Starling temperaments

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This page discusses miscellaneous rank-2 temperaments tempering out 126/125, the starling comma or septimal semicomma.

Temperaments discussed in families and clans are:

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

For the 5-limit version of this temperament, see High badness temperaments #Mynic.
Main article: Myna

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. 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 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.


Subgroup: 2.3.5.7

Comma list: 126/125, 1728/1715

Mapping: [1 9 9 8], 0 -10 -9 -7]]

Mapping generators: ~2, ~5/3

Wedgie⟨⟨10 9 7 -9 -17 -9]]

POTE generator: ~6/5 = 310.146

Minimax tuning:

[[1 0 0 0, [0 1 0 0, [9/10 9/10 0 0, [17/10 7/10 0 0]
Eigenmonzos (unchanged intervals): 2, 3

Vals27, 31, 58, 89

Badness: 0.027044

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 243/242

Mapping: [1 9 9 8 22], 0 -10 -9 -7 -25]]

POTE generator: ~6/5 = 310.144

Vals: 27e, 31, 58, 89

Badness: 0.016842

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 196/195

Mapping: [1 9 9 8 22 0], 0 -10 -9 -7 -25 5]]

POTE generator: ~6/5 = 310.276

Vals: 27e, 31, 58

Badness: 0.017125

Minah

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 126/125, 176/175

Mapping: [1 9 9 8 22 20], 0 -10 -9 -7 -25 -22]]

POTE generator: ~6/5 = 310.381

Vals: 27e, 31f, 58f

Badness: 0.027568

Maneh

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 126/125, 540/539

Mapping: [1 9 9 8 22 23], 0 -10 -9 -7 -25 -26]]

POTE generator: ~6/5 = 309.804

Vals: 27eff, 31

Badness: 0.029868

Myno

Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 385/384

Mapping: [1 9 9 8 -1], 0 -10 -9 -7 6]]

POTE generator: ~6/5 = 309.737

Vals: 27, 31

Badness: 0.033434

Coleto

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 1728/1715

Mapping: [1 9 9 8 2], 0 -10 -9 -7 2]]

POTE generator: ~6/5 = 310.853

Vals: 4, 23bc, 27e

Badness: 0.048687

Valentine

Main article: Valentine

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; 46EDO 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 "[t]he 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.

Subgroup: 2.3.5

Comma list: 1990656/1953125

Mapping: [1 1 2], 0 9 5]]

POTE generator: ~25/24 = 78.039

Vals15, 31, 46, 77, 123

Badness: 0.122765

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 1029/1024

Mapping: [1 1 2 3], 0 9 5 -3]]

Mapping generators: ~2, ~21/20

POTE generator: ~21/20 = 77.864

Minimax tuning:

[[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 (unchanged intervals): 2, 7/6
[[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 (unchanged intervals): 2, 9/7

Algebraic generator: smaller root of x2 - 89x + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.

Vals15, 31, 46, 77, 185, 262cd

Badness: 0.031056

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 176/175

Mapping: [1 1 2 3 3], 0 9 5 -3 7]]

Mapping generators: ~2, ~21/20

POTE generator: ~21/20 = 77.881

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 (unchanged intervals): 2, 11/7

Algebraic generator: positive root of 4x3 + 15x2 - 21, or else Gontrand2, the smallest positive root of 4x7 - 8x6 + 5.

Vals: 15, 31, 46, 77, 262cdee, 339cdeee

Badness: 0.016687

Dwynwen

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 126/125, 176/175

Mapping: [1 1 2 3 3 2], 0 9 5 -3 7 26]]

POTE generator: ~21/20 = 78.219

Vals: 15, 31f, 46

Badness: 0.023461

Lupercalia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 126/125

Mapping: [1 1 2 3 3 3], 0 9 5 -3 7 11]]

POTE generator: ~21/20 = 77.709

Vals: 15, 31, 77ff, 108eff, 139efff

Badness: 0.021328

Valentino

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 196/195

Mapping: [1 1 2 3 3 5], 0 9 5 -3 7 -20]]

POTE generator: ~21/20 = 77.958

Vals: 15f, 31, 46, 77

Badness: 0.020665

Semivalentine

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 169/168, 176/175

Mapping: [2 2 4 6 6 7], 0 9 5 -3 7 3]]

POTE generator: ~21/20 = 77.839

Vals: 16, 30, 46, 62, 108ef

Badness: 0.032749

Hemivalentine

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 343/338

Mapping: [1 1 2 3 3 4], 0 18 10 -6 14 -9]]

POTE generator: ~40/39 = 39.044

Vals: 30, 31, 61, 92f, 123f

Badness: 0.047059

Hemivalentino

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 1029/1024

Mapping: [1 1 2 3 2], 0 18 10 -6 45]]

POTE generator: ~45/44 = 38.921

Vals: 31, 92e, 123, 154, 185

Badness: 0.061275

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242, 1029/1024

Mapping: [1 1 2 3 2 5], 0 18 10 -6 45 -40]]

POTE generator: ~45/44 = 38.948

Vals: 31, 92e, 123, 154

Badness: 0.057919

Hemivalentoid

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 243/242, 343/338

Mapping: [1 1 2 3 2 4], 0 18 10 -6 45 -9]]

POTE generator: ~40/39 = 38.993

Vals: 31, 92ef, 123f

Badness: 0.057931

Casablanca

For the 5-limit version of this temperament, see High badness temperaments #Casablanca.

Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described 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.


Subgroup: 2.3.5.7

Comma list: 126/125, 589824/588245

Mapping: [1 12 10 5], 0 -19 -14 -4]]

Wedgie⟨⟨19 14 4 -22 -47 -30]]

POTE generator: ~35/24 = 657.818

Vals11b, 20b, 31, 104c, 135c, 166c

Badness: 0.101191

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2420/2401

Mapping: [1 12 10 5 4], 0 -19 -14 -4 -1]]

POTE generator: ~16/11 = 657.923

Vals: 11b, 20b, 31

Badness: 0.067291

Marrakesh

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: [1 12 10 5 21], 0 -19 -14 -4 -32]]

POTE generator: ~22/15 = 657.791

Vals: 31, 73, 104c, 135c

Badness: 0.040539

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: [1 12 10 5 21 -10], 0 -19 -14 -4 -32 25]]

POTE generator: ~22/15 = 657.756

Vals: 31, 73, 104c, 135c, 239ccf

Badness: 0.040774

Murakuc

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: [1 12 10 5 21 7], 0 -19 -14 -4 -32 -6]]

POTE generator: ~22/15 = 657.700

Vals: 31, 104cff, 135cff

Badness: 0.041395

Nusecond

For the 5-limit version of this temperament, see High badness temperaments #Nusecond.

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70. 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.


Subgroup: 2.3.5.7

Comma list: 126/125, 2430/2401

Mapping: [1 3 4 5], 0 -11 -13 -17]]

Mapping generators: ~2, ~49/45

Wedgie⟨⟨11 13 17 -5 -4 3]]

POTE generator: ~49/45 = 154.579

Minimax tuning:

[[1 0 0 0, [-5/13 0 11/13 0, [0 0 1 0, [-3/13 0 17/13 0]
Eigenmonzos (unchanged intervals): 2, 5
[[1 0 0 0, [0 1 0 0, [5/11 13/11 0 0, [4/11 17/11 0 0]
Eigenmonzos (unchanged intervals): 2, 3

Vals8d, 23d, 31, 101, 132c, 163c

Badness: 0.050389

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 126/125

Mapping: [1 3 4 5 5], 0 -11 -13 -17 -12]]

Mapping generators: ~2, ~11/10

POTE generator: ~11/10 = 154.645

Minimax tuning:

[[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 (unchanged intervals): 2, 11/9

Algebraic generator: positive root of 15x2 - 10x - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

Vals: 8d, 23de, 31, 101, 132ce, 163ce, 194cee

Badness: 0.025621

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: [1 3 4 5 5 5], 0 -11 -13 -17 -12 -10]]

POTE generator: ~11/10 = 154.478

Vals: 8d, 23de, 31, 70f, 101ff

Badness: 0.023323

Thuja

Subgroup: 2.3.5.7

Comma list: 126/125, 65536/64827

Mapping: [1 -4 0 7], 0 12 5 -9]]

Wedgie⟨⟨12 5 -9 -20 -48 -35]]

POTE generator: ~175/128 = 558.605

Vals15, 43, 58

Badness: 0.088441

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 1344/1331

Mapping: [1 -4 0 7 3], 0 12 5 -9 1]]

POTE generator: ~11/8 = 558.620

Vals: 15, 43, 58

Badness: 0.033078

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 364/363

Mapping: [1 -4 0 7 3 -7], 0 12 5 -9 1 23]]

POTE generator: ~11/8 = 558.589

Vals: 15, 43, 58

Badness: 0.022838

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 176/175, 221/220, 256/255

Mapping: [1 -4 0 7 3 -7 12], 0 12 5 -9 1 23 -17]]

POTE generator: ~11/8 = 558.509

Vals: 15, 43, 58

Badness: 0.022293

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220

Mapping: [1 -4 0 7 3 -7 12 1], 0 12 5 -9 1 23 -17 7]]

POTE generator: ~11/8 = 558.504

Vals: 15, 43, 58h

Badness: 0.018938

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: [1 -4 0 7 3 -7 12 1 5], 0 12 5 -9 1 23 -17 7 -1]]

POTE generator: ~11/8 = 558.522

Vals: 15, 43, 58hi

Badness: 0.016581

29-limit

The raison d'etre of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: [1 -4 0 7 3 -7 12 1 5 3], 0 12 5 -9 1 23 -17 7 -1 4]]

POTE generator: ~11/8 = 558.520

Vals: 15, 43, 58hi

Badness: 0.013762

Cypress

For the 5-limit version of this temperament, see High badness temperaments #Cypress.

