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]].
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]]. Important subsets of the 7-limit include the [[7-odd-limit]] and [[9-odd-limit]].


Rank-3 subgroups:
Rank-3 subgroups:
Line 6: Line 6:
* [[2.5.7 subgroup]]
* [[2.5.7 subgroup]]
* [[3.5.7 subgroup]]
* [[3.5.7 subgroup]]
{{Adv|The 7-limit includes the following odd harmonics below 256: 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105, 125, 135, 147, 175, 189, 225, 243, 245.}}
== Temperaments ==
Some important rank-2 full 7-limit temperaments:
* [[Superpyth]]
* Septimal [[Porcupine]]
* [[Garibaldi]]
* Septimal [[Meantone]]
* [[Meantone#Extensions|Flattone]]
* [[Pajara]]
* Septimal [[Magic]]
* [[Mothra]]
* [[Rodan]]
* [[Valentine]]
* [[Orwell]]
* [[Augene]]
* [[Blackwood]]
* [[Sensi]]
The most important rank-3 full 7-limit temperaments are
* [[Aberschismic]] ({{e|41}} & {{e|46}} & {{e|53}}): equates the 2.3.5 and 2.3.7 formal commas, 81/80 and 64/63; medium-high accuracy.
* [[Marvel]] ({{e|19}} & {{e|22}} & {{e|41}}): equates 25/16 and 14/9; medium accuracy.
== 7-limit interval qualities ==
The 7-limit thirds, in order of stability / consonance in triads (from most to least consonant), are 5/4, 7/6, 6/5, 9/7. Note that 7-limit L/MCJI offers four distinct interval qualities, whereas 12edo and simpler tuning systems tend to offer only two. The mapping of 12edo's interval qualities to the 7-limit's is best understood by considering each 12edo interval as "splitting" into two distinct 7-limit intervals, rather than the alternative approach of retaining 12edo's interval qualities and providing an additional neutral interval (characteristic of subgroups involving 11, like 2.3.11.19).
The four qualities of the 7-limit can be broken down into stable/unstable and bright/dark. The scheme used is the application of ADIN to systems like 22edo and 27edo; see [[Adaptive diatonic interval names#On "major" vs. "supermajor"]] for a further explanation.
{| class="wikitable"
!
!Stable
!Unstable
|-
!Bright
|Nearmajor (warm, pleasant, comforting)
|Supermajor (excited, animated, active)
|-
!Dark
|Subminor (depressive, sad, bluesy)
|Nearminor (angry, tense, stressful)
|}
These are, in fact, the basic "color qualities" provided by Kite: red (ru) = supermajor, yellow (yo) = nearmajor, green (gu) = nearminor, and blue (zo) = subminor. In [[keemic]] temperaments, they become equally spaced, incentivizing the metaphor of a "rainbow" of qualities promoted by Kite.


== Full 7-limit JI scales ==
== Full 7-limit JI scales ==
The scales are shown in [https://sw3.lumipakkanen.com/ Scale Workshop 3] format. Copy and paste into Scale Workshop 3 and you will be able to play the scale.
=== Mode 5 ===
=== Mode 5 ===
<pre>
<pre>
8:9:10:12:14:16
8:9:10:12:14:16
</pre>
</pre>
This scale is notably used in the music of the Wagogo people in Tanzania.
The simplest full 7-limit JI scale.
=== Rooted Mixolydian ===
 
=== /2 Mixolydian ===
<pre>
<pre>
16:18:20:21:24:27:28:32
16:18:20:21:24:27:28:32
</pre>
</pre>
Septimalydian is a mode of this.
 
