Marvel temperaments: Difference between revisions

Latest revision as of 12:32, 24 September 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 (+21/20 or 135/128) → Mavila family
Meantone (+81/80 or 126/125) → Meantone family
Garibaldi (+3125/3087) → Schismatic family
Pajara (+50/49 or 64/63) → Diaschismic family
Sharpie (+25/24 or 28/27) → Dicot family
Immune (+781250/750141) → Immunity family
August (+36/35 or 128/125) → Augmented family
Fog (+156250/151263) → Misty family
Bunya (+15625/15309) → Tetracot family
Negri (+49/48) → Semaphoresmic clan
Magic (+245/243) → Magic family
Passive (+256/245) → Passion family
Houborizic (+1250000/1240029) → Amity family
Qintosec (+2560000/2470629) → Quintosec family
Miracle (+1029/1024) → Gamelismic clan
Catakleismic (+4375/4374) → Kleismic family
Marvo (+78125000/78121827) → Gravity family
Orwell (+1728/1715) → Semicomma family
Snipes (+6125/5832) → Wesley family
Demibuzzard (+65536/64827) → Buzzardsmic clan
Escapist (+65625/65536) → Escapade family
Amavil (+17496/16807) → Mabila family
Betic (+1071875/1062882) → Sycamore family
Wizard (+118098/117649) → Stearnsmic clan
Raccoon (+41943040/40353607) → Vavoom family
Maquila (+30233088/28824005) → Maquila family
Gammy (+94143178827/91913281250) → Gammic family
Compton (+250047/250000) → Compton family

Considered below are tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing badness.

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.)

Tritonic

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

Tritonic tempers out 50421/50000 and may be described as the 29 & 31 temperament. It splits the 6th harmonic into five generators of ~10/7 tritones, hence the name. Its ploidacot is beta-pentacot. 60edo may be used as a tuning, which in the 11-limit entails the 60e val.

Subgroup: 2.3.5.7

Comma list: 225/224, 50421/50000

Mapping: [⟨1 -1 8 9], ⟨0 5 -11 -12]]

mapping generators: ~2, ~10/7

Optimal tunings:

WE: ~2 = 1201.3539 ¢, ~10/7 = 620.4131 ¢

error map: ⟨+1.354 -1.243 -0.027 -1.598]

CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6778 ¢

error map: ⟨0.000 -3.566 -2.769 -4.959]

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

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 8 9 5], ⟨0 5 -11 -12 -3]]

Optimal tunings:

WE: ~2 = 1201.7116 ¢, ~10/7 = 620.6166 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6890 ¢

Optimal ET sequence: 29, 31, 60e, 91e, 213bcdeee

Badness (Sintel): 0.782

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 8 9 5 13], ⟨0 5 -11 -12 -3 -18]]

Optimal tunings:

WE: ~2 = 1201.5355 ¢, ~10/7 = 620.6855 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8469 ¢

Optimal ET sequence: 29, 31, 60e

Badness (Sintel): 0.950

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -1 8 9 5 13 17], ⟨0 5 -11 -12 -3 -18 -25]]

Optimal tunings:

WE: ~2 = 1201.5260 ¢, ~10/7 = 620.7330 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8986 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 0.973

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 -1 8 9 5 13 17 12], ⟨0 5 -11 -12 -3 -18 -25 -15]]

Optimal tunings:

WE: ~2 = 1201.3100 ¢, ~10/7 = 620.6509 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9328 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 1.03

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 -1 8 9 5 13 17 12 4], ⟨0 5 -11 -12 -3 -18 -25 -15 1]]

Optimal tunings:

WE: ~2 = 1201.4074 ¢, ~10/7 = 620.7185 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9548 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 1.04

Tritoni

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 8 9 -11], ⟨0 5 -11 -12 28]]

Optimal tunings:

WE: ~2 = 1201.0888 ¢, ~10/7 = 620.1733 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6146 ¢

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

Badness (Sintel): 1.50

Septimin

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

Septimin may be described as the 41 & 50 temperament. It is generated by a septimal minor third (7/6), which gives rise to the name, but the generator can be taken to be the octave complement, 12/7, such that eleven of them octave reduced give the perfect fifth; its ploidacot is thus eta-hendecacot. 91edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping: [⟨1 -7 7 -5], ⟨0 11 -6 10]]

mapping generators: ~2, ~12/7

Optimal tunings:

WE: ~2 = 1201.2452 ¢, ~12/7 = 937.3394 ¢

error map: ⟨+1.245 +0.062 -1.633 -1.658]

CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.4036 ¢

error map: ⟨0.000 -1.516 -4.735 -4.790]

Optimal ET sequence: 9, 23, 32, 41, 91, 132d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 7 -5 -2], ⟨0 11 -6 10 7]]

Optimal tunings:

WE: ~2 = 1200.8059 ¢, ~12/7 = 936.9952 ¢
CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3906 ¢

Optimal ET sequence: 9, 23, 32, 41, 91, 223cdef

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -7 7 -5 -2 -8], ⟨0 11 -6 10 7 15]]

Optimal tunings:

WE: ~2 = 1200.5990 ¢, ~12/7 = 936.7670 ¢
CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3196 ¢

Optimal ET sequence: 9, 32f, 41, 91

Badness (Sintel): 0.955

Merman

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

Merman may be described as the 41 & 43 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, seven generator steps give the interval class of 3. The ploidacot for this temperament is gamma-heptacot.

The name was likely derived from Triton, which was in turn derived from tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 -2 10 11], ⟨0 7 -15 -16]]

mapping generators: ~2, ~10/7

Optimal tunings:

WE: ~2 = 1200.3898 ¢, ~10/7 = 614.6413 ¢

error map: ⟨+0.390 -0.435 -1.630 +1.634]

CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4073 ¢

error map: ⟨0.000 -1.104 -2.423 +0.657]

Optimal ET sequence: 41, 84, 125

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 10 11 5], ⟨0 7 -15 -16 -3]]

Optimal tunings:

WE: ~2 = 1199.9578 ¢, ~10/7 = 614.3720 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3943 ¢

Optimal ET sequence: 41, 84, 125e

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 10 11 5 -5], ⟨0 7 -15 -16 -3 17]]

Optimal tunings:

WE: ~2 = 1199.7422 ¢, ~10/7 = 614.2110 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3442 ¢

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

Badness (Sintel): 1.14

Mermaid

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 532400/531441

Mapping: [⟨1 -2 10 11 -16], ⟨0 7 -15 -16 38]]

Optimal tunings:

WE: ~2 = 1199.4973 ¢, ~10/7 = 614.7004 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4470 ¢

Optimal ET sequence: 41, 84e, 125, 166

Badness (Sintel): 1.46

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 10648/10647

Mapping: [⟨1 -2 10 11 22 32], ⟨0 7 -15 -16 38 58]]

Optimal tunings:

WE: ~2 = 1200.5126 ¢, ~10/7 = 614.7152 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4562 ¢

Optimal ET sequence: 41, 84ef, 125f, 166

Badness (Sintel): 1.47

Slender

Slender tempers out the hewuermera comma in addition to the marvel comma, and may be described as the 31 & 32 temperament. This temperament has a generator of 49/48, three of which equal marvel's 16/15 ~15/14, and ten generators give 5/4. Its ploidacot is omega-13-cot.

