Starling temperaments: Difference between revisions

Latest revision as of 12:40, 21 August 2025

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

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.

Main article: Myna

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 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

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⟩]

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

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

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

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: 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

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]]

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

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]]

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 tunings:

CTE: ~2 = 1\1, ~6/5 = 321.109
POTE: ~2 = 1\1, ~6/5 = 321.423

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

↑ Yahoo! Tuning Group | A rose by any other name . . .

[1] Yahoo! Tuning Group | A rose by any other name . . .

[1]

@@ Line 1: / Line 1: @@
-This page discusses miscellaneous rank-2 temperaments tempering out [[126/125]], the starling comma or septimal semicomma.
+{{Technical data page}}
+This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.
 Temperaments discussed in families and clans are:
-* ''[[Pater]]'', {16/15, 126/125} → [[Father family #Pater|Father family]]
+* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
-* ''[[Flat]]'', {21/20, 25/24} → [[Dicot family #Flat|Dicot family]]
+* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
-* ''[[Opossum]]'', {28/27, 126/125} → [[Trienstonic clan #Opossum|Trienstonic clan]]
+* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
-* ''[[Diminished]]'', {36/35, 50/49} → [[Dimipent family #Diminished|Dimipent family]] / [[Jubilismic clan #Diminished|jubilismic clan]]
+* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
-* [[Keemun]], {49/48, 126/125} → [[Kleismic family #Keemun|Kleismic family]]
+* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
-* ''[[Augene]]'', {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]
+* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
-* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
+* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
-* [[Mavila]], {126/125, 135/128} → [[Pelogic family #Mavila|Pelogic family]]
+* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
-* [[Sensi]], {126/125, 245/243}, [[Sensipent family #Sensi|Sensipent family]] / [[Sensamagic clan #Sensi|sensamagic clan]]
+* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
-* ''[[Gilead]]'', {126/125, 343/324} → [[Shibboleth family #Gilead|Shibboleth family]]
+* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
-* [[Muggles]], {126/125, 525/512} → [[Magic family #Muggles|Magic family]]
+* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
-* ''[[Diaschismic]]'', {126/125, 2048/2025} → [[Diaschismic family #Diaschismic|Diaschismic family]]
+* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
-* ''[[Wollemia]]'', {126/125, 2240/2187} → [[Tetracot family #Wollemia|Tetracot family]]
+* ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
-* ''[[Unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Unicorn|Unicorn family]]
+* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
-* ''[[Coblack]]'', {126/125, 16807/16384} → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
+* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
-* ''[[Grackle]]'', {126/125, 32805/32768} → [[Schismatic family #Grackle|Schismatic family]]
+* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
-* ''[[Worschmidt]]'', {126/125, 33075/32768} → [[Würschmidt family #Worschmidt|Würschmidt family]]
+* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
-* ''[[Passionate]]'', {126/125, 131072/127575} → [[Passion family #Passionate|Passion family]]
+* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
-* ''[[Vishnean]]'', {126/125, 540225/524288} → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
+* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
-* ''[[Ditonic]]'', {126/125, 8751645/8388608} → [[Ditonmic family #Ditonic|Ditonmic family]]
+* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
-* ''[[Muscogee]]'', {126/125, 33756345/33554432} → [[Mabila family #Muscogee|Mabila family]]
+* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
+* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
-Since (6/5)<sup>3</sup> = 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|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.
+Since {{nowrap|(6/5)<sup>3</sup> {{=}} 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 ==
@@ Line 30: / Line 32: @@
 {{Main| 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&amp;31 temperament. It has 6/5 as a generator, and [[58edo|58EDO]] can be used as a tuning, with [[89edo|89EDO]] being a better one, and fans of round amounts in cents may like [[120edo|120EDO]]. It is also possible to tune myna with pure fifths by taking 6<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
+-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.
-Subgroup: 2.3.5.7
+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 {{nowrap|27 &amp; 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<sup>1/10</sup> 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]]: [{{val| 1 9 9 8 }}, {{val| 0 -10 -9 -7 }}]
+{{Mapping|legend=1| 1 9 9 8 | 0 -10 -9 -7 }}
-Mapping generators: ~2, ~5/3
+: mapping generators: ~2, ~5/3
-{{Multival|legend=1| 10 9 7 -9 -17 -9 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 310.146
-[[POTE generator]]: ~6/5 = 310.146
 [[Minimax tuning]]:
 * 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 9/10 9/10 0 0 }}, {{monzo| 17/10 7/10 0 0 }}]
+: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
-: [[Eigenmonzo]]s (unchanged intervals): 2, 3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
-{{Val list|legend=1| 27, 31, 58, 89 }}
+{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
 [[Badness]]: 0.027044
@@ Line 58: / Line 60: @@
 Comma list: 126/125, 176/175, 243/242
-Mapping: [{{val| 1 9 9 8 22 }}, {{val| 0 -10 -9 -7 -25 }}]
+Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}
-POTE generator: ~6/5 = 310.144
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.144
-Optimal GPV sequence: {{Val list| 27e, 31, 58, 89 }}
+{{Optimal ET sequence|legend=1| 27e, 31, 58, 89 }}
 Badness: 0.016842
@@ Line 71: / Line 73: @@
 Comma list: 126/125, 144/143, 176/175, 196/195
-Mapping: [{{val| 1 9 9 8 22 0 }}, {{val| 0 -10 -9 -7 -25 5 }}]
+Mapping: {{mapping| 1 9 9 8 22 0 | 0 -10 -9 -7 -25 5 }}
-POTE generator: ~6/5 = 310.276
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.276
-Optimal GPV sequence: {{Val list| 27e, 31, 58 }}
+{{Optimal ET sequence|legend=1| 27e, 31, 58 }}
 Badness: 0.017125
@@ Line 84: / Line 86: @@
 Comma list: 78/77, 91/90, 126/125, 176/175
-Mapping: [{{val| 1 9 9 8 22 20 }}, {{val| 0 -10 -9 -7 -25 -22 }}]
+Mapping: {{mapping| 1 9 9 8 22 20 | 0 -10 -9 -7 -25 -22 }}
-POTE generator: ~6/5 = 310.381
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.381
-Optimal GPV sequence: {{Val list| 27e, 31f, 58f }}
+{{Optimal ET sequence|legend=1| 27e, 31f, 58f }}
 Badness: 0.027568
@@ Line 97: / Line 99: @@
 Comma list: 66/65, 105/104, 126/125, 540/539
-Mapping: [{{val| 1 9 9 8 22 23 }}, {{val| 0 -10 -9 -7 -25 -26 }}]
+Mapping: {{mapping| 1 9 9 8 22 23 | 0 -10 -9 -7 -25 -26 }}
-POTE generator: ~6/5 = 309.804
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 309.804
-Optimal GPV sequence: {{Val list| 27eff, 31 }}
+{{Optimal ET sequence|legend=1| 27eff, 31 }}
 Badness: 0.029868
@@ Line 110: / Line 112: @@
 Comma list: 99/98, 126/125, 385/384
-Mapping: [{{val| 1 9 9 8 -1 }}, {{val| 0 -10 -9 -7 6 }}]
+Mapping: {{mapping| 1 9 9 8 -1 | 0 -10 -9 -7 6 }}
-POTE generator: ~6/5 = 309.737
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 309.737
-Optimal GPV sequence: {{Val list| 27, 31 }}
+{{Optimal ET sequence|legend=1| 27, 31 }}
 Badness: 0.033434
@@ Line 123: / Line 125: @@
 Comma list: 56/55, 100/99, 1728/1715
