Starling temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular 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) → Father family
- Flattie (+21/20) → Dicot family
- Opossum (+28/27) → Trienstonic clan
- Diminished (+36/35) → Dimipent family
- Keemun (+49/48) → Kleismic family
- Augene (+64/63) → Augmented family
- Meantone (+81/80) → Meantone family
- Mavila (+135/128) → Pelogic family
- Sensi (+245/243), Sensipent family
- Muggles (+525/512) → Magic family
- Valentine (+1029/1024) → Gamelismic clan
- Diaschismic (+2048/2025) → Diaschismic family
- Wollemia (+2240/2187) → Tetracot family
- Unicorn (+10976/10935) → Unicorn family
- Coblack (+16807/16384) → Trisedodge family / cloudy clan
- Grackle (+32805/32768) → Schismatic family
- Worschmidt (+33075/32768) → Würschmidt family
- Thuja (+65536/64827) → Buzzardsmic clan
- Passionate (+131072/127575) → Passion family
- Vishnean (+540225/524288) → Vishnuzmic family
- Ditonic (+8751645/8388608) → Ditonmic family
- Muscogee (+33756345/33554432) → Mabila 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 actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.
Myna
- For the 5-limit version of this temperament, see High badness temperaments #Mynic.
7-limit myna is naturally found by establishing a structure of thirds, by making 7/6 - 6/5 - 49/40 - 5/4 - 9/7 all equidistant (the distances between which are 36/35, 49/48, and 50/49). 11-limit myna then arises from equating this neutral third to 11/9. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them (25/24) is twice the size of the interval between the pental and septimal thirds (36/35), leaving space for a neutral third in between. In that sense, it is opposed to keemic temperaments, where the chroma between the pental thirds is the same as the distance between the pental and septimal thirds.
In terms of commas tempered, in addition to 126/125, myna adds 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
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.146
- 7- and 9-odd-limit: ~6/5 = [1/10 1/10 0 0⟩
- [[1 0 0 0⟩, [0 1 0 0⟩, [9/10 9/10 0 0⟩, [17/10 7/10 0 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.3
Optimal ET sequence: 27, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.144
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.276
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.381
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 309.804
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 309.737
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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.853
Optimal ET sequence: 4, 23bc, 27e
Badness: 0.048687
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. Mosses 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
Optimal tuning (POTE): ~2 = 1\1, ~49/45 = 154.579
- 7-odd-limit: ~49/45 = [4/13 0 -1/13⟩
- [[1 0 0 0⟩, [-5/13 0 11/13 0⟩, [0 0 1 0⟩, [-3/13 0 17/13 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.5
- 9-odd-limit: ~49/45 = [3/11 -1/11⟩
- [[1 0 0 0⟩, [0 1 0 0⟩, [5/11 13/11 0 0⟩, [4/11 17/11 0 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.3
Optimal ET sequence: 8d, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.645
Minimax tuning:
- 11-odd-limit: ~11/10 = [1/10 -1/5 0 0 1/10⟩
- [[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⟩]
- unchanged-interval (eigenmonzo) basis: 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.
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.478
Optimal ET sequence: 8d, 23de, 31, 70f, 101ff
Badness: 0.023323
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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.679
Optimal ET sequence: 27, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.587
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.591
Optimal ET sequence: 27e, 77, 104c, 181c
Badness: 0.035582
Vines
- For the 5-limit version of this temperament, see High badness temperaments #Vines.
Subgroup: 2.3.5.7
Comma list: 126/125, 84035/82944
Mapping: [⟨2 7 8 8], ⟨0 -8 -7 -5]]
Optimal tuning (POTE): 1\2, ~6/5 = 312.602
Optimal ET sequence: 42, 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]]
Optimal tuning (POTE): 1\2, ~6/5 = 312.601
Optimal ET sequence: 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]]
Optimal tuning (POTE): 1\2, ~6/5 = 312.564
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.797
Optimal ET sequence: 16, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.898
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.961
Optimal ET sequence: 16, 27e, 43, 70e, 113cdee
Badness: 0.028920
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]]
Optimal tuning (POTE): ~2 = 1\1, ~135/98 = 541.828
Optimal ET sequence: 11cd, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.772
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.778
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~343/243 = 1\2, ~35/27 = 455.445
Optimal ET sequence: 50, 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~35/27 = 455.373
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~55/39 = 1\2, ~13/10 = 455.347
Optimal ET sequence: 50, 58, 166cef, 224ceeff
Badness: 0.023877
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 mosses 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.
Marrakesh, named by Herman Miller in 2011[1], is a more accurate 11-limit extension where the generator is identified with 22/15 as opposed to 16/11 in casablanca.
