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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, [[ | 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]. | ||
== Aberrismic theory == | == 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 == | |||
=== Aberrismic theory === | |||
:''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]] | :''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]] | ||
The fundamental pental aberrismic scale is [[blackdye]]. | The fundamental pental aberrismic scale is [[blackdye]]. | ||
{{Cat|JI groups}} | |||
Latest revision as of 04:20, 12 March 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 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
Aberrismic theory
- Main article: Aberrisma#Quasi-diatonic aberrismic scales
The fundamental pental aberrismic scale is blackdye.