Subgroup: 2.3.5.7

Comma list: 126/125, 19683/19208

Mapping: [1 7 10 15], 0 -12 -17 -27]]

Wedgie⟨⟨12 17 27 -1 9 15]]

POTE generator: ~135/98 = 541.828

Vals11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd

Badness: 0.099801

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 243/242

Mapping: [1 7 10 15 17], 0 -12 -17 -27 -30]]

POTE generator: ~15/11 = 541.772

Vals: 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde

Badness: 0.042719

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 7 10 15 17 15], 0 -12 -17 -27 -30 -25]]

POTE generator: ~15/11 = 541.778

Vals: 11cdeef, 20cdef, 31

Badness: 0.037849

Bisemidim

Subgroup: 2.3.5.7

Comma list: 126/125, 118098/117649

Mapping: [2 1 2 2], 0 9 11 15]]

Wedgie⟨⟨18 22 30 -7 -3 8]]

POTE generator: ~35/27 = 455.445

Vals50, 58, 108, 166c, 408ccc

Badness: 0.097786

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

Mapping: [2 1 2 2 5], 0 9 11 15 8]]

POTE generator: ~35/27 = 455.373

Vals: 50, 58, 108, 166ce, 224cee

Badness: 0.041190

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

Mapping: [2 1 2 2 5 5], 0 9 11 15 8 10]]

POTE generator: ~35/27 = 455.347

Vals: 50, 58, 166cef, 224ceeff

Badness: 0.023877

Vines

Subgroup: 2.3.5.7

Comma list: 126/125, 84035/82944

Mapping: [2 7 8 8], 0 -8 -7 -5]]

POTE generator: ~6/5 = 312.602

Vals42, 46, 96d, 142d, 238dd

Badness: 0.078049

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2401/2376

Mapping: [2 7 8 8 5], 0 -8 -7 -5 4]]

POTE generator: ~6/5 = 312.601

Vals: 42, 46, 96d, 142d, 238dd

Badness: 0.044499

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: [2 7 8 8 5 5], 0 -8 -7 -5 4 5]]

POTE generator: ~6/5 = 312.564

Vals: 42, 46, 96d, 238ddf

Badness: 0.029693

Kumonga

For the 5-limit version of this temperament, see High badness temperaments #Kumonga.

Subgroup: 2.3.5.7

Comma list: 126/125, 12288/12005

Mapping: [1 4 4 3], 0 -13 -9 -1]]

Wedgie⟨⟨13 9 1 -16 -35 -23]]

POTE generator: ~8/7 = 222.797

Vals16, 27, 43, 70, 167ccdd

Badness: 0.087500

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: [1 4 4 3 7], 0 -13 -9 -1 -19]]

POTE generator: ~8/7 = 222.898

Vals: 16, 27e, 43, 70e

Badness: 0.043336

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: [1 4 4 3 7 5], 0 -13 -9 -1 -19 -7]]

POTE generator: ~8/7 = 222.961

Vals: 16, 27e, 43, 70e, 113cdee

Badness: 0.028920

Amigo

See also: Sensamagic clan #Magus

Subgroup: 2.3.5.7

Comma list: 126/125, 2097152/2083725

Mapping: [1 -2 2 9], 0 11 1 -19]]

Wedgie⟨⟨11 1 -19 -24 -61 -47]]

POTE generator: ~5/4 = 391.094

Vals43, 46, 89, 135c, 359cc

Badness: 0.110873

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 16384/16335

Mapping: [1 -2 2 9 9], 0 11 1 -19 -17]]

POTE generator: ~5/4 = 391.075

Vals: 43, 46, 89, 135c, 224c

Badness: 0.043438

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 176/175, 364/363

Mapping: [1 -2 2 9 9 5], 0 11 1 -19 -17 -4]]

POTE generator: ~5/4 = 391.073

Vals: 43, 46, 89, 135cf, 224cf

Badness: 0.030666

Oolong

Main article: Oolong
For the 5-limit version of this temperament, see High badness temperaments #Oolong.

Subgroup: 2.3.5.7

Comma list: 126/125, 117649/116640

Mapping: [1 6 7 8], 0 -17 -18 -20]]

Wedgie⟨⟨17 18 20 -11 -16 -4]]

POTE generator: ~6/5 = 311.679

Vals27, 50, 77

Badness: 0.073509

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 26411/26244

Mapping: [1 6 7 8 18], 0 -17 -18 -20 -56]]

POTE generator: ~6/5 = 311.587

Vals: 27e, 77, 104c, 181c

Badness: 0.056915

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: [1 6 7 8 18 5], 0 -17 -18 -20 -56 -5]]

POTE generator: ~6/5 = 311.591

Vals: 27e, 77, 104c, 181c

Badness: 0.035582