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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stack() | stack() | ||
</pre> | </pre> | ||
=== | === Tetrachordal pental diatonic === | ||
A modified Zarlino, similar to Turkish Rast | A modified Zarlino, similar to Turkish Rast | ||
<pre> | <pre> | ||
let L = 9/8 | let L = 9/8 | ||
| Line 37: | Line 38: | ||
stack() | stack() | ||
</pre> | </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 === | === Aberrismic theory === | ||
:''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]] | :''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]] | ||
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()
