Marvel temperaments: Difference between revisions

Revision as of 11:05, 18 February 2026

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 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

Pelogic → Mavila family (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3)
Meantone → Meantone family (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3)
Garibaldi → Schismatic family (+3125/3087, generated by the fifth with 5/4 mapped to the d4)
Pajara → Diaschismic family (+50/49 or 64/63, generated by the fifth with a semioctave period)
Sharpie → Dicot family (+25/24 or 28/27, fifth sliced in two)
Immune → Immunity family (+781250/750141, twelfth sliced in two)
August → Augmented family (+36/35 or 128/125, generated by the fifth with a 1/3-octave period)
Fog → Misty family (+156250/151263, generated by the fifth with a 1/3-octave period)
Negri → Slendro clan (+49/48, fourth sliced in four)
Magic → Magic family (+245/243, twelfth sliced in five)
Passive → Passion family (+256/245, fourth sliced in five)
Quintapole → Quintaleap family (+7812500/7411887, fourth sliced in five)
Houborizic → Amity family (+1250000/1240029, eleventh sliced in five)
Qintosec → Quintosec family (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
Miracle → Gamelismic clan (+1029/1024, fifth sliced in six)
Catakleismic → Kleismic family (+4375/4374, twelfth sliced in six)
Marvo → Gravity family (+78125000/78121827, two octaves and a fifth sliced in six)
Orwell → Semicomma family (+1728/1715, twelfth sliced in seven)
Snipes → Wesley family (+6125/5832, two octaves and a fourth sliced in seven)
Submajor → Buzzardsmic clan (+65536/64827, two octaves and a fourth sliced in eight)
Escapist → Escapade family (+65625/65536, fourth sliced in nine)
Decic → Cloudy clan (+16807/16384, generated by the fifth with a 1/10-octave period)
Amavil → Mabila family (+17496/16807, four octaves and a fourth sliced in ten)
Betic → Sycamore family (+1071875/1062882, fifth sliced in eleven)
Hendeca → 11th-octave temperaments (+122880/117649, generated by the fifth with a 1/11-octave period)
Compton → Compton family (+250047/250000, generated by the classical major third with a 1/12-octave period)
Raccoon → Vavoom family (+41943040/40353607, twelfth sliced in seventeen)
Maquila → Maquila family (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
Gammy → Gammic family (+94143178827/91913281250, fifth sliced in twenty)

Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, alphorn, tertiosec, gwazy, and gracecordial.

Since (5/4)² = (225/224)⋅(14/9), these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. 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 two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.

The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.

Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)

Wizard

Main article: Wizard

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

Wizard has a semi-octave period and is generated by an interval that is best treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as 22 & 72. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the perfect twelfth. 72edo, 94edo, and especially 166edo are good tunings for it.

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tunings:

CTE: ~1225/864 = 600.000 ¢, ~245/216 = 216.919 ¢

error map: ⟨0.000 -0.443 -3.232 +0.361]

POTE: ~1225/864 = 600.000 ¢, ~245/216 = 216.744 ¢

error map: ⟨0.000 -1.492 -3.057 -1.388]

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c, 382c

Badness (Smith): 0.040846

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4000/3993

Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Optimal tunings:

CTE: ~99/70 = 600.000 ¢, ~25/22 = 216.900 ¢
POTE: ~99/70 = 600.000 ¢, ~25/22 = 216.768 ¢

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c

Badness (Smith): 0.018539

Lizard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Optimal tunings:

CTE: ~55/39 = 600.000 ¢, ~25/22 = 216.703 ¢
POTE: ~55/39 = 600.000 ¢, ~25/22 = 216.711 ¢

Optimal ET sequence: 22, 50, 72, 122, 194df

Badness (Smith): 0.021781

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Optimal tunings:

CTE: ~17/12 = 600.000 ¢, ~17/15 = 216.748 ¢
POTE: ~17/12 = 600.000 ¢, ~17/15 = 216.619 ¢

Optimal ET sequence: 22, 50, 72, 122g, 194dfg

Badness (Smith): 0.014536

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Optimal tunings:

CTE: ~17/12 = 600.000 ¢, ~17/15 = 216.677 ¢
POTE: ~17/12 = 600.000 ¢, ~17/15 = 216.523 ¢

Optimal ET sequence: 22h, 50, 72, 122g, 194dfg

Badness (Smith): 0.015702

Gizzard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Optimal tunings:

CTE: ~99/70 = 600.000 ¢, ~25/22 = 216.928 ¢
POTE: ~99/70 = 600.000 ¢, ~25/22 = 216.830 ¢

Optimal ET sequence: 22f, 72, 166, 238cf

Badness (Smith): 0.020252

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 289/288, 325/324, 375/374, 385/384

Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Optimal tunings:

CTE: ~17/12 = 600.000 ¢, ~17/15 = 216.943 ¢
POTE: ~17/12 = 600.000 ¢, ~17/15 = 216.825 ¢

Optimal ET sequence: 22f, 72, 166g, 238cfg

Badness (Smith): 0.013624

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594

Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Optimal tunings:

