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Starling comma

This page discusses some of the rank two temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess the starling tetrad, the 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.

Myna temperament

In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&31 temperament, or in terms of its wedgie <<10 9 7 -9 -17 -9||. It has 6/5 as a generator, and 58edo can be used as a tuning, with 89edo being a better one, and fans of round amounts in cents may like 120edo. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

5-limit (Mynic)

Comma: 10077696/9765625

POTE generator: ~6/5 = 310.140

Map: [<1 9 9|, <0 -10 -9|]
EDOs: 27, 31, 58, 89, 325c
Badness: 0.2500

7-limit

Commas: 126/125, 1728/1715

7 and 9 limit minimax
[|1 0 0 0>, |0 1 0 0 >, |9/10 9/10 0 0>, |17/10 7/10 0 0>]
Eigenmonzos: 2, 3

POTE generator: 310.146

Map: [<1 9 9 8|, <0 -10 -9 -7|]
Generators: 2, 5/3
EDOs: 27, 31, 58, 89
Badness: 0.0270

11-limit

Commas: 126/125, 176/175, 243/242

POTE generator: ~6/5 = 310.144

Map: [<1 9 9 8 22|, <0 -10 -9 -7 -25|]
EDOs: 31, 58, 89
Badness: 0.0168

13-limit

Commas: 126/125, 144/143, 176/175, 196/195

POTE generator: ~6/5 = 310.276

Map: [<1 9 9 8 22 0|, <0 -10 -9 -7 -25 5|]
EDOs: 27, 31, 58
Badness: 0.0171

Minah

Commas: 78/77, 91/90, 126/125, 176/175

POTE generator: ~6/5 = 310.381

Map: [<1 9 9 8 22 20|, <0 -10 -9 -7 -25 -22|]
EDOs: 27e, 31f, 58f, 116cef
Badness: 0.0276

Maneh

Commas: 66/65, 105/104, 126/125, 540/539

POTE generator: ~6/5 = 309.804

Map: [<1 9 9 8 22 23|, <0 -10 -9 -7 -25 -26|]
EDOs: 31
Badness: 0.0299

Myno

Commas: 99/98, 126/125, 385/384

POTE generator: ~6/5 = 309.737

Map: [<1 9 9 8 -1|, <0 -10 -9 -7 6|]
EDOs: 27, 31
Badness: 0.0334

Coleto

Commas: 56/55, 100/99, 1728/1715

POTE generator: ~6/5 = 310.853

Map: [<1 9 9 8 2|, <0 -10 -9 -7 2|]
EDOs: 23bc, 27e
Badness: 0.0487

Myna Music by Igliashon Jones

Sensi temperament

Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. 46edo is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.

Commas: 126/125, 245/243

7-limit minimax
[|1 0 0 0>, |1/13 0 0 7/13>, |5/13 0 0 9/13>, |0 0 0 1>]
Eigenmonzos: 2, 7

9-limit minimax
[|1 0 0 0>, |2/5 14/5 -7/5 0>,
|4/5 18/5 -9/5 0>, |3/5 26/5 -13/5 0>]
Eigenmonzos: 2, 9/5

POTE generator: ~9/7 = 443.383
Algebraic generator: Calista, the real root of x^7-2x^2-1, at 340.6467 cents.

Map: [<1 6 8 11|, <0 -7 -9 -13|]
Generators: 2, 14/9
EDOs: 19, 27, 46, 249, 295
Badness: 0.0256

Sensor

Commas: 126/125, 245/243, 385/384

POTE generator: ~9/7 = 443.294

Map: [<1 6 8 11 -6|, <0 -7 -9 -13 15|]
EDOs: 8, 19, 27, 46, 111, 157
Badness: 0.0379

13-limit

Commas: 91/90, 126/125, 169/168, 385/384

POTE generator: ~9/7 = 443.321

Map: [<1 6 8 11 -6 10|, <0 -7 -9 -13 15 -10|]
EDOs: 8, 19, 27, 46, 157
Badness: 0.0256

Sensis

Commas: 56/55, 100/99, 245/243

POTE generator: 443.962

Map: [<1 6 8 11 6|, <0 -7 -9 -13 -4|]
EDOs: 19, 27, 73, 100
Badness: 0.0287

13-limit

Commas: 56/55, 78/77, 91/90, 100/99

POTE generator: 443.945

Map: [<1 6 8 11 6 10|, <0 -7 -9 -13 -4 -10|]
EDOs: 19, 27, 73, 100
Badness: 0.0200

Sensus

Commas: 126/125, 176/175, 245/243

POTE generator: ~9/7 = 443.626

Map: [<1 6 8 11 23|, <0 -7 -9 -13 -31|]
EDOs: 8, 19, 27, 46, 165
Badness: 0.0295

13-limit

Commas: 91/90, 126/125, 169/168, 352/351

POTE generator: ~9/7 = 443.559

Map: [<1 6 8 11 23 10|, <0 -7 -9 -13 -31 -10|]
EDOs: 8, 19, 27, 46, 303
Badness: 0.0208

Valentine temperament

Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the lattice of 7-limit tetrads. Valentine can also be described as the 31&46 temperament, and 77edo, 108edo or 185edo make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as <<9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).

