3136/3125
Ratio | 3136/3125 |
Factorization | 2^{6} × 5^{-5} × 7^{2} |
Monzo | [6 0 -5 2⟩ |
Size in cents | 6.0832436¢ |
Names | hemimean comma, didacus comma |
Color name | zzg^{5}3, zozoquingu 3rd, Zozoquingu comma |
FJS name | [math]\text{ddd3}^{7,7}_{5,5,5,5,5}[/math] |
Special properties | reduced |
Tenney height (log_{2} nd) | 23.2244 |
Weil height (log_{2} max(n, d)) | 23.2294 |
Wilson height (sopfr (nd)) | 51 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.55514 bits |
Comma size | small |
open this interval in xen-calc |
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 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 both 126/125 and 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.