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) → Diminished 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

Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing badness.

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

Main article: Myna

For the 5-limit version, see Miscellaneous 5-limit 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). In that sense, it is opposed to keemic temperaments, in particular quasitemp, where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.

In terms of vanishing commas, 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, and has a ploidacot signature of beta-decacot. It has ~6/5 as a generator.

58edo can be used as a tuning, with 89edo being a better one, and fans of round cent values 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-limit.

Subgroup: 2.3.5.7

Comma list: 126/125, 1728/1715

Mapping: [⟨1 -1 0 1], ⟨0 10 9 7]]

mapping generators: ~2, ~6/5

Optimal tunings:

• WE: ~2 = 1199.3410 ¢, ~6/5 = 309.9756 ¢

error map: ⟨-0.659 -1.540 +3.467 +0.344]

• CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.0880 ¢

error map: ⟨0.000 -1.075 +4.479 +1.790]

Minimax tuning:

• 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, 236cc

Badness (Sintel): 0.684

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 0 1 -3], ⟨0 10 9 7 25]]

Optimal tunings:

• WE: ~2 = 1199.3441 ¢, ~6/5 = 309.9748 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.0982 ¢

Optimal ET sequence: 27e, 31, 58, 89, 236cce

Badness (Sintel): 0.557

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 0 1 -3 5], ⟨0 10 9 7 25 -5]]

Optimal tunings:

• WE: ~2 = 1198.6509 ¢, ~6/5 = 309.9273 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.2218 ¢

Optimal ET sequence: 27e, 31, 58, 205cceff, 263ccdeefff

Badness (Sintel): 0.708

Minah

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 0 1 -3 -2], ⟨0 10 9 7 25 22]]

Optimal tunings:

• WE: ~2 = 1199.1929 ¢, ~6/5 = 310.1724 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.3251 ¢

Optimal ET sequence: 27e, 31f, 58f

Badness (Sintel): 1.14

Maneh

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 126/125, 243/242

Mapping: [⟨1 -1 0 1 -3 -3], ⟨0 10 9 7 25 26]]

Optimal tunings:

• WE: ~2 = 1199.9109 ¢, ~6/5 = 309.7815 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 309.7987 ¢

Optimal ET sequence: 27eff, 31

Badness (Sintel): 1.23

Myno

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 0 1 5], ⟨0 10 9 7 -6]]

Optimal tunings:

• WE: ~2 = 1201.0652 ¢, ~6/5 = 310.0121 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 309.7812 ¢

Optimal ET sequence: 27, 31

Badness (Sintel): 1.11

Coleto

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 0 1 4], ⟨0 10 9 7 -2]]

Optimal tunings:

• WE: ~2 = 1196.1024 ¢, ~6/5 = 309.8434 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.6398 ¢

Optimal ET sequence: 4, 23bc, 27e

Badness (Sintel): 1.61

Nusecond

For the 5-limit version, see Miscellaneous 5-limit 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. Note that in the data below, the generator is its octave complement since eleven such generators octave reduced give the perfect fifth; its ploidacot is thus theta-hendecacot.

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 -8 -9 -12], ⟨0 11 13 17]]

mapping generators: ~2, ~49/27

Optimal tunings:

• WE: ~2 = 1199.6138 ¢, ~49/27 = 1045.0850 ¢

error map: ⟨-0.386 -2.931 +3.267 +2.253]

• CWE: ~2 = 1200.0000 ¢, ~49/27 = 1045.3909 ¢

error map: ⟨0.000 -2.655 +3.768 +2.819]

Minimax tuning:

• 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 (Sintel): 1.28

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -8 -9 -12 -7], ⟨0 11 13 17 12]]

Optimal tunings:

• WE: ~2 = 1200.3420 ¢, ~11/6 = 1045.6528 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/6 = 1045.3816 ¢

Minimax tuning:

• 11-odd-limit: ~11/6 = [9/10 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

Badness (Sintel): 0.847

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -8 -9 -12 -7 -5], ⟨0 11 13 17 12 10]]

Optimal tunings:

• WE: ~2 = 1198.9982 ¢, ~11/6 = 1044.6488 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/6 = 1045.4476 ¢

Optimal ET sequence: 8d, 23de, 31

Badness (Sintel): 0.964

Oolong

Main article: Oolong

For the 5-limit version, see Miscellaneous 5-limit temperaments #Oolong.