=== Rooted Ionian ===
=== /2 Ionian ===
<pre>
<pre>
16:18:20:21:24:27:30:32
16:18:20:21:24:27:30:32
</pre>
</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]] GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160).
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] ====
==== 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:
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] = [[cross-set]] of RH zarlino and 7/4
<pre>
<pre>
21/20
35/32; 9/8; 315/256; 5/4; 21/16; 45/32; 189/128; 3/2; 105/64; 27/16; 7/4; 15/8; 63/32; 2/1
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
</pre>
</pre>


LH zil[14]
LH zil[14] = cross-set of LH zarlino and 7/4
<pre>
<pre>
21/20
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
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
</pre>
</pre>


Line 66: Line 83:
Zil[24] is achiral. It has a 4×3×2 structure in the 7-limit lattice.
Zil[24] is achiral. It has a 4×3×2 structure in the 7-limit lattice.
<pre>
<pre>
525/512
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
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>
</pre>


=== 12:14:16:18:21:24 by 5/4 ===
=== Cross-set of 12:14:16:18:21:24 and 5/4 ===
A 10-note scale with an analogous structure to zil[14] (note that these are subsets of both zil[14] chiralities):
A 10-note scale with an analogous structure to zil[14] (note that these are subsets of both zil[14] chiralities):


RH
RH
<pre>
<pre>
35/32
35/32; 9/8; 5/4; 21/16; 45/32; 3/2; 105/64; 7/4; 15/8; 2/1
7/6
5/4
4/3
35/24
3/2
5/3
7/4
15/8
2/1
</pre>
</pre>


LH
LH
<pre>
<pre>
16/15
16/15; 8/7; 6/5; 4/3; 48/35; 3/2; 8/5; 12/7; 64/35; 2/1
8/7
</pre>
6/5
 
4/3
=== A Mothra[36] detemper ===
48/35
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
3/2
 
8/5
<pre>
12/7
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
64/35
</pre>
2/1
 
=== A Hemithirds[31] detemper ===
<pre>
525/512; 21/20; 16/15; 35/32; 28/25; 8/7; 75/64; 25/21; 128/105; 5/4; 32/25; 21/16; 4/3; 175/128; 7/5; 10/7; 256/175; 3/2; 32/21; 25/16; 8/5; 105/64; 42/25; 128/75; 7/4; 25/14; 64/35; 15/8; 40/21; 1024/525; 2/1
</pre>
 
=== An Aberschismic 34edo detemper ===
Contains multiple copies of [[aberrisma|aberrismic]] scales (diasem and blackdye); maps both 81/80 and 64/63 to one step.
<pre>
81/80; 21/20; 16/15; 35/32; 10/9; 9/8; 7/6; 189/160; 6/5; 315/256; 5/4; 35/27; 21/16; 4/3; 27/20; 7/5; 45/32; 35/24; 189/128; 3/2; 14/9; 63/40; 8/5; 105/64; 5/3; 27/16; 7/4; 16/9; 9/5; 28/15; 15/8; 35/18; 63/32; 2/1
</pre>
 
Use 81/64 instead of 35/27 to get a Pyth[7] subset:
<pre>
81/80; 21/20; 16/15; 35/32; 10/9; 9/8; 7/6; 189/160; 6/5; 315/256; 5/4; 81/64; 21/16; 4/3; 27/20; 7/5; 45/32; 35/24; 189/128; 3/2; 14/9; 63/40; 8/5; 105/64; 5/3; 27/16; 7/4; 16/9; 9/5; 28/15; 15/8; 35/18; 63/32; 2/1
</pre>
</pre>


