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The Archipelago: Difference between revisions

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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.
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


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


[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]


[[POTE generator]]: ~15/13 = 248.621
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621


{{Val list|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
{{Val list|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
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; Music
; 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]
* [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]
 
=== 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.
==== 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.


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
[[Subgroup]]: 2.3.13/5.19/5


[[Comma list]]: 676/675, 1216/1215
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}, 1216/1215 = {{monzo| 6 -5 0 1 }}


[[Sval]] [[mapping]]: [{{val| 1 0 -1 -7 }}, {{val| 0 2 3 10 }}]
[[Sval]] [[mapping]]: [{{val| 1 0 -1 -7 }}, {{val| 0 2 3 10 }}]


[[POTE generator]]: ~15/13 = 248.868
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.868


{{Val list|legend=1| 5, 24, 29, 53, 82, 111, 135 }}
{{Val list|legend=1| 5, 24, 29, 53, 82, 111, 135 }}
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Badness: ?
Badness: ?


=== Trinidad aka cata ===
=== Cata ===
{{Main| Catakleismic }}
{{Main| Catakleismic }}


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.
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: 2.3.5.13
[[Subgroup]]: 2.3.5.13


[[Comma list]]: 325/324, 625/624
[[Comma list]]: 325/324, 625/624
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[[Sval]] [[mapping]]: [{{val| 1 0 1 0 }}, {{val| 0 6 5 14 }}]
[[Sval]] [[mapping]]: [{{val| 1 0 1 0 }}, {{val| 0 6 5 14 }}]


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


{{Val list|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}
{{Val list|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}
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=== Tobago ===
=== Tobago ===
{{See also| Chromatic pairs #Tobago }}
{{See also| Chromatic pairs #Tobago }}
Tobago uses the semioctave period. It can be described as the 10 & 14 temperament and is related to [[neutral]] and [[barbados]].


Subgroup: 2.3.11.13/5
Subgroup: 2.3.11.13/5
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Comma list: 243/242, 676/675
Comma list: 243/242, 676/675


Sval name: 10&14
[[Sval]] [[mapping]]: [{{val| 2 0 -1 -2 }}, {{val| 0 2 5 3 }}]


Related temperaments: [[Chromatic_pairs#Neutral|neutral]], [[The_Archipelago#Barbados|barbados]]
[[Gencom]] [[mapping]]: [{{val| 2 4 -2 0 9 2 }}, {{val| 0 -2 3/2 0 -5 -3/2 }}]


[[Tp_tuning|POT2 generator]]: ~15/13 = 249.312
: [[gencom]]: [55/39 15/13; 243/242 676/675]


[[Gencom|Gencom]]: [55/39 15/13; 243/242 676/675]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 249.312
 
Gencom mapping: [<2 4 -2 0 9 2|, <0 -2 3/2 0 -5 -3/2|]
 
[[Sval]] [[mapping]]: [{{val| 2 0 -1 -2 }}, {{val| 0 2 5 3 }}]


{{Val list|legend=1| 10, 14, 24, 58, 82, 130 }}
{{Val list|legend=1| 10, 14, 24, 58, 82, 130 }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.3533 cents
[[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents


=== Parizekmic ===
=== 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.
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
[[Subgroup]]: 2.3.5.13


[[Comma list]]: 676/675
[[Comma list]]: 676/675
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; 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:Commatic realms]]
[[Category:Commatic realms]]
[[Category:Island]]
[[Category:Island]]
[[Category:Listen]]
[[Category:Listen]]
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