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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. [[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.
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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, 14cf, 15, 19, 24, 29, 34d, 43, 49f, 53, 58, 72, 87, 111, 121, 130, 183, 198, 270, 940, 1210f }}
{{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|940]]
[[Optimal patent val]]: [[940edo|940]]
Line 26: Line 27:
== Rank-4 temperaments ==
== Rank-4 temperaments ==
=== 1001/1000 ===
=== 1001/1000 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 676/675, 1001/1000
[[Comma list]]: 676/675, 1001/1000
Line 32: Line 33:
[[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 }}]
[[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 }}]


{{Val list|legend=1| 14cf, 15, 19, 29, 39df, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940, 1210f }}
{{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 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 49/48, 91/90
[[Comma list]]: 49/48, 91/90
Line 41: Line 42:
[[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 }}]
[[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 }}]


{{Val list|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff }}
{{Optimal ET sequence|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff }}


=== 1716/1715 ===
=== 1716/1715 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 676/675, 1716/1715
[[Comma list]]: 676/675, 1716/1715
Line 50: Line 51:
[[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 }}]
[[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 }}]


{{Val list|legend=1| 58, 72, 121, 130, 193, 198, 270, 940, 1210f }}
{{Optimal ET sequence|legend=1| 58, 72, 121, 130, 193, 198, 270, 940, 1210f }}


=== 364/363 ===
=== 364/363 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 364/363, 676/675
[[Comma list]]: 364/363, 676/675
Line 59: Line 60:
[[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 }}]
[[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 }}]


{{Val list|legend=1| 14cf, 15, 23deff, 24, 29, 34d, 43, 49f, 58, 72, 87, 121, 130, 193, 217, 289, 338e, 410e }}
{{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 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 351/350, 676/675
[[Comma list]]: 351/350, 676/675
Line 68: Line 69:
[[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 }}]
[[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 }}]


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


=== 352/351 ===
=== 352/351 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 352/351, 676/675
[[Comma list]]: 352/351, 676/675
Line 77: Line 78:
[[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 }}]
[[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 }}]


{{Val list|legend=1| 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf }}
{{Optimal ET sequence|legend=1| 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf }}


=== 540/539 ===
=== 540/539 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 540/539, 676/675
[[Comma list]]: 540/539, 676/675
Line 86: Line 87:
[[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 }}]
[[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 }}]


{{Val list|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49f, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
{{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 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 676/675, 847/845
[[Comma list]]: 676/675, 847/845
Line 95: Line 96:
[[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 }}]
[[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 }}]


{{Val list|legend=1| 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c }}
{{Optimal ET sequence|legend=1| 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c }}


== Rank-3 temperaments ==
== Rank-3 temperaments ==
Notable rank-3 temperaments of island include greenland, history, borneo, sumatra, madagascar, baffin, and kujuku, all shown below.
Notable rank-3 temperaments of island include:


=== Greenland ===
* [[Greenland]] → [[Breed family #Greenland|Breed family]]
{{See also| Breed family #Greenland }}
: +1001/1000, 1716/1715
* [[History (temperament)|History]] → [[Werckismic temperaments #History|Werckismic temperaments]]
: +364/363, 441/440
* [[Borneo]] → [[Landscape family #Borneo|Landscape family]]
: +1001/1000, 3025/3024
* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank-3 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]] → [[Vishdelismic clan #Baffin|Vishdelismic clan]]
: +1001/1000, 4096/4095
* [[Kujuku]] → [[Pentacircle clan #Kujuku|Pentacircle clan]]
: +352/351, 364/363
* [[Namaka]] → [[Aberschismic family #Namaka|Aberschismic family]]
: +352/351, 847/845


Subgroup: 2.3.5.7.11.13
== Rank-2 temperaments ==
Rank-2 temperaments tempering out 676/675 include the 13-limit versions of [[semaphoresmic clan #Negri|negri]], [[buzzardsmic clan #Buzzard|buzzard]], [[kleismic family #Catakleismic|catakleismic]], [[gravity family #Harry|harry]], [[kleismic family #Tritikleismic|tritikleismic]], [[quintosec family #Decoid|decoid]], [[kleismic family #Quadritikleismic|quadritikleismic]], [[landscape microtemperaments #Avicenna|avicenna]], [[septiennealimmal clan #Hemiennealimmal|hemiennealimmal]], [[wizmic microtemperaments #Tertiathirds|tertiathirds]], and [[aberschismic temperaments #Mystery|mystery]].


[[Comma list]]: 676/675, 1001/1000, 1716/1715
It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 2 in negri, 2 in buzzard, 3 in catakleismic, 6 in harry, 9 in tritikleismic, 10 in decoid, 12 in quadritikleismic, 12 in avicenna, 18 in hemiennealimmal, 26 in tertiathirds. 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.


