View source for 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]]

{{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 ==
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 ===
<pre>
8:9:10:12:14:16
</pre>
The simplest full 7-limit JI scale.

=== /2 Mixolydian ===
<pre>
16:18:20:21:24:27:28:32
</pre>

=== /2 Ionian ===
<pre>
16:18:20:21:24:27:30:32
</pre>
=== 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 [[interleaving|interleaved]] 5-limit [[zarlino]] copies. Note that Zil[14]'s lexicographical chirality label is the opposite of the lexicographical chirality of the Zarlino versions used.

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

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

==== Zil[24] ====
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>

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

LH
<pre>
16/15; 8/7; 6/5; 4/3; 48/35; 3/2; 8/5; 12/7; 64/35; 2/1
</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>

=== 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>

== 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]].
<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]] 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, 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}}