Schismatic family: Difference between revisions

Revision as of 18:05, 6 May 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.

The 5-limit parent comma for the schismatic (or schismic) family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the syntonic comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth.

Schismic, schismatic, a.k.a. helmholtz

Main article: Schismic

The 5-limit version of the temperament is a microtemperament, called schismic, schismatic, or helmholtz. The generator is a fifth, flattened by a fraction of a schisma, and 5/4 is represented by a diminished fourth. This defies the tradition of tertian harmony, as the just major triad on C is C–F♭–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.

As a 5-limit system, schismic is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismic, but you need 118edo if you want to get the full effect. In exact analogy with 1/4-comma meantone there is also 1/8 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 ¢, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.

Subgroup: 2.3.5

Comma list: 32805/32768

Mapping: [⟨1 0 15], ⟨0 1 -8]]

mapping generators: ~2, ~3

Optimal tunings:

WE: ~2 = 1200.0749 ¢, ~3/2 = 701.7797 ¢

error map: ⟨+0.075 -0.100 -0.027]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7308 ¢

error map: ⟨0.000 -0.224 -0.160]

Tuning ranges:

5-odd-limit diamond monotone: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
5-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)

Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc

Badness (Sintel): 0.0999

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Garibaldi adds [25 -14 0 -1⟩, grackle adds [-44 26 0 1⟩, pontiac adds [-59 39 0 -1⟩, and schism adds [6 -2 0 -1⟩. Those all have a fifth as generator.

Bischismic adds [-69 40 0 2⟩ and has a fifth generator with a half-octave period. Salsa adds [15 -13 0 2⟩ and has a hemififth generator. Hemischis adds [-34 25 0 -2⟩ and has a hemitwelfth generator. Guiron adds [-10 1 0 3⟩, with an ~8/7 generator, three of which give the fifth. Term adds [-94 54 0 3⟩ with a 1/3-octave period. Squirrel, tertiaschis, and countertertiaschis each has a generator that is 1/3 of the fourth. Quadrant adds [-119 68 0 4⟩ with a 1/4-octave period. Kleischismic adds [49 -38 0 4⟩ with a half-octave period and also a bisect generator. Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.

Temperaments involving larger splits include tsaharuk, quanharuk, quintilipyth, quintaschis, altinex, pogo, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths. Those split the schismic structure into five to thirteen parts.

Temperaments discussed elsewhere include:

Guiron (+1029/1024) → Gamelismic clan
Pogo (+118098/117649) → Stearnsmic clan

Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.

The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in #Subgroup extensions.

Garibaldi

Main article: Garibaldi

Garibaldi tempers out the garischisma, equating the septimal comma with both the syntonic comma and the Pythagorean comma. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its S-expression-based comma list is {S8/S9, S15}.

Subgroup: 2.3.5.7

Comma list: 225/224, 3125/3087

Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]

Optimal tunings:

WE: ~2 = 1200.1233 ¢, ~3/2 = 702.1573 ¢

error map: ⟨+0.123 +0.326 -2.709 +2.328]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0774 ¢

error map: ⟨0.000 +0.122 -2.933 +2.090]

Minimax tuning:

7-odd-limit: ~3/2 = [2/3 1/15 0 -1/15⟩

[[1 0 0 0⟩, [5/3 1/15 0 -1/15⟩, [5/3 -8/15 0 8/15⟩, [5/3 -14/15 0 14/15⟩]

unchanged-interval (eigenmonzo) basis: 2.7/3

9-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩

[[1 0 0 0⟩, [25/16 1/8 0 -1/16⟩, [5/2 -1 0 1/2⟩, [25/8 -7/4 0 7/8⟩]

unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]

Optimal ET sequence: 12, 29, 41, 53, 94

Badness (Sintel): 0.548

Cassandra

Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2200/2187

Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]

Optimal tunings:

WE: ~2 = 1200.3089 ¢, ~3/2 = 702.3377 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1562 ¢

Minimax tuning:

11-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩

unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]

Optimal ET sequence: 12e, 41, 53, 94, 229c

Badness (Sintel): 0.906

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 325/324, 385/384

Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]

