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{{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.


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. 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.
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{{val| 0 0 0 0 1 0 }}
{{val| 0 0 0 0 1 0 }}


{{Val list|legend=1| 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940 }}
{{Optimal ET sequence|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 34d, 43, 49f, 53, 58, 72, 87, 111, 121, 130, 183, 198, 270, 940, 1210f }}


[[Optimal patent val]]: [[940edo]]
[[Optimal patent val]]: [[940edo|940]]


== Rank-4 temperaments ==
== Rank-4 temperaments ==
=== 1001/1000 ===
=== 1001/1000 ===
Commas: 676/675, 1001/1000
[[Subgroup]]: 2.3.5.7.11.13


Map: [<1 0 0 0 4 -1|, <0 2 0 0 -3 3|, <0 0 1 0 2 1|, <0 0 0 1 -1 0|]
[[Comma list]]: 676/675, 1001/1000


EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940
[[Mapping]]: [{{val| 1 0 0 0 4 -1 }}, {{val| 0 2 0 0 -3 3 }}, {{val| 0 0 1 0 2 1 }}, {{val| 0 0 0 1 -1 0 }}]


[[Optimal_patent_val|Optimal patent val]]: [[940edo]]
{{Optimal ET sequence|legend=1| 14cf, 15, 19, 29, 39df, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940, 1210f }}


=== 49/48 ===
=== 49/48 ===
Commas: 49/48, 91/90
[[Subgroup]]: 2.3.5.7.11.13


Map: [<1 0 0 2 0 -1|, <0 2 0 1 0 3|, <0 0 1 0 0 1|, <0 0 0 0 1 0|]
[[Comma list]]: 49/48, 91/90


EDOs: 5, 9, 10, 15, 19, 24
[[Mapping]]: [{{val| 1 0 0 2 0 -1 }}, {{val| 0 2 0 1 0 3 }}, {{val| 0 0 1 0 0 1 }}, {{val| 0 0 0 0 1 0 }}]
 
{{Optimal ET sequence|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff }}


=== 1716/1715 ===
=== 1716/1715 ===
Commas: 676/675, 1716/1715
[[Subgroup]]: 2.3.5.7.11.13


Map: [<1 0 0 0 -1 -1|, <0 2 0 0 -5 3|, <0 0 1 0 0 1|, <0 0 0 1 3 0|]
[[Comma list]]: 676/675, 1716/1715


EDOs: 58, 72, 77, 121, 130, 140, 149, 198, 212, 270
[[Mapping]]: [{{val| 1 0 0 0 -1 -1 }}, {{val| 0 2 0 0 -5 3 }}, {{val| 0 0 1 0 0 1 }}, {{val| 0 0 0 1 3 0 }}]
 
{{Optimal ET sequence|legend=1| 58, 72, 121, 130, 193, 198, 270, 940, 1210f }}


=== 364/363 ===
=== 364/363 ===
Commas: 364/363, 676/675
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 364/363, 676/675


Map: [<1 0 0 -1 0 -1|, <0 2 0 1 1 3|, <0 0 1 1 1 1|, <0 0 0 2 1 0|]
[[Mapping]]: [{{val| 1 0 0 -1 0 -1 }}, {{val| 0 2 0 1 1 3 }}, {{val| 0 0 1 1 1 1 }}, {{val| 0 0 0 2 1 0 }}]


EDOs: 9, 15, 29, 43, 58, 72, 87, 121, 130
{{Optimal ET sequence|legend=1| 14cf, 15, 23deff, 24, 29, 34d, 43, 49f, 58, 72, 87, 121, 130, 193, 217, 289, 338e, 410e }}


=== 351/350 ===
=== 351/350 ===
Commas: 351/350, 676/675
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 351/350, 676/675


Map: [<1 0 0 -2 0 -1|, <0 2 0 9 0 3|, <0 0 1 -1 0 1|, <0 0 0 0 1 0|]
[[Mapping]]: [{{val| 1 0 0 -2 0 -1 }}, {{val| 0 2 0 9 0 3 }}, {{val| 0 0 1 -1 0 1 }}, {{val| 0 0 0 0 1 0 }}]


