5-limit: Difference between revisions

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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 [[EDO]]s that approximate these triads within 20 cents are [[12edo|12]], [[15edo|15]], [[19edo|19]], [[22edo|22]], etc.
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 [[EDO]]s that approximate these triads within 20 cents are [[12edo|12]], [[15edo|15]], [[19edo|19]], [[22edo|22]], etc.


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].  
{{Adv|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 ==
== Temperaments ==
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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.
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.


== Aberrismic theory ==
== JI scales ==
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.
=== Zarlino ===
Right-hand Zarlino:
<pre>
let L = 9/8
let m = 10/9
let s = 16/15
L;m;s;L;m;L;s;
stack()
</pre>
Left-hand Zarlino:
<pre>
let L = 9/8
let m = 10/9
let s = 16/15
m;L;s;L;m;L;s;
stack()
</pre>
=== Tetrachordal pental diatonic ===
A modified Zarlino, similar to Turkish Rast
<pre>
let L = 9/8
let m = 10/9
let s = 16/15
L;m;s;L;L;m;s;
stack()
</pre>
Minor version:
<pre>
let L = 9/8
let m = 10/9
let s = 16/15
L;s;m;L;L;s;m;
stack()
</pre>
 
=== Aberrismic theory ===
:''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]]
:''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]]
==== Blackdye ====
The fundamental pental aberrismic scale is [[blackdye]].
The fundamental pental aberrismic scale is [[blackdye]].
<pre>
let L = 10/9
let m = 16/15
let s = 81/80
s;L;m;L;s;L;m;L;s;L;
stack()
</pre>
==== Pinedye ====
{{Cat|JI groups}}

Latest revision as of 10:54, 13 April 2026

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.

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()

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

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