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-hand: sLmLsLmsL (50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
• Left-hand: 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.