EDOs: 19, 53, 58, 72, 77, 111, 130
{{Optimal ET sequence|legend=1| 14cf, 19, 24, 34d, 53, 58, 72, 111, 130, 183, 313, 462f }}


=== 352/351 ===
=== 352/351 ===
Commas: 352/351, 676/675
[[Subgroup]]: 2.3.5.7.11.13


Map: [<1 0 0 0 -6 -1|, <0 2 0 0 9 3|, <0 0 1 0 1 1|, <0 0 0 1 0 0|]
[[Comma list]]: 352/351, 676/675


EDOs: 29, 34, 53, 58, 63, 77, 87, 111, 121
[[Mapping]]: [{{val| 1 0 0 0 -6 -1 }}, {{val| 0 2 0 0 9 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 0 0 }}]
 
{{Optimal ET sequence|legend=1| 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf }}


=== 540/539 ===
=== 540/539 ===
Commas: 540/539, 676/675
[[Subgroup]]: 2.3.5.7.11.13


Map: [<1 0 0 0 2 -1|, <0 2 0 0 6 3|, <0 0 1 0 1 1|, <0 0 0 1 -2 0|]
[[Comma list]]: 540/539, 676/675


EDOs: 9, 19, 53, 58, 63, 72, 111, 121, 183
[[Mapping]]: [{{val| 1 0 0 0 2 -1 }}, {{val| 0 2 0 0 6 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 -2 0 }}]
 
{{Optimal ET sequence|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49f, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}


=== 847/845 ===
=== 847/845 ===
Commas: 676/675, 847/845
[[Subgroup]]: 2.3.5.7.11.13


Map: [<1 0 0 0 -1 -1|, <0 2 0 0 3 3|, <0 0 1 0 1 1|, <0 0 0 2 -1 0|]
[[Comma list]]: 676/675, 847/845


EDOs: 9, 29, 53, 58, 87, 111, 140, 149, 198
[[Mapping]]: [{{val| 1 0 0 0 -1 -1 }}, {{val| 0 2 0 0 3 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 2 -1 0 }}]
 
{{Optimal ET sequence|legend=1| 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c }}


== Rank-3 temperaments ==
== Rank-3 temperaments ==
=== [[Breed_family|Greenland]] ===
Notable rank-3 temperaments of island include:
Commas: 676/675, 1001/1000, 1716/1715
 
* [[Greenland]] → [[Breed family #Greenland|Breed family]]
: +1001/1000, 1716/1715
* [[History (temperament)|History]] → [[Werckismic temperaments #History|Werckismic temperaments]]
: +364/363, 441/440
* [[Borneo]] → [[Lehmerismic temperaments #Borneo|Lehmerismic temperaments]]
: +1001/1000, 3025/3024
* [[Enlil|Enlil aka sumatra]] → [[Kleismic rank three family #Enlil|Kleismic rank-3 family]]
: +325/324, 385/384
* [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]]
: +351/350, 540/539
* [[Hagrid]] → [[Cataharry family #Hagrid|Cataharry family]]
: +243/242, 351/350
* [[Baffin]] → [[Olympic clan #Baffin|Olympic clan]]
: +1001/1000, 4096/4095
* [[Kujuku]] → [[Pentacircle clan #Kujuku|Pentacircle clan]]
: +352/351, 364/363
 
== Rank-2 temperaments ==
Rank two temperaments tempering out 676/675 include the 13-limit versions of [[Ragismic microtemperaments #Hemiennealimmal|hemiennealimmal]], [[Breedsmic temperaments #Harry|harry]], [[Kleismic family #Tritikleismic|tritikleismic]], [[Kleismic family #Catakleismic|catakleimsic]], [[Marvel temperaments #Negri|negri]], [[Hemifamity temperaments #Mystery|mystery]], [[Hemifamity temperaments #Buzzard|buzzard]], [[Kleismic family #Quadritikleismic|quadritikleismic]].
 