-Mapping: [{{val| 1 9 9 8 2 }}, {{val| 0 -10 -9 -7 2 }}]
+Mapping: {{mapping| 1 9 9 8 2 | 0 -10 -9 -7 2 }}
-POTE generator: ~6/5 = 310.853
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.853
-Optimal GPV sequence: {{Val list| 4, 23bc, 27e }}
+{{Optimal ET sequence|legend=1| 4, 23bc, 27e }}
 Badness: 0.048687
-== Valentine ==
-{{Main| 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 [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the 31&amp;46 temperament, and [[77edo|77EDO]], [[108edo|108EDO]] or [[185edo|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)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as {{multival| 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)<sup>1/10</sup>.
-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]]: [{{val| 1 1 2 }}, {{val| 0 9 5 }}]
-[[POTE generator]]: ~25/24 = 78.039
-{{Val list|legend=1| 15, 31, 46, 77, 123 }}
-[[Badness]]: 0.122765
-=== 7-limit ===
-Subgroup: 2.3.5.7
-[[Comma list]]: 126/125, 1029/1024
-[[Mapping]]: [{{val| 1 1 2 3 }}, {{val| 0 9 5 -3 }}]
-Mapping generators: ~2, ~21/20
-[[POTE generator]]: ~21/20 = 77.864
-[[Minimax tuning]]:
-* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
-: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6
-* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
-: [[Eigenmonzo]]s (unchanged intervals): 2, 9/7
-[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.
-{{Val list|legend=1| 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: [{{val| 1 1 2 3 3 }}, {{val| 0 9 5 -3 7 }}]
-Mapping generators: ~2, ~21/20
-POTE generator: ~21/20 = 77.881
-Minimax tuning:
-* [[11-odd-limit]]: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
-: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
-: Eigenmonzos (unchanged intervals): 2, 11/7
-Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 1 2 3 3 2 }}, {{val| 0 9 5 -3 7 26 }}]
-POTE generator: ~21/20 = 78.219
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 1 2 3 3 3 }}, {{val| 0 9 5 -3 7 11 }}]
-POTE generator: ~21/20 = 77.709
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 1 2 3 3 5 }}, {{val| 0 9 5 -3 7 -20 }}]
-POTE generator: ~21/20 = 77.958
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 1 2 3 3 5 5 }}, {{val| 0 9 5 -3 7 -20 -14 }}]
-POTE generator: ~21/20 = 78.003
-Optimal GPV sequence: {{Val list| 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: [{{val| 2 2 4 6 6 7 }}, {{val| 0 9 5 -3 7 3 }}]
-POTE generator: ~21/20 = 77.839
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 1 2 3 3 4 }}, {{val| 0 18 10 -6 14 -9 }}]
-POTE generator: ~40/39 = 39.044
-Optimal GPV sequence: {{Val list| 30, 31, 61, 92f, 123f }}
-Badness: 0.047059
-=== Hemivalentino ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 243/242, 1029/1024
-Mapping: [{{val| 1 1 2 3 2 }}, {{val| 0 18 10 -6 45 }}]
-POTE generator: ~45/44 = 38.921
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 1 2 3 2 5 }}, {{val| 0 18 10 -6 45 -40 }}]
-POTE generator: ~45/44 = 38.948
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 1 2 3 2 4 }}, {{val| 0 18 10 -6 45 -9 }}]
-POTE generator: ~40/39 = 38.993
-Optimal GPV sequence: {{Val list| 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&amp;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|31EDO]] can be used as a tuning, or [[132edo|132EDO]] with a val which is the sum of the [[patent val]]s 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.
+Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as {{nowrap|31 &amp; 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 val]]s 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 2430/2401
-[[Mapping]]: [{{val| 1 3 4 5 }}, {{val| 0 -11 -13 -17 }}]
+{{Mapping|legend=1| 1 3 4 5 | 0 -11 -13 -17 }}
-Mapping generators: ~2, ~49/45
+: mapping generators: ~2, ~49/45
-{{Multival|legend=1| 11 13 17 -5 -4 3 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 154.579
-[[POTE generator]]: ~49/45 = 154.579
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
-: [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]
+: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
