The Archipelago - Revision history

BudjarnLambeth: /* Tobago */ correction

2026-04-29T13:13:05Z

Tobago: correction

← Older revision		Revision as of 13:13, 29 April 2026
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	: [[gencom]]: [55/39 15/13; 243/242 676/675]		: [[gencom]]: [55/39 15/13; 243/242 676/675]

	[[Optimal tuning]] ([[Tp tuning\|subgroup POTE]]): ~2 = 1\1, ~15/13 = 249.312		[[Optimal tuning]] ([[Tp tuning\|subgroup POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

	{{Optimal ET sequence\|legend=1\| 10, 14, 24, 58, 82, 130 }}		{{Optimal ET sequence\|legend=1\| 10, 14, 24, 58, 82, 130 }}

Overthink: /* Barbados */ + see also

2025-12-22T08:18:54Z

Barbados: + see also

← Older revision		Revision as of 08:18, 22 December 2025
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	== Subgroup temperaments ==		== Subgroup temperaments ==
	=== Barbados ===		=== Barbados ===
			{{See also\|Extraclassical tonality}}
	Perhaps the simplest method of making use of the barbados triad and other characteristic island harmonies is to strip things down to essentials by tempering the 2.3.13/5 [[just intonation subgroup]]. The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.		Perhaps the simplest method of making use of the barbados triad and other characteristic island harmonies is to strip things down to essentials by tempering the 2.3.13/5 [[just intonation subgroup]]. The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

Tristanbay: /* Rank-3 temperaments */Added Namaka temperament

2025-09-16T07:35:33Z

Rank-3 temperaments: Added Namaka temperament

← Older revision		Revision as of 07:35, 16 September 2025
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	* [[Kujuku]] → [[Pentacircle clan #Kujuku\|Pentacircle clan]]		* [[Kujuku]] → [[Pentacircle clan #Kujuku\|Pentacircle clan]]
	: +352/351, 364/363		: +352/351, 364/363
			* [[Namaka]] → [[Hemifamity family#Namaka\|Hemifamity family]]
			: +352/351, 1001/1000

	== Rank-2 temperaments ==		== Rank-2 temperaments ==

FloraC: Fix linking

2025-08-27T13:33:16Z

Fix linking

← Older revision		Revision as of 13:33, 27 August 2025
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	* [[Borneo]] → [[Lehmerismic temperaments #Borneo\|Lehmerismic temperaments]]		* [[Borneo]] → [[Lehmerismic temperaments #Borneo\|Lehmerismic temperaments]]
	: +1001/1000, 3025/3024		: +1001/1000, 3025/3024
	* [[Enlil\|Enlil aka sumatra]] → [[Kleismic rank ~~three~~ family #Enlil\|Kleismic rank-3 family]]		* [[Enlil\|Enlil aka sumatra]] → [[Kleismic rank-3 family #Enlil\|Kleismic rank-3 family]]
	: +325/324, 385/384		: +325/324, 385/384
	* [[Madagascar]] → [[Cataharry family #Madagascar\|Cataharry family]]		* [[Madagascar]] → [[Cataharry family #Madagascar\|Cataharry family]]

Tristanbay: /* Rank-3 temperaments */ Added Hagrid to notable rank-3 temperaments

2025-08-04T18:49:49Z

Rank-3 temperaments: Added Hagrid to notable rank-3 temperaments

← Older revision		Revision as of 18:49, 4 August 2025
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	* [[Madagascar]] → [[Cataharry family #Madagascar\|Cataharry family]]		* [[Madagascar]] → [[Cataharry family #Madagascar\|Cataharry family]]
	: +351/350, 540/539		: +351/350, 540/539
			* [[Hagrid]] → [[Cataharry family #Hagrid\|Cataharry family]]
			: +243/242, 351/350
	* [[Baffin]] → [[Olympic clan #Baffin\|Olympic clan]]		* [[Baffin]] → [[Olympic clan #Baffin\|Olympic clan]]
	: +1001/1000, 4096/4095		: +1001/1000, 4096/4095

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:47:58Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:47, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.		The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.

BudjarnLambeth: {{Technical data page}}

2025-04-13T06:42:40Z

{{Technical data page}}<br><br>

← Older revision		Revision as of 06:42, 13 April 2025
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			{{Technical data page}}<br><br>
	The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.		The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.

Lériendil: added pakkanian hemipyth; should probably move links to the Subgroup temperaments#Barbados onto here

2024-12-20T05:08:04Z

added pakkanian hemipyth; should probably move links to the Subgroup temperaments#Barbados onto here

← Older revision		Revision as of 05:08, 20 December 2024
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	[[Tp tuning#T2 tuning\|RMS error]]: 0.3533 cents		[[Tp tuning#T2 tuning\|RMS error]]: 0.3533 cents

			==== Pakkanian hemipyth ====

			[[Subgroup]]: 2.3.11.13/5.17

			[[Comma list]]: 221/220, 243/242, 289/288

			{{Mapping\|legend=2\| 2 0 -1 -2 5 \| 0 2 5 3 2 }}

			[[Optimal tuning]]s:
			* [[Tp tuning\|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
			* [[Tp tuning\|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

			{{Optimal ET sequence\|legend=1\| 10, 14, 24, 106, 130, 154, 178, 202 }}

			<nowiki>*</nowiki> wart for 13/5

	=== Cata ===		=== Cata ===

BudjarnLambeth: /* Avicenna */ add cte generator

2024-11-08T13:36:07Z

Avicenna: add cte generator

← Older revision		Revision as of 13:36, 8 November 2024
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	[[Mapping]]: [{{val\| 3 2 8 16 9 8 }}, {{val\| 0 8 -3 -22 4 9 }}]		[[Mapping]]: [{{val\| 3 2 8 16 9 8 }}, {{val\| 0 8 -3 -22 4 9 }}]

			[[CTE\|CTE generator]]: ~13/12 = 137.777

	[[POTE generator]]: ~13/12 = 137.777		[[POTE generator]]: ~13/12 = 137.777

TallKite at 10:57, 20 October 2024

2024-10-20T10:57:34Z

← Older revision		Revision as of 10:57, 20 October 2024
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	The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.		The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.

	The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer ~~dyad~~, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an ''ultramajor'' triad, with a third sharper even than the 9/7 supermajor third.		The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer interval, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an ''ultramajor'' triad, with a third sharper even than the 9/7 supermajor third.

	Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains ~~dyads~~ that are on average higher in complexity (9/7 vs 13/10 and 7/6 vs 15/13). Its inverse, however, is the ultraminor 26:30:39, which is far more complex than the 7-limit subminor 6:7:9. [[The Biosphere\|Temperaments in which 91/90 vanishes]] equate the two types of triads.		Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains intervals that are on average higher in complexity (9/7 vs 13/10 and 7/6 vs 15/13). Its inverse, however, is the ultraminor 26:30:39, which is far more complex than the 7-limit subminor 6:7:9. [[The Biosphere\|Temperaments in which 91/90 vanishes]] equate the two types of triads.

	[[24edo]] approximates this triad to within an error of four cents, and [[29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.		[[24edo]] approximates this triad to within an error of four cents, and [[29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.