The name was likely derived from slendro diesis, one of the names for the interval 49/48.

Subgroup: 2.3.5.7

Comma list: 225/224, 589824/588245

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

mapping generators: ~2, ~49/48

Optimal tunings:

WE: ~2 = 1200.3816 ¢, ~49/48 = 38.4256 ¢

error map: ⟨+0.382 -0.725 -1.295 +1.765]

CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.4079 ¢

error map: ⟨0.000 -1.257 -2.235 +0.727]

Optimal ET sequence: 31, 94, 125, 406c

Badness (Sintel): 1.44

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 tunings:

WE: ~2 = 1199.4983 ¢, ~49/48 = 38.4030 ¢
CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3775 ¢

Optimal ET sequence: 31, 63, 94, 125

Badness (Sintel): 0.838

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 tunings:

WE: ~2 = 1200.1728 ¢, ~49/48 = 38.3192 ¢
CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3129 ¢

Optimal ET sequence: 31, 63, 94

Badness (Sintel): 1.07

Triton

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Stump.

Triton may be described as the 19 & 21 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, three generator steps give the interval class of 3. The ploidacot for this temperament is alpha-tricot.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

mapping generators: ~2, ~10/7

Optimal tunings:

WE: ~2 = 1203.3828 ¢, ~10/7 = 632.9137 ¢

error map: ⟨+3.383 -3.214 +3.587 -8.457]

CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.9827 ¢

error map: ⟨0.000 -9.007 -3.192 -16.687]

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

Badness (Sintel): 1.50

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 tunings:

WE: ~2 = 1201.3875 ¢, ~10/7 = 631.5852 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.8007 ¢

Optimal ET sequence: 2, 17d, 19

Badness (Sintel): 1.51

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

mapping generators: ~2, ~21/20

Optimal tunings:

WE: ~2 = 1200.7714 ¢, ~21/20 = 83.4014 ¢

error map: ⟨+0.772 -0.820 -0.916 +0.381]

CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3640 ¢

error map: ⟨0.000 -2.139 -2.398 -1.362]

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

Badness (Sintel): 2.11

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 tunings:

WE: ~2 = 1200.7075 ¢, ~21/20 = 83.3888 ¢
CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3564 ¢

Optimal ET sequence: 29, 43, 72

Badness (Sintel): 0.958

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 tunings:

WE: ~2 = 1200.9467 ¢, ~21/20 = 83.3956 ¢
CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3516 ¢

Optimal ET sequence: 29, 43, 72

Badness (Sintel): 0.887

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 tunings:

WE: ~2 = 1200.9606 ¢, ~21/20 = 83.4030 ¢
CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3594 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.760

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 tunings:

WE: ~2 = 1200.7625 ¢, ~21/20 = 83.3895 ¢
CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3551 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.895

Decic

Main article: Decic

Named by Xenllium in 2021, decic tempers out 16807/16384, the cloudy comma, and [11 -10 -10 10⟩, the linus comma, in addition to the marvel comma. It may be described as the 50 & 60 temperament, with a period of 1/10 octave and a ploidacot signature of decaploid monocot. It is supported by 10-, 50-, and 60edo.

It can be extended to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 225/224, 16807/16384

Mapping: [⟨10 0 39 28], ⟨0 1 -1 0]]

mapping generators: ~15/14, ~3

Optimal tunings:

WE: ~15/14 = 120.1837 ¢, ~3/2 = 699.7654 ¢ (~49/48 = 21.3366 ¢)

error map: ⟨+1.837 -0.353 -0.753 -3.683]

CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.8236 ¢ (~49/48 = 21.1764 ¢)

error map: ⟨0.000 -3.131 -5.137 -8.826]

Optimal ET sequence: 10, 30b, 40, 50, 60, 110d, 170cdd

Badness (Sintel): 2.26

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 3087/3025

Mapping: [⟨10 0 39 28 3], ⟨0 1 -1 0 2]]

Optimal tunings:

WE: ~15/14 = 120.1406 ¢, ~3/2 = 697.6075 ¢ (~56/55 = 23.2360 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.0142 ¢ (~56/55 = 22.9858 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 2.11

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 2200/2197

Mapping: [⟨10 0 39 28 3 37], ⟨0 1 -1 0 2 0]]

Optimal tunings:

WE: ~15/14 = 120.1166 ¢, ~3/2 = 697.6705 ¢ (~78/77 = 23.0289 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.1492 ¢ (~78/77 = 22.8508 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 1.52

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 144/143, 170/169, 196/195, 221/220

Mapping: [⟨10 0 39 28 3 37 25], ⟨0 1 -1 0 2 0 1]]

Optimal tunings:

WE: ~15/14 = 120.1262 ¢, ~3/2 = 697.8185 ¢ (~78/77 = 22.9388 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.2757 ¢ (~78/77 = 22.7243 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 1.28

Splendecic

Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the fantastic rank-3 temperament, its name is a portmanteau of splendid and decic.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1617/1600, 2401/2376

Mapping: [⟨10 0 39 28 82], ⟨0 1 -1 0 -3]]

Optimal tunings:

WE: ~15/14 = 120.1874 ¢, ~3/2 = 699.6085 ¢ (~99/98 = 21.5156 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.3531 ¢ (~99/98 = 21.6469 ¢)

Optimal ET sequence: 10e, 40e, 50, 60e, 110de, 170cddee

Badness (Sintel): 1.98

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 1001/1000, 1188/1183

Mapping: [⟨10 0 39 28 82 37], ⟨0 1 -1 0 -3 0]]

Optimal tunings:

WE: ~15/14 = 120.1565 ¢, ~3/2 = 699.2756 ¢ (~91/90 = 21.6631 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.2480 ¢ (~91/90 = 21.7520 ¢)

Optimal ET sequence: 10e, 40e, 50, 60e, 110de

Badness (Sintel): 1.57

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 170/169, 196/195, 289/288, 375/374

Mapping: [⟨10 0 39 28 82 37 25], ⟨0 1 -1 0 -3 0 1]]

Optimal tunings:

WE: ~15/14 = 120.1571 ¢, ~3/2 = 699.2892 ¢ (~91/90 = 21.6536 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.3144 ¢ (~91/90 = 21.6856 ¢)

Optimal ET sequence: 10e, 50, 60e, 110deg

Badness (Sintel): 1.33

Prodecic

Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the prodigy rank-3 temperament, its name is a portmanteau of prodigy and decic.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 5929/5832

Mapping: [⟨10 0 39 28 -13], ⟨0 1 -1 0 3]]

Optimal tunings:

WE: ~15/14 = 120.2024 ¢, ~3/2 = 701.3908 ¢ (~55/54 = 19.8237 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.5235 ¢ (~55/54 = 19.4765 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 2.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 275/273, 5929/5832