It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 18 in hemiennealimmal, 6 in harry, 9 in tritikleismic, 3 in catakleismic, 2 in negri, 2 in buzzard, 12 in quadritikleismic. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to [[29edo]] alone.


Map: [<2 0 1 3 7 -1|, <0 2 1 1 -2 4|, <0 0 2 1 3 2|]
=== Decitonic aka decoid ===
{{see also| Breedsmic temperaments #Decoid}}


Edos: 58, 72, 130, 198, 270, 940
Subgroup: 2.3.5.7.11.13


[[Optimal_patent_val|Optimal patent val]]: [[940edo|940edo]]
[[Comma list]]: 676/675, 1001/1000, 1716/1715, 4096/4095


Badness: 0.000433
[[Mapping]]: [{{val| 10 0 47 36 98 37 }}, {{val| 0 2 -3 -1 -8 0 }}]


[[Spectrum_of_a_temperament|Spectrum]]: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9
[[POTE generator]]: ~15/13 = 248.917


=== [[Werckismic_temperaments|History]] ===
{{Optimal ET sequence|legend=1| 130, 270, 940, 1210f }}
Commas: 364/363, 441/440, 1001/1000


EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289
[[Badness]]: 0.013475


[[Optimal_patent_val|Optimal patent val]]: [[289edo|289edo]]
=== Avicenna ===
{{see also| Landscape microtemperaments #Avicenna }}


Badness: 0.000540
Subgroup: 2.3.5.7.11.13


Spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
[[Comma list]]: 676/675, 1001/1000, 3025/3024, 4096/4095


=== Borneo ===
[[Mapping]]: [{{val| 3 2 8 16 9 8 }}, {{val| 0 8 -3 -22 4 9 }}]
Commas: 676/675, 1001/1000, 3025/3024


Map: [<3 0 0 4 8 -3|, <0 2 0 -4 1 3|, <0 0 1 2 0 1|]
[[CTE|CTE generator]]: ~13/12 = 137.777


EDOs: 15, 72, 87, 111, 159, 183, 198, 270
[[POTE generator]]: ~13/12 = 137.777


[[Optimal_patent_val|Optimal patent val]]: [[270edo|270edo]]
{{Optimal ET sequence|legend=1| 87, 183, 270 }}


Badness: 0.000549
[[Badness]]: 0.015557


Spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7
=== Tertiathirds ===
{{see also| Wizmic microtemperaments #Tertiathirds }}


=== Sumatra ===
Subgroup: 2.3.5.7.11.13
Commas: 325/324, 385/384, 625/624


EDOs: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299
[[Comma list]]: 676/675, 1716/1715, 3025/3024, 4225/4224


Optimal patent val: [[299edo|299edo]]
[[Mapping]]: [{{val| 1 -4 2 -6 -9 -5 }}, {{val| 0 52 3 82 116 81 }}]


Badness: 0.000680
[[POTE generator]]: ~14/13 = 128.8902


=== [[Cataharry_family|Madagascar]] ===
{{Optimal ET sequence|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
Commas: 351/350, 540/539, 676/675


EDOs: 19, 53, 58, 72, 111, 130, 183, 313
[[Badness]]: 0.019494


[[Optimal_patent_val|Optimal patent val]]: [[313edo|313edo]]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.000560
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224


Spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11
Mapping: [{{val| 1 -4 2 -6 -9 -5 -3 }}, {{val| 0 52 3 82 116 81 66 }}]


[[madagascar19]]
POTE generator: ~14/13 = 128.8912


=== Baffin ===
{{Optimal ET sequence|legend=1| 121, 149, 270 }}
Commas: 676/675, 1001/1000, 4225/4224


Map: [<1 0 0 13 -9 1|, <0 2 0 -7 4 3|, <0 0 1 -2 4 1|]
Badness: 0.019107


EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940
== Subgroup temperaments ==
=== Barbados ===
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.