CTE: ~17/12 = 600.000 ¢, ~17/15 = 216.918 ¢
POTE: ~17/12 = 600.000 ¢, ~17/15 = 216.862 ¢

Optimal ET sequence: 72, 94, 166g

Badness (Smith): 0.014810

Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Optimal tunings:

CTE: ~77/54 = 600.000 ¢, ~55/48 = 217.247 ¢
POTE: ~77/54 = 600.000 ¢, ~55/48 = 216.876 ¢

Optimal ET sequence: 22, 50e, 72ee

Badness (Smith): 0.057799

Tritonic

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 50421/50000

Mapping: [⟨1 4 -3 -3], ⟨0 -5 11 12]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 580.286 ¢

Optimal ET sequence: 29, 31, 60, 91, 122, 213bcd

Badness (Smith): 0.047578

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 441/440

Mapping: [⟨1 4 -3 -3 2], ⟨0 -5 11 12 3]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 580.267 ¢

Optimal ET sequence: 29, 31, 60e

Badness (Smith): 0.023659

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 275/273

Mapping: [⟨1 4 -3 -3 2 -5], ⟨0 -5 11 12 3 18]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 580.108 ¢

Optimal ET sequence: 29, 31, 60e

Badness (Smith): 0.022993

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 273/272

Mapping: [⟨1 4 -3 -3 2 -5 -8], ⟨0 -5 11 12 3 18 25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 580.055 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Smith): 0.019093

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272

Mapping: [⟨1 4 -3 -3 2 -5 -8 -3], ⟨0 -5 11 12 3 18 25 15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 580.026 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Smith): 0.017010

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195

Mapping: [⟨1 4 -3 -3 2 -5 -8 -3 5], ⟨0 -5 11 12 3 18 25 15 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 580.009 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Smith): 0.014471

Tritoni

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 27783/27500

Mapping: [⟨1 4 -3 -3 17], ⟨0 -5 11 12 -28]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 580.389 ¢

Optimal ET sequence: 31, 91, 122, 153d

Badness (Smith): 0.045456

Septimin

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Septimin.

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping: [⟨1 4 1 5], ⟨0 -11 6 -10]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/6 = 263.632 ¢

Optimal ET sequence: 41, 91, 132d

Badness (Smith): 0.054502

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 385/384

Mapping: [⟨1 4 1 5 5], ⟨0 -11 6 -10 -7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/6 = 263.634 ¢

Optimal ET sequence: 41, 91, 223cdef

Badness (Smith): 0.031309

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 245/242

Mapping: [⟨1 4 1 5 5 7], ⟨0 -11 6 -10 -7 -15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/6 = 263.700 ¢

Optimal ET sequence: 41, 91

Badness (Smith): 0.023117

Merman

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Merman.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 5 -5 -5], ⟨0 -7 15 16]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 585.585 ¢

Optimal ET sequence: 41, 84, 125

Badness (Smith): 0.055078

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1344/1331

Mapping: [⟨1 5 -5 -5 2], ⟨0 -7 15 16 3]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 585.606 ¢

Optimal ET sequence: 41, 84, 125e

Badness (Smith): 0.036383

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 225/224, 364/363, 441/440

Mapping: [⟨1 5 -5 -5 2 12], ⟨0 -7 15 16 3 -17]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 585.657 ¢

Optimal ET sequence: 41, 84, 125e, 209ef, 293ef

Badness (Smith): 0.027544

Slender

Slender (31 & 32) tempers out the hewuermera comma in addition to the marvel comma. This temperament has a generator of 49/48, 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.

Subgroup: 2.3.5.7

Comma list: 225/224, 589824/588245

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 38.413 ¢

Optimal ET sequence: 31, 94, 125

Badness (Smith): 0.056934

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1331/1323

Mapping: [⟨1 2 2 3 4], ⟨0 -13 10 -6 -17]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 38.387 ¢

Optimal ET sequence: 31, 63, 94, 125

Badness (Smith): 0.025342

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 385/384, 1331/1323

Mapping: [⟨1 2 2 3 4 3], ⟨0 -13 10 -6 -17 22]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 38.314 ¢

Optimal ET sequence: 31, 63, 94

Badness (Smith): 0.025913

Triton

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Stump.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 568.865 ¢

Optimal ET sequence: 2, 17d, 19, 78bd, 97bd

Badness (Smith): 0.059245

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 1029/1000

Mapping: [⟨1 0 6 7 4], ⟨0 3 -7 -8 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 569.144 ¢

Optimal ET sequence: 2, 17d, 19, 59bde, 78bde, 97bde

Badness (Smith): 0.045675

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.348 ¢

Optimal ET sequence: 29, 43, 72, 619bcd, 691bcd

Badness (Smith): 0.083338

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4000/3993

Mapping: [⟨1 2 1 1 2], ⟨0 -6 19 26 21]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.340 ¢

Optimal ET sequence: 29, 43, 72

Badness (Smith): 0.028965

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 441/440

Mapping: [⟨1 2 1 1 2 3], ⟨0 -6 19 26 21 10]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.330 ¢

Optimal ET sequence: 29, 43, 72, 115f

Badness (Smith): 0.021470

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 364/363, 441/440

Mapping: [⟨1 2 1 1 2 3 2], ⟨0 -6 19 26 21 10 30]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.330 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Smith): 0.014924