-: [[Eigenmonzo]]s (unchanged intervals): 2, 5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 * [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]
+: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
-: [[Eigenmonzo]]s (unchanged intervals): 2, 3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
-{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
+{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
 [[Badness]]: 0.050389
@@ Line 348: / Line 165: @@
 Comma list: 99/98, 121/120, 126/125
-Mapping: [{{val| 1 3 4 5 5 }}, {{val| 0 -11 -13 -17 -12 }}]
+Mapping: {{mapping| 1 3 4 5 5 | 0 -11 -13 -17 -12 }}
-Mapping generators: ~2, ~11/10
+Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.645
-POTE generator: ~11/10 = 154.645
 Minimax tuning:
 * [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
-: Eigenmonzos (unchanged intervals): 2, 11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
-Optimal GPV sequence: {{Val list| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
+{{Optimal ET sequence|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
 Badness: 0.025621
@@ Line 370: / Line 185: @@
 Comma list: 66/65, 99/98, 121/120, 126/125
-Mapping: [{{val| 1 3 4 5 5 5 }}, {{val| 0 -11 -13 -17 -12 -10 }}]
+Mapping: {{mapping| 1 3 4 5 5 5 | 0 -11 -13 -17 -12 -10 }}
-POTE generator: ~11/10 = 154.478
+Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.478
-Optimal GPV sequence: {{Val list| 8d, 23de, 31, 70f, 101ff }}
+{{Optimal ET sequence|legend=1| 8d, 23de, 31, 70f, 101ff }}
 Badness: 0.023323
@@ Line 382: / Line 197: @@
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Oolong]].''
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 117649/116640
-[[Mapping]]: [{{val| 1 6 7 8 }}, {{val| 0 -17 -18 -20 }}]
+{{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}
-{{Multival|legend=1| 17 18 20 -11 -16 -4 }}
-[[POTE generator]]: ~6/5 = 311.679
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 311.679
-{{Val list|legend=1| 27, 50, 77 }}
+{{Optimal ET sequence|legend=1| 27, 50, 77 }}
 [[Badness]]: 0.073509
@@ Line 401: / Line 214: @@
 Comma list: 126/125, 176/175, 26411/26244
-Mapping: [{{val| 1 6 7 8 18 }}, {{val| 0 -17 -18 -20 -56 }}]
+Mapping: {{mapping| 1 6 7 8 18 | 0 -17 -18 -20 -56 }}
-POTE generator: ~6/5 = 311.587
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.587
-Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}
+{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}
 Badness: 0.056915
@@ Line 414: / Line 227: @@
 Comma list: 126/125, 176/175, 196/195, 13013/12960
-Mapping: [{{val| 1 6 7 8 18 5 }}, {{val| 0 -17 -18 -20 -56 -5 }}]
+Mapping: {{mapping| 1 6 7 8 18 5 | 0 -17 -18 -20 -56 -5 }}
-POTE generator: ~6/5 = 311.591
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.591
-Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}
+{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}
 Badness: 0.035582
@@ Line 425: / Line 238: @@
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Vines]].''
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 84035/82944
-[[Mapping]]: [{{val| 2 7 8 8 }}, {{val| 0 -8 -7 -5 }}]
+{{Mapping|legend=1| 2 7 8 8 | 0 -8 -7 -5 }}
-[[POTE generator]]: ~6/5 = 312.602
+[[Optimal tuning]] ([[POTE]]): 1\2, ~6/5 = 312.602
-{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}
+{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}
 [[Badness]]: 0.078049
@@ Line 442: / Line 255: @@
 Comma list: 126/125, 385/384, 2401/2376
-Mapping: [{{val| 2 7 8 8 5 }}, {{val| 0 -8 -7 -5 4 }}]
+Mapping: {{mapping| 2 7 8 8 5 | 0 -8 -7 -5 4 }}
-POTE generator: ~6/5 = 312.601
+Optimal tuning (POTE): 1\2, ~6/5 = 312.601
-Optimal GPV sequence: {{Val list| 42, 46, 96d, 142d, 238dd }}
+{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}
 Badness: 0.044499
@@ Line 455: / Line 268: @@
 Comma list: 126/125, 196/195, 364/363, 385/384
-Mapping: [{{val| 2 7 8 8 5 5 }}, {{val| 0 -8 -7 -5 4 5 }}]
+Mapping: {{mapping| 2 7 8 8 5 5 | 0 -8 -7 -5 4 5 }}
-POTE generator: ~6/5 = 312.564
+Optimal tuning (POTE): 1\2, ~6/5 = 312.564
-Optimal GPV sequence: {{Val list| 42, 46, 96d, 238ddf }}