[[Optimal_patent_val|Optimal patent val]]: [[940edo|940edo]]
[[Subgroup]]: 2.3.13/5


Badness: 0.000604
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}


Spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]


=== Kujuku ===
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
Commas: 352/351, 364/363, 676/675


Map: [<1 0 0 -13 -6 -1|, <0 2 0 17 9 3|, <0 0 1 1 1 1|]
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}


EDOs: 24, 29, 58, 87, 121, 145, 208, 266ef, 474bef
[[Badness]]: 0.002335


[[Optimal_patent_val|Optimal patent val]]: [[208edo|208edo]]
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]


Badness: 0.001060
==== Pinkan ====
Pinkan adds the [[19/10]] major seventh to the mix to form a fundamental over-5 tetrad of 10:13:15:19, whose bright, fruity and tropical sound might recall the idyllic landscapes of Pinkan Island and its namesake berry. By contrast, utonal takes on this chord, while still somewhat bright due to the bounding 19/10, have a more turbulent and "swirling" sound, recalling the whirlpools that surround the island. Given the added complexity involved in building its chords, Pinkan may benefit from a "constrained melody, free harmony" approach, where a scale of lower cardinality like (5 or 9) is used for melody, but resides within a larger gamut of tones (like 24 or 29) that allow for facile use of the expanded harmony.


Spectrum: 15/13, 4/3, 13/10, 9/8, 13/11, 15/11, 12/11, 11/9, 11/8, 14/11, 16/13, 16/15, 11/10, 13/12, 9/7, 5/4, 18/13, 7/6, 6/5, 8/7, 10/9, 14/13, 15/14, 7/5
The combination of 676/675 and 1216/1215 also implies yet another essential tempering comma of [[1521/1520]].


== Rank-2 temperaments ==
[[Subgroup]]: 2.3.13/5.19/5
Rank two temperaments tempering out 676/675 include the 13-limit versions of [[Ragismic_microtemperaments|hemiennealimmal]], [[Breedsmic_temperaments|harry]], [[Kleismic_family|tritikleismic]], [[Kleismic_family|catakleimsic]], [[Marvel_temperaments|negri]], [[Hemifamity_temperaments|mystery]], [[Hemifamity_temperaments|buzzard]], [[Kleismic_family|quadritikleismic]].


It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 18 in hemiennealimmal, 6 in harry, 9 in tritikleismic, 3 in catakleismic, 2 in negri, 2 in buzzard, 12 in quadritikleismic. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to [[29edo|29EDO]] alone.
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}, 1216/1215 = {{monzo| 6 -5 0 1 }}


=== Decitonic (aka Decoid) ===
[[Sval]] [[mapping]]: [{{val| 1 0 -1 -7 }}, {{val| 0 2 3 10 }}]
{{see also|Breedsmic temperaments#Decoid}}


[[Comma list]]: 676/675, 1001/1000, 1716/1715, 4225/4224
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.868


[[Mapping]]: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]
{{Optimal ET sequence|legend=1| 5, 24, 29, 53, 82, 111, 135 }}


[[POTE_tuning|POTE generator]]: ~15/13 = 248.917
[[Badness]]: ?


{{Val list|legend=1| 130, 270, 940, 1210f, 1480cf }}
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


[[Badness]]: 0.013475
Tobago uses the semioctave period. It can be described as the 10 & 14 temperament and is related to [[neutral]] and [[barbados]].  