Mapping: [⟨10 0 39 28 -13 37], ⟨0 1 -1 0 3 0]]

Optimal tunings:

WE: ~15/14 = 120.1654 ¢, ~3/2 = 701.4683 ¢ (~91/90 = 19.5242 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.7175 ¢ (~91/90 = 19.2825 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 1.73

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 170/169, 196/195, 289/288

Mapping: [⟨10 0 39 28 -13 37 25], ⟨0 1 -1 0 3 0 1]]

Optimal tunings:

WE: ~15/14 = 120.1577 ¢, ~3/2 = 701.3950 ¢ (~91/90 = 19.5514 ¢)
CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.6932 ¢ (~91/90 = 19.3068 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 1.41

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 tunings:

WE: ~2592/2401 = 133.4174 ¢, ~12005/6912 = 950.7667 ¢ (~1728/1715 = 16.8452 ¢)

error map: ⟨+0.756 -0.422 -1.395 +0.298]

CWE: ~2592/2401 = 133.3333 ¢, ~12005/6912 = 950.2969 ¢ (~1728/1715 = 16.9636 ¢)

error map: ⟨0.000 -1.361 -3.277 -1.565]

Optimal ET sequence: 9, 54, 63, 72, 495bccd, 567bcccd

Badness (Sintel): 2.37

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 tunings:

WE: ~121/112 = 133.4071 ¢, ~210/121 = 950.7131 ¢ (~99/98 = 16.8633 ¢)
CWE: ~121/112 = 133.3333 ¢, ~210/121 = 950.2994 ¢ (~99/98 = 16.9661 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 1.01

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 tunings:

WE: ~14/13 = 133.4245 ¢, ~26/15 = 950.9362 ¢ (~105/104 = 16.9650 ¢)
CWE: ~14/13 = 133.3333 ¢, ~26/15 = 950.4364 ¢ (~99/98 = 17.1031 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 0.922

Gracecordial

For the 5-limit version, 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]]

mapping generators: ~2, ~3

Optimal tunings:

WE: ~2 = 1200.4904 ¢, ~3/2 = 701.1103 ¢

error map: ⟨+0.490 -0.354 -1.655 +1.241]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8112 ¢

error map: ⟨0.000 -1.144 -2.537 +0.349]

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

Badness (Sintel): 2.44

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 tunings:

WE: ~2 = 1200.5571 ¢, ~3/2 = 701.1589 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8328 ¢

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

Badness (Sintel): 2.96

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 tunings:

WE: ~2 = 1200.6282 ¢, ~3/2 = 701.2080 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8421 ¢

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

Badness (Sintel): 2.16

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 tunings:

WE: ~2 = 1200.5058 ¢, ~3/2 = 701.1360 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8414 ¢

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

Badness (Sintel): 1.96

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 tunings:

WE: ~2 = 1200.4418 ¢, ~3/2 = 701.0999 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8425 ¢

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

Badness (Sintel): 1.71

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 tunings:

WE: ~2 = 1200.4641 ¢, ~3/2 = 701.1145 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8444 ¢

Optimal ET sequence: 12e, 113, 238cfi

Badness (Sintel): 1.57

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 tunings:

WE: ~2 = 1200.4400 ¢, ~3/2 = 701.0986 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8428 ¢

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

Badness (Sintel): 1.50

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 tunings:

WE: ~2 = 1200.4178 ¢, ~3/2 = 701.0822 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8396 ¢

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

Badness (Sintel): 1.53

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 tunings:

WE: ~2 = 1200.6064 ¢, ~3/2 = 701.2398 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8718 ¢

Optimal ET sequence: 12, …, 101cd, 113

Badness (Sintel): 2.21

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 tunings:

WE: ~2 = 1200.6225 ¢, ~3/2 = 701.2539 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8781 ¢

Optimal ET sequence: 12f, …, 101cdf, 113

Badness (Sintel): 1.83

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 tunings:

WE: ~2 = 1200.3308 ¢, ~3/2 = 701.0632 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8654 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.87

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 tunings:

WE: ~2 = 1200.2658 ¢, ~3/2 = 701.0213 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8629 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.68

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 -7 5 -9], ⟨0 16 -5 22]]

mapping generators: ~2, ~35/24

Optimal tunings:

WE: ~2 = 1201.3004 ¢, ~35/24 = 644.4767 ¢

error map: ⟨+1.300 +0.569 -2.195 -2.043]

CWE: ~2 = 1200.3333 ¢, ~35/24 = 643.8137 ¢

error map: ⟨0.000 -0.936 -5.382 -4.924]

Optimal ET sequence: 13d, 28d, 41, 151cd, 192cdd, 233ccdd

Badness (Sintel): 3.27

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -7 5 -9 4], ⟨0 16 -5 22 -1]]

Optimal tunings:

WE: ~2 = 1200.5123 ¢, ~16/11 = 644.1307 ¢
CWE: ~2 = 1200.0000 ¢, ~16/11 = 643.8662 ¢

Optimal ET sequence: 13d, 28d, 41

Badness (Sintel): 2.43

Misneb

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

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 -12 15 1], ⟨0 15 -14 2]]

mapping generators: ~2, ~15/8

Optimal tunings:

WE: ~2 = 1199.7642 ¢, ~15/8 = 1086.5513 ¢

error map: ⟨-0.236 -0.856 -1.569 +4.041]

CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.7633 ¢

error map: ⟨0.000 -0.506 -0.999 +4.701]

Optimal ET sequence: 21, 32, 53

Badness (Sintel): 3.57

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -12 15 1 27], ⟨0 15 -14 2 -26]]

Optimal tunings:

WE: ~2 = 1200.1654 ¢, ~15/8 = 1086.8269 ¢
CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6766 ¢

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

Badness (Sintel): 2.82

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -12 15 1 27 20], ⟨0 15 -14 2 -26 -18]]

Optimal tunings:

WE: ~2 = 1200.1687 ¢, ~15/8 = 1086.8295 ¢
CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6757 ¢

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

Badness (Sintel): 1.88

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 tunings:

WE: ~2 = 1200.0839 ¢, ~15/8 = 1086.9343 ¢
CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.8593 ¢

Optimal ET sequence: 21e, 32, 53

Badness (Sintel): 2.89

Untriton

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

Named by Petr Pařízek in 2011^[1], untriton may be described as the 51 & 53 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 -3 12 13], ⟨0 9 -19 -20]]

mapping generators: ~2, ~10/7

Optimal tunings:

WE: ~2 = 1199.8275 ¢, ~10/7 = 611.2710 ¢

error map: ⟨-0.172 +0.002 -2.533 +3.511]

CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3614 ¢

error map: ⟨0.000 +0.298 -2.181 +3.946]

Optimal ET sequence: 51, 53

Badness (Sintel): 3.64

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 12 13 6], ⟨0 9 -19 -20 -5]]

Optimal tunings:

WE: ~2 = 1200.3591 ¢, ~10/7 = 611.5569 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3690 ¢