Valentine is very closely related to Carlos Alpha, the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in Beauty in the Beast suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.

Commas: 1029/1024, 126/125

Minimax tuning:
7-limit: [|1 0 0 0>, |5/2 3/4 0 -3/4>,
|17/6 5/12 0 -5/12>, [5/2 -1/4 0 1/4>]
Eigenmonzos: 2, 7/6

9-limit: [|1 0 0 0>, |10/7 6/7 0 -3/7>,
|47/21 10/21 0 -5/21>, |20/7 -2/7 0 1/7>]
Eigenmonzos: 2, 9/7

POTE generator: 77.864

Algebraic generator: smaller root of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.

Map: [<1 1 2 3|, <0 9 5 -3|]
Generators: 2, 21/20
EDOs: 15, 31, 46, 77, 185, 262
Badness: 0.0311

11-limit

Commas: 121/120, 126/125, 176/175

Minimax tuning:
[|1 0 0 0 0>, |1 0 0 -9/10 9/10>,
|2 0 0 -1/2 1/2>, |3 0 0 3/10 -3/10>, |3 0 0 -7/10 7/10>]
Eigenmonzos: 2, 11/7

Minimax generator: (11/7)^(1/10) = 78.249
POTE generator: 77.881

Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.

Map: [<1 1 2 3 3|, <0 9 5 -3 7|]
Edos: 15, 31, 46, 77, 108, 185
Badness: 0.0167

See also: Chords of valentine

Dwynwen

Commas: 91/90, 121/120, 126/125, 176/175

POTE generator: ~21/20 = 78.219

Map: [<1 1 2 3 3 2|, <0 9 5 -3 7 26|]
EDOs: 15, 46
Badness: 0.0235

Lupercalia

Commas: 66/65, 105/104, 121/120, 126/125

POTE generator: ~22/21 = 77.709

Map: [<1 1 2 3 3 3|, <0 9 5 -3 7 11|]
EDOs: 15, 31, 108, 139
Badness: 0.0213

Valentino

Commas: 121/120, 126/125, 176/175, 196/195

POTE generator: ~22/21 = 77.958

Map: [<1 1 2 3 3 5|, <0 9 5 -3 7 -20|]
EDOs: 15, 31, 46, 77, 431
Badness: 0.0207

Semivalentine

Commas: 121/120, 126/125, 169/168, 176/175

POTE generator: ~22/21 = ~21/20 = 77.839

Map: [<2 2 4 6 6 7|, <0 9 5 -3 7 3|]
EDOs: 16, 30, 46, 62, 108ef
Badness: 0.0327

Alicorn temperament

Commas: 126/125, 10976/10935

POTE generator: ~28/27 = 62.278

Map: [<1 2 3 4|, <0 -8 -13 -23|]
Wedgie: <<8 13 23 2 14 17||
EDOs: 19, 58, 77, 96
Badness: 0.0409

11-limit

Commas: 126/125, 540/539, 896/891

POTE generator: ~28/27 = 62.101

Map: [<1 2 3 4 3|, <0 -8 -13 -23 9|]
EDOs: 19, 58
Badness: 0.0392

13-limit

Commas: 126/125, 144/143, 196/195, 676/675

POTE generator: ~28/27 = 62.119

Map: [<1 2 3 4 3 5|, <0 -8 -13 -23 9 -25|]
EDOs: 19, 58

Badness: 0.0237

Camahueto

Commas: 126/125, 10976/10935, 385/384

POTE generator: ~28/27 = 62.431

Map: [<1 2 3 4 2|, <0 -8 -13 -23 28|]
EDOs: 19, 58, 77, 96
Badness: 0.0659

13-limit

Commas: 126/125, 196/195, 385/384, 676/675

POTE generator: ~28/27 = 62.434

Map: [<1 2 3 4 2 5|, <0 -8 -13 -23 28 -25|]
EDOs: 19, 58, 77
Badness: 0.0362


Casablanca temperament

Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described by its wedgie, <<19 14 4 -22 -47 -30||, or as 31&73. 74/135 or 91/166 supply good tunings for the generator, and 20 and 31 note MOS are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a hexany and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.