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440

Mapping: [⟨1 2 1 1 2 3 2 3], ⟨0 -6 19 26 21 10 30 18]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.330 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Smith): 0.014722

Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

Mapping generators: ~2592/2401, ~12005/6912

Optimal tuning (POTE): ~2592/2401 = 133.3333 ¢, ~12005/6912 = 950.1680 ¢ (~1728/1715 = 16.8347 ¢)

Optimal ET sequence: 9, 63, 72, 495bcd

Badness (Smith): 0.093679

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12005/11979

Mapping: [⟨9 0 28 11 24], ⟨0 2 -1 2 1]]

Optimal tuning (POTE): ~121/112 = 133.3333 ¢, ~210/121 = 950.1873 ¢ (~99/98 = 16.8540 ¢)

Optimal ET sequence: 9, 63, 72, 423cd, 495bcd

Badness (Smith): 0.030426

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 1716/1715

Mapping: [⟨9 0 28 11 24 19], ⟨0 2 -1 2 1 2]]

Optimal tuning (POTE): ~14/13 = 133.3333 ¢, ~26/15 = 950.2867 ¢ (~105/104 = 16.9534 ¢)

Optimal ET sequence: 9, 63, 72, 279cf

Badness (Smith): 0.022322

Gracecordial

For the 5-limit version of this temperament, see Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit).

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.824 ¢

Optimal ET sequence: 12, 113, 125, 238c, 363c

Badness (Smith): 0.096279

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 236328125/234365481

Mapping: [⟨1 0 34 63 -90], ⟨0 1 -20 -38 59]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.834 ¢

Optimal ET sequence: 12e, 101cde, 113, 125, 238c

Badness (Smith): 0.089588

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 831875/830466

Mapping: [⟨1 0 34 63 -90 -66], ⟨0 1 -20 -38 59 44]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.841 ¢

Optimal ET sequence: 12e, 101cde, 113, 125f, 238cf

Badness (Smith): 0.052235

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913

Mapping: [⟨1 0 34 63 -90 -66 -7], ⟨0 1 -20 -38 59 44 7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.841 ¢

Optimal ET sequence: 12e, 101cde, 113, 125f, 238cf

Badness (Smith): 0.038565

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444

Mapping: [⟨1 0 34 63 -90 -66 -7 9], ⟨0 1 -20 -38 59 44 7 -3]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.842 ¢

Optimal ET sequence: 12e, 101cde, 113, 125f, 238cf

Badness (Smith): 0.028165

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43], ⟨0 1 -20 -38 59 44 7 -3 30]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.843 ¢

Optimal ET sequence: 12e, 101cde, 113, 238cfi

Badness (Smith): 0.021879

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49], ⟨0 1 -20 -38 59 44 7 -3 30 34]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.842 ¢

Optimal ET sequence: 12e, 101cde, 113, 125f, 238cfi

Badness (Smith): 0.018011

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49 -79], ⟨0 1 -20 -38 59 44 7 -3 30 34 53]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.838 ¢

Optimal ET sequence: 12e, 101cdek, 113, 125f, 238cfi

Badness (Smith): 0.016007

Gracecord

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 109375/107811

Mapping: [⟨1 0 34 63 89], ⟨0 1 -20 -38 -54]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.885 ¢

Optimal ET sequence: 12, 101cd, 113

Badness (Smith): 0.066964

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 441/440, 6125/6084

Mapping: [⟨1 0 34 63 89 113], ⟨0 1 -20 -38 -54 -69]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.890 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Smith): 0.044196

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989

Mapping: [⟨1 0 34 63 89 113 -7], ⟨0 1 -20 -38 -54 -69 7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.870 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Smith): 0.036637

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663

Mapping: [⟨1 0 34 63 89 113 -7 9], ⟨0 1 -20 -38 -54 -69 7 -3]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 700.866 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Smith): 0.027559

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 9 0 13], ⟨0 -16 5 -22]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~48/35 = 556.221 ¢

Optimal ET sequence: 28d, 41, 151cd, 192cd, 233cd

Badness (Smith): 0.129258

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12250/11979

Mapping: [⟨1 9 0 13 3], ⟨0 -16 5 -22 1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/8 = 556.144 ¢

Optimal ET sequence: 28d, 41, 315cde

Badness (Smith): 0.073459

Misneb

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Misneb.

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 3 1 3], ⟨0 -15 14 -2]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/15 = 113.235 ¢

Optimal ET sequence: 21, 32, 53

Badness (Smith): 0.140970

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1310720/1294139

Mapping: [⟨1 3 1 3 1], ⟨0 -15 14 -2 26]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/15 = 113.323 ¢

Optimal ET sequence: 21, 32e, 53, 127, 180de

Badness (Smith): 0.085390

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 640/637, 847/845

Mapping: [⟨1 3 1 3 1 2], ⟨0 -15 14 -2 26 18]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/15 = 113.323 ¢

Optimal ET sequence: 21, 32e, 53, 127, 180de

Badness (Smith): 0.045569

Musneb

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 66550/64827

Mapping: [⟨1 3 1 3 6], ⟨0 -15 14 -2 -27]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/15 = 113.142 ¢

Optimal ET sequence: 32, 53, 191de, 244cddee, 297cddee

Badness (Smith): 0.087333

Untriton

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Untriton.

