7-limit

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:

• 5-limit
• 2.3.7 subgroup
• 2.5.7 subgroup
• 3.5.7 subgroup

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. 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] = RH zarlino by 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] = LH zarlino by 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

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; 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

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@