Optimal ET sequence: 51, 53

Badness (Sintel): 2.46

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -3 12 13 6 20], ⟨0 9 -19 -20 -5 -32]]

Optimal tunings:

WE: ~2 = 1200.4078 ¢, ~10/7 = 611.5536 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3392 ¢

Optimal ET sequence: 51f, 53

Badness (Sintel): 1.96

Naiadical

Named by Xenllium in 2026, naiadical may be described as the 21 & 29 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 823543/800000

Mapping: [⟨1 -4 11 9], ⟨0 9 -14 -10]]

mapping generators: ~2, ~32/21

Optimal tunings:

WE: ~2 = 1202.1198 ¢, ~32/21 = 745.4675 ¢

error map: ⟨+2.120 -1.227 +0.459 -4.423]

CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1318 ¢

error map: ⟨0.000 -4.769 -4.159 -10.144]

Optimal ET sequence: 21, 29, 50, 79d, 129cdd, 179bcddd

Badness (Sintel): 3.67

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -4 11 9 14], ⟨0 9 -14 -10 -17]]

Optimal tunings:

WE: ~2 = 1201.9008 ¢, ~21/16 = 745.3867 ¢
CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1777 ¢

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

Badness (Sintel): 2.00

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -4 11 9 14 13], ⟨0 9 -14 -10 -17 -15]]

Optimal tunings:

WE: ~2 = 1201.7863 ¢, ~20/13 = 745.3344 ¢
CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1931 ¢

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

Badness (Sintel): 1.43

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -4 11 9 14 13 14], ⟨0 9 -14 -10 -17 -15 -16]]

Optimal tunings:

WE: ~2 = 1201.9208 ¢, ~20/13 = 745.3976 ¢
CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1669 ¢

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

Badness (Sintel): 1.26

Quintannic

Named by Scott Dakota, quintannic may be described as the 43 & 60 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]

mapping generators: ~2, ~10000/9261

Optimal tunings:

WE: ~2 = 1200.9803 ¢, ~10000/9261 = 139.9522 ¢

error map: ⟨+0.980 -1.214 -0.313 -0.243]

CWE: ~2 = 1200.0000 ¢, ~10000/9261 = 139.8184 ¢

error map: ⟨0.000 -2.863 -2.136 -2.287]

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

Badness (Sintel): 3.81

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 tunings:

WE: ~2 = 1201.0031 ¢, ~320/297 = 139.9435 ¢
CWE: ~2 = 1200.0000 ¢, ~320/297 = 139.8053 ¢

Optimal ET sequence: 43, 60e, 103, 369bcdeee, 472bbcddeee

Badness (Sintel): 1.74

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 tunings:

WE: ~2 = 1200.8354 ¢, ~13/12 = 139.9095 ¢
CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.7997 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.35

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 tunings:

WE: ~2 = 1200.7402 ¢, ~13/12 = 139.9015 ¢
CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.8038 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.17

Hendeca

Not to be confused with Hendec.

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

Hendeca tempers out the same 5-limit comma as hendecatonic, and has a period of 1/11 octave. However, in this temperament, nine periods represent 7/4, the same as undeka. It can be tuned to 22edo or 33edo using the patent val, or 55edo using the 55d val. It was named by Xenllium in 2025 as a low-accuracy variant of hendecatonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 122880/117649

Mapping: [⟨11 0 43 31], ⟨0 1 -1 0]]

mapping generators: ~16/15, ~3

Optimal tunings:

WE: ~16/15 = 108.9526 ¢, ~3/2 = 702.4215 ¢

error map: ⟨-1.521 -1.055 -2.252 +8.705]

CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.2071 ¢

error map: ⟨0.000 +1.252 +1.388 +12.992]

Optimal ET sequence: 22, 55d, 77d, 99dd

Badness (Sintel): 4.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 352/343

Mapping: [⟨11 0 43 31 38], ⟨0 1 -1 0 0]]

Optimal tunings:

WE: ~16/15 = 109.0109 ¢, ~3/2 = 702.5746 ¢
CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.0395 ¢

Optimal ET sequence: 22, 55d

Badness (Sintel): 2.46

Gwazy

For the 5-limit version, see Very high accuracy temperaments #Kwazy.

Named by Petr Pařízek in 2011^[1], gwazy may be described as the 22 & 74 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]

mapping generators: ~2401/1728, ~35/32

Optimal tunings:

WE: ~2401/1728 = 599.7132 ¢, ~35/32 = 162.5806 ¢

error map: ⟨-0.574 -1.597 -0.937 +5.510]

CWE: ~2401/1728 = 600.0000 ¢, ~35/32 = 162.6388 ¢

error map: ⟨0.000 -0.844 +0.492 +7.007]

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

Badness (Sintel): 4.53

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 tunings:

WE: ~363/256 = 599.8517 ¢, ~11/10 = 162.5518 ¢
CWE: ~363/256 = 600.0000 ¢, ~11/10 = 162.5863 ¢

Optimal ET sequence: 22, 74, 96

Badness (Sintel): 2.26

Tertiosec

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

Tertiosec may be described as the 21 & 75 temperament. It was initially named tertiomar by Petr Pařízek in 2011^[1], but was changed to tertiosec in 2012^[2].

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 -1 12 7], ⟨0 8 -7 2]]

mapping generators: ~3072/2401, ~2048/1715

Optimal tunings:

WE: ~3072/2401 = 399.8257 ¢, ~2048/1715 = 287.5920 ¢

error map: ⟨-0.523 -1.044 -1.549 +5.138]

CWE: ~3072/2401 = 400.0000 ¢, ~2048/1715 = 287.7088 ¢

error map: ⟨0.000 -0.284 -0.276 +6.592]

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

Badness (Sintel): 10.9

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 -1 12 7 14], ⟨0 8 -7 2 -5]]

Optimal tunings:

WE: ~44/35 = 399.6550 ¢, ~33/28 = 287.5803 ¢
CWE: ~44/35 = 400.0000 ¢, ~33/28 = 287.8224 ¢

Optimal ET sequence: 21, 54, 75e

Badness (Sintel): 5.74

References

[petr's_long_post-1] 1.0 ^1.1 ^1.2 Yahoo! Tuning Group | Suggested names for the unclasified temperaments

[2] Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780

[1]

[2]