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 6 -7 -7], ⟨0 -9 19 20]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 588.641 ¢

Optimal ET sequence: 51, 53, 316cd, 369cdd, 422cdd

Badness (Smith): 0.143976

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 22000/21609

Mapping: [⟨1 6 -7 -7 1], ⟨0 -9 19 20 5]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 588.626 ¢

Optimal ET sequence: 51, 53

Badness (Smith): 0.074295

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 1040/1029

Mapping: [⟨1 6 -7 -7 1 -12], ⟨0 -9 19 20 5 32]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~7/5 = 588.654 ¢

Optimal ET sequence: 51f, 53

Badness (Smith): 0.047441

Naiadical

Subgroup: 2.3.5.7

Comma list: 225/224, 823543/800000

Mapping: [⟨1 5 -3 -1], ⟨0 -9 14 10]]

Optimal tunings:

WE: ~2 = 1202.120 ¢, ~21/16 = 456.652 ¢
CWE: ~2 = 1200.000 ¢, ~21/16 = 455.868 ¢

Optimal ET sequence: 21, 29, 50, 79d

Badness (Sintel): 3.667

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 1617/1600

Mapping: [⟨1 5 -3 -1 -3], ⟨0 -9 14 10 17]]

Optimal tunings:

WE: ~2 = 1201.901, ~21/16 ¢ = 456.514 ¢
CWE: ~2 = 1200.000, ~21/16 ¢ = 455.822 ¢

Optimal ET sequence: 21, 29, 50, 79d

Badness (Sintel): 1.998

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/242, 1001/1000

Mapping: [⟨1 5 -3 -1 -3 -2], ⟨0 -9 14 10 17 15]]

Optimal tunings:

WE: ~2 = 1201.786 ¢, ~13/10 = 456.452 ¢
CWE: ~2 = 1200.000 ¢, ~13/10 = 455.807 ¢

Optimal ET sequence: 21, 29, 50, 79d

Badness (Sintel): 1.430

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 170/169, 196/195, 221/220, 245/242

Mapping: [⟨1 5 -3 -1 -3 -2 -2], ⟨0 -9 14 10 17 15 16]]

Optimal tunings:

WE: ~2 = 1201.921 ¢, ~13/10 = 456.523 ¢
CWE: ~2 = 1200.000 ¢, ~13/10 = 455.833 ¢

Optimal ET sequence: 21, 29g, 50, 79dg

Badness (Sintel): 1.260

Quintannic

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~10000/9261 = 139.838 ¢

Optimal ET sequence: 43, 60, 103, 266bcd, 369bcd

Badness (Smith): 0.150565

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 43923/43750

Mapping: [⟨1 1 5 7 8], ⟨0 5 -23 -36 -39]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~320/297 = 139.827 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Smith): 0.052590

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 441/440, 1001/1000, 1188/1183

Mapping: [⟨1 1 5 7 8 3], ⟨0 5 -23 -36 -39 6]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~13/12 = 139.812 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Smith): 0.032730

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 375/374, 441/440, 891/884

Mapping: [⟨1 1 5 7 8 3 7], ⟨0 5 -23 -36 -39 6 -25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~13/12 = 139.815 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Smith): 0.023038

Gwazy

See also: Very high accuracy temperaments § Kwazy

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]

Optimal tuning (POTE): ~2401/1728 = 600.000 ¢, ~35/32 = 162.658 ¢

Optimal ET sequence: 22, 74, 96, 118d

Badness (Smith): 0.178826

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 65536/65219

Mapping: [⟨2 1 6 4 8], ⟨0 8 -5 6 -4]]

Optimal tuning (POTE): ~363/256 = 600.000 ¢, ~11/10 = 162.592 ¢

Optimal ET sequence: 22, 74, 96, 118d

Badness (Smith): 0.068410

Tertiosec

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

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 7 5 9], ⟨0 -8 7 -2]]

Optimal tuning (POTE): ~3072/2401 = 400.000 ¢, ~15/14 = 112.283 ¢

Optimal ET sequence: 21, 54, 75, 96, 171d

Badness (Smith): 0.431636

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 3840/3773, 12005/11979

Mapping: [⟨3 7 5 9 9], ⟨0 -8 7 -2 5]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~15/14 = 112.171 ¢