Line 126: Line 125:
=== Superpyth[12] ===
=== 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]].
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]].
<pre>
let L = 2187/2048
let s = 256/243
L;s;L;s;L;s;s;L;s;L;s;s;
stack()
27@
</pre>
=== Pajara ===
=== Pajara ===
[[Pajara]] can be used as an interpretation of 2L8s and 10L2s or their modifications. Pajara works best in [[22edo]].
[[Pajara]] can be used as an interpretation of 2L8s and 10L2s or their modifications. Pajara works best in [[22edo]].
==== Pajara[10] ====
<pre>
let L = 10/9
let s = 16/15
s;s;L;s;s;s;s;L;s;s;
stack()
22@
</pre>
==== Pentachordal Pajara[10] ====
<pre>
let L = 10/9
let s = 16/15
s;s;s;s;s;L;s;s;s;L;
stack()
22@
</pre>
==== Pajara[12] ====
<pre>
let L = 16/15
let s = 25/24
L;L;L;L;L;s;L;L;L;L;L;s;
stack()
22@
</pre>
==== Hexachordal Pajara[12] ====
<pre>
let L = 16/15
let s = 25/24
L;L;L;L;s;L;L;L;L;L;L;s;
stack()
22@
</pre>
=== 7-limit diachrome ===
=== 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]].
7-limit diachrome, an [[aberrismic]] scale, is constructed by taking a 6+6 (for achiral diachrome) or 7+5 (for chiral diachrome) fifth chain structure and tempering out [[5120/5103]]. The scales are shown below in 41edo tuning, but they work in any Aberschismic tuning such as [[46edo]] and [[53edo]].
==== 5sC ====
<pre>
let L = 10/9
let m = 256/243
let s = 81/80
L;s;L;s;L;m;s;L;s;L;s;m;
stack()
41@
</pre>
==== 5sL ====
<pre>
let L = 10/9
let m = 256/243
let s = 81/80
L;s;L;s;L;m;L;s;L;s;m;
stack()
41@
</pre>
==== 5sR ====
<pre>
let L = 10/9
let m = 256/243
let s = 81/80
L;m;s;L;s;L;s;L;m;s;L;s;
stack()
41@
</pre>
 
=== Aberschismic whitedye ===
Aberschismic whitedye is constructed by taking a diatonic scale and offsetting it by 64/63~81/80, tempering out [[5120/5103]].
 
Shown below in 41edo tuning.
<pre>
let L = 10/9
let m = 28/27
let s = 81/80
L;s;L;s;L;s;m;s;L;s;L;s;m;s;
stack()
41@
</pre>


{{Cat|JI groups}}
{{Cat|JI groups}}

Latest revision as of 22:48, 22 May 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. Important subsets of the 7-limit include the 7-odd-limit and 9-odd-limit.

Rank-3 subgroups:

The 7-limit includes the following odd harmonics below 256: 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105, 125, 135, 147, 175, 189, 225, 243, 245.

Temperaments

Some important rank-2 full 7-limit temperaments:

The most important rank-3 full 7-limit temperaments are

  • Aberschismic (​41 & ​46 & ​53): equates the 2.3.5 and 2.3.7 formal commas, 81/80 and 64/63; medium-high accuracy.
  • Marvel (​19 & ​22 & ​41): equates 25/16 and 14/9; medium accuracy.

7-limit interval qualities

The 7-limit thirds, in order of stability / consonance in triads (from most to least consonant), are 5/4, 7/6, 6/5, 9/7. Note that 7-limit L/MCJI offers four distinct interval qualities, whereas 12edo and simpler tuning systems tend to offer only two. The mapping of 12edo's interval qualities to the 7-limit's is best understood by considering each 12edo interval as "splitting" into two distinct 7-limit intervals, rather than the alternative approach of retaining 12edo's interval qualities and providing an additional neutral interval (characteristic of subgroups involving 11, like 2.3.11.19).

The four qualities of the 7-limit can be broken down into stable/unstable and bright/dark. The scheme used is the application of ADIN to systems like 22edo and 27edo; see Adaptive diatonic interval names#On "major" vs. "supermajor" for a further explanation.

Stable Unstable
Bright Nearmajor (warm, pleasant, comforting) Supermajor (excited, animated, active)
Dark Subminor (depressive, sad, bluesy) Nearminor (angry, tense, stressful)

These are, in fact, the basic "color qualities" provided by Kite: red (ru) = supermajor, yellow (yo) = nearmajor, green (gu) = nearminor, and blue (zo) = subminor. In keemic temperaments, they become equally spaced, incentivizing the metaphor of a "rainbow" of qualities promoted by Kite.

Full 7-limit JI scales

The scales are shown in Scale Workshop 3 format. Copy and paste into Scale Workshop 3 and you will be able to play the scale.

