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== Subgroup temperaments ==
== Subgroup temperaments ==
=== Barbados ===
=== 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 of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
Subgroup: 2.3.13/5
Subgroup: 2.3.13/5


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


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 }}, {{val| 0 2 3 }}]


[[POTE generator]]: ~15/13 = 248.621
[[POTE generator]]: ~15/13 = 248.621


[[Sval]] [[map]]: [<1 0 -1|, <0 2 3|]
{{Val list|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}


EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
[[Badness]]: 0.002335


Badness: 0.002335
; 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]


; Music
* [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 ===
=== Pinkan ===
Subgroup: 2.3.13/5.19/5
Commas: 676/675, 1216/1215


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.
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.


The combination of 676/675 and 1216/1215 also implies yet another essential tempering comma of [[1521/1520]].
The combination of 676/675 and 1216/1215 also implies yet another essential tempering comma of [[1521/1520]].
Subgroup: 2.3.13/5.19/5
[[Comma list]]: 676/675, 1216/1215
[[Sval]] [[mapping]]: [{{val| 1 0 -1 -7 }}, {{val| 0 2 3 10 }}]


[[POTE generator]]: ~15/13 = 248.868
[[POTE generator]]: ~15/13 = 248.868


[[Sval]] [[map]]: [<1 0 -1 -7|, <0 2 3 10|]
{{Val list|legend=1| 5, 24, 29, 53, 82, 111, 135 }}
 
Badness: ?


EDOs: 5, 24, 29, 53, 82, 111, 135
=== Trinidad aka cata ===
{{Main| Catakleismic }}


Badness: ?
Trinidad may be viewed as the reduction of [[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.


=== Trinidad ===
Subgroup: 2.3.5.13
Subgroup: 2.3.5.13


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


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.
[[Sval]] [[mapping]]: [{{val| 1 0 1 0 }}, {{val| 0 6 5 14 }}]


[[POTE generator]]: 317.076
[[POTE generator]]: ~6/5 = 317.076


[[Sval]] [[map]]: [<1 0 1 0 |, <0 6 5 14|]
{{Val list|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}


EDOs: 15, 19, 34, 53, 87, 140, 193, 246
=== Tobago ===
{{See also| Chromatic pairs #Tobago }}


=== [[Chromatic pairs #Tobago|Tobago]] ===
Subgroup: 2.3.11.13/5
Subgroup: 2.3.11.13/5


Commas: 243/242, 676/675
Comma list: 243/242, 676/675
 
[[Sval]] [[mapping]]: [{{val| 2 0 -1 -2 }}, {{val| 0 2 5 3 }}]


POT2 generator: ~15/13 = 249.312
POT2 generator: ~15/13 = 249.312


Map: [<2 0 -1 -2], <0 2 5 3]]
{{Val list|legend=1| 10, 14, 24, 58, 82, 130 }}


EDOs: 10, 14, 24, 58, 82, 130
=== Parizekmic ===
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.


=== Parizekmic ===
Subgroup: 2.3.5.13
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]] [[map]]: [&lt;1 0 0 -1|, &lt;0 2 0 3|, &lt;0 0 1 1|]
[[Sval]] [[map]]: [&lt;1 0 0 -1|, &lt;0 2 0 3|, &lt;0 0 1 1|]


EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270
{{Val list|legend=1| 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270 }}


; Music
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