@@ Line 3: / Line 3: @@
 Temperaments considered in families and clans are:
-* ''[[Pelogic]]'' → [[Mavila family #Pelogic|Mavila family]] (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3)
+* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
-* [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3)
+* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
-* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+3125/3087, generated by the fifth with 5/4 mapped to the d4)
+* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
-* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]] (+50/49 or 64/63, generated by the fifth with a semioctave period)
+* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
-* ''[[Sharpie]]'' → [[Dicot family #Sharpie|Dicot family]] (+25/24 or 28/27, fifth sliced in two)
+* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
-* ''[[Immune]]'' → [[Immunity family #Immune|Immunity family]] (+781250/750141, twelfth sliced in two)
+* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
-* ''[[August]]'' → [[Augmented family #August|Augmented family]] (+36/35 or 128/125, generated by the fifth with a 1/3-octave period)
+* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
-* ''[[Fog]]'' → [[Misty family #Fog|Misty family]] (+156250/151263, generated by the fifth with a 1/3-octave period)
+* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
-* [[Negri]] → [[Slendro clan #Negri|Slendro clan]] (+49/48, fourth sliced in four)
+* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
-* [[Magic]] → [[Magic family #Magic|Magic family]] (+245/243, twelfth sliced in five)
+* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
-* ''[[Passive]]'' → [[Passion family #Passive|Passion family]] (+256/245, fourth sliced in five)
+* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
-* ''[[Quintapole]]'' → [[Quintaleap family #Quintapole|Quintaleap family]] (+7812500/7411887, fourth sliced in five)
+* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
-* ''[[Houborizic]]'' → [[Amity family #Houborizic|Amity family]] (+1250000/1240029, eleventh sliced in five)
+* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
-* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
+* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
-* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six)
+* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
-* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six)
+* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
-* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six)
+* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
-* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven)
+* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
-* ''[[Snipes]]''  → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven)
+* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
-* ''[[Submajor]]'' → [[Buzzardsmic clan #Submajor|Buzzardsmic clan]] (+65536/64827, two octaves and a fourth sliced in eight)
+* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
-* ''[[Escapist]]'' → [[Escapade family #Escapist|Escapade family]] (+65625/65536, fourth sliced in nine)
+* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
-* ''[[Decic]]'' → [[Cloudy clan #Decic|Cloudy clan]] (+16807/16384, generated by the fifth with a 1/10-octave period)
+* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
-* ''[[Amavil]]'' → [[Mabila family #Amavil|Mabila family]] (+17496/16807, four octaves and a fourth sliced in ten)
+* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
-* ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] (+1071875/1062882, fifth sliced in eleven)
+* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
-* ''[[Hendeca]]'' → [[11th-octave temperaments #Hendeca|11th-octave temperaments]] (+122880/117649, generated by the fifth with a 1/11-octave period)
+* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
-* ''[[Compton]]'' → [[Compton family #Compton|Compton family]] (+250047/250000, generated by the classical major third with a 1/12-octave period)
+* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
-* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen)
+* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
-* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
+* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
-* ''[[Gammy]]'' → [[Gammic family #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.
+Considered below are tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].
 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.
@@ Line 41: / Line 40: @@
 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 {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
-== Wizard ==
+== Tritonic ==
-{{Main| Wizard }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
-: ''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 & 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.
+Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 118098/117649
+[[Comma list]]: 225/224, 50421/50000
-{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
+{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
-: mapping generators: ~1225/864, ~245/216
+: mapping generators: ~2, ~10/7
 [[Optimal tuning]]s:
-* [[CTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.919{{c}}
+* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
-: [[error map]]: {{val| 0.000 -0.443 -3.232 +0.361 }}
+: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
-* [[POTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.744{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
-: error map: {{val| 0.000 -1.492 -3.057 -1.388 }}
+: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
+{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
-[[Badness]] (Smith): 0.040846
+[[Badness]] (Sintel): 1.20
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 4000/3993
+Comma list: 121/120, 225/224, 441/440
-Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
+{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
 Optimal tunings:
-* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.900{{c}}
+* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
-* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.768{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
-{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
-Badness (Smith): 0.018539
+Badness (Sintel): 0.782
-==== Lizard ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 351/350, 364/363, 385/384
+Comma list: 105/104, 121/120, 196/195, 275/273
-Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
 Optimal tunings:
-* CTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.703{{c}}
+* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
-* POTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.711{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
-{{Optimal ET sequence|legend=0| 22, 50, 72, 122, 194df }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e }}
-Badness (Smith): 0.021781
+Badness (Sintel): 0.950
-===== 17-limit =====
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
+Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
-Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.748{{c}}
+* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
-* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.619{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
-{{Optimal ET sequence|legend=0| 22, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness (Smith): 0.014536
+Badness (Sintel): 0.973
-===== 19-limit =====
+==== 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
+Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
-Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.677{{c}}
+* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
-* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.523{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
-{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness (Smith): 0.015702
+Badness (Sintel): 1.03
-==== Gizzard ====
+==== 23-limit ====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 225/224, 325/324, 385/384, 1573/1568
+Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
-Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
 Optimal tunings:
-* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.928{{c}}
+* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
-* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.830{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
-{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness (Smith): 0.020252
+Badness (Sintel): 1.04
-===== 17-limit =====
+=== Tritoni ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
-Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
+Comma list: 225/224, 385/384, 27783/27500
-Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
+{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.943{{c}}
+* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
-* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.825{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
-{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
+{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
-Badness (Smith): 0.013624
+Badness (Sintel): 1.50
-===== 19-limit =====
+== Septimin ==
-Subgroup: 2.3.5.7.11.13.17.19
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
+Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 225/224, 84035/82944
+{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
+: mapping generators: ~2, ~12/7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
+: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
+: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}
+{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}
+[[Badness]] (Sintel): 1.38
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
+Comma list: 225/224, 245/242, 385/384
-Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
+{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.918{{c}}
+* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
-* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.862{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
-{{Optimal ET sequence|legend=0| 72, 94, 166g }}
+{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
-Badness (Smith): 0.014810
+Badness (Sintel): 1.04
-=== Mage ===
+=== 13-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
-Comma list: 99/98, 176/175, 1331/1296
+Comma list: 105/104, 144/143, 196/195, 245/242
-Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
+{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
 Optimal tunings:
-* CTE: ~77/54 = 600.000{{c}}, ~55/48 = 217.247{{c}}
+* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