+{{Optimal ET sequence|legend=1| 42, 46, 96d, 238ddf }}
 Badness: 0.029693
@@ Line 466: / Line 279: @@
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Kumonga]].''
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 12288/12005
-[[Mapping]]: [{{val| 1 4 4 3 }}, {{val| 0 -13 -9 -1 }}]
+{{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}
-{{Multival|legend=1| 13 9 1 -16 -35 -23 }}
-[[POTE generator]]: ~8/7 = 222.797
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 222.797
-{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}
+{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}
 [[Badness]]: 0.087500
@@ Line 485: / Line 296: @@
 Comma list: 126/125, 176/175, 864/847
-Mapping: [{{val| 1 4 4 3 7 }}, {{val| 0 -13 -9 -1 -19 }}]
+Mapping: {{mapping| 1 4 4 3 7 | 0 -13 -9 -1 -19 }}
-POTE generator: ~8/7 = 222.898
+Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.898
-Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e }}
+{{Optimal ET sequence|legend=1| 16, 27e, 43, 70e }}
 Badness: 0.043336
@@ Line 498: / Line 309: @@
 Comma list: 78/77, 126/125, 144/143, 176/175
-Mapping: [{{val| 1 4 4 3 7 5 }}, {{val| 0 -13 -9 -1 -19 -7 }}]
+Mapping: {{mapping| 1 4 4 3 7 5 | 0 -13 -9 -1 -19 -7 }}
-POTE generator: ~8/7 = 222.961
+Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.961
-Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e, 113cdee }}
+{{Optimal ET sequence|legend=1| 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]]: [{{val| 1 -4 0 7 }}, {{val| 0 12 5 -9 }}]
-{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}
-[[POTE generator]]: ~175/128 = 558.605
-{{Val list|legend=1| 15, 43, 58 }}
-[[Badness]]: 0.088441
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 176/175, 1344/1331
-Mapping: [{{val| 1 -4 0 7 3 }}, {{val| 0 12 5 -9 1 }}]
-POTE generator: ~11/8 = 558.620
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 -4 0 7 3 -7 }}, {{val| 0 12 5 -9 1 23 }}]
-POTE generator: ~11/8 = 558.589
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 -4 0 7 3 -7 12 }}, {{val| 0 12 5 -9 1 23 -17 }}]
-POTE generator: ~11/8 = 558.509
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 -4 0 7 3 -7 12 1 }}, {{val| 0 12 5 -9 1 23 -17 7 }}]
-POTE generator: ~11/8 = 558.504
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 -4 0 7 3 -7 12 1 5 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 }}]
-POTE generator: ~11/8 = 558.522
-Optimal GPV sequence: {{Val list| 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: [{{val| 1 -4 0 7 3 -7 12 1 5 3 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 4 }}]
-POTE generator: ~11/8 = 558.520
-Optimal GPV sequence: {{Val list| 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 19683/19208
-[[Mapping]]: [{{val| 1 7 10 15 }}, {{val| 0 -12 -17 -27 }}]
+{{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}
-{{Multival|legend=1| 12 17 27 -1 9 15 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~135/98 = 541.828
-[[POTE generator]]: ~135/98 = 541.828
+{{Optimal ET sequence|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}
-{{Val list|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}
 [[Badness]]: 0.099801
@@ Line 625: / Line 337: @@
 Comma list: 99/98, 126/125, 243/242
-Mapping: [{{val| 1 7 10 15 17 }}, {{val| 0 -12 -17 -27 -30 }}]
+Mapping: {{mapping| 1 7 10 15 17 | 0 -12 -17 -27 -30 }}
-POTE generator: ~15/11 = 541.772
+Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.772
-Optimal GPV sequence: {{Val list| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
+{{Optimal ET sequence|legend=1| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
 Badness: 0.042719
@@ Line 636: / Line 348: @@
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 99/98. 126/125, 243/242
+Comma list: 66/65, 99/98, 126/125, 243/242
-Mapping: [{{val| 1 7 10 15 17 15 }}, {{val| 0 -12 -17 -27 -30 -25 }}]
+Mapping: {{mapping| 1 7 10 15 17 15 | 0 -12 -17 -27 -30 -25 }}
-POTE generator: ~15/11 = 541.778
+Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.778
-Optimal GPV sequence: {{Val list| 11cdeef, 20cdef, 31 }}
+{{Optimal ET sequence|legend=1| 11cdeef, 20cdef, 31 }}