Subgroup: 2.3.5.7

Comma list: 126/125, 117649/116640

Mapping: [⟨1 -11 -11 -12], ⟨0 17 18 20]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1199.9188 ¢, ~5/3 = 888.2606 ¢

error map: ⟨-0.081 -0.632 +3.269 -2.640]

• CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.3163 ¢

error map: ⟨0.000 -0.578 +3.379 -2.500]

Optimal ET sequence: 23d, 27, 50, 77

Badness (Sintel): 1.86

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -11 -11 -12 -38], ⟨0 17 18 20 56]]

Optimal tunings:

• WE: ~2 = 1198.9982 ¢, ~5/3 = 888.0239 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.3941 ¢

Optimal ET sequence: 27e, 50e, 77, 104c

Badness (Sintel): 1.88

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -11 -11 -12 -38 0], ⟨0 17 18 20 56 5]]

Optimal tunings:

• WE: ~2 = 1199.5177 ¢, ~5/3 = 888.0521 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.3959 ¢

Optimal ET sequence: 27e, 50e, 77, 104c

Badness (Sintel): 1.47

Vines

For the 5-limit version, see Miscellaneous 5-limit temperaments #Vines.

Vines may be described as the 46 & 50 temperament. It has a semi-octave period and a ~6/5 generator. Eight generators minus three periods give the perfect fifth, so the ploidacot for the temperament is diploid gamma-octacot. 96edo in the 96d val may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 126/125, 84035/82944

Mapping: [⟨2 -1 1 3], ⟨0 8 7 5]]

mapping generators: ~343/240, ~6/5

Optimal tunings:

• WE: ~343/240 = 600.2436 ¢, ~6/5 = 312.7294 ¢

error map: ⟨+0.487 -0.363 +3.036 -4.448]

• CWE: ~343/240 = 600.0000 ¢, ~6/5 = 312.6547 ¢

error map: ⟨0.000 -0.717 +2.269 -5.552]

Optimal ET sequence: 46, 96d, 142d

Badness (Sintel): 1.98

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -1 1 3 9], ⟨0 8 7 5 -4]]

Optimal tunings:

• WE: ~99/70 = 600.2454 ¢, ~6/5 = 312.7293 ¢
• CWE: ~99/70 = 600.0000 ¢, ~6/5 = 312.6282 ¢

Optimal ET sequence: 46, 96d, 142d

Badness (Sintel): 1.47

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -1 1 3 9 10], ⟨0 8 7 5 -4 -5]]

Optimal tunings:

• WE: ~55/39 = 600.3065 ¢, ~6/5 = 312.7240 ¢
• CWE: ~55/39 = 600.0000 ¢, ~6/5 = 312.5836 ¢

Optimal ET sequence: 46, 96d

Badness (Sintel): 1.23

Xenial

Main article: Xenial

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Xenial.

Named by Xenllium in 2026, xenial may be described as the 19 & 70 temperament, splitting the perfect eleventh into nine equal parts, each for ~10/9. Equivalently, a stack of nine 9/5s is equated with the perfect fifth above 7 octaves, so the ploidacot for the temperament is zeta-enneacot, and from this it derives its name.

Subgroup: 2.3.5.7

Comma list: 126/125, 177147/175616

Mapping: [⟨1 -6 -12 -25], ⟨0 9 17 33]]

mapping generators: ~2, ~9/5

Optimal tunings:

• WE: ~2 = 1200.0095 ¢, ~9/5 = 1011.1532 ¢

error map: ⟨+0.010 -1.634 +3.176 -1.009]

• CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.1456 ¢

error map: ⟨0.000 -1.644 +3.162 -1.021]

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 2.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 16384/16335

Mapping: [⟨1 -6 -12 -25 22], ⟨0 9 17 33 -22]]

Optimal tunings:

• WE: ~2 = 1199.6137 ¢, ~9/5 = 1010.8717 ¢
• CWE: ~2 = 1200.000 ¢, ~9/5 = 1011.1915 ¢

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 2.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 729/728

Mapping: [⟨1 -6 -12 -25 22 -14], ⟨0 9 17 33 -22 21]]

Optimal tunings:

