3136/3125
| Interval information |
didacus comma
Zozoquingu comma
(Shannon, [math]\sqrt{nd}[/math])
3136/3125, the hemimean comma or didacus comma, is a small 7-limit 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 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). Examining the latter expression we can observe that this gives us a relatively simple S-expression of (S4/S5)/(S5/S7) which can be rearranged to S4*S7/S52. Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = S6*S7*S8 = (6/5)/(9/8); substituting this in and simplifying yields: S6*S72*S8/S52, from which we can obtain an alternative equivalence 3136/3125 = (49/45)/(25/24)2, meaning we split 49/45 into two 25/24's in the resulting temperament.
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 representing 28/25. See hemimean clan for extensions of didacus.
Hemimean (2.3.5.7)
Tempering out this comma in the full 7-limit leads to the rank-3 hemimean temperament, which splits the syntonic comma into two equal parts, each representing 126/125~225/224. See hemimean family for the family of rank-3 temperaments where it is tempered out.
Note that if we temper 126/125 and/or 225/224 we get septimal meantone.
Orion
As 28/25 is close to 19/17 and as the latter is the mediant of 9/8 and 10/9 (which together make 5/4), it is natural to temper (28/25)/(19/17) = 476/475, or equivalently stated, 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
Sval mapping: [⟨1 0 -3 0 -1], ⟨0 2 5 0 1], ⟨0 0 0 1 1]]
- sval mapping generators: ~2, ~56/25, ~17
Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434
Optimal ET sequence: 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch
Badness: 0.0150
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 a strong extension of hemimean and weak extension of orion, as we have a ~3/2 generator slicing the second generator of orion into five.
See Hemimean family #Hemimean orion.
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.
See Hemimean family #Semiorion.
Etymology
This comma was first named as parahemwuer by Gene Ward Smith in 2005 as a contraction of parakleismic and hemiwürschmidt[1]. It is not clear how it later became hemimean, but the root of hemimean is obvious, being a contraction of hemiwürschmidt and meantone.