Starling temperaments
This page discusses miscellaneous rank-2 temperaments tempering out 126/125, the starling comma or septimal semicomma.
Temperaments discussed in families and clans are:
- pater ({16/15, 126/125}, father family)
- flat ({21/20, 25/24}, dicot family)
- opossum ({28/27, 126/125}, trienstonic clan)
- diminished ({36/35, 50/49}, dimipent family / jubilismic clan)
- keemun ({49/48, 126/125}, kleismic family / slendro clan)
- augene ({64/63, 126/125}, augmented family / archytas clan)
- septimal meantone ({81/80, 126/125}, meantone family)
- mavila ({126/125, 135/128}, pelogic family)
- sensi ({126/125, 245/243}, sensipent family / sensamagic clan)
- gilead ({126/125, 343/324}, shibboleth family)
- muggles ({126/125, 525/512}, magic family)
- diaschismic ({126/125, 2048/2025}, diaschismic family)
- wollemia ({126/125, 2240/2187}, tetracot family)
- coblack ({126/125, 16807/16384}, cloudy clan)
- grackle ({126/125, 32805/32768}, schismatic family)
- worschmidt ({126/125, 33075/32768}, würschmidt family)
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
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.
5-limit (mynic)
Subgroup: 2.3.5
Comma list: 10077696/9765625
Mapping: [⟨1 9 9], ⟨0 -10 -9]]
POTE generator: ~6/5 = 310.140
Badness: 0.2500
7-limit
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
- 7- and 9-odd-limit
- [[1 0 0 0⟩, [0 1 0 0⟩, [9/10 9/10 0 0⟩, [17/10 7/10 0 0⟩]
- Eigenmonzos: 2, 3
Badness: 0.0270
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
Badness: 0.0168
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
Badness: 0.0171
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
Badness: 0.0276
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
Badness: 0.0299
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
Badness: 0.0334
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
Badness: 0.0487
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.
5-limit
Subgroup: 2.3.5
Comma list: 1990656/1953125
Mapping: [⟨1 1 2], ⟨0 9 5]]
POTE generator: ~25/24 = 78.039
Badness: 0.1228
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
- [[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
- [[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
Algebraic generator: smaller root of x2 - 89x + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.
Badness: 0.0311
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:
- 11-odd-limit
- [[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
Algebraic generator: positive root of 4x3 + 15x2 - 21, or else Gontrand2, the smallest positive root of 4x7 - 8x6 + 5.
Badness: 0.0167
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
Badness: 0.0235
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
Badness: 0.0213
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
Badness: 0.0207
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
Badness: 0.0327
Alicorn
Subgroup: 2.3.5.7
Comma list: 126/125, 10976/10935
Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]
Wedgie: ⟨⟨ 8 13 23 2 14 17 ]]
POTE generator: ~28/27 = 62.278
Badness: 0.0409
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 540/539, 896/891
Mapping: [⟨1 2 3 4 3], ⟨0 -8 -13 -23 9]]
POTE generator: ~28/27 = 62.101
Badness: 0.0392
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 196/195, 676/675
Mapping: [⟨1 2 3 4 3 5], ⟨0 -8 -13 -23 9 -25]]
POTE generator: ~28/27 = 62.119
Badness: 0.0237
Camahueto
Subgroup: 2.3.5.7.11
Comma list: 126/125, 385/384, 10976/10935
Mapping: [⟨1 2 3 4 2], ⟨0 -8 -13 -23 28]]
POTE generator: ~28/27 = 62.431
Badness: 0.0659
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 385/384, 676/675
Mapping: [⟨1 2 3 4 2 5], ⟨0 -8 -13 -23 28 -25]]
POTE generator: ~28/27 = 62.434
Badness: 0.0362
Qilin
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 10976/10935
Mapping: [⟨1 2 3 4 6], ⟨0 -8 -13 -23 -49]]
POTE generator: ~28/27 = 62.196
Badness: 0.0414
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 196/195, 2200/2197
Mapping: [⟨1 2 3 4 6 5], ⟨0 -8 -13 -23 -49 -25]]
POTE generator: ~28/27 = 62.197
Badness: 0.0228
Monocerus
Subgroup: 2.3.5.7.11
Comma list: 126/125, 243/242, 5488/5445
Mapping: [⟨2 4 6 8 9], ⟨0 -8 -13 -23 -20]]
POTE generator: ~28/27 = 62.292
Badness: 0.0528
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 364/363, 676/675
Mapping: [⟨2 4 6 8 9 10], ⟨0 -8 -13 -23 -20 -25]]
POTE generator: ~28/27 = 62.301
Badness: 0.0288
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
Badness: 0.1012
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
Badness: 0.0623
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
Badness: 0.0405
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
Badness: 0.0408
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
Badness: 0.0414
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.
