5-limit

Not to be confused with the 5-odd-limit.

The 5-limit (aka pental) consists of all ratios in just intonation whose ratios contain no prime factors greater than 5; for example 5/4, 16/15, and 81/80. The 5-limit is considered the basis of western harmony, with the ratios of the 5-odd-limit being considered consonances. The main consonant chords of the 5-limit are the 4:5:6 major triad and the 10:12:15 minor triad. The smallest EDOs that approximate these triads within 20 cents are 12, 15, 19, 22, etc.

The 5-limit includes the following odd harmonics below 256: 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 125, 135, 225, 243.

The 5-limit is associated with the 7-form, though this is a weaker match than the 2.3.7 subgroup and the 5-form. (The fundamental 5-limit diatonic temperament, Meantone, is not a 7-cluster temperament but rather is tuned optimally near a golden generator, unlike with Archy). The main other 5-limit form is the 12-form, as in Meantone[12], Augmented[12], Diminished[12], and Diaschismic[12].

Temperaments

The 5-limit is the unique full prime-limit in which a rank-2 temperament is defined by tempering out a single comma. Historical meantone temperament was chiefly 5-limit, hence the conception of a regular temperament as tempering out some set of commas.

A large number of temperaments are defined in the 5-limit. This is partially a result of the legacy of the Xenharmonic Wiki, wherein the 5-limit was considered the 'default' subgroup for tempering before extensions and other subgroup restrictions were considered. But also, due to its non-clustering characteristic, there are justifications for many more temperament archetypes in the 5-limit than in groups like 2.3.7.

Some important rank-2 5-limit temperaments:

• Porcupine
• Schismic
• Kleismic
• Meantone
• Diaschismic
• Magic
• Tetracot
• Würschmidt
• Negri
• Augmented
• Diminished

JI scales

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

Zarlino

Right-hand Zarlino:

let L = 9/8
let m = 10/9
let s = 16/15
L;m;s;L;m;L;s;
stack()

Left-hand Zarlino:

let L = 9/8
let m = 10/9
let s = 16/15
m;L;s;L;m;L;s;
stack()

Tetrachordal pental diatonic

A modified Zarlino, similar to Turkish Rast

let L = 9/8
let m = 10/9
let s = 16/15
L;m;s;L;L;m;s;
stack()

Minor version:

let L = 9/8
let m = 10/9
let s = 16/15
L;s;m;L;L;s;m;
stack()

Duodene

The duodene scale forms a 4 (in the 3 dimension) × 3 (in the 5 dimension) parallelogram in the 2.3.5 lattice. It is considered the canonical 5-limit JI detempering of 12edo.

16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1

Aberrismic theory

Main article: Aberrisma#Quasi-diatonic aberrismic scales

Blackdye

The fundamental pental aberrismic scale is blackdye.

let L = 10/9
let m = 16/15
let s = 81/80
s;L;m;L;s;L;m;L;s;L;
stack()

Pinedye

1sC (achiral):

let l = 10/9
let m = 27/25
let s = 81/80
l;l;s;l;l;m;l;m;
stack()

1sR (chiral):

let l = 10/9
let m = 27/25
let s = 81/80
l;l;m;l;l;m;l;s;
stack()

1sL (chiral):

let l = 10/9
let m = 27/25
let s = 81/80
l;l;m;l;l;s;l;m;
stack()