Marvel temperaments

This page discusses miscellaneous rank-2 temperaments tempering out [-5 2 2 -1⟩ = 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

• Pelogic → Pelogic 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)
• Sharp → 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 → Qintosec 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, eleventh sliced in seven)
• Snipes → Wesley family (+6125/5832, two octaves and a fourth sliced in seven)
• Escapade → 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)
• Compton → Compton family (+250047/250000, generated by the classical major third with a 1/12-octave period)
• 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, submajor, 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 of this temperament, see High badness temperaments #Wizard.

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 tuning (POTE): ~1225/864 = 1\2, ~5/4 = 383.256 (~245/216 = 216.744)

Wedgie: ⟨⟨ 12 -2 20 -31 -2 52 ]]

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

Badness: 0.040846

Scales: wizard22

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 tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.232 (~25/22 = 216.768)

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

Badness: 0.018539

Scales: wizard22

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 tuning (POTE): ~55/39 = 1\2, ~5/4 = 383.389 (~25/22 = 216.711)

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

Badness: 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 tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.381 (~17/15 = 216.619)

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

Badness: 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 tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.477 (~17/15 = 216.523)

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

Badness: 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 tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.170 (~25/22 = 216.830)

Optimal ET sequence: 72, 166, 238cf

Badness: 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 tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.175 (~25/22 = 216.825)

Optimal ET sequence: 72, 166g, 238cfg

Badness: 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 tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.138 (~17/15 = 216.862)

Optimal ET sequence: 72, 94, 166g

Badness: 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 tuning (POTE): ~77/54 = 1\2, ~5/4 = 383.124 (~55/48 = 216.876)

Optimal ET sequence: 22, 50e, 72ee, 94ee

Badness: 0.057799

Tritonic

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

Subgroup: 2.3.5.7

Comma list: 225/224, 50421/50000

Mapping: [⟨1 4 -3 -3], ⟨0 -5 11 12]]

Wedgie: ⟨⟨ 5 -11 -12 -29 -33 3 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.286

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

Badness: 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 = 1\1, ~7/5 = 580.267

Optimal ET sequence: 29, 31, 60e

Badness: 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 = 1\1, ~7/5 = 580.108

Optimal ET sequence: 29, 31, 60e, 151cde

Badness: 0.022993

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 = 1\1, ~7/5 = 580.389

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

Badness: 0.045456

Septimin

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

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping: [⟨1 4 1 5], ⟨0 -11 6 -10]]

Wedgie: ⟨⟨ 11 -6 10 -35 -15 40 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.632

Optimal ET sequence: 41, 91, 132d

Badness: 0.054502

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2401/2376

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

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.634

Optimal ET sequence: 41, 91, 223cdef

Badness: 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 = 1\1, ~7/6 = 263.700

Optimal ET sequence: 41, 91

Badness: 0.023117

Merman

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

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 5 -5 -5], ⟨0 -7 15 16]]

Wedgie: ⟨⟨ 7 -15 -16 -40 -45 5 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.585

Optimal ET sequence: 41, 84, 125

Badness: 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 = 1\1, ~7/5 = 585.606

Optimal ET sequence: 41, 84, 125e

Badness: 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 = 1\1, ~7/5 = 585.657

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

Badness: 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]]

Wedgie: ⟨⟨ 13 -10 6 -46 -27 42 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.413

Optimal ET sequence: 31, 94, 125

Badness: 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 = 1\1, ~49/48 = 38.387

Optimal ET sequence: 31, 63, 94, 125

Badness: 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 = 1\1, ~49/48 = 38.314

Optimal ET sequence: 31, 63, 94

Badness: 0.025913

Triton

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

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

Wedgie: ⟨⟨ 3 -7 -8 -18 -21 1 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 568.865

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

Badness: 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 = 1\1, ~7/5 = 569.144

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

Badness: 0.045675

Submajor

Subgroup: 2.3.5

Comma list: 69198046875/68719476736

Mapping: [⟨1 4 -1], ⟨0 -8 11]]

Optimal tuning (POTE): ~2 = 1\1, ~10125/8192 = 362.321

Optimal ET sequence: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc

Badness: 0.130236

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 51200/50421

Mapping: [⟨1 4 -1 1], ⟨0 -8 11 6]]

Wedgie: ⟨⟨ 8 -11 -6 -36 -32 17 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.255

Optimal ET sequence: 10, 33, 43, 53

Badness: 0.060533

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 6655/6561

Mapping: [⟨1 4 -1 1 11], ⟨0 -8 11 6 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101

Optimal ET sequence: 10, 43e, 53, 116, 169de, 285cde

Badness: 0.050582

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 275/273, 385/384

Mapping: [⟨1 4 -1 1 11 4], ⟨0 -8 11 6 -25 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105

Optimal ET sequence: 10, 43e, 53, 116, 169de, 285cdef

Badness: 0.027689

Interpental

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 51200/50421

Mapping: [⟨1 4 -1 1 -5], ⟨0 -8 11 6 28]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418

