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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

ultramajortriad, 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.

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.

## Parent Temperaments

## Island

Comma: 676/675Map:

<1 0 0 0 0 -1|

<0 2 0 0 0 3|

<0 0 1 0 0 1|

<0 0 0 1 0 0|

<0 0 0 0 1 0|

EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940

Optimal patent val: 940edo

## Barbados

Subgroup: 2.3.13/5Commas: 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 intontation subgroup. The minimax tuning for this makes the generator 2/sqrt(3), or 249.0225 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.

POTE generator: ~15/13 = 248.621

Sval map: [<1 0 -1|, <0 2 3|]

EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362

Badness: 0.002335

## Rank four temperaments

## 1001/1000

Commas: 676/675, 1001/1000EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940

Optimal patent val: 940edo

## 49/48

Commas: 49/48, 91/90## 1716/1715

Commas: 676/675, 1716/1715## 364/363

Commas: 364/363, 676/675## 351/350

Commas: 351/350, 676/675## Rank three temperaments

## Greenland

Commas: 676/675, 1001/1000, 1716/1715Map: [<2 0 1 3 7 -1|, <0 2 1 1 -2 4|, <0 0 2 1 3 2|]

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

Optimal patent val: 940edo

Badness: 0.000433

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

## History

Commas: 364/363, 441/440, 1001/1000EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289

Optimal patent val: 289edo

Badness: 0.000540

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

## Borneo

Commas: 676/675, 1001/1000, 3025/3024Map: [<3 0 0 4 8 -3|, <0 2 0 -4 1 3|, <0 0 1 2 0 1|]

EDOs: 15, 72, 87, 111, 159, 183, 198, 270

Optimal patent val: 270edo

Badness: 0.000549

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

## Sumatra

Commas: 325/324, 385/384, 625/624EDOs: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299

Optimal patent val: 299edo

Badness: 0.000680

## Madagascar

Commas: 351/350, 540/539, 676/675EDOs: 19, 53, 58, 72, 111, 130, 183, 313

Optimal patent val: 313edo

Badness: 0.000560

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

madagascar19

## Baffin

Commas: 676/675, 1001/1000, 4225/4224Map: [<1 0 0 13 -9 1|, <0 2 0 -7 4 3|, <0 0 1 -2 4 1|]

EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940

Optimal patent val: 940edo

Badness: 0.000604

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

## Kujuku

Commas: 352/351, 364/363, 676/675Map: [<1 0 0 -13 -6 -1|, <0 2 0 17 9 3|, <0 0 1 1 1 1|]

EDOs: 24, 29, 58, 87, 121, 145, 208, 266ef, 474bef

Optimal patent val: 208edo

Badness: 0.001060

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 two temperaments

Rank two temperaments tempering out 676/675 include the 13-limit versions of hemiennealimmal, harry, tritikleismic, catakleimsic, negri, mystery, buzzard, 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

Commas: 676/675, 1001/1000, 1716/1715, 4225/4224POTE generator: ~15/13 = 248.917

Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]

EDOs: 130, 270, 940, 1480

Badness: 0.0135

## Avicenna

Commas: 676/675, 1001/1000, 3025/3024, 4096/4095POTE generator: ~13/12 = 137.777

Map: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|]

EDOs: 87, 183, 270

Badness: 0.0156

## Subgroup temperaments

## Barbados

Subgroup: 2.3.13/5Commas: 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 intontation subgroup. The minimax tuning for this makes the generator 2/sqrt(3), or 249.0225 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.

POTE generator: ~15/13 = 248.621

Sval map: [<1 0 -1|, <0 2 3|]

EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362

Badness: 0.002335

## Music

Desert Island Rain in 313et tuned Barbados[9], by Sevish## Trinidad

Subgroup: 2.3.5.13Commas: 325/324, 625/624

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.

POTE generator: 317.076

Sval map: [<1 0 1 0 |, <0 6 5 14|]

EDOs: 15, 19, 34, 53, 87, 140, 193, 246

## Tobago

## Parizekmic

Subgroup: 2.3.5.13Commas: 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

<1 0 0 -1|

<0 2 0 3|

<0 0 1 1|

## Music

Petr's Pump, a comma pump based ditty in Parizekmic temperament.EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270