2.3.7 subgroup: Difference between revisions

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[[File:2.3.7 intervals.png|thumb|678x678px|21-odd-limit 2.3.7 intervals (tuned in 36edo), showing their closeness to the 5-form.]]
[[File:2.3.7 intervals.png|thumb|678x678px|21-odd-limit 2.3.7 intervals (tuned in 36edo), showing their closeness to the 5-form.]]
The '''2.3.7 subgroup''' (aka '''septal''', or in color notation, '''za''') is the subgroup of [[just intonation]] comprising the intervals reachable by stacking [[2/1]], [[3/2]], and [[7/4]], with the exclusion of [[5/4]] (adding which would result in the full [[7-limit]]).  
The '''2.3.7 subgroup''' (aka '''septal''', or in [[color notation]], '''za''') is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[3/2]], and [[7/4]], with the exclusion of [[5/4]] (adding which would result in the full [[7-limit]]).  


Notable intervals include 7/4 itself (the septimal subminor seventh), [[9/7]] and [[7/6]] (the septimal supermajor and subminor thirds), and [[21/16]] (the septimal subfourth). More intervals can be seen at [[List of just intonation intervals#2.3.7]]. Generically, 2.3.7 intervals and scales that include prime 7 are called ''septal'' (to distinguish them from "septimal", referring to the 7-limit including prime 5).
Notable intervals include 7/4 itself (the septimal subminor seventh), [[9/7]] and [[7/6]] (the septimal supermajor and subminor thirds), and [[21/16]] (the septimal subfourth). More intervals can be seen at [[List of just intonation intervals#2.3.7]]. Generically, 2.3.7 intervals and scales that include prime 7 are called ''septal'' (to distinguish them from "septimal", referring to the 7-limit including prime 5).
The 2.3.7 subgroup includes the following odd harmonics below 256: 1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 243.


Any 2.3.7 interval is separated by a number of [[64/63]] dieses from a [[Pythagorean tuning|Pythagorean]] interval, which serve as the [[formal comma]] for 7 in Pythagorean-spine notation systems, and can be tempered out to equate septal intervals with Pythagorean intervals, resulting in [[Archy]] temperament.   
Any 2.3.7 interval is separated by a number of [[64/63]] dieses from a [[Pythagorean tuning|Pythagorean]] interval, which serve as the [[formal comma]] for 7 in Pythagorean-spine notation systems, and can be tempered out to equate septal intervals with Pythagorean intervals, resulting in [[Archy]] temperament.   
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The fundamental 2.3.7 (super)major triad (14:18:21) is more otonally complex than the (sub)minor triad (6:7:9), which is therefore generally more stable - the opposite situation to 2.3.5 harmony. Additionally, the thirds are further apart, and so may be played in the same chord without harsh clashes.  
The fundamental 2.3.7 (super)major triad (14:18:21) is more otonally complex than the (sub)minor triad (6:7:9), which is therefore generally more stable - the opposite situation to 2.3.5 harmony. Additionally, the thirds are further apart, and so may be played in the same chord without harsh clashes.  


The 2.3.7 subgroup is analyzable under the 5-form, much as the 2.3.5 subgroup is under the 7-form, and one can consider [[Equipentatonic#Just equipentatonic scale|12:14:16:18:21:24]] the basic pentatonic for the 2.3.7 subgroup. From there, [1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1] is a reasonable [[aberrismic]] extension, with an aberrisma of (7/6)/(8/7) = 49/48. Note that this implies fourth-bounded triads as the basic unit of harmony, rather than fifth-bounded ones.  
The 2.3.7 subgroup is analyzable under the 5-form, much as the 2.3.5.11 subgroup is under the 7-form, and one can consider [[Equipentatonic#Just equipentatonic scale|12:14:16:18:21:24]] the basic pentatonic for the 2.3.7 subgroup. From there, [1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1] is a reasonable [[aberrismic]] extension, with an aberrisma of (7/6)/(8/7) = 49/48. Note that this implies fourth-bounded triads as the basic unit of harmony, rather than fifth-bounded ones. However, it is also a variant of a [[diatonic]] scale, [1/1 8/7 64/49 4/3 3/2 12/7 96/49 2/1], and becomes [[mosdiatonic]] in Archy temperament.
== JI scales ==
=== Archylino ===
Right-hand Archylino:
<pre>
let L = 8/7
let m = 9/8
let s = 28/27
L;m;s;m;L;m;s;
stack()
</pre>
Left-hand Archylino:
<pre>
let L = 8/7
let m = 9/8
let s = 28/27
m;L;s;m;L;m;s;
stack()
</pre>
=== Tetrachordal septal diatonic ===
Minor version
<pre>
let L = 8/7
let m = 9/8
let s = 28/27
m;s;L;m;m;s;L;
stack()
</pre>
 