Optimal ET sequence: 43, 53, 96, 149d

Badness: 0.051806

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 640/637

Mapping: [⟨1 4 -1 1 -5 4], ⟨0 -8 11 6 28 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402

Optimal ET sequence: 43, 53, 96, 149d

Badness: 0.029680

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

Wedgie: ⟨⟨ 6 -19 -26 -44 -58 -7 ]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.348

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

Badness: 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 = 1\1, ~21/20 = 83.340

Optimal ET sequence: 29, 43, 72

Badness: 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 = 1\1, ~21/20 = 83.330

Optimal ET sequence: 29, 43, 72, 115f

Badness: 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 = 1\1, ~21/20 = 83.330

Optimal ET sequence: 29g, 43, 72

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 = 1\1, ~21/20 = 83.330

Optimal ET sequence: 29g, 43, 72

Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

Wedgie: ⟨⟨ 18 -9 18 -56 -22 67 ]]

Optimal tuning (POTE): ~2592/2401 = 1\9, ~5/4 = 383.165

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

Badness: 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 = 1\9, ~5/4 = 383.146

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

Badness: 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 = 1\9, ~5/4 = 383.047

Optimal ET sequence: 9, 63, 72, 279cf

Badness: 0.022322

Gracecordial

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

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]

Wedgie: ⟨⟨ 1 -20 -38 -34 -63 -32 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.824

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

Badness: 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 = 1\1, ~3/2 = 700.834

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

Badness: 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 = 1\1, ~3/2 = 700.841

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

Badness: 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 = 1\1, ~3/2 = 700.841

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

Badness: 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 = 1\1, ~3/2 = 700.842

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

Badness: 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 = 1\1, ~3/2 = 700.843

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

Badness: 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 = 1\1, ~3/2 = 700.842

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

Badness: 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 = 1\1, ~3/2 = 700.838

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

Badness: 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 = 1\1, ~3/2 = 700.885

Optimal ET sequence: 12, 101cd, 113

Badness: 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 = 1\1, ~3/2 = 700.890

Optimal ET sequence: 12f, 101cdf, 113

Badness: 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 = 1\1, ~3/2 = 700.870

Optimal ET sequence: 12f, 101cdf, 113

Badness: 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 = 1\1, ~3/2 = 700.866

Optimal ET sequence: 12f, 101cdf, 113

Badness: 0.027559

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 9 0 13], ⟨0 -16 5 -22]]

Wedgie: ⟨⟨ 16 -5 22 -45 -10 65 ]]

Optimal tuning (POTE): ~2 = 1\1, ~48/35 = 556.221

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

Badness: 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 = 1\1, ~11/8 = 556.144

Optimal ET sequence: 28d, 41, 315cde

Badness: 0.073459

Misneb

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

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 3 1 3], ⟨0 -15 14 -2]]

Wedgie: ⟨⟨ 15 -14 2 -57 -39 44 ]]

Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.235

Optimal ET sequence: 21, 32, 53

Badness: 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 = 1\1, ~16/15 = 113.323

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

Badness: 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 = 1\1, ~16/15 = 113.323

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

Badness: 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 = 1\1, ~16/15 = 113.142

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

Badness: 0.087333

Untriton

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

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 6 -7 -7], ⟨0 -9 19 20]]

Wedgie: ⟨⟨ 9 -19 -20 -51 -57 7 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.641

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

Badness: 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 = 1\1, ~7/5 = 588.626

Optimal ET sequence: 51, 53

Badness: 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 = 1\1, ~7/5 = 588.654

Optimal ET sequence: 51f, 53

Badness: 0.047441

Quintannic

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]

Wedgie: ⟨⟨ 5 -23 -36 -48 -71 -19 ]]

Optimal tuning (POTE): ~2 = 1\1, ~10000/9261 = 139.838

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

Badness: 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 = 1\1, ~320/297 = 139.827

Optimal ET sequence: 43, 60e, 103

Badness: 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 = 1\1, ~13/12 = 139.812

Optimal ET sequence: 43, 60e, 103

Badness: 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 = 1\1, ~13/12 = 139.815

Optimal ET sequence: 43, 60e, 103

Badness: 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]]

Wedgie: ⟨⟨ 16 -10 12 -53 -26 56 ]]

Optimal tuning (POTE): 1\2, ~35/32 = 162.658

Optimal ET sequence: 22, 74, 96, 118d

Badness: 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): 1\2, ~11/10 = 162.592

Optimal ET sequence: 22, 74, 96, 118d

Badness: 0.068410

Tertiosec

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

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 7 5 9], ⟨0 -8 7 -2]]

Wedgie: ⟨⟨ 24 -21 6 -89 -58 73 ]]

Optimal tuning (POTE): 1\3, ~15/14 = 112.283

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

Badness: 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): 1\3, ~15/14 = 112.171

Optimal ET sequence: 21, 54, 75e

Badness: 0.173485