2.5.7 subgroup: Difference between revisions
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== JI scales == | == JI scales == | ||
The fundamental 2.5.7 [[aberrismic]] scale is 4L2m3s, L = 28/25, m = 35/32, s = 50/49 | The fundamental 2.5.7 [[aberrismic]] scale is prelembic (tentative name) or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or citric, via adding three 50/49 steps. | ||
* Achiral: LsmLsLmsL (28/25 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1) | * 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) | * Right-handed: sLmLsLmsL (50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1) | ||
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|25/14 | |25/14 | ||
|} | |} | ||
== Tempered scales == | == Tempered scales == | ||
=== [[Didacus]][6] === | === [[Didacus]][6] === | ||
Revision as of 17:53, 13 July 2026
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 prelembic (tentative name) or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or citric, via adding three 50/49 steps.
- 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 among 2.5.7 temperaments. 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).
- Mabilic (16 & 25): Generated by a flat armotonic fifth around 672 cents.
- 3edo.7 (6 & 15): Generates 3L3s with generator ~7/4. Supported by Augmented edos such as 15edo, 21edo, and 27edo.
- Sidewalk (21 & 46): Generated by a neominor third, two of which make a flattened 7/5; reaches 8/7 after 5 generator steps and 8/5 after 7 generator steps. Accurately extends to primes 11, 13, 17, and 23.
