The Archipelago
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.
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.
Rank five temperament
Island
Comma: 676/675
Map:
<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
Rank four temperaments
1001/1000
Commas: 676/675, 1001/1000
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|]
EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940
49/48
Commas: 49/48, 91/90
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|]
EDOs: 5, 9, 10, 15, 19, 24
1716/1715
Commas: 676/675, 1716/1715
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|]
EDOs: 58, 72, 77, 121, 130, 140, 149, 198, 212, 270
364/363
Commas: 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|]
EDOs: 9, 15, 29, 43, 58, 72, 87, 121, 130
351/350
Commas: 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|]
EDOs: 19, 53, 58, 72, 77, 111, 130
352/351
Commas: 352/351, 676/675
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|]
EDOs: 29, 34, 53, 58, 63, 77, 87, 111, 121
540/539
Commas: 540/539, 676/675
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|]
EDOs: 9, 19, 53, 58, 63, 72, 111, 121, 183
847/845
Commas: 676/675, 847/845
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|]
EDOs: 9, 29, 53, 58, 87, 111, 140, 149, 198
Rank three temperaments
Greenland
Commas: 676/675, 1001/1000, 1716/1715
Map: [<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
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, 1001/1000
EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289
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, 3025/3024
Map: [<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
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, 625/624
EDOs: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299
Optimal patent val: 299edo
Badness: 0.000680
Madagascar
Commas: 351/350, 540/539, 676/675
EDOs: 19, 53, 58, 72, 111, 130, 183, 313
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
Baffin
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|]
EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940
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, 676/675
Map: [<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
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 (aka Decoid)
Comma list: 676/675, 1001/1000, 1716/1715, 4225/4224
Mapping: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]
POTE generator: ~15/13 = 248.917
Badness: 0.013475
Avicenna
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
Mapping: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|]
POTE generator: ~13/12 = 137.777
Badness: 0.015557
Tertiathirds
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224
Mapping: [<1 -4 2 -6 -9 -5|, <0 52 3 82 116 81|]
POTE generator: ~14/13 = 128.8902
Badness: 0.019494
17-limit
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224
Mapping: [<1 -4 2 -6 -9 -5 -3|, <0 52 3 82 116 81 66|]
POTE generator: ~14/13 = 128.8912
Vals: Template:Val list
Badness: 0.019107
Subgroup temperaments
Barbados
Subgroup: 2.3.13/5
Commas: 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 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.
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: 2.3.5.13
Commas: 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
Subgroup: 2.3.11.13/5
Commas: 243/242, 676/675
POT2 generator: ~15/13 = 249.312
Map: [<2 0 -1 -2], <0 2 5 3]]
EDOs: 10, 14, 24, 58, 82, 130
Parizekmic
Subgroup: 2.3.5.13
Commas: 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|]
EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270
Music
- Petr's Pump, a comma pump based ditty in Parizekmic temperament.