[[Mapping]]: [{{val| 2 0 1 3 7 -1 }}, {{val| 0 2 1 1 -2 4 }}, {{val| 0 0 2 1 3 2 }}]
== Subgroup temperaments ==
=== 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.


{{Val list|legend=1| 58, 72, 130, 198, 270, 940, 1210f }}
[[Subgroup]]: 2.3.13/5


[[Badness]]: 0.433 × 10<sup>-3</sup>
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}


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


=== History ===
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
{{See also| Werckismic temperaments #History }}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}


[[Comma list]]: 364/363, 441/440, 676/675
[[Badness]]: 0.002335


[[Mapping]]: [{{val| 1 2 0 0 1 2 }}, {{val| 0 -6 0 7 2 -9 }}, {{val| 0 0 1 1 1 1 }}]
; 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]


{{Val list|legend=1| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}
==== 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.


[[Badness]]: 0.540 × 10<sup>-3</sup>
The combination of 676/675 and 1216/1215 also implies yet another essential tempering comma of [[1521/1520]].


[[Complexity 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
[[Subgroup]]: 2.3.13/5.19/5


=== Borneo ===
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}, 1216/1215 = {{monzo| 6 -5 0 1 }}
{{See also| Lehmerismic temperaments #Borneo }}


Subgroup: 2.3.5.7.11.13
[[Sval]] [[mapping]]: [{{val| 1 0 -1 -7 }}, {{val| 0 2 3 10 }}]


[[Comma list]]: 676/675, 1001/1000, 3025/3024
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.868


[[Mapping]]: [{{val| 3 0 0 4 8 -3 }}, {{val| 0 2 0 -4 1 3 }}, {{val| 0 0 1 2 0 1 }}]
{{Optimal ET sequence|legend=1| 5, 24, 29, 53, 82, 111, 135 }}


{{Val list|legend=1| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}
[[Badness]]: ?


[[Badness]]: 0.549 × 10<sup>-3</sup>
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


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


=== Sumatra ===
[[Subgroup]]: 2.3.11.13/5
{{See also| Kleismic rank three family #13-limit (aka sumatra) }}


Subgroup: 2.3.5.7.11.13
[[Comma list]]: 243/242 = {{monzo| -1 5 -2 }}, 676/675 = {{monzo| 2 -3 0 2 }}


[[Comma list]]: 325/324, 385/384, 625/624
[[Sval]] [[mapping]]: [{{val| 2 0 -1 -2 }}, {{val| 0 2 5 3 }}]


[[Mapping]]: [{{val| 1 0 1 0 6 0 }}, {{val| 0 6 5 0 1 14 }}, {{val| 0 0 0 1 -1 0 }}]
[[Gencom]] [[mapping]]: [{{val| 2 4 -2 0 9 2 }}, {{val| 0 -2 3/2 0 -5 -3/2 }}]


{{Val list|legend=1| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]


[[Badness]]: 0.680 × 10<sup>-3</sup>
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~55/39 = 1\2, ~15/13 = 249.312


=== Madagascar ===
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
{{See also| Cataharry family #Madagascar }}


Subgroup: 2.3.5.7.11.13
[[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents


[[Comma list]]: 351/350, 540/539, 676/675
==== Pakkanian hemipyth ====


[[Mapping]]: [{{val| 1 0 0 -2 6 -1 }}, {{val| 0 2 0 9 -12 3 }}, {{val| 0 0 1 -1 3 1 }}]
[[Subgroup]]: 2.3.11.13/5.17


{{Val list|legend=1| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}
[[Comma list]]: 221/220, 243/242, 289/288


[[Badness]]: 0.560 × 10<sup>-3</sup>
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}


[[Complexity 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
[[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)


Scales: [[madagascar19]]
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}


=== Baffin ===
<nowiki>*</nowiki> wart for 13/5
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 676/675, 1001/1000, 4096/4095
=== Cata ===
{{Main| Catakleismic }}
{{See also| Kleismic family #Cata }}


[[Mapping]]: [{{val| 1 0 0 13 -9 1 }}, {{val| 0 2 0 -7 4 3 }}, {{val| 0 0 1 -2 4 1 }}]
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.