Commas: 126/125, 589824/588245

POTE generator: ~35/24 = 657.818

Map: [<1 12 10 5|, <0 -19 -14 -4|]
EDOs: 9bc, 11b, 31, 135c, 166c
Badness: 0.1012

11-limit

Commas: 126/125, 385/384, 2420/2401

POTE generator: ~16/11 = 657.923

Map: [<1 12 10 5 4|, |0 -19 -14 -4 -1>]
EDOs: 9bc, 11b, 31, 259bce, 549bce
Badness: 0.0623

Marrakesh

Commas: 126/125, 176/175, 14641/14580

POTE generator: ~22/15 = 657.791

Map: [<1 12 10 5 21|, |0 -19 -14 -4 -32>]
EDOs: 9bce, 11be, 20be, 31, 42e, 73
Badness: 0.0405

13-limit

126/125, 176/175, 196/195, 17303/17280

POTE generator: ~22/15 = 657.756

Map: [<1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25>]
EDOs: 31, 73, 104c, 135c, 239cf
Badness: 0.0408

Murakuc

Commas: 126/125, 144/143, 176/175, 1540/1521

POTE generator: ~22/15 = 657.700

Map: [<1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6>]
EDOs: 31, 73f, 104cf
Badness: 0.0414

Nusecond temperament

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70, or in terms of its wedgie as <<11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. 31edo can be used as a tuning, or 132edo with a val which is the sum of the patent vals for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.

5-limit

Comma: 51018336/48828125

POTE generator: ~3125/2916 = 154.523

Map: [<1 3 4|, <0 -11 -13|]
EDOs: 8, 23, 31, 70, 101, 132c, 233c, 365bc
Badness: 0.4665

7-limit

Commas: 126/125, 2430/2401

7-limit minimax
[|1 0 0 0>, |-5/13 0 11/13 0>, |0 0 1 0>, |-3/13 0 17/13 0>]
Eigenmonzos: 2, 5

9-limit minimax
[|1 0 0 0>, |0 1 0 0>, |5/11 13/11 0 0>, |4/11 17/11 0 0>]
Eigenmonzos: 2, 3

POTE generator: 154.579

Map: [<1 3 4 5|, <0 -11 -13 -17|]
Generators: 2, 49/45
EDOs: 7, 8, 31, 101, 132, 163
Badness: 0.0504

11-limit

Commas: 99/98, 121/120, 126/125

11-limit minimax
[|1 0 0 0 0>, |19/10 11/5 0 0 -11/10>,
|27/10 13/5 0 0 -13/10>, |33/10 17/5 0 0 -17/10>,
|19/5 12/5 0 0 -6/5>]
Eigenmonzos: 2, 11/9

POTE generator: ~11/10 = 154.645
Algebraic generator: positive root of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.

Map: [<1 3 4 5 5|, <0 -11 -13 -17 -12|]
Generators: 2, 11/10
EDOs: 7, 8, 31, 101, 194
Badness: 0.0256

13-limit

Commas: 66/65 99/98 121/120 126/125

POTE generator: ~11/10 = 154.478

Map: [<1 3 4 5 5 5|, <0 -11 -13 -17 -12 -10|]
EDOs: 31, 70f, 101f
Badness: 0.0233

Thuja

Commas: 126/125, 65536/64827

POTE generator: ~175/128 = 558.605

Map: [<1 8 5 -2|, <0 -12 -5 9|]
Wedgie: <<12 5 -9 -20 -48 -35||
EDOs: 15, 43, 58
Badness: 0.0884

11-limit

Commas: 126/125, 176/175, 1344/1331

POTE generator: ~11/8 = 558.620

Map: [<1 8 5 -2 4|, <0 -12 -5 9 -1|]
EDOs: 13, 15, 28, 43, 58
Badness: 0.0331

13-limit

Commas: 126/125, 144/143, 176/175, 364/363

POTE generator: ~11/8 = 558.589

Map: [<1 8 5 -2 4 16|, <0 -12 -5 9 -1 -23|]
EDOs: 15, 43, 58
Badness: 0.0228

29-limit

POTE generator: ~11/8 = 558.520

Map: [<1 -4 0 7 3 -7 12 1 5 3|, <0 12 5 -9 1 23 -17 7 -1 4|]
EDOs: 43, 58
(Raisin d'etre of this entry being the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.)

