Turkish maqam music temperaments

Yarman temperament

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 mapping:

[<1 2 3 2 4 4 4 5|, <0 -33 -54 64 -43 -24 7 -60|]
vs.
[<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. A Pythagorean tuning, i.e. one with pure fifths, is also possible.

80&159 temperament

Commas: 10976/10935, 244140625/243045684

POTE generator: ~126/125 = 15.0667

Map: [<1 2 3 4|, <0 -33 -54 -95|]

EDOs: 79d, 80, 159, 239

Badness: 0.193315

11-limit yarman

Commas: 3025/3024, 4000/3993, 10976/10935

POTE generator: ~121/120 = 15.0658

Map: [<1 2 3 4 4|, <0 -33 -54 -95 -43|]

EDOs: 79d, 80, 159, 239

Badness: 0.049170

13-limit yarman

Commas: 325/324, 364/363, 1001/1000, 10976/10935

POTE generator: ~121/120 = 15.0752

Map: [<1 2 3 4 4 4|, <0 -33 -54 -95 -43 -24|]

EDOs: 79d, 80, 159, 239

Badness: 0.040929

79&159 temperament

Commas: 5359375/5308416, 390625000/387420489

POTE generator: ~6144/6125 = 15.1062

Map: [<1 2 3 2|, <0 -33 -54 64|]

EDOs: 79, 80d, 159, 238c, 239d

Badness: 0.655487

11-limit 79&159

Commas: 385/384, 4000/3993, 78121827/77948684

POTE generator: ~121/120 = 15.1071

Map: [<1 2 3 2 4|, <0 -33 -54 64 -43|]

EDOs: 79, 80d, 159, 238c, 239d

Badness: 0.143477

13-limit 79&159

Commas: 325/324, 385/384, 1575/1573, 85683/85184

POTE generator: ~121/120 = 15.1071

Map: [<1 2 3 2 4 4|, <0 -33 -54 64 -43 -24|]

EDOs: 79, 80d, 159, 238cf, 239d

Badness: 0.068150

Karadeniz temperament

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

Commas: 225/224, 243/242, 3125/3087

POTE generator: ~11/9 = 350.994

Map: [<1 1 7 11 2|, <0 2 -16 -28 5|]

EDOs: 41, 106, 147

Badness: 0.041562

13-limit

Commas: 225/224, 243/242, 325/324, 640/637

POTE generator: ~11/9 = 351.014

Map: [<1 1 7 11 2 -8|, <0 2 -16 -28 5 40|]

EDOs: 41, 106, 147

Badness: 0.042564