3136/3125

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 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).

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. (The temperament orion is described next on this page.)

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 2 2 4 4], ⟨0 2 5 0 1], ⟨0 0 0 1 1]]

Mapping generators: ~2, ~28/25, ~17/16

Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434

Template:Val list

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