 Badness: 0.037849
 == Bisemidim ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 118098/117649
-[[Mapping]]: [{{val| 2 1 2 2 }}, {{val| 0 9 11 15 }}]
+{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
-{{Multival|legend=1| 18 22 30 -7 -3 8 }}
-[[POTE generator]]: ~35/27 = 455.445
+[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~35/27 = 455.445
-{{Val list|legend=1| 50, 58, 108, 166c, 408ccc }}
+{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
 [[Badness]]: 0.097786
@@ Line 666: / Line 376: @@
 Comma list: 126/125, 540/539, 1344/1331
-Mapping: [{{val| 2 1 2 2 5 }}, {{val| 0 9 11 15 8 }}]
+Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
-POTE generator: ~35/27 = 455.373
+Optimal tuning (POTE): ~99/70 = 1\2, ~35/27 = 455.373
-Optimal GPV sequence: {{Val list| 50, 58, 108, 166ce, 224cee }}
+{{Optimal ET sequence|legend=1| 50, 58, 108, 166ce, 224cee }}
 Badness: 0.041190
@@ Line 679: / Line 389: @@
 Comma list: 126/125, 144/143, 196/195, 364/363
-Mapping: [{{val| 2 1 2 2 5 5 }}, {{val| 0 9 11 15 8 10 }}]
+Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
-POTE generator: ~35/27 = 455.347
+Optimal tuning (POTE): ~55/39 = 1\2, ~13/10 = 455.347
-Optimal GPV sequence: {{Val list| 50, 58, 166cef, 224ceeff }}
+{{Optimal ET sequence|legend=1| 50, 58, 166cef, 224ceeff }}
 Badness: 0.023877
@@ Line 690: / Line 400: @@
 : ''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&amp;73. 74\135 or 91\166 supply good tunings for the generator, and 20 and 31 note MOS are available.
+Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described as {{nowrap|31 &amp; 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.
-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<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 589824/588245
-[[Mapping]]: [{{val| 1 12 10 5 }}, {{val| 0 -19 -14 -4 }}]
+{{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}
-{{Multival|legend=1| 19 14 4 -22 -47 -30 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/24 = 657.818
-[[POTE generator]]: ~35/24 = 657.818
+{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
-{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
 [[Badness]]: 0.101191
@@ Line 713: / Line 423: @@
 Comma list: 126/125, 385/384, 2420/2401
-Mapping: [{{val| 1 12 10 5 4 }}, {{val| 0 -19 -14 -4 -1 }}]
+Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}
-POTE generator: ~16/11 = 657.923
+Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923
-Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}
+{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}
 Badness: 0.067291
@@ Line 726: / Line 436: @@
 Comma list: 126/125, 196/195, 385/384, 2420/2401
-Mapping: [{{val| 1 12 10 5 4 7 }}, {{val| 0 -19 -14 -4 -1 -6 }}]
+Mapping: {{mapping| 1 12 10 5 4 7 | 0 -19 -14 -4 -1 -6 }}
-POTE generator: ~16/11 = 657.854
+Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.854
-Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}
+{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}
 === Marrakesh ===
@@ Line 737: / Line 447: @@
 Comma list: 126/125, 176/175, 14641/14580
-Mapping: [{{val| 1 12 10 5 21 }}, {{val| 0 -19 -14 -4 -32 }}]
+Mapping: {{mapping| 1 12 10 5 21 | 0 -19 -14 -4 -32 }}
-POTE generator: ~22/15 = 657.791
+Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791
-Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c }}
+{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c }}
 Badness: 0.040539
@@ Line 750: / Line 460: @@
 Comma list: 126/125, 176/175, 196/195, 14641/14580
-Mapping: [{{val| 1 12 10 5 21 -10 }}, {{val| 0 -19 -14 -4 -32 25 }}]
+Mapping: {{mapping| 1 12 10 5 21 -10 | 0 -19 -14 -4 -32 25 }}
-POTE generator: ~22/15 = 657.756
+Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.756
-Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c, 239ccf }}
+{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c, 239ccf }}
 Badness: 0.040774
@@ Line 763: / Line 473: @@
 Comma list: 126/125, 144/143, 176/175, 1540/1521
-Mapping: [{{val| 1 12 10 5 21 7 }}, {{val| 0 -19 -14 -4 -32 -6 }}]
+Mapping: {{mapping| 1 12 10 5 21 7 | 0 -19 -14 -4 -32 -6 }}