Mode 5

8:9:10:12:14:16

The simplest full 7-limit JI scale.

/2 Mixolydian

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

/2 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] = cross-set of RH zarlino and 7/4 

35/32; 9/8; 315/256; 5/4; 21/16; 45/32; 189/128; 3/2; 105/64; 27/16; 7/4; 15/8; 63/32; 2/1

LH zil[14] = cross-set of LH zarlino and 7/4 

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

Cross-set of 12:14:16:18:21:24 and 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; 9/8; 5/4; 21/16; 45/32; 3/2; 105/64; 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

A Hemithirds[31] detemper

525/512; 21/20; 16/15; 35/32; 28/25; 8/7; 75/64; 25/21; 128/105; 5/4; 32/25; 21/16; 4/3; 175/128; 7/5; 10/7; 256/175; 3/2; 32/21; 25/16; 8/5; 105/64; 42/25; 128/75; 7/4; 25/14; 64/35; 15/8; 40/21; 1024/525; 2/1

An Aberschismic 34edo detemper

Contains multiple copies of aberrismic scales (diasem and blackdye); maps both 81/80 and 64/63 to one step. 

81/80; 21/20; 16/15; 35/32; 10/9; 9/8; 7/6; 189/160; 6/5; 315/256; 5/4; 35/27; 21/16; 4/3; 27/20; 7/5; 45/32; 35/24; 189/128; 3/2; 14/9; 63/40; 8/5; 105/64; 5/3; 27/16; 7/4; 16/9; 9/5; 28/15; 15/8; 35/18; 63/32; 2/1

Use 81/64 instead of 35/27 to get a Pyth[7] subset: 

81/80; 21/20; 16/15; 35/32; 10/9; 9/8; 7/6; 189/160; 6/5; 315/256; 5/4; 81/64; 21/16; 4/3; 27/20; 7/5; 45/32; 35/24; 189/128; 3/2; 14/9; 63/40; 8/5; 105/64; 5/3; 27/16; 7/4; 16/9; 9/5; 28/15; 15/8; 35/18; 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. 

let L = 2187/2048
let s = 256/243
L;s;L;s;L;s;s;L;s;L;s;s;
stack()
27@

Pajara

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

Pajara[10]

let L = 10/9
let s = 16/15
s;s;L;s;s;s;s;L;s;s;
stack()
22@

Pentachordal Pajara[10]

let L = 10/9
let s = 16/15
s;s;s;s;s;L;s;s;s;L;
stack()
22@

Pajara[12]

let L = 16/15
let s = 25/24
L;L;L;L;L;s;L;L;L;L;L;s;
stack()
22@

Hexachordal Pajara[12]

let L = 16/15
let s = 25/24
L;L;L;L;s;L;L;L;L;L;L;s;
stack()
22@

7-limit diachrome

7-limit diachrome, an aberrismic scale, is constructed by taking a 6+6 (for achiral diachrome) or 7+5 (for chiral diachrome) fifth chain structure and tempering out 5120/5103. The scales are shown below in 41edo tuning, but they work in any Aberschismic tuning such as 46edo and 53edo.

5sC

let L = 10/9
let m = 256/243
let s = 81/80
L;s;L;s;L;m;s;L;s;L;s;m;
stack()
41@

5sL

let L = 10/9
let m = 256/243
let s = 81/80
L;s;L;s;L;m;L;s;L;s;m;
stack()
41@

5sR

let L = 10/9
let m = 256/243
let s = 81/80
L;m;s;L;s;L;s;L;m;s;L;s;
stack()
41@

Aberschismic whitedye

Aberschismic whitedye is constructed by taking a diatonic scale and offsetting it by 64/63~81/80, tempering out 5120/5103.

Shown below in 41edo tuning. 

let L = 10/9
let m = 28/27
let s = 81/80
L;s;L;s;L;s;m;s;L;s;L;s;m;s;
stack()
41@
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