The Archipelago

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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

Optimal patent val: 940edo


Rank four temperaments

1001/1000

Commas: 676/675, 1001/1000

EDOs: 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, 1716/1715

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

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, 1001/1000

EDOs: 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, 3025/3024

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

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, 625/624

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

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, 4225/4224

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

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, 676/675

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

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, 4225/4224

POTE 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, 4096/4095

POTE 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: 2.3.13/5

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 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


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.