Major version
<pre>
let L = 8/7
let m = 9/8
let s = 28/27
m;L;s;m;m;L;s;
stack()
</pre>
 
=== Interseptimal diatonic ===
<pre>
let L = 8/7
let m = 9/8
let s = 49/48
L;s;L;m;L;s;L;
stack()
</pre>
=== Aberrismic theory ===
:''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]]
The fundamental septal aberrismic scales are septal diasem, and (if Slendric is not used) diaslen. If Slendric is used then diaslen becomes the MOS Slendric[11] (5L6s).
==== Diasem ====
Right-handed diasem:
<pre>
let L = 9/8
let m = 28/27
let s = 64/63
L;m;L;s;L;m;L;s;L;
stack()
</pre>
 
Left-handed diasem:
<pre>
let L = 9/8
let m = 28/27
let s = 64/63
L;s;L;m;L;s;L;m;L;
stack()
</pre>
==== Diaslen ====
Achiral diaslen:
<pre>
let L = 9/8
let m = 49/48
let s = 64/63
s;L;m;L;s;L;s;L;m;L;s;
stack()
</pre>
Right-handed diaslen:
<pre>
let L = 9/8
let m = 49/48
let s = 64/63
s;L;s;L;m;L;s;L;s;L;m;
stack()
</pre>
Left-handed diaslen:
<pre>
let L = 9/8
let m = 49/48
let s = 64/63
m;L;s;L;s;L;m;L;s;L;s;
stack()
</pre>


The '''septal lattice''' is the JI lattice for the 2.3.7 subgroup.
== Tempered 2.3.7 scales ==
=== [[Slendric]][11] ===
(In [[36edo]] tuning)
<pre>
let L = 9/8
let s = 64/63
s;L;s;L;s;L;s;L;s;L;s;
stack()
36@
</pre>


== Aberrismic theory ==
=== [[Archy]] diatonic ===
The fundamental septal aberrismic scales are septal diasem and diaslen.
(In [[22edo]] tuning)
<pre>
let L = 9/8
let s = 256/243
L;s;L;L;s;L;L;
stack()
22@
</pre>


== Septal tuning ==
(In [[37edo]] tuning)
Septal tuning refers to tuning intervals to the 2.3.7 subgroup, regardless of their interpretation. For example, one may temper out [[81/80]] in 2.3.5.7 (meantone.7 temperament) and the resulting structure may be called septal, as it is generated by an approximate 3/2 and 7/4. This is analogous to how schismic (or more loosely, any monocot temperament) may be thought of as [[Pythagorean tuning|Pythagorean]].
<pre>
let L = 9/8
let s = 256/243
L;s;L;L;s;L;L;
stack()
37@
</pre>


===Septimal meantone===
== Temperaments ==
Common rank-2 temperaments in 2.3.7 (i.e. temperaments that interpret intervals as 2.3.7 JI ratios):
* [[Slendric]] ({{e|31}} & {{e|36}}): Best accuracy-simplicity tradeoff, but lacks canonical extension to other primes. Splits into Mothra (soft of 36edo) and Rodan (hard of 36edo).
* [[Archy]] ({{e|22}} & {{e|27}}): A simple monocot temperament, but less accurate. In the full 7-limit, splits into Superpyth (between 22edo and 27edo) and Ultrapyth (essentially 37edo).
* [[Buzzard]] ({{e|48}} & {{e|53}}): More complex; exaggerates the difference between 21/16 and 64/49.
* [[Semaphore]] = ({{e|19}} & {{e|24}}): Almost an exotemperment; conflates 7/6 and 8/7.


Here I need to put a chart of the continuum with 43, 74 (37x2), 31, 81 (27x3), and 50edo (25x2).
{{Cat|JI groups}}

Latest revision as of 03:44, 13 April 2026

21-odd-limit 2.3.7 intervals (tuned in 36edo), showing their closeness to the 5-form.

The 2.3.7 subgroup (aka septal, or in color notation, za) is the subgroup of just intonation consisting of the intervals reachable by stacking 2/1, 3/2, and 7/4, with the exclusion of 5/4 (adding which would result in the full 7-limit).

Notable intervals include 7/4 itself (the septimal subminor seventh), 9/7 and 7/6 (the septimal supermajor and subminor thirds), and 21/16 (the septimal subfourth). More intervals can be seen at List of just intonation intervals#2.3.7. Generically, 2.3.7 intervals and scales that include prime 7 are called septal (to distinguish them from "septimal", referring to the 7-limit including prime 5).