• WE: ~2 = 1199.8559 ¢, ~9/5 = 1011.0911 ¢
• CWE: ~2 = 1200.000 ¢, ~9/5 = 1011.2102 ¢

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 1.98

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

Mapping: [⟨1 -6 -12 -25 22 -14 26], ⟨0 9 17 33 -22 21 -26]]

Optimal tunings:

• WE: ~2 = 1199.6970 ¢, ~9/5 = 1010.9792 ¢
• CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2323 ¢

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 2.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

Mapping: [⟨1 -6 -12 -25 22 -14 26 27], ⟨0 9 17 33 -22 21 -26 -27]]

Optimal tunings:

• WE: ~2 = 1199.7741 ¢, ~9/5 = 1011.0334 ¢
• CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2230 ¢

Optimal ET sequence: 19, 51cdh, 70, 89

Badness (Sintel): 2.03

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230

Mapping: [⟨1 -6 -12 -25 22 -14 26 27 2], ⟨0 9 17 33 -22 21 -26 -27 3]]

Optimal tunings:

• WE: ~2 = 1199.6628 ¢, ~9/5 = 1010.9415 ¢
• CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2245 ¢

Optimal ET sequence: 19, 51cdh, 70, 89

Badness (Sintel): 1.93

Kumonga

For the 5-limit version, see Miscellaneous 5-limit temperaments #Kumonga.

Subgroup: 2.3.5.7

Comma list: 126/125, 12288/12005

Mapping: [⟨1 -9 -5 2], ⟨0 13 9 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

• WE: ~2 = 1198.0653 ¢, ~7/4 = 975.6277 ¢

error map: ⟨-1.935 -1.382 +4.009 +2.932]

• CWE: ~2 = 1200.0000 ¢, ~7/4 = 977.1096 ¢

error map: ⟨0.000 +0.470 +7.673 +8.284]

Optimal ET sequence: 16, 27, 43, 70, 167ccdd

Badness (Sintel): 2.21

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -9 -5 2 -12], ⟨0 13 9 1 19]]

Optimal tunings:

• WE: ~2 = 1197.9101 ¢, ~7/4 = 975.4007 ¢
• CWE: ~2 = 1200.0000 ¢, ~7/4 = 976.9964 ¢

Optimal ET sequence: 16, 27e, 43, 70e

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -9 -5 2 -12 -2], ⟨0 13 9 1 19 7]]

Optimal tunings:

• WE: ~2 = 1198.4987 ¢, ~7/4 = 975.8162 ¢
• CWE: ~2 = 1200.0000 ¢, ~7/4 = 976.9677 ¢

Optimal ET sequence: 16, 27e, 43, 70e, 113cdee

Badness (Sintel): 1.19

Paraguay

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Parakleismic.

Named by Xenllium in 2026, paraguay tempers out 12005/11664 and may be described as the 19 & 61 temperament. It is a variant of parakleismic, mapping 7th harmonic to 16 generators.

Subgroup: 2.3.5.7

Comma list: 126/125, 12005/11664

Mapping: [⟨1 -8 -8 -9], ⟨0 13 14 16]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1200.6421 ¢, ~5/3 = 885.3232 ¢

error map: ⟨+0.642 +2.110 +3.074 -9.434]

• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8949 ¢

error map: ⟨0.000 +1.678 +2.214 -10.508]

Optimal ET sequence: 19, 61, 80d, 99d

Badness (Sintel): 2.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 12005/11664

Mapping: [⟨1 -8 -8 -9 2], ⟨0 13 14 16 2]]

Optimal tunings:

• WE: ~2 = 1197.7783 ¢, ~5/3 = 883.6140 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.1383 ¢

Optimal ET sequence: 19, 42e, 61e

Badness (Sintel): 2.49

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99, 343/338

Mapping: [⟨1 -8 -8 -9 2 -14], ⟨0 13 14 16 2 24]]

Optimal tunings:

• WE: ~2 = 1197.7848 ¢, ~5/3 = 883.6431 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.1623 ¢

Optimal ET sequence: 19, 42ef, 61e

Badness (Sintel): 1.86

Uruguay

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 100/99, 1183/1152

Mapping: [⟨1 -8 -8 -9 2 0], ⟨0 13 14 16 2 5]]

Optimal tunings:

• WE: ~2 = 1199.6132 ¢, ~5/3 = 884.7325 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.0005 ¢

Optimal ET sequence: 19, 42e

Badness (Sintel): 2.51

Bisemidim

Bisemidim tempers out 118098/117649 and may be described as the 50 & 58 temperament. It has a semi-octave period and a ~49/45 generator. Nine generators minus a period give the perfect fifth, so the ploidacot for the temperament is diploid alpha-enneacot. 108edo and 166edo in the 166cef val may be recommended as tunings.

