7-limit: Difference between revisions

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The '''7-limit''' or the '''2.3.5.7 subgroup''' is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[3/2]], [[5/4]], and [[7/4]].
Rank-3 subgroups:
* [[5-limit]]
* [[2.3.7 subgroup]]
* [[2.5.7 subgroup]]
* [[3.5.7 subgroup]]


== Full 7-limit JI scales ==
== Full 7-limit JI scales ==
=== Mode 5 ===
<pre>
8:9:10:12:14:16
</pre>
The simplest full 7-limit JI scale. This scale is notably used in the music of the Wagogo people in Tanzania.
=== Rooted Mixolydian ===
<pre>
16:18:20:21:24:27:28:32
</pre>
=== Rooted Ionian ===
<pre>
16:18:20:21:24:27:30:32
</pre>
=== Zil ===
=== Zil ===
Zil (from the temperament Godzilla which the zil series serves as a detempering of) is a series of [[7-limit]] JI scales created from a [[generator sequence]]. The most discussed of the zil scales is zil[14] which is chiral depending on the chirality of the [[interleaving|interleaved]] 5-limit [[zarlino]] copies:
Zil (from the temperament Godzilla which the zil series serves as a detempering of) is a series of 7-limit JI scales created from a [[generator sequence]] GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160).
==== Zil[14] ====
The most discussed of the zil scales is zil[14] which is chiral depending on the chirality of the [[interleaving|interleaved]] 5-limit [[zarlino]] copies:


RH zil[14]
RH zil[14]
Line 40: Line 64:
</pre>
</pre>


=== Jar ===
==== Zil[24] ====
Jar is a detempering of Pajara[10] with an analogous structure to zil[14]:
Zil[24] is achiral. It has a 4×3×2 structure in the 7-limit lattice.
<pre>
525/512
135/128
35/32
9/8
4725/4096
75/64
315/256
5/4
21/16
675/512
175/128
45/32
189/128
3/2
1575/1024
25/16
105/64
27/16
7/4
225/128
945/512
15/8
63/32
2/1
</pre>


RH jar
=== 12:14:16:18:21:24 by 5/4 ===
A 10-note scale with an analogous structure to zil[14] (note that these are subsets of both zil[14] chiralities):
 
RH
<pre>
<pre>
35/32
35/32
Line 57: Line 110:
</pre>
</pre>


LH jar
LH
<pre>
<pre>
16/15
16/15
Line 70: Line 123:
2/1
2/1
</pre>
</pre>
=== A Mothra[36] detemper ===
GS(8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 245/216)[36]; 4×3×3 generator structure
<pre>
33075/32768
525/512
135/128
2205/2048
35/32
9/8
147/128
4725/4096
75/64
1225/1024
315/256
5/4
1323/1024
21/16
675/512
11025/8192
175/128
45/32
735/512
189/128
3/2
49/32
1575/1024
25/16
6615/4096
105/64
27/16
441/256
7/4
225/128
3675/2048
945/512
15/8
245/128
63/32
2/1
</pre>
</pre>
== Full 7-limit tempered scales ==
=== Superpyth[12] ===
Superpyth[12] is constructed by applying [[Superpyth]] temperament (2.3.5.7[22 & 27]; equivalently tempering out 64/63 and 245/243) to a 12-note chain of fifths. It contains Superpyth-tempered 5-limit [[blackdye]].
=== Pajara ===
[[Pajara]] can be used as an interpretation of 2L8s and 10L2s or their modifications. Pajara works best in [[22edo]].
=== 7-limit diachrome ===
7-limit diachrome, an [[aberrismic]] scale, is constructed by taking a 6+4 or 5+5 fifth chain structure and tempering out [[5120/5103]].
{{Cat|JI groups}}

Latest revision as of 16:20, 11 April 2026

The 7-limit or the 2.3.5.7 subgroup is the subgroup of just intonation consisting of the intervals reachable by stacking 2/1, 3/2, 5/4, and 7/4.

Rank-3 subgroups:

Full 7-limit JI scales

Mode 5

8:9:10:12:14:16

The simplest full 7-limit JI scale. This scale is notably used in the music of the Wagogo people in Tanzania.

Rooted Mixolydian

16:18:20:21:24:27:28:32

Rooted Ionian

16:18:20:21:24:27:30:32

Zil

Zil (from the temperament Godzilla which the zil series serves as a detempering of) is a series of 7-limit JI scales created from a generator sequence GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160).

Zil[14]

The most discussed of the zil scales is zil[14] which is chiral depending on the chirality of the interleaved 5-limit zarlino copies:

RH zil[14] 

21/20
9/8
189/160
6/5
21/16
27/20
7/5
3/2
63/40
8/5
7/4
9/5
63/32
2/1

LH zil[14] 

21/20
9/8
7/6
6/5
21/16
4/3
7/5
3/2
63/40
8/5
7/4
9/5
63/32
2/1

Zil[24]

Zil[24] is achiral. It has a 4×3×2 structure in the 7-limit lattice. 

525/512
135/128
35/32
9/8
4725/4096
75/64
315/256
5/4
21/16
675/512
175/128
45/32
189/128
3/2
1575/1024
25/16
105/64
27/16
7/4
225/128
945/512
15/8
63/32
2/1

12:14:16:18:21:24 by 5/4

A 10-note scale with an analogous structure to zil[14] (note that these are subsets of both zil[14] chiralities):

RH 

35/32
7/6
5/4
4/3
35/24
3/2
5/3
7/4
15/8
2/1

LH 

16/15
8/7
6/5
4/3
48/35
3/2
8/5
12/7
64/35
2/1

A Mothra[36] detemper

GS(8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 245/216)[36]; 4×3×3 generator structure 

33075/32768
525/512
135/128
2205/2048
35/32
9/8
147/128
4725/4096
75/64
1225/1024
315/256
5/4
1323/1024
21/16
675/512
11025/8192
175/128
45/32
735/512
189/128
3/2
49/32
1575/1024
25/16
6615/4096
105/64
27/16
441/256
7/4
225/128
3675/2048
945/512
15/8
245/128
63/32
2/1

Full 7-limit tempered scales

Superpyth[12]

Superpyth[12] is constructed by applying Superpyth temperament (2.3.5.7[22 & 27]; equivalently tempering out 64/63 and 245/243) to a 12-note chain of fifths. It contains Superpyth-tempered 5-limit blackdye.

Pajara

Pajara can be used as an interpretation of 2L8s and 10L2s or their modifications. Pajara works best in 22edo.

7-limit diachrome

7-limit diachrome, an aberrismic scale, is constructed by taking a 6+4 or 5+5 fifth chain structure and tempering out 5120/5103.

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