Optimal ET sequence: 21, 54, 75e

Badness (Smith): 0.173485

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-This page discusses miscellaneous rank-2 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.
+This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.
 Temperaments considered in families and clans are:
@@ Line 35: / Line 35: @@
 Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, alphorn, tertiosec, gwazy, and gracecordial.
-Since {{nowrap|(5/4)<sup>2</sup> {{=}} 225/224 × 14/9}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. 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 two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
+Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. 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 two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
 The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
@@ Line 43: / Line 43: @@
 == Wizard ==
 {{Main| Wizard }}
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Wizard]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
-Wizard has a semi-octave period and is generated by an interval that is best treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as {{nowrap|22 &amp; 72}}. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
+Wizard has a semi-octave period and is generated by an interval that is best treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as {{nowrap| 22 & 72 }}. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
 [[Subgroup]]: 2.3.5.7
@@ Line 52: / Line 52: @@
 {{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
 : mapping generators: ~1225/864, ~245/216
 [[Optimal tuning]]s:
-* [[CTE]]: ~1225/864 = 600.000, ~245/216 = 216.919
+* [[CTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.919{{c}}
 : [[error map]]: {{val| 0.000 -0.443 -3.232 +0.361 }}
-* [[POTE]]: ~1225/864 = 600.000, ~245/216 = 216.744
+* [[POTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.744{{c}}
 : error map: {{val| 0.000 -1.492 -3.057 -1.388 }}
@@ Line 73: / Line 72: @@
 Optimal tunings:
-* CTE: ~99/70 = 600.000, ~25/22 = 216.900
+* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.900{{c}}
-* POTE: ~99/70 = 600.000, ~25/22 = 216.768
+* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.768{{c}}
 {{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
@@ Line 88: / Line 87: @@
 Optimal tunings:
-* CTE: ~55/39 = 600.000, ~25/22 = 216.703
+* CTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.703{{c}}
-* POTE: ~55/39 = 600.000, ~25/22 = 216.711
+* POTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.711{{c}}
 {{Optimal ET sequence|legend=0| 22, 50, 72, 122, 194df }}
@@ Line 103: / Line 102: @@
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~17/15 = 216.748
+* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.748{{c}}
-* POTE: ~17/12 = 600.000, ~17/15 = 216.619
+* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.619{{c}}
 {{Optimal ET sequence|legend=0| 22, 50, 72, 122g, 194dfg }}
@@ Line 118: / Line 117: @@
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~17/15 = 216.677
+* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.677{{c}}
-* POTE: ~17/12 = 600.000, ~17/15 = 216.523
+* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.523{{c}}
 {{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
@@ Line 133: / Line 132: @@
 Optimal tunings:
-* CTE: ~99/70 = 600.000, ~25/22 = 216.928
+* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.928{{c}}
-* POTE: ~99/70 = 600.000, ~25/22 = 216.830
+* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.830{{c}}
 {{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
@@ Line 148: / Line 147: @@
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~17/15 = 216.943
+* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.943{{c}}
-* POTE: ~17/12 = 600.000, ~17/15 = 216.825
+* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.825{{c}}
 {{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
@@ Line 163: / Line 162: @@
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~17/15 = 216.918
+* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.918{{c}}
-* POTE: ~17/12 = 600.000, ~17/15 = 216.862
+* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.862{{c}}
 {{Optimal ET sequence|legend=0| 72, 94, 166g }}
@@ Line 178: / Line 177: @@
 Optimal tunings:
-* CTE: ~77/54 = 600.000, ~55/48 = 217.247
+* CTE: ~77/54 = 600.000{{c}}, ~55/48 = 217.247{{c}}
-* POTE: ~77/54 = 600.000, ~55/48 = 216.876
+* POTE: ~77/54 = 600.000{{c}}, ~55/48 = 216.876{{c}}
 {{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
@@ Line 194: / Line 193: @@
 {{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 580.286
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 580.286{{c}}
 {{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
-[[Badness]]: 0.047578
+[[Badness]] (Smith): 0.047578
 === 11-limit ===
@@ Line 207: / Line 206: @@
 Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.267
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.267{{c}}
-{{Optimal ET sequence|legend=1| 29, 31, 60e }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e }}
-Badness: 0.023659
+Badness (Smith): 0.023659
 ==== 13-limit ====
@@ Line 220: / Line 219: @@
 Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.108
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.108{{c}}
-{{Optimal ET sequence|legend=1| 29, 31, 60e }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e }}
-Badness: 0.022993
+Badness (Smith): 0.022993
 ==== 17-limit ====
@@ Line 233: / Line 232: @@
 Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 | 0 -5 11 12 3 18 25 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.055
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.055{{c}}
-{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness: 0.019093
+Badness (Smith): 0.019093
 ==== 19-limit ====
@@ Line 246: / Line 245: @@
 Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 | 0 -5 11 12 3 18 25 15 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.026
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.026{{c}}
-{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness: 0.017010
+Badness (Smith): 0.017010
 ==== 23-limit ====
@@ Line 259: / Line 258: @@
 Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 5 | 0 -5 11 12 3 18 25 15 -1 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.009
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.009{{c}}
-{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness: 0.014471
+Badness (Smith): 0.014471
 === Tritoni ===
@@ Line 272: / Line 271: @@
 Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.389
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.389{{c}}
-{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}
+{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
-Badness: 0.045456
+Badness (Smith): 0.045456
 == Septimin ==
@@ Line 287: / Line 286: @@
 {{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/6 = 263.632
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/6 = 263.632{{c}}
 {{Optimal ET sequence|legend=1| 41, 91, 132d }}
-[[Badness]]: 0.054502
+[[Badness]] (Smith): 0.054502
 === 11-limit ===
@@ Line 300: / Line 299: @@
 Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.634
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.634{{c}}
-{{Optimal ET sequence|legend=1| 41, 91, 223cdef }}
+{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
-Badness: 0.031309
+Badness (Smith): 0.031309
 === 13-limit ===
@@ Line 313: / Line 312: @@
 Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.700
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.700{{c}}
-{{Optimal ET sequence|legend=1| 41, 91 }}
+{{Optimal ET sequence|legend=0| 41, 91 }}
-Badness: 0.023117
+Badness (Smith): 0.023117
 == Merman ==
@@ Line 328: / Line 327: @@
 {{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 585.585
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 585.585{{c}}
 {{Optimal ET sequence|legend=1| 41, 84, 125 }}
-[[Badness]]: 0.055078
+[[Badness]] (Smith): 0.055078
 === 11-limit ===
@@ Line 341: / Line 340: @@
 Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 585.606
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.606{{c}}
-{{Optimal ET sequence|legend=1| 41, 84, 125e }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e }}
-Badness: 0.036383
+Badness (Smith): 0.036383
 === 13-limit ===
@@ Line 354: / Line 353: @@
 Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 585.657
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.657{{c}}
-{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
-Badness: 0.027544
+Badness (Smith): 0.027544
 == Slender ==
-Slender ({{nowrap|31 &amp; 32}}) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma. This temperament has a generator of [[49/48]], 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.
+Slender ({{nowrap| 31 & 32 }}) tempers out the [[hewuermera comma]] in addition to the marvel comma. This temperament has a generator of [[49/48]], 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.
 [[Subgroup]]: 2.3.5.7
@@ Line 369: / Line 368: @@
 {{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 38.413
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 38.413{{c}}
 {{Optimal ET sequence|legend=1| 31, 94, 125 }}
-[[Badness]]: 0.056934
+[[Badness]] (Smith): 0.056934
 === 11-limit ===
@@ Line 382: / Line 381: @@
 Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.387
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.387{{c}}
-{{Optimal ET sequence|legend=1| 31, 63, 94, 125 }}
+{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
-Badness: 0.025342
+Badness (Smith): 0.025342
 === 13-limit ===
@@ Line 395: / Line 394: @@
 Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.314
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.314{{c}}
-{{Optimal ET sequence|legend=1| 31, 63, 94 }}
+{{Optimal ET sequence|legend=0| 31, 63, 94 }}
-Badness: 0.025913
+Badness (Smith): 0.025913
 == Triton ==
@@ Line 410: / Line 409: @@
 {{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 568.865
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 568.865{{c}}
 {{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
-[[Badness]]: 0.059245
+[[Badness]] (Smith): 0.059245
 === 11-limit ===
@@ Line 423: / Line 422: @@
 Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 569.144
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 569.144{{c}}
-{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}
+{{Optimal ET sequence|legend=0| 2, 17d, 19, 59bde, 78bde, 97bde }}
-Badness: 0.045675
+Badness (Smith): 0.045675
 == Marvolo ==
@@ Line 436: / Line 435: @@
 {{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 83.348
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 83.348{{c}}
 {{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}
-[[Badness]]: 0.083338
+[[Badness]] (Smith): 0.083338
 === 11-limit ===
@@ Line 449: / Line 448: @@
 Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.340
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.340{{c}}
-{{Optimal ET sequence|legend=1| 29, 43, 72 }}
+{{Optimal ET sequence|legend=0| 29, 43, 72 }}
-Badness: 0.028965
+Badness (Smith): 0.028965
 === 13-limit ===
@@ Line 462: / Line 461: @@
 Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
-{{Optimal ET sequence|legend=1| 29, 43, 72, 115f }}
+{{Optimal ET sequence|legend=0| 29, 43, 72, 115f }}
-Badness: 0.021470
+Badness (Smith): 0.021470
 === 17-limit ===
@@ Line 475: / Line 474: @@
 Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
-{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-Badness: 0.014924
+Badness (Smith): 0.014924
 === 19-limit ===
@@ Line 488: / Line 487: @@
 Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
-{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-Badness: 0.014722