-POTE generator: ~22/15 = 657.700
+Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.700
-Optimal GPV sequence: {{Val list| 31, 104cff, 135cff }}
+{{Optimal ET sequence|legend=1| 31, 104cff, 135cff }}
 Badness: 0.041395
 == Amigo ==
-{{see also| High badness temperaments #Magus }}
+{{See also| High badness temperaments #Magus }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 2097152/2083725
-[[Mapping]]: [{{val| 1 -2 2 9 }}, {{val| 0 11 1 -19 }}]
+{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
-{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.094
-[[POTE generator]]: ~5/4 = 391.094
+{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}
-{{Val list|legend=1| 43, 46, 89, 135c, 359cc }}
 [[Badness]]: 0.110873
@@ Line 793: / Line 501: @@
 Comma list: 126/125, 176/175, 16384/16335
-Mapping: [{{val| 1 -2 2 9 9 }}, {{val| 0 11 1 -19 -17 }}]
+Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}
-POTE generator: ~5/4 = 391.075
+Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.075
-Optimal GPV sequence: {{Val list| 43, 46, 89, 135c, 224c }}
+{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 224c }}
 Badness: 0.043438
@@ Line 806: / Line 514: @@
 Comma list: 126/125, 169/168, 176/175, 364/363
-Mapping: [{{val| 1 -2 2 9 9 5 }}, {{val| 0 11 1 -19 -17 -4 }}]
+Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}
-POTE generator: ~5/4 = 391.073
+Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.073
-Optimal GPV sequence: {{Val list| 43, 46, 89, 135cf, 224cf }}
+{{Optimal ET sequence|legend=1| 43, 46, 89, 135cf, 224cf }}
 Badness: 0.030666
+== Gilead ==
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 343/324
+{{Mapping|legend=1| 1 4 5 6 | 0 -9 -10 -12 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1\1, ~6/5 = 321.109
+* [[POTE]]: ~2 = 1\1, ~6/5 = 321.423
+{{Optimal ET sequence|legend=1| 11cd, 15, 41dd, 56dd }}
+[[Badness]]: 0.115292
 == Supersensi ==
-Supersensi (8d &amp; 43) has supermajor third as a generator like [[sensi]], but the no-fives comma 17496/16807 rather than 245/243 tempered out.
+Supersensi ({{nowrap|8d &amp; 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 17496/16807
-[[Mapping]]: [{{val| 1 -4 -4 -5 }}, {{val| 0 15 17 21 }}]
+{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
-{{Multival|legend=1| 15 17 21 -8 -9 1 }}
-[[POTE generator]]: ~343/270 = 446.568
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/270 = 446.568
-{{Val list|legend=1| 8d, 35, 43 }}
+{{Optimal ET sequence|legend=1| 8d, 35, 43 }}
 [[Badness]]: 0.148531
@@ Line 836: / Line 557: @@
 Comma list: 99/98, 126/125, 864/847
-Mapping: [{{val| 1 -4 -4 -5 -1 }}, {{val| 0 15 17 21 12 }}]
+Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
-POTE generator: ~72/55 = 446.616
+Optimal tuning (POTE): ~2 = 1\1, ~72/55 = 446.616
-Optimal GPV sequence: {{Val list| 8d, 35, 43 }}
+{{Optimal ET sequence|legend=1| 8d, 35, 43 }}
 Badness: 0.059449
@@ Line 849: / Line 570: @@
 Comma list: 78/77, 99/98, 126/125, 144/143
-Mapping: [{{val| 1 -4 -4 -5 -1 -3 }}, {{val| 0 15 17 21 12 18 }}]
+Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
-POTE generator: ~13/10 = 446.598
+Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 446.598
-Optimal GPV sequence: {{Val list| 8d, 35f, 43 }}
+{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}
 Badness: 0.035258
@@ Line 862: / Line 583: @@
 Comma list: 78/77, 99/98, 120/119, 126/125, 144/143
-Mapping: [{{val| 1 -4 -4 -5 -1 -3 0 }}, {{val| 0 15 17 21 12 18 11 }}]
+Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
-POTE generator: ~13/10 = 446.631
+Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 446.631
-Optimal GPV sequence: {{Val list| 8d, 35f, 43 }}
+{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}
 Badness: 0.025907
 == Cobalt ==
-The name of ''cobalt temperament'' comes from Cobalt, the 27th element.
+The name of the cobalt temperament comes from the 27th element.
-Cobalt (27 &amp; 81) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]].
+Cobalt ({{nowrap|27 &amp; 81}}) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]].
