5-limit - Revision history

Lériendil at 00:34, 14 May 2026

2026-05-14T00:34:44Z

← Older revision		Revision as of 00:34, 14 May 2026
Line 1:		Line 1:
	:''Not to be confused with the [[5-odd-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 [[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. In that framework, the main consonances of the 5-limit are the opposing pair of 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.

	{{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.}}		{{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.}}
Line 10:		Line 10:
	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.		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 subgroup\|2.3.7]].		A large number of temperaments are defined basally 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 subgroup\|2.3.7]].

	Some important rank-2 5-limit temperaments:		Some important rank-2 5-limit temperaments:

Inthar: /* Aberrismic theory */

2026-04-15T13:08:28Z

Aberrismic theory

@@ Line 91: / Line 91: @@
 </pre>
 ==== Pinedye ====
 {{Cat|JI groups}}

Inthar: /* Duodene */

2026-04-15T00:45:33Z

Duodene

← Older revision		Revision as of 00:45, 15 April 2026
Line 63:		Line 63:

	=== Duodene ===		=== Duodene ===
	The duodene scale forms a 4 (in the 3 dimension) × 3 (in the 5 dimension) parallelogram in the 2.3.5 lattice.		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.
	<pre>		<pre>
	16/15		16/15

Inthar: /* Duodene */

2026-04-15T00:44:50Z

Duodene

← Older revision		Revision as of 00:44, 15 April 2026
Line 63:		Line 63:

	=== Duodene ===		=== Duodene ===
			The duodene scale forms a 4 (in the 3 dimension) × 3 (in the 5 dimension) parallelogram in the 2.3.5 lattice.
	<pre>		<pre>
	16/15		16/15

Inthar: /* Tetrachordal pental diatonic */

2026-04-15T00:44:05Z

Tetrachordal pental diatonic

@@ Line 60: / Line 60: @@
 L;s;m;L;L;s;m;
 stack()
 </pre>

Inthar at 18:37, 13 April 2026

2026-04-13T18:37:36Z

← Older revision		Revision as of 18:37, 13 April 2026
Line 1:		Line 1:
			:''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 [[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.

Inthar: /* Temperaments */

2026-04-13T18:36:08Z

Temperaments

← Older revision		Revision as of 18:36, 13 April 2026
Line 8:		Line 8:
	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.		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.		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 subgroup\|2.3.7]].

	Some important rank-2 5-limit temperaments:		Some important rank-2 5-limit temperaments:

Inthar: /* Temperaments */

2026-04-13T18:35:44Z

Temperaments

← Older revision		Revision as of 18:35, 13 April 2026
Line 20:		Line 20:
	* [[Würschmidt]]		* [[Würschmidt]]
	* [[Negri]]		* [[Negri]]
	* [[Augmented]]		* [[Augmented (temperament)\|Augmented]]
	* [[Diminished]]		* [[Diminished (temperament)\|Diminished]]

	== JI scales ==		== JI scales ==

Inthar: /* Temperaments */

2026-04-13T18:35:14Z

Temperaments

← Older revision		Revision as of 18:35, 13 April 2026
Line 19:		Line 19:
	* [[Tetracot (temperament)\|Tetracot]]		* [[Tetracot (temperament)\|Tetracot]]
	* [[Würschmidt]]		* [[Würschmidt]]
			* [[Negri]]
			* [[Augmented]]
			* [[Diminished]]

	== JI scales ==		== JI scales ==

Inthar: /* Temperaments */

2026-04-13T18:32:16Z

Temperaments

← Older revision		Revision as of 18:32, 13 April 2026
Line 17:		Line 17:
	* [[Diaschismic]]		* [[Diaschismic]]
	* [[Magic]]		* [[Magic]]
	* [[Tetracot]]		* [[Tetracot (temperament)\|Tetracot]]
	* [[Würschmidt]]		* [[Würschmidt]]