2.5.7 subgroup: Difference between revisions

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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 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)
* 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)
* Right-handed: 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)
* 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.
Didacus tempering sets L = m + s.
=== Interval matrices ===
==== Achiral ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!2/1
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!28/25
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|625/392
|25/14
|-
!8/7
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|49/25
|-
!5/4
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|224/125
|64/35
|-
!7/5
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|8/5
|80/49
|25/14
|-
!10/7
|28/25
|49/40
|'''5/4'''
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!8/5
|35/32
|125/112
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!7/4
|50/49
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!25/14
|28/25
|784/625
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}
==== Right-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!2/1
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!50/49
|28/25
|49/40
|343/250
|7/5
|196/125
|343/200
|'''7/4'''
|49/25
|-
!8/7
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|25/14
|-
!5/4
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|80/49
|64/35
|-
!7/5
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|500/343
|80/49
|25/14
|-
!10/7
|28/25
|49/40
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!8/5
|35/32
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!7/4
|50/49
|8/7
|400/343
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!25/14
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}
==== Left-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!2/1
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!28/25
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!8/7
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|343/200
|'''7/4'''
|49/25
|-
!5/4
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|64/35
|-
!7/5
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|80/49
|25/14
|-
!10/7
|28/25
|49/40
|343/250
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!8/5
|35/32
|49/40
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!7/4
|28/25
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!49/25
|50/49
|8/7
|400/343
|125/98
|10/7
|500/343
|80/49
|25/14
|}
{{Cat|JI groups}}
{{Cat|JI groups}}

Revision as of 13:06, 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
2/1 28/25 8/7 5/4 7/5 10/7 8/5 7/4 25/14
28/25 50/49 125/112 5/4 125/98 10/7 25/16 625/392 25/14
8/7 35/32 49/40 5/4 7/5 49/32 25/16 7/4 49/25
5/4 28/25 8/7 32/25 7/5 10/7 8/5 224/125 64/35
7/5 50/49 8/7 5/4 125/98 10/7 8/5 80/49 25/14
10/7 28/25 49/40 5/4 7/5 196/125 8/5 7/4 49/25
8/5 35/32 125/112 5/4 7/5 10/7 25/16 7/4 25/14
7/4 50/49 8/7 32/25 64/49 10/7 8/5 80/49 64/35
25/14 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
2/1 50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14
50/49 28/25 49/40 343/250 7/5 196/125 343/200 7/4 49/25
8/7 35/32 49/40 5/4 7/5 49/32 25/16 7/4 25/14
5/4 28/25 8/7 32/25 7/5 10/7 8/5 80/49 64/35
7/5 50/49 8/7 5/4 125/98 10/7 500/343 80/49 25/14
10/7 28/25 49/40 5/4 7/5 10/7 8/5 7/4 49/25
8/5 35/32 125/112 5/4 125/98 10/7 25/16 7/4 25/14
7/4 50/49 8/7 400/343 64/49 10/7 8/5 80/49 64/35
25/14 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
2/1 28/25 8/7 5/4 7/5 10/7 8/5 7/4 49/25
28/25 50/49 125/112 5/4 125/98 10/7 25/16 7/4 25/14
8/7 35/32 49/40 5/4 7/5 49/32 343/200 7/4 49/25
5/4 28/25 8/7 32/25 7/5 196/125 8/5 224/125 64/35
7/5 50/49 8/7 5/4 7/5 10/7 8/5 80/49 25/14
10/7 28/25 49/40 343/250 7/5 196/125 8/5 7/4 49/25
8/5 35/32 49/40 5/4 7/5 10/7 25/16 7/4 25/14
7/4 28/25 8/7 32/25 64/49 10/7 8/5 80/49 64/35
49/25 50/49 8/7 400/343 125/98 10/7 500/343 80/49 25/14
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