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 40353607/40310784
-[[Mapping]]: [{{val|27 43 63 76}}, {{val|0 -1 -1 -1}}]
+{{Mapping|legend=1| 27 43 63 76 | 0 -1 -1 -1 }}
-[[POTE generator]]: ~3/2 = 701.244
+[[Optimal tuning]] ([[POTE]]): 1\27, ~3/2 = 701.244
-{{Val list|legend=1| 27, 81, 108, 135c, 243c }}
+{{Optimal ET sequence|legend=1| 27, 81, 108, 135c, 243c }}
 [[Badness]]: 0.173308
@@ Line 892: / Line 613: @@
 Comma list: 126/125, 540/539, 21609/21296
-Mapping: [{{val|27 43 63 76 94}}, {{val|0 -1 -1 -1 -2}}]
+Mapping: {{mapping| 27 43 63 76 94 | 0 -1 -1 -1 -2 }}
-POTE generator: ~3/2 = 700.001
+Optimal tuning (POTE): 1\27, ~3/2 = 700.001
-Optimal GPV sequence: {{Val list| 27e, 81, 108 }}
+{{Optimal ET sequence|legend=1| 27e, 81, 108 }}
 Badness: 0.078060
@@ Line 905: / Line 626: @@
 Comma list: 126/125, 144/143, 196/195, 21609/21296
-Mapping: [{{val|27 43 63 76 94 100}}, {{val|0 -1 -1 -1 -2 0}}]
+Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 0 }}
-POTE generator: ~3/2 = 700.867
+Optimal tuning (POTE): 1\27, ~3/2 = 700.867
-Optimal GPV sequence: {{Val list| 27e, 81, 108, 243ceef }}
+{{Optimal ET sequence|legend=1| 27e, 81, 108, 243ceef }}
 Badness: 0.057145
@@ Line 918: / Line 639: @@
 Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445
-Mapping: [{{val|27 43 63 76 94 100 111}}, {{val|0 -1 -1 -1 -2 0 -2}}]
+Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -2 }}
-POTE generator: ~3/2 = 700.397
+Optimal tuning (POTE): 1\27, ~3/2 = 700.397
-Optimal GPV sequence: {{Val list| 27eg, 81, 108g }}
+{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}
 Badness: 0.042106
@@ Line 931: / Line 652: @@
 Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968
-Mapping: [{{val|27 43 63 76 94 100 111 115}}, {{val|0 -1 -1 -1 -2 0 -2 -1}}]
+Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -2 -1 }}
-POTE generator: ~3/2 = 700.429
+Optimal tuning (POTE): 1\27, ~3/2 = 700.429
-Optimal GPV sequence: {{Val list| 27eg, 81, 108g }}
+{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}
 Badness: 0.030415
@@ Line 944: / Line 665: @@
 Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968
-Mapping: [{{val|27 43 63 76 94 100 111}}, {{val|0 -1 -1 -1 -2 0 -3}}]
+Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -3 }}
-POTE generator: ~3/2 = 701.595
+Optimal tuning (POTE): 1\27, ~3/2 = 701.595
-Optimal GPV sequence: {{Val list| 27eg, 81gg, 108, 135ce }}
+{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ce }}
 Badness: 0.047163
@@ Line 957: / Line 678: @@
 Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083
-Mapping: [{{val|27 43 63 76 94 100 111 115}}, {{val|0 -1 -1 -1 -2 0 -3 -1}}]
+Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -3 -1 }}
-POTE generator: ~3/2 = 701.673
+Optimal tuning (POTE): 1\27, ~3/2 = 701.673
-Optimal GPV sequence: {{Val list| 27eg, 81gg, 108, 135ceh }}
+{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ceh }}
 Badness: 0.034176
@@ Line 970: / Line 691: @@
 Comma list: 126/125, 169/168, 540/539, 975/968
-Mapping: [{{val|27 43 63 76 94 100}}, {{val|0 -1 -1 -1 -2 -1}}]
+Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 -1 }}
-POTE generator: ~3/2 = 699.179
+Optimal tuning (POTE): 1\27, ~3/2 = 699.179
-Optimal GPV sequence: {{Val list| 27e, 54bdef, 81f, 108f }}
+{{Optimal ET sequence|legend=1| 27e, 54bdef, 81f, 108f }}
 Badness: 0.052732
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Starling temperaments| ]] <!-- main article -->
 [[Category:Myna]]
 [[Category:Rank 2]]

Starling temperaments: Difference between revisions

Latest revision as of 12:40, 21 August 2025

Myna

11-limit

13-limit

Minah

Maneh

Myno

Coleto

Nusecond

11-limit

13-limit

Oolong

11-limit

13-limit

Vines

11-limit

13-limit

Kumonga

11-limit

13-limit

Cypress

11-limit

13-limit

Bisemidim

11-limit

13-limit

Casablanca

11-limit

13-limit

Marrakesh

13-limit

Murakuc

Amigo

11-limit

13-limit

Gilead

Supersensi

11-limit

13-limit

17-limit

Cobalt

11-limit

13-limit

Cobaltous

19-limit

Cobaltic

19-limit

Cobaltite

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