2.5.7 subgroup: Difference between revisions

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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]]).  
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), and 35/32 (the septimal neutral second).
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)


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.
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.
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* 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)
* Left-handed: LsmLsLmLs (28/25 8/7 5/4 7/5 10/7 8/5 7/4 49/25 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.
It sounds like soft [[citric]] (4L2s) with aberrismas.
 
Didacus tempering sets L = m + s. 37edo equates 49/40 to 16/13.


=== Interval matrices ===
=== Interval matrices ===
Line 28: Line 37:
!8
!8
|-
|-
!2/1
!LsmLsLmsL
|28/25
|28/25
|8/7
|8/7
Line 38: Line 47:
|25/14
|25/14
|-
|-
!28/25
!smLsLmsLL
|50/49
|50/49
|125/112
|125/112
Line 48: Line 57:
|25/14
|25/14
|-
|-
!8/7
!mLsLmsLLs
|35/32
|35/32
|49/40
|49/40
Line 58: Line 67:
|49/25
|49/25
|-
|-
!5/4
!LsLmsLLsm
|28/25
|28/25
|8/7
|8/7
Line 68: Line 77:
|64/35
|64/35
|-
|-
!7/5
!sLmsLLsmL
|50/49
|50/49
|8/7
|8/7
Line 78: Line 87:
|25/14
|25/14
|-
|-
!10/7
!LmsLLsmLs
|28/25
|28/25
|49/40
|49/40
Line 88: Line 97:
|49/25
|49/25
|-
|-
!8/5
!msLLsmLsL
|35/32
|35/32
|125/112
|125/112
Line 98: Line 107:
|25/14
|25/14
|-
|-
!7/4
!sLLsmLsLm
|50/49
|50/49
|8/7
|8/7
Line 108: Line 117:
|64/35
|64/35
|-
|-
!25/14
!LLsmLsLms
|28/25
|28/25
|784/625
|784/625
Line 131: Line 140:
!8
!8
|-
|-
!2/1
!sLmLsLmsL
|50/49
|50/49
|8/7
|8/7
Line 141: Line 150:
|25/14
|25/14
|-
|-
!50/49
!LmLsLmsLs
|28/25
|28/25
|49/40
|49/40
Line 151: Line 160:
|49/25
|49/25
|-
|-
!8/7
!mLsLmsLsL
|35/32
|35/32
|49/40
|49/40
Line 161: Line 170:
|25/14
|25/14
|-
|-
!5/4
!LsLmsLsLm
|28/25
|28/25
|8/7
|8/7
Line 171: Line 180:
|64/35
|64/35
|-
|-
!7/5
!sLmsLsLmL
|50/49
|50/49
|8/7
|8/7
Line 181: Line 190:
|25/14
|25/14
|-
|-
!10/7
!LmsLsLmLs
|28/25
|28/25
|49/40
|49/40
Line 191: Line 200:
|49/25
|49/25
|-
|-
!8/5
!msLsLmLsL
|35/32
|35/32
|125/112
|125/112
Line 201: Line 210:
|25/14
|25/14
|-
|-
!7/4
!sLsLmLsLm
|50/49
|50/49
|8/7
|8/7
Line 211: Line 220:
|64/35
|64/35
|-
|-
!25/14
!LsLmLsLms
|28/25
|28/25
|8/7
|8/7
Line 234: Line 243:
!8
!8
|-
|-
!2/1
!LsmLsLmLs
|28/25
|28/25
|8/7
|8/7
Line 244: Line 253:
|49/25
|49/25
|-
|-
!28/25
!smLsLmLsL
|50/49
|50/49
|125/112
|125/112
Line 254: Line 263:
|25/14
|25/14
|-
|-
!8/7
!mLsLmLsLs
|35/32
|35/32
|49/40
|49/40
Line 264: Line 273:
|49/25
|49/25
|-
|-
!5/4
!LsLmLsLsm
|28/25
|28/25
|8/7
|8/7
Line 274: Line 283:
|64/35
|64/35
|-
|-
!7/5
!sLmLsLsmL
|50/49
|50/49
|8/7
|8/7
Line 284: Line 293:
|25/14
|25/14
|-
|-
!10/7
!LmLsLsmLs
|28/25
|28/25
|49/40
|49/40
Line 294: Line 303:
|49/25
|49/25
|-
|-
!8/5
!mLsLsmLsL
|35/32
|35/32
|49/40
|49/40
Line 304: Line 313:
|25/14
|25/14
|-
|-
!7/4
!LsLsmLsLm
|28/25
|28/25
|8/7
|8/7
Line 314: Line 323:
|64/35
|64/35
|-
|-
!49/25
!sLsmLsLmL
|50/49
|50/49
|8/7
|8/7

Latest revision as of 02:28, 11 March 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)

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)

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
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