{{Val list|legend=1| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}
[[Subgroup]]: 2.3.5.13


[[Badness]]: 0.604 × 10<sup>-3</sup>
[[Comma list]]: 325/324, 625/624


[[Complexity 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 0 }}, {{val| 0 6 5 14 }}]
 
=== Kujuku ===
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 352/351, 364/363, 676/675
 
[[Mapping]]: [{{val| 1 0 0 -13 -6 -1 }}, {{val| 0 2 0 17 9 3 }}, {{val| 0 0 1 1 1 1 }}]
 
{{Val list|legend=1| 24, 29, 58, 87, 121, 145, 208, 266ef, 474bef }}
 
[[Badness]]: 1.060 × 10<sup>-3</sup>
 
[[Complexity 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
 
== 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.


=== Decitonic aka decoid ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.076
{{see also| Breedsmic temperaments #Decoid}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}


[[Comma list]]: 676/675, 1001/1000, 1716/1715, 4096/4095
[[Badness]]: 0.00394


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


[[POTE generator]]: ~15/13 = 248.917
[[Subgroup]]: 2.3.5.13


{{Val list|legend=1| 130, 270, 940, 1210f }}
[[Comma list]]: 676/675, 32805/32768


[[Badness]]: 0.013475
[[Sval]] [[mapping]]: [{{val| 1 0 15 14 }}, {{val| 0 2 -16 -13 }}]


=== Avicenna ===
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8331
{{see also| Landscape microtemperaments #Avicenna }}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}


[[Comma list]]: 676/675, 1001/1000, 3025/3024, 4096/4095
[[Badness]]: 0.0100


[[Mapping]]: [{{val| 3 2 8 16 9 8 }}, {{val| 0 8 -3 -22 4 9 }}]
==== Dakota ====
[[Subgroup]]: 2.3.5.13.19


[[POTE generator]]: ~13/12 = 137.777
[[Comma list]]: 361/360, 513/512, 676/675


{{Val list|legend=1| 87, 183, 270 }}
[[Sval]] [[mapping]]: [{{val| 1 0 15 14 9 }}, {{val| 0 2 -16 -13 -6 }}]


[[Badness]]: 0.015557
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8199


=== Tertiathirds ===
{{Optimal ET sequence|legend=1| 24, 29, 53, 130, 183, 236h, 289h }}
{{see also| Wizmic microtemperaments #Tertiathirds }}


Subgroup: 2.3.5.7.11.13
[[Badness]]: 0.00575


[[Comma list]]: 676/675, 1716/1715, 3025/3024, 4225/4224
===== 2.3.5.13.19.37 subgroup =====
[[Subgroup]]: 2.3.5.13.19.37


[[Mapping]]: [{{val| 1 -4 2 -6 -9 -5 }}, {{val| 0 52 3 82 116 81 }}]
[[Comma list]]: 361/360, 481/480, 513/512, 676/675


[[POTE generator]]: ~14/13 = 128.8902
[[Sval]] [[mapping]]: [{{val| 1 0 15 14 9 6 }}, {{val| 0 2 -16 -13 -6 -1 }}]


{{Val list|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8187


[[Badness]]: 0.019494
{{Optimal ET sequence|legend=1| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}


==== 17-limit ====
[[Badness]]: 0.00357
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224
 
Mapping: [{{val| 1 -4 2 -6 -9 -5 -3 }}, {{val| 0 52 3 82 116 81 66 }}]
 
POTE generator: ~14/13 = 128.8912
 
Vals: {{Val list| 121, 149, 270 }}
 
Badness: 0.019107
 
== 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 of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
 
Subgroup: 2.3.13/5
 
[[Comma list]]: 676/675
 
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
 
[[POTE generator]]: ~15/13 = 248.621
 
{{Val list|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
 
[[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]
 
=== 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]].
 
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
 
{{Val list|legend=1| 5, 24, 29, 53, 82, 111, 135 }}
 
Badness: ?
 
=== Trinidad aka cata ===
{{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.
 
Subgroup: 2.3.5.13
 
[[Comma list]]: 325/324, 625/624
 
[[Sval]] [[mapping]]: [{{val| 1 0 1 0 }}, {{val| 0 6 5 14 }}]
 
[[POTE generator]]: ~6/5 = 317.076
 
{{Val list|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}
 
=== Tobago ===
{{See also| Chromatic pairs #Tobago }}
 
Subgroup: 2.3.11.13/5
 
Comma list: 243/242, 676/675
 
[[Sval]] [[mapping]]: [{{val| 2 0 -1 -2 }}, {{val| 0 2 5 3 }}]
 
POT2 generator: ~15/13 = 249.312
 
{{Val list|legend=1| 10, 14, 24, 58, 82, 130 }}


=== 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
Line 338: Line 267:
[[Sval]] [[mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 0 3 }}, {{val| 0 0 1 1 }}]
[[Sval]] [[mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 0 3 }}, {{val| 0 0 1 1 }}]


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