Subgroup: 2.3.5.7

Comma list: 126/125, 118098/117649

Mapping: [⟨2 1 2 2], ⟨0 9 11 15]]

mapping generators: ~343/243, ~49/45

Optimal tunings:

• WE: ~343/243 = 599.8915 ¢, ~49/45 = 144.5293 ¢

error map: ⟨-0.217 -1.299 +3.292 -1.103]

• CWE: ~343/243 = 600.0000 ¢, ~49/45 = 144.5351 ¢

error map: ⟨0.000 -1.139 +3.572 -0.799]

Optimal ET sequence: 50, 58, 108, 166c, 408ccc

Badness (Sintel): 2.47

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 tunings:

• WE: ~99/70 = 599.6360 ¢, ~12/11 = 144.5388 ¢
• CWE: ~99/70 = 600.0000 ¢, ~12/11 = 144.5623 ¢

Optimal ET sequence: 50, 58, 108, 166ce, 224cee

Badness (Sintel): 1.36

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 tunings:

• WE: ~55/39 = 599.5217 ¢, ~12/11 = 144.5375 ¢
• CWE: ~55/39 = 600.0000 ¢, ~12/11 = 144.5698 ¢

Optimal ET sequence: 50, 58, 166cef, 224ceeff

Badness (Sintel): 0.987

Cypress

For the 5-limit version, see Miscellaneous 5-limit temperaments #Cypress.

Subgroup: 2.3.5.7

Comma list: 126/125, 19683/19208

Mapping: [⟨1 -5 -7 -12], ⟨0 12 17 27]]

Optimal tunings:

• WE: ~2 = 1200.1652 ¢, ~196/135 = 658.2622 ¢

error map: ⟨+0.165 -3.634 +2.988 +2.272]

• CWE: ~2 = 1200.0000 ¢, ~196/135 = 658.1814 ¢

error map: ⟨0.000 -3.779 +2.769 +2.071]

Optimal ET sequence: 11cd, 20cd, 31

Badness (Sintel): 2.53

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -5 -7 -12 -13], ⟨0 12 17 27 30]]

Optimal tunings:

• WE: ~2 = 1200.1117 ¢, ~22/15 = 658.2892 ¢
• CWE: ~2 = 1200.0000 ¢, ~22/15 = 658.2345 ¢

Optimal ET sequence: 11cdee, 20cde, 31, 144cd

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -5 -7 -12 -13 -10], ⟨0 12 17 27 30 25]]

Optimal tunings:

• WE: ~2 = 1199.4328 ¢, ~22/15 = 657.9111 ¢
• CWE: ~2 = 1200.0000 ¢, ~22/15 = 658.1886 ¢

Optimal ET sequence: 11cdeef, 20cdef, 31

Badness (Sintel): 1.56

Casablanca

For the 5-limit version, see Miscellaneous 5-limit temperaments #Casablanca.

Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may be described as 31 & 73 with a ploidacot signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note mos scales 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 ~48/35 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.

If we add 385/384 to the list of commas, 48/35 is identified with 11/8, 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, is a more accurate 11-limit extension where the generator is identified with 15/11 as opposed to 11/8 in casablanca.

Subgroup: 2.3.5.7

Comma list: 126/125, 589824/588245

Mapping: [⟨1 -7 -4 1], ⟨0 19 14 4]]

mapping generators: ~2, ~48/35

Optimal tunings:

• WE: ~2 = 1199.6286 ¢, ~48/35 = 542.0141 ¢

error map: ⟨-0.371 -1.087 +3.370 -1.141]

• CWE: ~2 = 1200.0000 ¢, ~48/35 = 542.1684 ¢

error map: ⟨0.000 -0.756 +4.044 -0.152]

Optimal ET sequence: 11b, 20b, 31, 104c, 135c, 166c

Badness (Sintel): 2.56

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 -4 1 3], ⟨0 19 14 4 1]]