+Badness (Smith): 0.014722
 == Enneaportent ==
@@ Line 500: / Line 499: @@
 {{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
 : Mapping generators: ~2592/2401, ~12005/6912
-[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)
+[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.1680{{c}} (~1728/1715 = 16.8347{{c}})
 {{Optimal ET sequence|legend=1| 9, 63, 72, 495bcd }}
-[[Badness]]: 0.093679
+[[Badness]] (Smith): 0.093679
 === 11-limit ===
@@ Line 516: / Line 514: @@
 Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
-Optimal tuning (POTE): ~121/112 = 133.3333, ~210/121 = 950.1873 (~99/98 = 16.8540)
+Optimal tuning (POTE): ~121/112 = 133.3333{{c}}, ~210/121 = 950.1873{{c}} (~99/98 = 16.8540{{c}})
-{{Optimal ET sequence|legend=1| 9, 63, 72, 423cd, 495bcd }}
+{{Optimal ET sequence|legend=0| 9, 63, 72, 423cd, 495bcd }}
-Badness: 0.030426
+Badness (Smith): 0.030426
 === 13-limit ===
@@ Line 529: / Line 527: @@
 Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
-Optimal tuning (POTE): ~14/13 = 133.3333, ~26/15 = 950.2867 (~105/104 = 16.9534)
+Optimal tuning (POTE): ~14/13 = 133.3333{{c}}, ~26/15 = 950.2867{{c}} (~105/104 = 16.9534{{c}})
-{{Optimal ET sequence|legend=1| 9, 63, 72, 279cf }}
+{{Optimal ET sequence|legend=0| 9, 63, 72, 279cf }}
-Badness: 0.022322
+Badness (Smith): 0.022322
 == Gracecordial ==
@@ Line 544: / Line 542: @@
 {{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 700.824
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.824{{c}}
 {{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
-[[Badness]]: 0.096279
+[[Badness]] (Smith): 0.096279
 === 11-limit ===
@@ Line 557: / Line 555: @@
 Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.834
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.834{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125, 238c }}
+{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125, 238c }}
-Badness: 0.089588
+Badness (Smith): 0.089588
 ==== 13-limit ====
@@ Line 570: / Line 568: @@
 Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}
-Badness: 0.052235
+Badness (Smith): 0.052235
 ==== 17-limit ====
@@ Line 583: / Line 581: @@
 Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}
-Badness: 0.038565
+Badness (Smith): 0.038565
 ==== 19-limit ====
@@ Line 596: / Line 594: @@
 Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.842{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}
-Badness: 0.028165
+Badness (Smith): 0.028165
 ==== 23-limit ====
@@ Line 609: / Line 607: @@
 Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.843
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.843{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 238cfi }}
-Badness: 0.021879
+Badness (Smith): 0.021879
 ==== 29-limit ====
@@ Line 622: / Line 620: @@
 Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.842{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cfi }}
-Badness: 0.018011
+Badness (Smith): 0.018011
 ==== 31-limit ====
@@ Line 635: / Line 633: @@
 Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.838
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.838{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cdek, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 101cdek, 113, 125f, 238cfi }}
-Badness: 0.016007
+Badness (Smith): 0.016007
 === Gracecord ===
@@ Line 648: / Line 646: @@
 Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.885
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.885{{c}}
-{{Optimal ET sequence|legend=1| 12, 101cd, 113 }}
+{{Optimal ET sequence|legend=0| 12, 101cd, 113 }}
-Badness: 0.066964
+Badness (Smith): 0.066964
 ==== 13-limit ====
@@ Line 661: / Line 659: @@
 Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.890
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.890{{c}}
-{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness: 0.044196
+Badness (Smith): 0.044196
 ==== 17-limit ====
@@ Line 674: / Line 672: @@
 Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.870
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.870{{c}}
-{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness: 0.036637
+Badness (Smith): 0.036637
 ==== 19-limit ====
@@ Line 687: / Line 685: @@
 Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.866
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.866{{c}}
-{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness: 0.027559
+Badness (Smith): 0.027559
 == Alphorn ==
@@ Line 700: / Line 698: @@
 {{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 556.221
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~48/35 = 556.221{{c}}
 {{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
-[[Badness]]: 0.129258
+[[Badness]] (Smith): 0.129258
 === 11-limit ===
@@ Line 713: / Line 711: @@
 Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 556.144
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 556.144{{c}}
-{{Optimal ET sequence|legend=1| 28d, 41, 315cde }}
+{{Optimal ET sequence|legend=0| 28d, 41, 315cde }}
-Badness: 0.073459
+Badness (Smith): 0.073459
 == Misneb ==
@@ Line 728: / Line 726: @@
 {{Mapping|legend=1| 1 3 1 3 | 0 -15 14 -2 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16/15 = 113.235
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/15 = 113.235{{c}}
 {{Optimal ET sequence|legend=1| 21, 32, 53 }}
-[[Badness]]: 0.140970
+[[Badness]] (Smith): 0.140970
 === 11-limit ===
@@ Line 741: / Line 739: @@
 Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}
-{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127, 180de }}
-Badness: 0.085390
+Badness (Smith): 0.085390
 ==== 13-limit ====
@@ Line 754: / Line 752: @@
 Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}
-{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127, 180de }}
-Badness: 0.045569
+Badness (Smith): 0.045569
 === Musneb ===
@@ Line 767: / Line 765: @@
 Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.142