-* POTE: ~77/54 = 600.000{{c}}, ~55/48 = 216.876{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
-{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
+{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
-Badness (Smith): 0.057799
+Badness (Sintel): 0.955
-== Tritonic ==
+== Merman ==
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
+Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
+The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 50421/50000
+[[Comma list]]: 225/224, 2500000/2470629
-{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}
+{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
+: mapping generators: ~2, ~10/7
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 580.286{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
+: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
+: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}
-{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
+{{Optimal ET sequence|legend=1| 41, 84, 125 }}
-[[Badness]] (Smith): 0.047578
+[[Badness]] (Sintel): 1.39
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 121/120, 225/224, 441/440
+Comma list: 225/224, 441/440, 1344/1331
-Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}
+{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.267{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
-{{Optimal ET sequence|legend=0| 29, 31, 60e }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e }}
-Badness (Smith): 0.023659
+Badness (Sintel): 1.20
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 121/120, 196/195, 275/273
+Comma list: 144/143, 225/224, 364/363, 441/440
+{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
+Optimal tunings:
+* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
-Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.108{{c}}
+Badness (Sintel): 1.14
-{{Optimal ET sequence|legend=0| 29, 31, 60e }}
+=== Mermaid ===
+Subgroup: 2.3.5.7.11
-Badness (Smith): 0.022993
+Comma list: 225/224, 385/384, 532400/531441
-==== 17-limit ====
+{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
+Optimal tunings:
+* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
-Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 | 0 -5 11 12 3 18 25 }}
+{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.055{{c}}
+Badness (Sintel): 1.46
-{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness (Smith): 0.019093
+Comma list: 225/224, 325/324, 385/384, 10648/10647
-==== 19-limit ====
+{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
+Optimal tunings:
+* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
-Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 | 0 -5 11 12 3 18 25 15 }}
+{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.026{{c}}
+Badness (Sintel): 1.47
-{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
+== Slender ==
+Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.
-Badness (Smith): 0.017010
+The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
-==== 23-limit ====
+[[Subgroup]]: 2.3.5.7
-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
+[[Comma list]]: 225/224, 589824/588245
-Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 5 | 0 -5 11 12 3 18 25 15 -1 }}
+{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
+: mapping generators: ~2, ~49/48
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.009{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
+: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
+: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}
-{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
+{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
-Badness (Smith): 0.014471
+[[Badness]] (Sintel): 1.44
-=== Tritoni ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 27783/27500
+Comma list: 225/224, 385/384, 1331/1323
-Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}
+{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.389{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
-{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
+{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
-Badness (Smith): 0.045456
+Badness (Sintel): 0.838
-== Septimin ==
+=== 13-limit ===
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Septimin]].''
+Subgroup: 2.3.5.7.11.13
-[[Subgroup]]: 2.3.5.7
+Comma list: 225/224, 275/273, 385/384, 1331/1323
-[[Comma list]]: 225/224, 84035/82944
+{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
-{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}
+Optimal tunings:
+* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/6 = 263.632{{c}}
+{{Optimal ET sequence|legend=0| 31, 63, 94 }}
-{{Optimal ET sequence|legend=1| 41, 91, 132d }}
+Badness (Sintel): 1.07
-[[Badness]] (Smith): 0.054502
+== Triton ==
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
-=== 11-limit ===
+Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.
-Subgroup: 2.3.5.7.11
-Comma list: 225/224, 245/242, 385/384
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}
+[[Comma list]]: 225/224, 1029/1000
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.634{{c}}
+{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
+: mapping generators: ~2, ~10/7
-{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
+: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
+: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}
-Badness (Smith): 0.031309
+{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
-=== 13-limit ===
+[[Badness]] (Sintel): 1.50
-Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 144/143, 196/195, 245/242
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}
+Comma list: 45/44, 56/55, 1029/1000
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.700{{c}}
+{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
-{{Optimal ET sequence|legend=0| 41, 91 }}
+Optimal tunings:
+* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
-Badness (Smith): 0.023117
+{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
-== Merman ==
+Badness (Sintel): 1.51
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Merman]].''
+== Marvolo ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 2500000/2470629
+[[Comma list]]: 225/224, 156250000/155649627
-{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}
+{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
+: mapping generators: ~2, ~21/20
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 585.585{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
+: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
+: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}
-{{Optimal ET sequence|legend=1| 41, 84, 125 }}
+{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
-[[Badness]] (Smith): 0.055078
+[[Badness]] (Sintel): 2.11
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 441/440, 1344/1331
+Comma list: 225/224, 441/440, 4000/3993
-Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}
+{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.606{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}
-{{Optimal ET sequence|legend=0| 41, 84, 125e }}
+{{Optimal ET sequence|legend=0| 29, 43, 72 }}
-Badness (Smith): 0.036383
+Badness (Sintel): 0.958
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 144/143, 225/224, 364/363, 441/440
+Comma list: 169/168, 225/224, 364/363, 441/440
+{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
+Optimal tunings:
+* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
+{{Optimal ET sequence|legend=0| 29, 43, 72 }}
+Badness (Sintel): 0.887
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
+{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
+Optimal tunings:
+* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
+Badness (Sintel): 0.760
+=== 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|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
-Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}
+Optimal tunings:
+* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.657{{c}}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
+Badness (Sintel): 0.895
+== Decic ==
+{{Main| Decic }}
-Badness (Smith): 0.027544
+Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].
-== Slender ==
+It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.
-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
-[[Comma list]]: 225/224, 589824/588245
+[[Comma list]]: 225/224, 16807/16384
-{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
+{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
+: mapping generators: ~15/14, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 38.413{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
+: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
+* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
+: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}
-{{Optimal ET sequence|legend=1| 31, 94, 125 }}
+{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
-[[Badness]] (Smith): 0.056934
+[[Badness]] (Sintel): 2.26
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 1331/1323
+Comma list: 225/224, 385/384, 3087/3025
-Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
+{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.387{{c}}
+Optimal tunings:
+* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
-{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-Badness (Smith): 0.025342
+Badness (Sintel): 2.11
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 275/273, 385/384, 1331/1323
+Comma list: 105/104, 144/143, 196/195, 2200/2197
-Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
+{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.314{{c}}
+Optimal tunings:
+* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})
-{{Optimal ET sequence|legend=0| 31, 63, 94 }}
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-Badness (Smith): 0.025913
+Badness (Sintel): 1.52
-== Triton ==
+==== 17-limit ====
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Stump]].''
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
-[[Comma list]]: 225/224, 1029/1000
+{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
-{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 568.865{{c}}
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
+Badness (Sintel): 1.28
-[[Badness]] (Smith): 0.059245
+=== Splendecic ===
+Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 45/44, 56/55, 1029/1000
+Comma list: 225/224, 1617/1600, 2401/2376
-Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
+{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 569.144{{c}}
+Optimal tunings:
+* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})
-{{Optimal ET sequence|legend=0| 2, 17d, 19, 59bde, 78bde, 97bde }}
+{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}