Optimal tunings:

• WE: ~2 = 1200.6404 ¢, ~11/8 = 542.3659 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/8 = 542.0945 ¢

Optimal ET sequence: 11b, 20b, 31

Badness (Sintel): 2.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 2420/2401

Mapping: [⟨1 -7 -4 1 3 1], ⟨0 19 14 4 1 6]]

Optimal tunings:

• WE: ~2 = 1199.7367 ¢, ~11/8 = 542.0269 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/8 = 542.1392 ¢

Optimal ET sequence: 11b, 20b, 31

Badness (Sintel): 2.31

Marrakesh

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 -4 1 -11], ⟨0 19 14 4 32]]

Optimal tunings:

• WE: ~2 = 1199.6315 ¢, ~15/11 = 542.0428 ¢
• CWE: ~2 = 1200.0000 ¢, ~15/11 = 542.1958 ¢

Optimal ET sequence: 31, 73, 104c, 135c

Badness (Sintel): 1.34

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -7 -4 1 -11 15], ⟨0 19 14 4 32 -25]]

Optimal tunings:

• WE: ~2 = 1199.3741 ¢, ~15/11 = 541.9613 ¢
• CWE: ~2 = 1200.0000 ¢, ~15/11 = 542.2361 ¢

Optimal ET sequence: 31, 73, 104c, 135c, 239ccf

Badness (Sintel): 1.68

Murakuc

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -7 -4 1 -11 1], ⟨0 19 14 4 32 6]]

Optimal tunings:

• WE: ~2 = 1198.6578 ¢, ~15/11 = 541.6930 ¢
• CWE: ~2 = 1200.0000 ¢, ~15/11 = 542.2577 ¢

Optimal ET sequence: 31, 73f, 104cff

Badness (Sintel): 1.71

Amigo

For the 5-limit version, see Miscellaneous 5-limit temperaments #Magus.

Subgroup: 2.3.5.7

Comma list: 126/125, 2097152/2083725

Mapping: [⟨1 -2 2 9], ⟨0 11 1 -19]]

mapping generators: ~2, ~5/4

Optimal tunings:

• WE: ~2 = 1199.4354 ¢, ~5/4 = 390.9104 ¢

error map: ⟨-0.565 -0.811 +3.467 -1.206]

• CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.0937 ¢

error map: ⟨0.000 +0.076 +4.780 +0.393]

Optimal ET sequence: 43, 46, 89, 135c, 359cc

Badness (Sintel): 2.81

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 tunings:

• WE: ~2 = 1199.5267 ¢, ~5/4 = 390.9211 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.0783 ¢

Optimal ET sequence: 43, 46, 89, 135c, 224c

Badness (Sintel): 1.44

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 tunings:

• WE: ~2 = 1199.8174 ¢, ~5/4 = 391.0130 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.0737 ¢

Optimal ET sequence: 43, 46, 89

Badness (Sintel): 1.27

Gilead

For the 5-limit version, see Miscellaneous 5-limit temperaments #Shibboleth.

Subgroup: 2.3.5.7

Comma list: 126/125, 343/324

Mapping: [⟨1 -5 -5 -6], ⟨0 9 10 12]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1201.4516 ¢, ~5/3 = 879.6394 ¢

error map: ⟨+1.452 +7.542 +2.823 -21.862]

• CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.7223 ¢

error map: ⟨0.000 +6.545 +0.909 -24.159]

Optimal ET sequence: 11cd, 15, 41dd

Badness (Sintel): 2.92

Supersensi

Named by Xenllium in 2022, supersensi tempers out the no-fives comma 17496/16807, and may be described as 8d & 43. It has a ultramajor third generator, which is sharper than the generator for sensi, hence the name. Its ploidacot is epsilon-15-cot.