5-limit
Subgroup: 2.3.5
Comma list: 51018336/48828125
Mapping: [⟨1 3 4], ⟨0 -11 -13]]
POTE generator: ~3125/2916 = 154.523
Badness: 0.4665
7-limit
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
- [[1 0 0 0⟩, [-5/13 0 11/13 0⟩, [0 0 1 0⟩, [-3/13 0 17/13 0⟩]
- Eigenmonzos: 2, 5
- [[1 0 0 0⟩, [0 1 0 0⟩, [5/11 13/11 0 0⟩, [4/11 17/11 0 0⟩]
- Eigenmonzos: 2, 3
Badness: 0.0504
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:
- 11-odd-limit
- [[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
Algebraic generator: positive root of 15x2 - 10x - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
Badness: 0.0256
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
Badness: 0.0233
Thuja
Subgroup: 2.3.5.7
Comma list: 126/125, 65536/64827
Mapping: [⟨1 8 5 -2], ⟨0 -12 -5 9]]
Wedgie: ⟨⟨ 12 5 -9 -20 -48 -35 ]]
POTE generator: ~175/128 = 558.605
Badness: 0.0884
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 1344/1331
Mapping: [⟨1 8 5 -2 4], ⟨0 -12 -5 9 -1]]
POTE generator: ~11/8 = 558.620
Badness: 0.0331
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 176/175, 364/363
Mapping: [⟨1 8 5 -2 4 16], ⟨0 -12 -5 9 -1 -23]]
POTE generator: ~11/8 = 558.589
Badness: 0.0228
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
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
Cypress
5-limit
Subgroup: 2.3.5
Comma list: 258280326/244140625
Mapping: [⟨1 7 10], ⟨0 -12 -17]]
POTE generator: ~4374/3125 = 541.726
Badness: 0.8166
7-limit
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
Badness: 0.0998
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
Badness: 0.0427
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
Badness: 0.0378
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
Badness: 0.0978
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
Badness: 0.0412
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
Badness: 0.0239
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
Badness: 0.0780
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
Badness: 0.0445
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
Badness: 0.0297
Kumonga
5-limit
Subgroup: 2.3.5
Comma list: 1289945088/1220703125
Mapping: [⟨1 4 4], ⟨0 -13 -9]]
POTE generator: ~144/125 = 222.912
Badness: 0.7296
7-limit
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
Badness: 0.0875
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
Badness: 0.0433
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
Badness: 0.0289
Amigo
Subgroup: 2.3.5.7
Comma list: 126/125, 2097152/2083725
Mapping: [⟨1 9 3 -10], ⟨0 -11 -1 19]]
POTE generator: ~5/4 = 391.094
Badness: 0.1109
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 16384/16335
Mapping: [⟨1 9 3 -10 -8], ⟨0 -11 -1 19 17]]
POTE generator: ~5/4 = 391.075
Badness: 0.0434
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 169/168, 176/175, 364/363
Mapping: [⟨1 9 3 -10 -8 1], ⟨0 -11 -1 19 17 4]]
POTE generator: ~5/4 = 391.072
Badness: 0.0307
Oolong
5-limit
Subgroup: 2.3.5
Comma list: [11 18 -17>
Mapping: [⟨1 6 7], ⟨0 -17 -18]]
POTE generator: ~6/5 = 311.6942
Badness: 0.9428
7-limit
Subgroup: 2.3.5.7
Comma list: 126/125, 117649/116640
Mapping: [⟨1 6 7 8], ⟨0 -17 -18 -20]]
POTE generator: ~6/5 = 311.6793
Badness: 0.0735
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 = 311.5873
Badness: 0.0569
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 = 311.5908
Badness: 0.0356