Marvel temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
This page discusses miscellaneous rank-2 temperaments tempering out 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
- 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
- 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 ¢
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 ¢
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 ¢
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 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
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