2.5.7 subgroup
The 2.5.7 subgroup is the subgroup of just intonation comprising the intervals reachable by stacking 2/1, 5/4, and 7/4, with the exclusion of 3/2 (adding which would result in the full 7-limit).
Notable intervals include 5/4 (the pental major third), 7/4 (the septimal subminor seventh), 7/5 (the lesser septimal tritone), 10/7 (the greater septimal tritone), 28/25 (the septimal quasi-meantone), and 35/32 (the septimal neutral second).
An especially efficient temperament in 2.5.7 is Didacus, 2.5.7[25 & 31], which is generated by a tempered 28/25 and tempers out 3136/3125, the interval between a stack of two 7/5 tritones and three 5/4 major thirds. Didacus is a 6-form cluster temperament.
31edo is a particularly accurate 2.5.7 system, but 37edo is more accurate for extensions to larger subgroups such as 2.5.7.11.13.
Aberrismic theory
The fundamental 2.5.7 aberrismic scale is 4L2m3s, L = 28/25, m = 35/32, s = 50/49:
- Achiral: LsmLsLmsL (28/25 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
- Right-handed: sLmLsLmsL (50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
- Left-handed: LsmLsLmLs (28/25 8/7 5/4 7/5 10/7 8/5 7/4 49/25 2/1)
Didacus tempering sets L = m + s.
Interval matrices
Achiral
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|
| 2/1 | 28/25 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 25/14 |
| 28/25 | 50/49 | 125/112 | 5/4 | 125/98 | 10/7 | 25/16 | 625/392 | 25/14 |
| 8/7 | 35/32 | 49/40 | 5/4 | 7/5 | 49/32 | 25/16 | 7/4 | 49/25 |
| 5/4 | 28/25 | 8/7 | 32/25 | 7/5 | 10/7 | 8/5 | 224/125 | 64/35 |
| 7/5 | 50/49 | 8/7 | 5/4 | 125/98 | 10/7 | 8/5 | 80/49 | 25/14 |
| 10/7 | 28/25 | 49/40 | 5/4 | 7/5 | 196/125 | 8/5 | 7/4 | 49/25 |
| 8/5 | 35/32 | 125/112 | 5/4 | 7/5 | 10/7 | 25/16 | 7/4 | 25/14 |
| 7/4 | 50/49 | 8/7 | 32/25 | 64/49 | 10/7 | 8/5 | 80/49 | 64/35 |
| 25/14 | 28/25 | 784/625 | 32/25 | 7/5 | 196/125 | 8/5 | 224/125 | 49/25 |
Right-handed
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|
| 2/1 | 50/49 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 25/14 |
| 50/49 | 28/25 | 49/40 | 343/250 | 7/5 | 196/125 | 343/200 | 7/4 | 49/25 |
| 8/7 | 35/32 | 49/40 | 5/4 | 7/5 | 49/32 | 25/16 | 7/4 | 25/14 |
| 5/4 | 28/25 | 8/7 | 32/25 | 7/5 | 10/7 | 8/5 | 80/49 | 64/35 |
| 7/5 | 50/49 | 8/7 | 5/4 | 125/98 | 10/7 | 500/343 | 80/49 | 25/14 |
| 10/7 | 28/25 | 49/40 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 49/25 |
| 8/5 | 35/32 | 125/112 | 5/4 | 125/98 | 10/7 | 25/16 | 7/4 | 25/14 |
| 7/4 | 50/49 | 8/7 | 400/343 | 64/49 | 10/7 | 8/5 | 80/49 | 64/35 |
| 25/14 | 28/25 | 8/7 | 32/25 | 7/5 | 196/125 | 8/5 | 224/125 | 49/25 |
Left-handed
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|
| 2/1 | 28/25 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 49/25 |
| 28/25 | 50/49 | 125/112 | 5/4 | 125/98 | 10/7 | 25/16 | 7/4 | 25/14 |
| 8/7 | 35/32 | 49/40 | 5/4 | 7/5 | 49/32 | 343/200 | 7/4 | 49/25 |
| 5/4 | 28/25 | 8/7 | 32/25 | 7/5 | 196/125 | 8/5 | 224/125 | 64/35 |
| 7/5 | 50/49 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 80/49 | 25/14 |
| 10/7 | 28/25 | 49/40 | 343/250 | 7/5 | 196/125 | 8/5 | 7/4 | 49/25 |
| 8/5 | 35/32 | 49/40 | 5/4 | 7/5 | 10/7 | 25/16 | 7/4 | 25/14 |
| 7/4 | 28/25 | 8/7 | 32/25 | 64/49 | 10/7 | 8/5 | 80/49 | 64/35 |
| 49/25 | 50/49 | 8/7 | 400/343 | 125/98 | 10/7 | 500/343 | 80/49 | 25/14 |