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.142{{c}}
-{{Optimal ET sequence|legend=1| 32, 53, 191de, 244cddee, 297cddee }}
+{{Optimal ET sequence|legend=0| 32, 53, 191de, 244cddee, 297cddee }}
-Badness: 0.087333
+Badness (Smith): 0.087333
 == Untriton ==
@@ Line 782: / Line 780: @@
 {{Mapping|legend=1| 1 6 -7 -7 | 0 -9 19 20 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 588.641
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 588.641{{c}}
 {{Optimal ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
-[[Badness]]: 0.143976
+[[Badness]] (Smith): 0.143976
 === 11-limit ===
@@ Line 795: / Line 793: @@
 Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.626
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.626{{c}}
-{{Optimal ET sequence|legend=1| 51, 53 }}
+{{Optimal ET sequence|legend=0| 51, 53 }}
-Badness: 0.074295
+Badness (Smith): 0.074295
 === 13-limit ===
@@ Line 808: / Line 806: @@
 Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.654
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.654{{c}}
-{{Optimal ET sequence|legend=1| 51f, 53 }}
+{{Optimal ET sequence|legend=0| 51f, 53 }}
-Badness: 0.047441
+Badness (Smith): 0.047441
 == Naiadical ==
@@ Line 822: / Line 820: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1202.120, ~21/16 = 456.652
+* [[WE]]: ~2 = 1202.120{{c}}, ~21/16 = 456.652{{c}}
-* [[CWE]]: ~2 = 1200.000, ~21/16 = 455.868
+* [[CWE]]: ~2 = 1200.000{{c}}, ~21/16 = 455.868{{c}}
 {{Optimal ET sequence|legend=1| 21, 29, 50, 79d }}
@@ Line 837: / Line 835: @@
 Optimal tunings:
-* WE: ~2 = 1201.901, ~21/16 = 456.514
+* WE: ~2 = 1201.901, ~21/16{{c}} = 456.514{{c}}
-* CWE: ~2 = 1200.000, ~21/16 = 455.822
+* CWE: ~2 = 1200.000, ~21/16{{c}} = 455.822{{c}}
 {{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
@@ Line 852: / Line 850: @@
 Optimal tunings:
-* WE: ~2 = 1201.786, ~13/10 = 456.452
+* WE: ~2 = 1201.786{{c}}, ~13/10 = 456.452{{c}}
-* CWE: ~2 = 1200.000, ~13/10 = 455.807
+* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.807{{c}}
 {{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
@@ Line 867: / Line 865: @@
 Optimal tunings:
-* WE: ~2 = 1201.921, ~13/10 = 456.523
+* WE: ~2 = 1201.921{{c}}, ~13/10 = 456.523{{c}}
-* CWE: ~2 = 1200.000, ~13/10 = 455.833
+* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.833{{c}}
 {{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
@@ Line 881: / Line 879: @@
 {{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~10000/9261 = 139.838
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10000/9261 = 139.838{{c}}
 {{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
-[[Badness]]: 0.150565
+[[Badness]] (Smith): 0.150565
 === 11-limit ===
@@ Line 894: / Line 892: @@
 Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~320/297 = 139.827
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~320/297 = 139.827{{c}}
-{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness: 0.052590
+Badness (Smith): 0.052590
 === 13-limit ===
@@ Line 907: / Line 905: @@
 Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.812
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.812{{c}}
-{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness: 0.032730
+Badness (Smith): 0.032730
 === 17-limit ===
@@ Line 920: / Line 918: @@
 Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.815
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.815{{c}}
-{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness: 0.023038
+Badness (Smith): 0.023038
 == Gwazy ==
@@ Line 935: / Line 933: @@
 {{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
-[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000, ~35/32 = 162.658
+[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000{{c}}, ~35/32 = 162.658{{c}}
 {{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
-[[Badness]]: 0.178826
+[[Badness]] (Smith): 0.178826
 === 11-limit ===
@@ Line 948: / Line 946: @@
 Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
-Optimal tuning (POTE): ~363/256 = 600.000, ~11/10 = 162.592
+Optimal tuning (POTE): ~363/256 = 600.000{{c}}, ~11/10 = 162.592{{c}}
-{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
+{{Optimal ET sequence|legend=0| 22, 74, 96, 118d }}
-Badness: 0.068410
+Badness (Smith): 0.068410
 == Tertiosec ==
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 963: / Line 961: @@
 {{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}
-[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 400.000, ~15/14 = 112.283
+[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 400.000{{c}}, ~15/14 = 112.283{{c}}
 {{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
-[[Badness]]: 0.431636
+[[Badness]] (Smith): 0.431636
 === 11-limit ===
@@ Line 976: / Line 974: @@
 Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}
-Optimal tuning (POTE): ~44/35 = 400.000, ~15/14 = 112.171
+Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~15/14 = 112.171{{c}}
-{{Optimal ET sequence|legend=1| 21, 54, 75e }}
+{{Optimal ET sequence|legend=0| 21, 54, 75e }}
-Badness: 0.173485
+Badness (Smith): 0.173485
 [[Category:Temperament collections]]

Marvel temperaments: Difference between revisions

Revision as of 11:05, 18 February 2026

Wizard

11-limit

Lizard

17-limit

19-limit

Gizzard

17-limit

19-limit

Mage

Tritonic

11-limit

13-limit

17-limit

19-limit

23-limit

Tritoni

Septimin

11-limit

13-limit

Merman

11-limit

13-limit

Slender

11-limit

13-limit

Triton

11-limit

Marvolo

11-limit

13-limit

17-limit

19-limit

Enneaportent

11-limit

13-limit

Gracecordial

11-limit

13-limit

17-limit

19-limit

23-limit

29-limit

31-limit

Gracecord

13-limit

17-limit

19-limit

Alphorn

11-limit

Misneb

11-limit

13-limit

Musneb

Untriton

11-limit

13-limit

Naiadical

11-limit

13-limit

17-limit

Quintannic

11-limit

13-limit

17-limit

Gwazy

11-limit

Tertiosec

11-limit

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