]). | ||
Revision as of 17:29, 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.
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