=== Avicenna ===
[[Subgroup]]: 2.3.11.13/5
{{see also|Landscape microtemperaments#Avicenna}}


[[Comma list]]: 676/675, 1001/1000, 3025/3024, 4096/4095
[[Comma list]]: 243/242 = {{monzo| -1 5 -2 }}, 676/675 = {{monzo| 2 -3 0 2 }}


[[Mapping]]: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|]
[[Sval]] [[mapping]]: [{{val| 2 0 -1 -2 }}, {{val| 0 2 5 3 }}]


[[POTE_tuning|POTE generator]]: ~13/12 = 137.777
[[Gencom]] [[mapping]]: [{{val| 2 4 -2 0 9 2 }}, {{val| 0 -2 3/2 0 -5 -3/2 }}]


{{Val list|legend=1| 87, 183, 270 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]


[[Badness]]: 0.015557
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 249.312


=== Tertiathirds ===
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
{{see also|Wizmic microtemperaments#Tertiathirds}}


[[Comma list]]: 676/675, 1716/1715, 3025/3024, 4225/4224
[[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents


[[Mapping]]: [<1 -4 2 -6 -9 -5|, <0 52 3 82 116 81|]
==== Pakkanian hemipyth ====


[[POTE_tuning|POTE generator]]: ~14/13 = 128.8902
[[Subgroup]]: 2.3.11.13/5.17


{{Val list|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
[[Comma list]]: 221/220, 243/242, 289/288


[[Badness]]: 0.019494
{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}


==== 17-limit ====
[[Optimal tuning]]s:
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224
* [[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)


Mapping: [<1 -4 2 -6 -9 -5 -3|, <0 52 3 82 116 81 66|]
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}


POTE generator: ~14/13 = 128.8912
<nowiki>*</nowiki> wart for 13/5


Vals: {{Val list| 121, 149, 270 }}
=== Cata ===
{{Main| Catakleismic }}
{{See also| Kleismic family #Cata }}


Badness: 0.019107
Cata may be viewed as the [[restriction|"reduction"]] of [[catakleismic]] to the 2.3.5.13 subgroup. Another way to put it is that it is the rank-2 2.3.5.13 subgroup temperament tempering out 325/324, 625/624 and hence also 676/675.


== Subgroup temperaments ==
[[Subgroup]]: 2.3.5.13
=== Barbados ===
Subgroup: 2.3.13/5


Commas: 676/675
[[Comma list]]: 325/324, 625/624


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_subgroups|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|24edo]], [[29edo|29edo]], [[53edo|53edo]] and [[111edo|111edo]], with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
[[Sval]] [[mapping]]: [{{val| 1 0 1 0 }}, {{val| 0 6 5 14 }}]


[[POTE generator]]: ~15/13 = 248.621
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.076


[[Sval]] [[map]]: [&lt;1 0 -1|, &lt;0 2 3|]
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}


EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
[[Badness]]: 0.00394


Badness: 0.002335
=== Taylor ===
Taylor is the "reduction" of [[hemischis]] to the 2.3.5.13 subgroup, tempering out the [[schisma]] in addition to 676/675. It can be reasonably extended to include harmonic 19 like most schismic temperaments, but even better, the hemifourth may be interpreted as an octave-reduced harmonic 37 ([[37/32]]). The extension is dubbed ''dakota'' (not to be confused with [[595/594 #Temperaments|dakotismic and/or dakotic]]).  


; Music
[[Subgroup]]: 2.3.5.13
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 Desert Island Rain] in 313et tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]
=== Pinkan ===
Subgroup: 2.3.13/5.19/5


Commas: 676/675, 1216/1215
[[Comma list]]: 676/675, 32805/32768


Pinkan adds the 19/10 major seventh to the mix to form a fundamental over-5 tetrad of 10:13:15:19, whose bright, fruity and tropical sound might recall the idyllic landscapes of Pinkan Island and its namesake berry. By contrast, utonal takes on this chord sound dark and stormy, perhaps recalling the whirlpools that surround the island. Given the added complexity involved in building its chords, Pinkan may benefit from a "constrained melody, free harmony" approach, where a scale of lower cardinality like (5 or 9) is used for melody, but resides within a larger gamut of tones (like 24 or 29) that allow for facile use of the expanded harmony.
[[Sval]] [[mapping]]: [{{val| 1 0 15 14 }}, {{val| 0 2 -16 -13 }}]


