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 odd 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.
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 s = 28/25 let L = 7/4 s;s;s;s;s;L; stack() 31@
Didacus[13]
let L = 35/32 let s = 50/49 s;L;s;L;s;L;s;L;s;L;s;L;s; stack() 31@
Jubilismic[6]
let L = 7/4 let s = 35/32 L;L;s;L;L;s; stack() 16@
Temperaments
Common rank-2 temperaments in 2.5.7 (i.e. temperaments that interpret intervals as 2.5.7 JI ratios):
- Didacus (25 & 31): Best accuracy-simplicity tradeoff. Generates 5L1s generated by ~28/25.
- Jubilismic (16 & 22): Less accurate, identifying 7/5 and 10/7. Has citric (4L2s, LLsLLs) MOS scales (L/s = 3/2 in 16edo, L/s = 4/3 in 22edo).
- 3edo.7 (6 & 15): Generates 3L3s with generator ~7/4. Supported by Augmented edos such as 15edo, 21edo, and 27edo.