-Badness (Smith): 0.045675
+Badness (Sintel): 1.98
-== Marvolo ==
+==== 13-limit ====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13
+Comma list: 105/104, 196/195, 1001/1000, 1188/1183
-[[Comma list]]: 225/224, 156250000/155649627
+{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
-{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 83.348{{c}}
+{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}
-{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}
+Badness (Sintel): 1.57
-[[Badness]] (Smith): 0.083338
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-=== 11-limit ===
+Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
-Subgroup: 2.3.5.7.11
-Comma list: 225/224, 441/440, 4000/3993
+{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
-Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.340{{c}}
+{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}
-{{Optimal ET sequence|legend=0| 29, 43, 72 }}
+Badness (Sintel): 1.33
-Badness (Smith): 0.028965
+=== Prodecic ===
+Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.
-=== 13-limit ===
+Subgroup: 2.3.5.7.11
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 225/224, 364/363, 441/440
+Comma list: 225/224, 441/440, 5929/5832
-Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
+{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
+Optimal tunings:
+* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})
-{{Optimal ET sequence|legend=0| 29, 43, 72, 115f }}
+{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
-Badness (Smith): 0.021470
+Badness (Sintel): 2.20
-=== 17-limit ===
+==== 13-limit ====
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
+Comma list: 105/104, 196/195, 275/273, 5929/5832
-Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
+{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
+Optimal tunings:
+* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})
-{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
+{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
-Badness (Smith): 0.014924
+Badness (Sintel): 1.73
-=== 19-limit ===
+==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
+Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
-Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
+{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}
+Optimal tunings:
+* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})
-{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
+{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
-Badness (Smith): 0.014722
+Badness (Sintel): 1.41
 == Enneaportent ==
@@ Line 499: / Line 624: @@
 {{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
-: Mapping generators: ~2592/2401, ~12005/6912
+: mapping generators: ~2592/2401, ~12005/6912
-[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.1680{{c}} (~1728/1715 = 16.8347{{c}})
+[[Optimal tuning]]s:
+* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
+: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
+* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
+: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}
-{{Optimal ET sequence|legend=1| 9, 63, 72, 495bcd }}
+{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
-[[Badness]] (Smith): 0.093679
+[[Badness]] (Sintel): 2.37
 === 11-limit ===
@@ Line 512: / Line 641: @@
 Comma list: 225/224, 385/384, 12005/11979
-Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
+{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}
-Optimal tuning (POTE): ~121/112 = 133.3333{{c}}, ~210/121 = 950.1873{{c}} (~99/98 = 16.8540{{c}})
+Optimal tunings:
+* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
+* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})
-{{Optimal ET sequence|legend=0| 9, 63, 72, 423cd, 495bcd }}
+{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
-Badness (Smith): 0.030426
+Badness (Sintel): 1.01
 === 13-limit ===
@@ Line 525: / Line 656: @@
 Comma list: 169/168, 225/224, 364/363, 1716/1715
-Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
+{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
-Optimal tuning (POTE): ~14/13 = 133.3333{{c}}, ~26/15 = 950.2867{{c}} (~105/104 = 16.9534{{c}})
+Optimal tunings:
+* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
+* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})
-{{Optimal ET sequence|legend=0| 9, 63, 72, 279cf }}
+{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
-Badness (Smith): 0.022322
+Badness (Sintel): 0.922
 == Gracecordial ==
-: ''For the 5-limit version of this temperament, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''
+: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 541: / Line 674: @@
 {{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
+: mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.824{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
+: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
+: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}
-{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
+{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
-[[Badness]] (Smith): 0.096279
+[[Badness]] (Sintel): 2.44
 === 11-limit ===
@@ Line 553: / Line 691: @@
 Comma list: 225/224, 385/384, 236328125/234365481
-Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
+{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.834{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}
-{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125, 238c }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}
-Badness (Smith): 0.089588
+Badness (Sintel): 2.96
 ==== 13-limit ====
@@ Line 566: / Line 706: @@
 Comma list: 225/224, 325/324, 385/384, 831875/830466
-Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}
-{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness (Smith): 0.052235
+Badness (Sintel): 2.16
 ==== 17-limit ====
@@ Line 579: / Line 721: @@
 Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}
-{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness (Smith): 0.038565
+Badness (Sintel): 1.96
 ==== 19-limit ====
@@ Line 592: / Line 736: @@
 Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.842{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}
-{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness (Smith): 0.028165
+Badness (Sintel): 1.71
 ==== 23-limit ====
@@ Line 605: / Line 751: @@
 Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.843{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}
-{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
-Badness (Smith): 0.021879
+Badness (Sintel): 1.57
 ==== 29-limit ====
@@ Line 618: / Line 766: @@
 Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
+{{Mapping|legend=0| 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{{c}}, ~3/2 = 700.842{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}
-{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
-Badness (Smith): 0.018011
+Badness (Sintel): 1.50
 ==== 31-limit ====
@@ Line 631: / Line 781: @@
 Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
+{{Mapping|legend=0| 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{{c}}, ~3/2 = 700.838{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}
-{{Optimal ET sequence|legend=0| 12e, 101cdek, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
-Badness (Smith): 0.016007
+Badness (Sintel): 1.53
 === Gracecord ===
@@ Line 644: / Line 796: @@
 Comma list: 225/224, 441/440, 109375/107811
-Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
+{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.885{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}
-{{Optimal ET sequence|legend=0| 12, 101cd, 113 }}
+{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
-Badness (Smith): 0.066964
+Badness (Sintel): 2.21
 ==== 13-limit ====
@@ Line 657: / Line 811: @@
 Comma list: 225/224, 364/363, 441/440, 6125/6084
-Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
+{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.890{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}
-{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
-Badness (Smith): 0.044196
+Badness (Sintel): 1.83
 ==== 17-limit ====
@@ Line 670: / Line 826: @@
 Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
-Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
+{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.870{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}
 {{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness (Smith): 0.036637
+Badness (Sintel): 1.87
 ==== 19-limit ====
@@ Line 683: / Line 841: @@
 Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
-Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
+{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.866{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}
 {{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness (Smith): 0.027559
+Badness (Sintel): 1.68
 == Alphorn ==
@@ Line 696: / Line 856: @@
 [[Comma list]]: 225/224, 5764801/5668704
-{{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}
+{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
+: mapping generators: ~2, ~35/24
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~48/35 = 556.221{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
+: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
+* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
+: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}
-{{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
+{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
-[[Badness]] (Smith): 0.129258
+[[Badness]] (Sintel): 3.27
 === 11-limit ===
@@ Line 709: / Line 874: @@
 Comma list: 225/224, 385/384, 12250/11979
-Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}
+{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 556.144{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}
-{{Optimal ET sequence|legend=0| 28d, 41, 315cde }}
+{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}
-Badness (Smith): 0.073459
+Badness (Sintel): 2.43
 == Misneb ==
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Misneb]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 724: / Line 891: @@
 [[Comma list]]: 225/224, 4194304/4117715
-{{Mapping|legend=1| 1 3 1 3 | 0 -15 14 -2 }}
+{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
+: mapping generators: ~2, ~15/8
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/15 = 113.235{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
+: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
+: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}
 {{Optimal ET sequence|legend=1| 21, 32, 53 }}
-[[Badness]] (Smith): 0.140970
+[[Badness]] (Sintel): 3.57
 === 11-limit ===
@@ Line 737: / Line 909: @@
