Turkish maqam music temperaments

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This is a collection of some proposed temperaments for Turkish maqam music.

Yarman I

Ozan Yarman has proposed defining the tuning of Turkish maqam music using a MOS of 79 or 80 notes out of 159. This means a generator of 2\159, which suggests the 19-limit mappings:

[1 2 3 2 4 4 4 5], 0 -33 -54 64 -43 -24 7 -60]]

[1 2 3 4 4 4 4 5], 0 -33 -54 -95 -43 -24 7 -60]]

The first mapping may be called 79&159 in terms of patent vals, and the second 80&159. In any event both mappings can be used inconsistently, and both temperaments are weak 7-limit extensions of quartonic temperament. A Pythagorean tuning, i.e. one with pure fifths, is also possible.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 244140625/243045684

Mapping: [1 2 3 4], 0 -33 -54 -95]]

Wedgie⟨⟨33 54 95 9 58 69]]

POTE generator: ~126/125 = 15.0667

Optimal GPV sequence79d, 80, 159, 239

Badness: 0.193315

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 10976/10935

Mapping: [1 2 3 4 4], 0 -33 -54 -95 -43]]

POTE generator: ~121/120 = 15.0658

Optimal GPV sequence: 79d, 80, 159, 239

Badness: 0.049170

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 1001/1000, 10976/10935

Mapping: [1 2 3 4 4 4], 0 -33 -54 -95 -43 -24]]

POTE generator: ~121/120 = 15.0752

Optimal GPV sequence: 79d, 80, 159, 239

Badness: 0.040929

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 595/594, 1001/1000, 10976/10935

Mapping: [1 2 3 4 4 4 4], 0 -33 -54 -95 -43 -24 7]]

POTE generator: ~120/119 = 15.0715

Optimal GPV sequence: 79d, 80, 159, 239

Badness: 0.031015

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 361/360, 364/363, 595/594, 1001/1000, 1521/1520

Mapping: [1 2 3 4 4 4 4 5], 0 -33 -54 -95 -43 -24 7 -60]]

POTE generator: ~120/119 = 15.0713

Optimal GPV sequence: 79dh, 80, 159, 239

Badness: 0.023193

Yarman II

Subgroup: 2.3.5.7

Comma list: 5359375/5308416, 390625000/387420489

Mapping: [1 2 3 2], 0 -33 -54 64]]

POTE generator: ~6144/6125 = 15.1062

Optimal GPV sequence79, 80d, 159

Badness: 0.655487

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 78121827/77948684

Mapping: [1 2 3 2 4], 0 -33 -54 64 -43]]

POTE generator: ~121/120 = 15.1071

Optimal GPV sequence: 79, 80d, 159

Badness: 0.143477

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 85683/85184

Mapping: [1 2 3 2 4 4], 0 -33 -54 64 -43 -24]]

POTE generator: ~105/104 = 15.1071

Optimal GPV sequence: 79, 80d, 159

Badness: 0.068150

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 385/384, 1575/1573, 4928/4913

Mapping: [1 2 3 2 4 4 4], 0 -33 -54 64 -43 -24 7]]

POTE generator: ~105/104 = 15.1037

Optimal GPV sequence: 79, 80d, 159

Badness: 0.051019

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 273/272, 325/324, 385/384, 665/663, 969/968, 1575/1573

Mapping: [1 2 3 2 4 4 4 5], 0 -33 -54 64 -43 -24 7 -60]]

POTE generator: ~105/104 = 15.1013

Optimal GPV sequence: 79h, 159

Badness: 0.038430

Karadeniz

See also: Schismatic family #Garibaldi

K. E. Karadeniz proposed a 41-note MOS with generator 31/106, giving a "hemigaribaldi" type of tuning, with an 11/9 neutral third generator. It's more plausible as an 11-limit system than 13-limit; the 13-limit wedgie is:

⟨⟨2 -16 -28 5 40 -30 -50 1 56 -20 67 152 111 216 120]]

which in the 11-limit becomes:

⟨⟨2 -16 -28 5 -30 -50 1 -20 67 111]]

It tempers out 3125/3087, 4000/3969, 243/242, 5120/5103, 225/224, and 3025/3024, and can also be called 41&106. Aside from 31/106, 43/147 or 74/253 can be recommended as generators.

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3125/3087

Mapping: [1 1 7 11 2], 0 2 -16 -28 5]]

POTE generator: ~11/9 = 350.994

Optimal GPV sequence41, 106, 147

Badness: 0.041562

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 640/637

Mapping: [1 1 7 11 2 -8], 0 2 -16 -28 5 40]]

POTE generator: ~11/9 = 351.014

Optimal GPV sequence: 41, 106, 147

Badness: 0.042564