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 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 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. 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-5 temperaments
Island
Subgroup: 2.3.5.7.11.13
Mapping:
⟨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]
Optimal ET sequence: 5, 9, 10, 14cf, 15, 19, 24, 29, 34d, 43, 49f, 53, 58, 72, 87, 111, 121, 130, 183, 198, 270, 940, 1210f
Rank-4 temperaments
1001/1000
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000
Mapping: [⟨1 0 0 0 4 -1], ⟨0 2 0 0 -3 3], ⟨0 0 1 0 2 1], ⟨0 0 0 1 -1 0]]
Optimal ET sequence: 14cf, 15, 19, 29, 39df, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940, 1210f
49/48
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 91/90
Mapping: [⟨1 0 0 2 0 -1], ⟨0 2 0 1 0 3], ⟨0 0 1 0 0 1], ⟨0 0 0 0 1 0]]
Optimal ET sequence: 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff
1716/1715
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1716/1715
Mapping: [⟨1 0 0 0 -1 -1], ⟨0 2 0 0 -5 3], ⟨0 0 1 0 0 1], ⟨0 0 0 1 3 0]]
Optimal ET sequence: 58, 72, 121, 130, 193, 198, 270, 940, 1210f
364/363
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 676/675
Mapping: [⟨1 0 0 -1 0 -1], ⟨0 2 0 1 1 3], ⟨0 0 1 1 1 1], ⟨0 0 0 2 1 0]]
Optimal ET sequence: 14cf, 15, 23deff, 24, 29, 34d, 43, 49f, 58, 72, 87, 121, 130, 193, 217, 289, 338e, 410e
351/350
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 676/675
Mapping: [⟨1 0 0 -2 0 -1], ⟨0 2 0 9 0 3], ⟨0 0 1 -1 0 1], ⟨0 0 0 0 1 0]]
Optimal ET sequence: 14cf, 19, 24, 34d, 53, 58, 72, 111, 130, 183, 313, 462f
352/351
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 676/675
Mapping: [⟨1 0 0 0 -6 -1], ⟨0 2 0 0 9 3], ⟨0 0 1 0 1 1], ⟨0 0 0 1 0 0]]
Optimal ET sequence: 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf
540/539
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 676/675
Mapping: [⟨1 0 0 0 2 -1], ⟨0 2 0 0 6 3], ⟨0 0 1 0 1 1], ⟨0 0 0 1 -2 0]]
Optimal ET sequence: 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49f, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de
847/845
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 847/845
Mapping: [⟨1 0 0 0 -1 -1], ⟨0 2 0 0 3 3], ⟨0 0 1 0 1 1], ⟨0 0 0 2 -1 0]]
Optimal ET sequence: 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c
Rank-3 temperaments
Notable rank-3 temperaments of island include:
- +1001/1000, 1716/1715
- +364/363, 441/440
- +1001/1000, 3025/3024
- +325/324, 385/384
- +351/350, 540/539
- +1001/1000, 4096/4095
- +352/351, 364/363
Rank-2 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
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]
POTE generator: ~15/13 = 248.917
Optimal ET sequence: 130, 270, 940, 1210f
Badness: 0.013475
Avicenna
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
Mapping: [⟨3 2 8 16 9 8], ⟨0 8 -3 -22 4 9]]
CTE generator: ~13/12 = 137.777
POTE generator: ~13/12 = 137.777
Optimal ET sequence: 87, 183, 270
Badness: 0.015557
Tertiathirds
Subgroup: 2.3.5.7.11.13
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
Optimal ET sequence: 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff
Badness: 0.019494
17-limit
Subgroup: 2.3.5.7.11.13.17
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
Optimal ET sequence: 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 scales 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 = [2 -3 2⟩
Sval mapping: [⟨1 0 -1], ⟨0 2 3]]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.621
Optimal ET sequence: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
Badness: 0.002335
- Music
- Desert Island Rain in 313edo tuned Barbados[9], by 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 = [2 -3 2⟩, 1216/1215 = [6 -5 0 1⟩
Sval mapping: [⟨1 0 -1 -7], ⟨0 2 3 10]]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.868
Optimal ET sequence: 5, 24, 29, 53, 82, 111, 135
Badness: ?
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
Comma list: 243/242 = [-1 5 -2⟩, 676/675 = [2 -3 0 2⟩
Sval mapping: [⟨2 0 -1 -2], ⟨0 2 5 3]]
Gencom mapping: [⟨2 4 -2 0 9 2], ⟨0 -2 3/2 0 -5 -3/2]]
- gencom: [55/39 15/13; 243/242 676/675]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 249.312
Optimal ET sequence: 10, 14, 24, 58, 82, 130
RMS error: 0.3533 cents
Cata
Cata may be viewed as the "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
Comma list: 325/324, 625/624
Sval mapping: [⟨1 0 1 0], ⟨0 6 5 14]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.076
Optimal ET sequence: 15, 19, 34, 53, 87, 140, 193, 246
Badness: 0.00394
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 dakotismic and/or dakotic).
Subgroup: 2.3.5.13
Comma list: 676/675, 32805/32768
Sval mapping: [⟨1 0 15 14], ⟨0 2 -16 -13]]
Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 950.8331
Optimal ET sequence: 24, 53, 130, 183, 236, 525f, 761ff
Badness: 0.0100
Dakota
Subgroup: 2.3.5.13.19
Comma list: 361/360, 513/512, 676/675
Sval mapping: [⟨1 0 15 14 9], ⟨0 2 -16 -13 -6]]
Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 950.8199
Optimal ET sequence: 24, 29, 53, 130, 183, 236h, 289h
Badness: 0.00575
2.3.5.13.19.37 subgroup
Subgroup: 2.3.5.13.19.37
Comma list: 361/360, 481/480, 513/512, 676/675
Sval mapping: [⟨1 0 15 14 9 6], ⟨0 2 -16 -13 -6 -1]]
Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 950.8187
Optimal ET sequence: 24, 29, 53, 183, 236h, 289hl, 631fhhll
Badness: 0.00357
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.
Subgroup: 2.3.5.13
Comma list: 676/675
Sval mapping: [⟨1 0 0 -1], ⟨0 2 0 3], ⟨0 0 1 1]]
Optimal ET sequence: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270
Badness: 0.00811 × 10-3
- Music
- Petr's Pump, a comma pump based ditty in Parizekmic temperament.