Subgroup: 2.3.5.7
Comma list: 126/125, 589824/588245
Mapping: [⟨1 12 10 5], ⟨0 -19 -14 -4]]
Optimal tuning (POTE): ~2 = 1\1, ~35/24 = 657.818
Optimal ET sequence: 11b, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923
Optimal ET sequence: 11b, 20b, 31
Badness: 0.067291
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 385/384, 2420/2401
Mapping: [⟨1 12 10 5 4 7], ⟨0 -19 -14 -4 -1 -6]]
Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.854
Optimal ET sequence: 11b, 20b, 31
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]]
Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.756
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.700
Optimal ET sequence: 31, 104cff, 135cff
Badness: 0.041395
Amigo
Subgroup: 2.3.5.7
Comma list: 126/125, 2097152/2083725
Mapping: [⟨1 -2 2 9], ⟨0 11 1 -19]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.094
Optimal ET sequence: 43, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.075
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.073
Optimal ET sequence: 43, 46, 89, 135cf, 224cf
Badness: 0.030666
Gilead
Subgroup: 2.3.5.7
Comma list: 126/125, 343/324
Mapping: [⟨1 4 5 6], ⟨0 -9 -10 -12]]
Optimal ET sequence: 11cd, 15, 41dd, 56dd
Badness: 0.115292
Supersensi
Supersensi (8d & 43) has supermajor third as a generator like sensi, but the no-fives comma 17496/16807 rather than 245/243 tempered out.
Subgroup: 2.3.5.7
Comma list: 126/125, 17496/16807
Mapping: [⟨1 -4 -4 -5], ⟨0 15 17 21]]
Optimal tuning (POTE): ~2 = 1\1, ~343/270 = 446.568
Optimal ET sequence: 8d, 35, 43
Badness: 0.148531
11-limit
Subgroup: 2.3.5.7.11
Comma list: 99/98, 126/125, 864/847
Mapping: [⟨1 -4 -4 -5 -1], ⟨0 15 17 21 12]]
Optimal tuning (POTE): ~2 = 1\1, ~72/55 = 446.616
Optimal ET sequence: 8d, 35, 43
Badness: 0.059449
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 99/98, 126/125, 144/143
Mapping: [⟨1 -4 -4 -5 -1 -3], ⟨0 15 17 21 12 18]]
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 446.598
Optimal ET sequence: 8d, 35f, 43
Badness: 0.035258
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 78/77, 99/98, 120/119, 126/125, 144/143
Mapping: [⟨1 -4 -4 -5 -1 -3 0], ⟨0 15 17 21 12 18 11]]
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 446.631
Optimal ET sequence: 8d, 35f, 43
Badness: 0.025907
Cobalt
The name of the cobalt temperament comes from the 27th element.
Cobalt (27 & 81) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the aplonis temperament.
Subgroup: 2.3.5.7
Comma list: 126/125, 40353607/40310784
Mapping: [⟨27 43 63 76], ⟨0 -1 -1 -1]]
Optimal tuning (POTE): 1\27, ~3/2 = 701.244
Optimal ET sequence: 27, 81, 108, 135c, 243c
Badness: 0.173308
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 540/539, 21609/21296
Mapping: [⟨27 43 63 76 94], ⟨0 -1 -1 -1 -2]]
Optimal tuning (POTE): 1\27, ~3/2 = 700.001
Optimal ET sequence: 27e, 81, 108
Badness: 0.078060
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 196/195, 21609/21296
Mapping: [⟨27 43 63 76 94 100], ⟨0 -1 -1 -1 -2 0]]
Optimal tuning (POTE): 1\27, ~3/2 = 700.867
Optimal ET sequence: 27e, 81, 108, 243ceef
Badness: 0.057145
Cobaltous
Subgroup: 2.3.5.7.11.13.17
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445
Mapping: [⟨27 43 63 76 94 100 111], ⟨0 -1 -1 -1 -2 0 -2]]
Optimal tuning (POTE): 1\27, ~3/2 = 700.397
Optimal ET sequence: 27eg, 81, 108g
Badness: 0.042106
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968
Mapping: [⟨27 43 63 76 94 100 111 115], ⟨0 -1 -1 -1 -2 0 -2 -1]]
Optimal tuning (POTE): 1\27, ~3/2 = 700.429
Optimal ET sequence: 27eg, 81, 108g
Badness: 0.030415
Cobaltic
Subgroup: 2.3.5.7.11.13.17
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968
Mapping: [⟨27 43 63 76 94 100 111], ⟨0 -1 -1 -1 -2 0 -3]]
Optimal tuning (POTE): 1\27, ~3/2 = 701.595
Optimal ET sequence: 27eg, 81gg, 108, 135ce
Badness: 0.047163
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083
Mapping: [⟨27 43 63 76 94 100 111 115], ⟨0 -1 -1 -1 -2 0 -3 -1]]
Optimal tuning (POTE): 1\27, ~3/2 = 701.673
Optimal ET sequence: 27eg, 81gg, 108, 135ceh
Badness: 0.034176
Cobaltite
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 169/168, 540/539, 975/968
Mapping: [⟨27 43 63 76 94 100], ⟨0 -1 -1 -1 -2 -1]]
Optimal tuning (POTE): 1\27, ~3/2 = 699.179
Optimal ET sequence: 27e, 54bdef, 81f, 108f
Badness: 0.052732