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-5 temperaments

Island

Subgroup: 2.3.5.7.11.13

Comma list: 676/675

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

Optimal patent val: 940

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

See also: Breedsmic temperaments #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

See also: Landscape microtemperaments #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]]

POTE generator: ~13/12 = 137.777

Optimal ET sequence: 87, 183, 270

Badness: 0.015557

Tertiathirds

See also: Wizmic microtemperaments #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

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

See also: Chromatic pairs #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

Main article: Catakleismic
See also: Kleismic family #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.
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