2.5.7 subgroup
The 2.5.7 subgroup is the subgroup of just intonation consisting of 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)
- 35/32 (the septimal neutral second)
- 49/40 (a neutral third)
The 2.5.7 subgroup includes the following harmonics below 256: 1, 5, 7, 25, 35, 49, 125, 175, 245.
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.
Odd harmonics below 256
1, 5, 7, 25, 35, 49, 125, 175, 245
JI scales
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)
It sounds like soft citric (4L2s) with aberrismas.
Didacus tempering sets L = m + s. 37edo equates 49/40 to 16/13.
Interval matrices
Achiral
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|
| LsmLsLmsL | 28/25 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 25/14 |
| smLsLmsLL | 50/49 | 125/112 | 5/4 | 125/98 | 10/7 | 25/16 | 625/392 | 25/14 |
| mLsLmsLLs | 35/32 | 49/40 | 5/4 | 7/5 | 49/32 | 25/16 | 7/4 | 49/25 |
| LsLmsLLsm | 28/25 | 8/7 | 32/25 | 7/5 | 10/7 | 8/5 | 224/125 | 64/35 |
| sLmsLLsmL | 50/49 | 8/7 | 5/4 | 125/98 | 10/7 | 8/5 | 80/49 | 25/14 |
| LmsLLsmLs | 28/25 | 49/40 | 5/4 | 7/5 | 196/125 | 8/5 | 7/4 | 49/25 |
| msLLsmLsL | 35/32 | 125/112 | 5/4 | 7/5 | 10/7 | 25/16 | 7/4 | 25/14 |
| sLLsmLsLm | 50/49 | 8/7 | 32/25 | 64/49 | 10/7 | 8/5 | 80/49 | 64/35 |
| LLsmLsLms | 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 | |
|---|---|---|---|---|---|---|---|---|
| sLmLsLmsL | 50/49 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 25/14 |
| LmLsLmsLs | 28/25 | 49/40 | 343/250 | 7/5 | 196/125 | 343/200 | 7/4 | 49/25 |
| mLsLmsLsL | 35/32 | 49/40 | 5/4 | 7/5 | 49/32 | 25/16 | 7/4 | 25/14 |
| LsLmsLsLm | 28/25 | 8/7 | 32/25 | 7/5 | 10/7 | 8/5 | 80/49 | 64/35 |
| sLmsLsLmL | 50/49 | 8/7 | 5/4 | 125/98 | 10/7 | 500/343 | 80/49 | 25/14 |
| LmsLsLmLs | 28/25 | 49/40 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 49/25 |
| msLsLmLsL | 35/32 | 125/112 | 5/4 | 125/98 | 10/7 | 25/16 | 7/4 | 25/14 |
| sLsLmLsLm | 50/49 | 8/7 | 400/343 | 64/49 | 10/7 | 8/5 | 80/49 | 64/35 |
| LsLmLsLms | 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 | |
|---|---|---|---|---|---|---|---|---|
| LsmLsLmLs | 28/25 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 7/4 | 49/25 |
| smLsLmLsL | 50/49 | 125/112 | 5/4 | 125/98 | 10/7 | 25/16 | 7/4 | 25/14 |
| mLsLmLsLs | 35/32 | 49/40 | 5/4 | 7/5 | 49/32 | 343/200 | 7/4 | 49/25 |
| LsLmLsLsm | 28/25 | 8/7 | 32/25 | 7/5 | 196/125 | 8/5 | 224/125 | 64/35 |
| sLmLsLsmL | 50/49 | 8/7 | 5/4 | 7/5 | 10/7 | 8/5 | 80/49 | 25/14 |
| LmLsLsmLs | 28/25 | 49/40 | 343/250 | 7/5 | 196/125 | 8/5 | 7/4 | 49/25 |
| mLsLsmLsL | 35/32 | 49/40 | 5/4 | 7/5 | 10/7 | 25/16 | 7/4 | 25/14 |
| LsLsmLsLm | 28/25 | 8/7 | 32/25 | 64/49 | 10/7 | 8/5 | 80/49 | 64/35 |
| sLsmLsLmL | 50/49 | 8/7 | 400/343 | 125/98 | 10/7 | 500/343 | 80/49 | 25/14 |
Tempered scales
Didacus[6]
let L = 28/25 let s = 7/4 L;L;L;L;L;s; stack() 31@
Temperaments
Common rank-2 temperaments in 2.3.7 (i.e. temperaments that interpret intervals as 2.3.7 JI ratios):
