3136/3125
| 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 five classic major thirds (5/4) and two subminor sevenths (7/4); it is also the difference between the septimal semicomma (126/125) and the septimal kleisma (225/224). Perhaps most importantly, it is (28/25)2/(5/4) and (because 28/25 = (7/5)/(5/4)) therefore also (28/25)3/(7/5) which means that its square is equal to the difference between (28/25)5 and 7/4. This 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.
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 a precise approximation of half 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 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:
[⟨1 2 2 4 4]
⟨0 2 5 0 1]
⟨0 0 0 1 1]]
CTE generators: 2/1, ~28/25 = 193.642, ~17/16 = 104.434
2.3.5.7.17.19
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 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:
[⟨1 1 2 2 1 1]
⟨0 1 0 0 5 5]
⟨0 0 2 5 1 2]]
CTE generators: 2/1, ~3/2 = 702.132, ~28/25 = 193.647
Semiorion
As 1445/1444 = S17/S19 we can alternatively extend this temperament 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 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:
[⟨2 2 4 4 7 7]
⟨0 1 0 0 1 1]
⟨0 0 2 5 0 1]]
CTE generators: ~17/12 = 600.0, ~3/2 = 702.509, ~28/25 = 193.669