3136/3125
| Interval information |
didacus comma
Zozoquingu comma
(Shannon, [math]\displaystyle{ \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.
The name didacus seems to be first attested in September 2016 (here), and the name was created by Gene Ward Smith. It is unclear what the origin of this name is; St. Didacus was a Spanish missionary after whom the city of San Diego was named, but there seems to be no relation between this individual and musical temperament.