5-limit - Revision history

Inthar at 10:54, 13 April 2026

2026-04-13T10:54:37Z

← Older revision		Revision as of 10:54, 13 April 2026
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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 includes the following odd harmonics below 256: 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 125, 135, 225, 243.		{{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].		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].

Inthar at 01:01, 13 April 2026

2026-04-13T01:01:56Z

← Older revision		Revision as of 01:01, 13 April 2026
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	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 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].		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 ==

Inthar at 01:01, 13 April 2026

2026-04-13T01:01:43Z

← Older revision		Revision as of 01:01, 13 April 2026
Line 1:		Line 1:
	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 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].		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].

Inthar: /* Tetrachordal pental diatonic */

2026-04-12T14:30:57Z

Tetrachordal pental diatonic

← Older revision		Revision as of 14:30, 12 April 2026
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	stack()		stack()
	</pre>		</pre>
			Minor version:
	<pre>		<pre>
	let L = 9/8		let L = 9/8

Inthar: /* Tetrachordal pental diatonic */

2026-04-12T14:30:45Z

Tetrachordal pental diatonic

← Older revision		Revision as of 14:30, 12 April 2026
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	let s = 16/15		let s = 16/15
	L;m;s;L;L;m;s;		L;m;s;L;L;m;s;
			stack()
			</pre>

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

Inthar: /* Dunno what to call this */

2026-04-12T14:29:57Z

Dunno what to call this

← Older revision		Revision as of 14:29, 12 April 2026
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	stack()		stack()
	</pre>		</pre>
	=== ~~Dunno what to call this~~ ===		=== Tetrachordal pental diatonic ===
	A modified Zarlino, similar to Turkish Rast		A modified Zarlino, similar to Turkish Rast
	<pre>		<pre>

Inthar: /* Dunno what to call this */

2026-04-12T09:40:57Z

Dunno what to call this

← Older revision		Revision as of 09:40, 12 April 2026
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	=== Dunno what to call this ===		=== Dunno what to call this ===
	A modified Zarlino, similar to Turkish Rast		A modified Zarlino, similar to Turkish Rast
	~~Right-hand Zarlino:~~
	<pre>		<pre>
	let L = 9/8		let L = 9/8
Line 37:		Line 36:
	stack()		stack()
	</pre>		</pre>

	=== Aberrismic theory ===		=== Aberrismic theory ===
	:''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]]		:''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]]

Inthar: /* Aberrismic theory */

2026-04-12T09:40:44Z

Aberrismic theory

← Older revision		Revision as of 09:40, 12 April 2026
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	=== Aberrismic theory ===		=== 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>		<pre>
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	stack()		stack()
	</pre>		</pre>
			==== Pinedye ====

	{{Cat\|JI groups}}		{{Cat\|JI groups}}

Inthar: /* JI scales */

2026-04-12T09:39:39Z

JI scales

@@ Line 9: / Line 9: @@
 == JI scales ==
 === Aberrismic theory ===
 :''Main article: [[Aberrisma#Quasi-diatonic aberrismic scales]]
 The fundamental pental aberrismic scale is [[blackdye]].
 {{Cat|JI groups}}

Inthar: /* Aberrismic theory */

2026-03-12T04:20:42Z

Aberrismic theory

← Older revision		Revision as of 04:20, 12 March 2026
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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 ==
			=== 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}}		{{Cat\|JI groups}}