 Comma list: 99/98, 176/175, 1310720/1294139
-Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}
+{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}
-{{Optimal ET sequence|legend=0| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
-Badness (Smith): 0.085390
+Badness (Sintel): 2.82
 ==== 13-limit ====
@@ Line 750: / Line 924: @@
 Comma list: 99/98, 176/175, 640/637, 847/845
-Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}
+{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}
-{{Optimal ET sequence|legend=0| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
-Badness (Smith): 0.045569
+Badness (Sintel): 1.88
 === Musneb ===
@@ Line 763: / Line 939: @@
 Comma list: 225/224, 385/384, 66550/64827
-Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
+{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.142{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}
-{{Optimal ET sequence|legend=0| 32, 53, 191de, 244cddee, 297cddee }}
+{{Optimal ET sequence|legend=0| 21e, 32, 53 }}
-Badness (Smith): 0.087333
+Badness (Sintel): 2.89
 == Untriton ==
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Untriton]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
+Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 778: / Line 958: @@
 [[Comma list]]: 225/224, 125000000/121060821
-{{Mapping|legend=1| 1 6 -7 -7 | 0 -9 19 20 }}
+{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
+: mapping generators: ~2, ~10/7
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 588.641{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
+: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
+: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}
-{{Optimal ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
+{{Optimal ET sequence|legend=1| 51, 53 }}
-[[Badness]] (Smith): 0.143976
+[[Badness]] (Sintel): 3.64
 === 11-limit ===
@@ Line 791: / Line 976: @@
 Comma list: 121/120, 225/224, 22000/21609
-Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}
+{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.626{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}
 {{Optimal ET sequence|legend=0| 51, 53 }}
-Badness (Smith): 0.074295
+Badness (Sintel): 2.46
 === 13-limit ===
@@ Line 804: / Line 991: @@
 Comma list: 121/120, 225/224, 275/273, 1040/1029
-Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}
+{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.654{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}
 {{Optimal ET sequence|legend=0| 51f, 53 }}
-Badness (Smith): 0.047441
+Badness (Sintel): 1.96
 == Naiadical ==
+Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 823543/800000
-{{Mapping|legend=1| 1 5 -3 -1 | 0 -9 14 10 }}
+{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
+: mapping generators: ~2, ~32/21
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1202.120{{c}}, ~21/16 = 456.652{{c}}
+* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~21/16 = 455.868{{c}}
+: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
+: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}
-{{Optimal ET sequence|legend=1| 21, 29, 50, 79d }}
+{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
-[[Badness]] (Sintel): 3.667
+[[Badness]] (Sintel): 3.67
 === 11-limit ===
@@ Line 832: / Line 1,026: @@
 Comma list: 225/224, 245/242, 1617/1600
-Mapping: {{Mapping| 1 5 -3 -1 -3 | 0 -9 14 10 17 }}
+Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}
 Optimal tunings:
-* WE: ~2 = 1201.901, ~21/16{{c}} = 456.514{{c}}
+* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
-* CWE: ~2 = 1200.000, ~21/16{{c}} = 455.822{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}
 {{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
-Badness (Sintel): 1.998
+Badness (Sintel): 2.00
 === 13-limit ===
@@ Line 847: / Line 1,041: @@
 Comma list: 105/104, 196/195, 245/242, 1001/1000
-Mapping: {{Mapping| 1 5 -3 -1 -3 -2 | 0 -9 14 10 17 15 }}
+Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
 Optimal tunings:
-* WE: ~2 = 1201.786{{c}}, ~13/10 = 456.452{{c}}
+* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.807{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}
 {{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
-Badness (Sintel): 1.430
+Badness (Sintel): 1.43
 === 17-limit ===
@@ Line 862: / Line 1,056: @@
 Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
-Mapping: {{Mapping| 1 5 -3 -1 -3 -2 -2 | 0 -9 14 10 17 15 16 }}
+Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
 Optimal tunings:
-* WE: ~2 = 1201.921{{c}}, ~13/10 = 456.523{{c}}
+* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.833{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}
 {{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
-Badness (Sintel): 1.260
+Badness (Sintel): 1.26
 == Quintannic ==
+Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 878: / Line 1,074: @@
 {{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
+: mapping generators: ~2, ~10000/9261
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10000/9261 = 139.838{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
+: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
+: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}
 {{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
-[[Badness]] (Smith): 0.150565
+[[Badness]] (Sintel): 3.81
 === 11-limit ===
@@ Line 890: / Line 1,091: @@
 Comma list: 225/224, 441/440, 43923/43750
-Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
+{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~320/297 = 139.827{{c}}
+Optimal tunings:
+* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}
-{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}
-Badness (Smith): 0.052590
+Badness (Sintel): 1.74
 === 13-limit ===
@@ Line 903: / Line 1,106: @@
 Comma list: 225/224, 441/440, 1001/1000, 1188/1183
-Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
+{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.812{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}
 {{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness (Smith): 0.032730
+Badness (Sintel): 1.35
 === 17-limit ===
@@ Line 916: / Line 1,121: @@
 Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
-Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
+{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.815{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}
 {{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness (Smith): 0.023038
+Badness (Sintel): 1.17
+== Hendeca ==
+{{Distinguish| Hendec }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
+Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 225/224, 122880/117649
+{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
+: mapping generators: ~16/15, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
+: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
+* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
+: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}
+{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
+[[Badness]] (Sintel): 4.37
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 121/120, 225/224, 352/343
+{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}
+Optimal tunings:
+* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
+* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}
+{{Optimal ET sequence|legend=0| 22, 55d }}
+Badness (Sintel): 2.46
 == Gwazy ==
-{{See also| Very high accuracy temperaments #Kwazy }}
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
+Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 932: / Line 1,179: @@
 {{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
+: mapping generators: ~2401/1728, ~35/32
-[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000{{c}}, ~35/32 = 162.658{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
+: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
+* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
+: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}
 {{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
-[[Badness]] (Smith): 0.178826
+[[Badness]] (Sintel): 4.53
 === 11-limit ===
@@ Line 944: / Line 1,196: @@
 Comma list: 99/98, 176/175, 65536/65219
-Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
+{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}
-Optimal tuning (POTE): ~363/256 = 600.000{{c}}, ~11/10 = 162.592{{c}}
+Optimal tunings:
+* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
+* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}
-{{Optimal ET sequence|legend=0| 22, 74, 96, 118d }}
+{{Optimal ET sequence|legend=0| 22, 74, 96 }}
-Badness (Smith): 0.068410
+Badness (Sintel): 2.26
 == Tertiosec ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''
+Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.
 [[Subgroup]]: 2.3.5.7
@@ Line 959: / Line 1,215: @@
 [[Comma list]]: 225/224, 14495514624/13841287201
-{{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}
+{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
+: mapping generators: ~3072/2401, ~2048/1715
-[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 400.000{{c}}, ~15/14 = 112.283{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
+: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
+* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
+: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}
 {{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
-[[Badness]] (Smith): 0.431636
+[[Badness]] (Sintel): 10.9
 === 11-limit ===
@@ Line 972: / Line 1,233: @@
 Comma list: 225/224, 3840/3773, 12005/11979
-Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}
+{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
-Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~15/14 = 112.171{{c}}
+Optimal tunings:
+* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}
 {{Optimal ET sequence|legend=0| 21, 54, 75e }}
-Badness (Smith): 0.173485
+Badness (Sintel): 5.74
+== References ==
+[[Category:Marvel temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Marvel temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Marvel| ]] <!-- key article -->
-[[Category:Rank 2]]

Marvel temperaments: Difference between revisions

Latest revision as of 12:32, 24 September 2026

Tritonic

11-limit

13-limit

17-limit

19-limit

23-limit

Tritoni

Septimin

11-limit

13-limit

Merman

11-limit

13-limit

Mermaid

13-limit

Slender

11-limit

13-limit

Triton

11-limit

Marvolo

11-limit

13-limit

17-limit

19-limit

Decic

11-limit

13-limit

17-limit

Splendecic

13-limit

17-limit

Prodecic

13-limit

17-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

Hendeca

11-limit

Gwazy

11-limit

Tertiosec

11-limit

References

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