[[POTE generator]]: ~15/13 = 248.868
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8331


[[Sval]] [[map]]: [&lt;1 0 -1 -7|, &lt;0 2 3 10|]
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}


EDOs: 5, 24, 29, 53, 82, 111, 135
[[Badness]]: 0.0100


Badness: ?
==== Dakota ====
[[Subgroup]]: 2.3.5.13.19


=== Trinidad ===
[[Comma list]]: 361/360, 513/512, 676/675
Subgroup: 2.3.5.13


Commas: 325/324, 625/624
[[Sval]] [[mapping]]: [{{val| 1 0 15 14 9 }}, {{val| 0 2 -16 -13 -6 }}]


Trinidad may be viewed as the reduction of [[Kleismic_family|catakleismic temperament]] to the 2.3.5.13 subgroup. Another way to put it is that it is the rank two 2.3.5.13 subgroup temperament tempering out 325/324, 625/624 and hence also 676/675.
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8199


[[POTE generator]]: 317.076
{{Optimal ET sequence|legend=1| 24, 29, 53, 130, 183, 236h, 289h }}


[[Sval]] [[map]]: [&lt;1 0 1 0 |, &lt;0 6 5 14|]
[[Badness]]: 0.00575


EDOs: 15, 19, 34, 53, 87, 140, 193, 246
===== 2.3.5.13.19.37 subgroup =====
[[Subgroup]]: 2.3.5.13.19.37


=== [[Chromatic pairs #Tobago|Tobago]] ===
[[Comma list]]: 361/360, 481/480, 513/512, 676/675
Subgroup: 2.3.11.13/5


Commas: 243/242, 676/675
[[Sval]] [[mapping]]: [{{val| 1 0 15 14 9 6 }}, {{val| 0 2 -16 -13 -6 -1 }}]


POT2 generator: ~15/13 = 249.312
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8187


Map: [<2 0 -1 -2], <0 2 5 3]]
{{Optimal ET sequence|legend=1| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}


EDOs: 10, 14, 24, 58, 82, 130
[[Badness]]: 0.00357


=== Parizekmic ===
=== Parizekmic ===
Subgroup: 2.3.5.13
Closely related to barbados temperament is parizekmic, the rank-3 2.3.5.13 subgroup temperament tempering out 676/675. This is generated by ~2, ~5, and ~15/13, where the minimax tuning makes 2 and 5 pure, and 15/13 sharp by sqrt (676/675), or 1.28145 cents. This is, in other words, the same sqrt (4/3) generator as the minimax tuning for barbados, and it gives parizekmic a just 5-limit, with barbados triads where the 13/10 is a cent flat.
 
[[Subgroup]]: 2.3.5.13


Commas: 676/675
[[Comma list]]: 676/675


Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. This is generated by 2, 5, and 15/13, where the minimax tuning makes 2 and 5 pure, and 15/13 sharp by sqrt(676/675), or 1.28145 cents. This is, in other words, the same sqrt(4/3) generator as the minimax tuning for barbados, and it gives parizekmic a just 5-limit, with barbados triads where the 13/10 is a cent flat.
[[Sval]] [[mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 0 3 }}, {{val| 0 0 1 1 }}]


[[Sval]] [[map]]: [&lt;1 0 0 -1|, &lt;0 2 0 3|, &lt;0 0 1 1|]
{{Optimal ET sequence|legend=1| 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270 }}


EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270
[[Badness]]: 0.00811 × 10<sup>-3</sup>


; Music
; Music
* [http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3 Petr's Pump], a comma pump based ditty in Parizekmic temperament.
* [http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3 ''Petr's Pump''], a comma pump based ditty in Parizekmic temperament.


[[Category:Regular temperament theory]]
[[Category:Commatic realms]]
[[Category:Commatic realm]]
[[Category:Island]]
[[Category:Island]]
[[Category:Listen]]
[[Category:Listen]]
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