Subgroup: 2.3.5.7

Comma list: 126/125, 17496/16807

Mapping: [⟨1 -4 -4 -5], ⟨0 15 17 21]]

mapping generators: ~2, ~343/270

Optimal tunings:

• WE: ~2 = 1199.1406 ¢, ~343/270 = 446.2478 ¢

error map: ⟨-0.859 -4.800 +3.337 +6.675]

• CWE: ~2 = 1200.0000 ¢, ~343/270 = 446.5163 ¢

error map: ⟨0.000 -4.210 +4.464 +8.017]

Optimal ET sequence: 8d, …, 35, 43

Badness (Sintel): 3.76

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 tunings:

• WE: ~2 = 1198.6099 ¢, ~72/55 = 446.0983 ¢
• CWE: ~2 = 1200.0000 ¢, ~72/55 = 446.5381 ¢

Optimal ET sequence: 8d, …, 35, 43

Badness (Sintel): 1.97

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 tunings:

• WE: ~2 = 1198.9947 ¢, ~13/10 = 446.2243 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/10 = 446.5420 ¢

Optimal ET sequence: 8d, …, 35f, 43

Badness (Sintel): 1.46

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 tunings:

• WE: ~2 = 1198.7070 ¢, ~13/10 = 446.1493 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/10 = 446.5645 ¢

Optimal ET sequence: 8d, …, 35f, 43

Badness (Sintel): 1.32

Cobalt

For the 5-limit version, see 27th-octave temperaments #Cobalt.

Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the aplonis temperament. It may be described as 27 & 81.

Cobalt was named by Xenllium in 2022 after the 27th element.

Subgroup: 2.3.5.7

Comma list: 126/125, 40353607/40310784

Mapping: [⟨27 0 20 33], ⟨0 1 1 1]]

mapping generators: ~36/35, ~3

Optimal tunings:

• WE: ~36/35 = 44.4363 ¢, ~3/2 = 701.1154 ¢

error map: ⟨-0.221 -1.060 +3.307 -1.534]

• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 701.0414 ¢

error map: ⟨0.000 -0.914 +3.617 -1.118]

Optimal ET sequence: 27, 81, 108, 135c

Badness (Sintel): 4.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 21609/21296

Mapping: [⟨27 0 20 33 8], ⟨0 1 1 1 2]]

Optimal tunings:

• WE: ~36/35 = 44.4418 ¢, ~3/2 = 699.9594 ¢
• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 699.9386 ¢

Optimal ET sequence: 27e, 81, 108

Badness (Sintel): 2.58

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 21609/21296

Mapping: [⟨27 0 20 33 8 100], ⟨0 1 1 1 2 0]]

Optimal tunings:

• WE: ~36/35 = 44.4250 ¢, ~3/2 = 700.5606 ¢
• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 700.5524 ¢

Optimal ET sequence: 27e, 81, 108, 243ceef

Badness (Sintel): 2.36

Cobaltous

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445

Mapping: [⟨27 0 20 33 8 100 79], ⟨0 1 1 1 2 0 2]]

Optimal tunings:

• WE: ~36/35 = 44.4237 ¢, ~3/2 = 700.0699 ¢
• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 700.0569 ¢

Optimal ET sequence: 27eg, 81, 108g

Badness (Sintel): 2.14

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 0 20 33 8 100 79 99], ⟨0 1 1 1 2 0 2 1]]

Optimal tunings:

• WE: ~36/35 = 44.4227 ¢, ~3/2 = 700.0859 ¢
• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 700.0852 ¢

Optimal ET sequence: 27eg, 81, 108g

Badness (Sintel): 1.85

Cobaltic

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968

Mapping: [⟨27 0 20 33 8 100 -18], ⟨0 1 1 1 2 0 3]]

Optimal tunings:

• WE: ~36/35 = 44.4203 ¢, ~3/2 = 701.2133 ¢
• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 701.2530 ¢

Optimal ET sequence: 27eg, 108, 135ce

Badness (Sintel): 2.40

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 0 20 33 8 100 -18 72], ⟨0 1 1 1 2 0 3 1]]

Optimal tunings:

• WE: ~36/35 = 44.4177 ¢, ~3/2 = 701.2519 ¢
• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 701.3143 ¢

Optimal ET sequence: 27eg, 108, 135ceh

Badness (Sintel): 2.08

Cobaltite

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 975/968

Mapping: [⟨27 0 20 33 8 57], ⟨0 1 1 1 2 1]]

Optimal tunings:

• WE: ~36/35 = 44.4177 ¢, ~3/2 = 699.5121 ¢
• CWE: ~36/35 = 44.4444 ¢, ~3/2 = 699.6606 ¢

Optimal ET sequence: 27e, 54bdef, 81f

Badness (Sintel): 2.18