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) → Father family
• Flat (+21/20) → Dicot family
• Opossum (+28/27) → Trienstonic clan
• Diminished (+36/35) → Dimipent family / jubilismic clan
• 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 / sensamagic clan
• Gilead (+343/324) → Shibboleth family
• Muggles (+525/512) → Magic family
• 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
• 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 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 6^1/10 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]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.146

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⟩]

eigenmonzo (unchanged-interval) 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

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.853

Optimal ET sequence: 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-1 non-octave 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-1 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. Mosses 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]]

Optimal tuning (POTE): ~2 = 1\1, ~25/24 = 78.039

Optimal ET sequence: 15, 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

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.864

Minimax tuning:

• 7-odd-limit: ~21/20 = [1/6 1/12 0 -1/12⟩

[[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⟩]

eigenmonzo (unchanged-interval) basis: 2.7/3

• 9-odd-limit: ~21/20 = [1/21 2/21 0 -1/21⟩

[[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⟩]

eigenmonzo (unchanged-interval) basis: 2.9/7

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

Optimal ET sequence: 15, 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

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.881

Minimax tuning:

• 11-odd-limit: ~21/20 = [0 0 0 -1/10 1/10⟩

[[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⟩]

eigenmonzo (unchanged-interval) basis: 2.11/7

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

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.219

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.709

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.958

Optimal ET sequence: 15f, 31, 46, 77

Badness: 0.020665

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 126/125, 154/153, 176/175, 196/195

Mapping: [⟨1 1 2 3 3 5 5], ⟨0 9 5 -3 7 -20 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.003

Optimal ET sequence: 15f, 31, 46, 77, 123e, 200ceg

Badness: 0.016768

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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.839

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 39.044

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.921

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.948

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 38.993

Optimal ET sequence: 31, 92ef, 123f

Badness: 0.057931

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

Wedgie: ⟨⟨11 13 17 -5 -4 3]]

Optimal tuning (POTE): ~2 = 1\1, ~49/45 = 154.579

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⟩]

eigenmonzo (unchanged-interval) 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⟩]

eigenmonzo (unchanged-interval) 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⟩]

eigenmonzo (unchanged-interval) 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

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]]

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]]

Wedgie: ⟨⟨13 9 1 -16 -35 -23]]

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

Thuja

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

Optimal tuning (POTE): ~2 = 1\1, ~175/128 = 558.605

Optimal ET sequence: 15, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.620

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.589

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.509

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.504

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.522

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.520

Optimal ET sequence: 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]]

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]]

Wedgie: ⟨⟨18 22 30 -7 -3 8]]

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, 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]]

Wedgie: ⟨⟨19 14 4 -22 -47 -30]]

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

See also: High badness temperaments #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]]

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

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]]

Wedgie: ⟨⟨15 17 21 -8 -9 1]]

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