The 2.3.7 subgroup includes the following odd harmonics below 256: 1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 243.

Any 2.3.7 interval is separated by a number of 64/63 dieses from a Pythagorean interval, which serve as the formal comma for 7 in Pythagorean-spine notation systems, and can be tempered out to equate septal intervals with Pythagorean intervals, resulting in Archy temperament.

Another important diesis is 49/48, which separates 8/7 from 7/6, and 9/7 from 21/16. Equating these two gives Slendric temperament. 36edo, as an example of an EDO that supports Slendric and represents the perfect fifth well, provides a great tempering of the 2.3.7 subgroup.

The fundamental 2.3.7 (super)major triad (14:18:21) is more otonally complex than the (sub)minor triad (6:7:9), which is therefore generally more stable - the opposite situation to 2.3.5 harmony. Additionally, the thirds are further apart, and so may be played in the same chord without harsh clashes.

The 2.3.7 subgroup is analyzable under the 5-form, much as the 2.3.5.11 subgroup is under the 7-form, and one can consider 12:14:16:18:21:24 the basic pentatonic for the 2.3.7 subgroup. From there, [1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1] is a reasonable aberrismic extension, with an aberrisma of (7/6)/(8/7) = 49/48. Note that this implies fourth-bounded triads as the basic unit of harmony, rather than fifth-bounded ones. However, it is also a variant of a diatonic scale, [1/1 8/7 64/49 4/3 3/2 12/7 96/49 2/1], and becomes mosdiatonic in Archy temperament.

JI scales

Archylino

Right-hand Archylino: 

let L = 8/7
let m = 9/8
let s = 28/27
L;m;s;m;L;m;s;
stack()

Left-hand Archylino: 

let L = 8/7
let m = 9/8
let s = 28/27
m;L;s;m;L;m;s;
stack()

Tetrachordal septal diatonic

Minor version 

let L = 8/7
let m = 9/8
let s = 28/27
m;s;L;m;m;s;L;
stack()

Major version 

let L = 8/7
let m = 9/8
let s = 28/27
m;L;s;m;m;L;s;
stack()

Interseptimal diatonic

let L = 8/7
let m = 9/8
let s = 49/48
L;s;L;m;L;s;L;
stack()

Aberrismic theory

Main article: Aberrisma#Quasi-diatonic aberrismic scales

The fundamental septal aberrismic scales are septal diasem, and (if Slendric is not used) diaslen. If Slendric is used then diaslen becomes the MOS Slendric[11] (5L6s).

Diasem

Right-handed diasem: 

let L = 9/8
let m = 28/27
let s = 64/63
L;m;L;s;L;m;L;s;L;
stack()

Left-handed diasem: 

let L = 9/8
let m = 28/27
let s = 64/63
L;s;L;m;L;s;L;m;L;
stack()

Diaslen

Achiral diaslen: 

let L = 9/8
let m = 49/48
let s = 64/63
s;L;m;L;s;L;s;L;m;L;s;
stack()

Right-handed diaslen: 

let L = 9/8
let m = 49/48
let s = 64/63
s;L;s;L;m;L;s;L;s;L;m;
stack()

Left-handed diaslen: 

let L = 9/8
let m = 49/48
let s = 64/63
m;L;s;L;s;L;m;L;s;L;s;
stack()

Tempered 2.3.7 scales

Slendric[11]

(In 36edo tuning) 

let L = 9/8
let s = 64/63
s;L;s;L;s;L;s;L;s;L;s;
stack()
36@

Archy diatonic

(In 22edo tuning) 

let L = 9/8
let s = 256/243
L;s;L;L;s;L;L;
stack()
22@

(In 37edo tuning) 

let L = 9/8
let s = 256/243
L;s;L;L;s;L;L;
stack()
37@

Temperaments

Common rank-2 temperaments in 2.3.7 (i.e. temperaments that interpret intervals as 2.3.7 JI ratios):

  • Slendric (31 & 36): Best accuracy-simplicity tradeoff, but lacks canonical extension to other primes. Splits into Mothra (soft of 36edo) and Rodan (hard of 36edo).
  • Archy (22 & 27): A simple monocot temperament, but less accurate. In the full 7-limit, splits into Superpyth (between 22edo and 27edo) and Ultrapyth (essentially 37edo).
  • Buzzard (48 & 53): More complex; exaggerates the difference between 21/16 and 64/49.
  • Semaphore = (19 & 24): Almost an exotemperment; conflates 7/6 and 8/7.
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