Cypress

Comma: 258280326/244140625

POTE generator: ~4374/3125 = 541.726

Map: [<1 7 10|, <0 -12 -17|]
EDOs: 20c, 31, 113c, 144c, 175c, 381bc
Badness: 0.8166

7-limit

Commas: 126/125, 19683/19208

POTE generator: ~135/98 = 541.828

Map: [<1 7 10 15|, <0 -12 -17 -27|]
Wedgie: <<12 17 27 -1 9 15||
EDOs: 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bcd
Badness: 0.0998

11-limit

Commas: 99/98, 126/125, 243/242

POTE generator: ~15/11 = 541.772

Map: [<1 7 10 15 17|, <0 -12 -17 -27 -30|]
EDOs: 31, 144cd, 175cd, 206bcde, 237bcde
Badness: 0.0427

13-limit

Commas: 66/65, 99/98. 126/125, 243/242

POTE generator: ~15/11 = 541.778

Map: [<1 7 10 15 17 15|, <0 -12 -17 -27 -30 -25|]
EDOs: 31
Badness: 0.0378

Bisemidim

Commas: 126/125, 118098/117649

POTE generator: ~35/27 = 455.445

Map: [<2 1 2 2|, <0 9 11 15|]
Wedgie: <<18 22 30 -7 -3 8||
EDOs: 50, 58, 108, 166c, 408c
Badness: 0.0978

11-limit

Commas: 126/125, 540/539, 1344/1331

POTE generator: ~35/27 = 455.373

Map: [<2 1 2 2 5|, <0 9 11 15 8|]
EDOs: 50, 58, 108, 166ce, 224ce
Badness: 0.0412

13-limit

Commas: 126/125, 144/143, 196/195, 364/363

POTE generator: ~35/27 = 455.347

Map: [<2 1 2 2 5 5|, <0 9 11 15 8 10|]
EDOs: 50, 58, 166cef, 224cef
Badness: 0.0239

Vines

Commas: 126/125, 84035/82944

POTE generator: ~6/5 = 312.602

Map: [<2 7 8 8|, <0 -8 -7 -5|]
EDOs: 4, 42, 46, 96d, 142d, 238d
Badness: 0.0780

11-limit

Commas: 126/125, 385/384, 2401/2376

POTE generator: ~6/5 = 312.601

Map: [<2 7 8 8 5|, <0 -8 -7 -5 4|]
EDOs: 4, 42, 46, 96d, 142d, 238d
Badness: 0.0445

13-limit

Commas: 126/125, 196/195, 364/363, 385/384

POTE generator: ~6/5 = 312.564

Map: [<2 7 8 8 5 5|, <0 -8 -7 -5 4 5|]
EDOs: 4, 42, 46, 96d, 238df
Badness: 0.0297

Kumonga

Comma: 1289945088/1220703125

POTE generator: ~144/125 = 222.912

Map: [<1 4 4|, <0 -13 -9|]
EDOs: 16, 27, 43, 70, 183c
Badness: 0.7296

7-limit

Commas: 126/125, 12288/12005

POTE generator: ~8/7 = 222.797

Map: [<1 4 4 3|, <0 -13 -9 -1|]
Wedgie: <<13 9 1 -16 -35 -23||
EDOs: 16, 27, 43, 70, 167cd
Badness: 0.0875

11-limit

Commas: 126/125, 176/175, 864/847

POTE generator: ~8/7 = 222.898

Map: [<1 4 4 3 7|, <0 -13 -9 -1 -19|]
EDOs: 16, 27e, 43, 70e
Badness: 0.0433

13-limit

Commas: 78/77, 126/125, 144/143, 176/175

POTE generator: ~8/7 = 222.961

Map: [<1 4 4 3 7 5|, <0 -13 -9 -1 -19 -7|]
EDOs: 16, 27e, 43, 70e, 113cde
Badness: 0.0289

Amigo

Commas: 126/125, 2097152/2083725

POTE generator: ~5/4 = 391.094

Map: [<1 9 3 -10|, <0 -11 -1 19|]
EDOs: 43, 46, 89, 135c, 359c
Badness: 0.1109

11-limit

Commas: 126/125, 176/175, 16384/16335

POTE generator: ~5/4 = 391.075

Map: [<1 9 3 -10 -8|, <0 -11 -1 19 17|]
EDOs: 43, 46, 89, 135c, 224c
Badness: 0.0434

13-limit

Commas: 126/125, 169/168, 176/175, 364/363

POTE generator: ~5/4 = 391.072

Map: [<1 9 3 -10 -8 1|, <0 -11 -1 19 17 4|]
EDOs: 43, 46, 89, 135cf, 224cf
Badness: 0.0307