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 itself (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)
 
{{Adv|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.
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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[[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.
[[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 ==
== 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 prejubilic or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or soft 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-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. 37edo equates 49/40 to 16/13.
 
=== Interval matrices ===
 
==== Achiral ====
{| class="wikitable"
!
!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 ====
{| class="wikitable"
!
!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 ====
{| class="wikitable"
!
!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] ===
<pre>
let s = 28/25
let L = 7/4
s;s;s;s;s;L;
stack()
31@
</pre>
=== Didacus[13] ===
<pre>
let L = 35/32
let s = 50/49
s;L;s;L;s;L;s;L;s;L;s;L;s;
stack()
31@
</pre>
=== Jubilismic[6] ===
<pre>
let L = 7/4
let s = 35/32
L;L;s;L;L;s;
stack()
16@
</pre>
 
== Temperaments ==
Common rank-2 temperaments in 2.5.7 (i.e. temperaments that interpret intervals as 2.5.7 JI ratios):
* [[Didacus]] ({{e|25}} & {{e|31}}): Best accuracy-simplicity tradeoff among 2.5.7 temperaments. Generates 5L1s generated by ~28/25.
* [[Jubilismic]] ({{e|16}} & {{e|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]] ({{e|16}} & {{e|25}}): Generated by a flat armotonic fifth around 672 cents.
* 3edo.7 (6 & 15): Generates 3L3s with generator ~7/4. Supported by [[Augmented (temperament)|Augmented]] edos such as [[15edo]], [[21edo]], and [[27edo]].
{{Cat|JI groups}}
{{Cat|JI groups}}
* [[Sidewalk]] ({{e|21}} & {{e|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.

Latest revision as of 02:55, 17 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 prejubilic or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or soft 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)

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.
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