2.5.7 subgroup: Difference between revisions

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Revision as of 13:12, 10 March 2026

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
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
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