Optimal tunings:

WE: ~2 = 1200.1703 ¢, ~3/2 = 702.2122 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1135 ¢

Minimax tuning:

13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34⟩

unchanged-interval (eigenmonzo) basis: 2.13/7

Tuning ranges:

13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]

Optimal ET sequence: 41, 53, 94, 429ccdeef, 523ccdeef

Badness (Sintel): 0.854

Cassie

Subgroup: 2.3.5.7.11.13.17

Comma list: 120/119, 154/153, 225/224, 273/272, 325/324

Mapping: [⟨1 0 15 25 -33 -28 -7], ⟨0 1 -8 -14 23 20 7]]

Optimal tunings:

WE: ~2 = 1199.8140 ¢, ~3/2 = 701.9833 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0909 ¢

Optimal ET sequence: 12e, 41, 53, 94g

Badness (Sintel): 1.19

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272

Mapping: [⟨1 0 15 25 -33 -28 -7 9], ⟨0 1 -8 -14 23 20 7 -3]]

Optimal tunings:

WE: ~2 = 1199.9556 ¢, ~3/2 = 702.0530 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0787 ¢

Optimal ET sequence: 12e, 41, 53

Badness (Sintel): 1.11

Cassandric

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 275/273, 325/324, 375/374, 385/384

Mapping: [⟨1 0 15 25 -33 -28 77], ⟨0 1 -8 -14 23 20 -46]]

Optimal tunings:

WE: ~2 = 1200.0046 ¢, ~3/2 = 702.2167 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0962 ¢

Optimal ET sequence: 41g, 53, 94

Badness (Sintel): 1.18

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374

Mapping: [⟨1 0 15 25 -33 -28 77 9], ⟨0 1 -8 -14 23 20 -46 -3]]

Optimal tunings:

WE: ~2 = 1200.2910 ¢, ~3/2 = 702.2681 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0967 ¢

Optimal ET sequence: 41g, 53, 94

Badness (Sintel): 1.07

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 209/208, 225/224, 253/252, 275/273, 325/324, 375/374

Mapping: [⟨1 0 15 25 -33 -28 77 9 60], ⟨0 1 -8 -14 23 20 -46 -3 -35]]

Optimal tunings:

WE: ~2 = 1200.2970 ¢, ~3/2 = 702.2697 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0943 ¢

Optimal ET sequence: 41g, 53, 94

Badness (Sintel): 1.08

Cassander

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 225/224, 275/273, 325/324, 385/384

Mapping: [⟨1 0 15 25 -33 -28 -72], ⟨0 1 -8 -14 23 20 48]]

Optimal tunings:

WE: ~2 = 1200.1986 ¢, ~3/2 = 702.2598 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1455 ¢

Optimal ET sequence: 41, 53g, 94

Badness (Sintel): 1.14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324

Mapping: [⟨1 0 15 25 -33 -28 -72 9], ⟨0 1 -8 -14 23 20 48 -3]]

Optimal tunings:

WE: ~2 = 1200.3057 ¢, ~3/2 = 702.3138 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1373 ¢

Optimal ET sequence: 41, 53g, 94

Badness (Sintel): 1.07

Andromeda

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/242

Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]

Optimal tunings:

WE: ~2 = 1200.1917 ¢, ~3/2 = 702.4836 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3599 ¢

Minimax tuning:

11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20⟩

unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:

11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]

Optimal ET sequence: 12, 29, 41

Badness (Sintel): 0.779

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195, 245/242

Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]

Optimal tunings:

WE: ~2 = 1200.3031 ¢, ~3/2 = 702.7368 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5420 ¢

Minimax tuning:

13- and 15-odd-limit: ~3/2 = [14/23 2/23 0 0 0 -1/23⟩

unchanged-interval (eigenmonzo) basis: 2.13/9

Tuning ranges:

13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]

Optimal ET sequence: 12f, 29, 41

Badness (Sintel): 0.857

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 189/187, 196/195

Mapping: [⟨1 0 15 25 32 37 -7], ⟨0 1 -8 -14 -18 -21 7]]

Optimal tunings:

WE: ~2 = 1199.1984 ¢, ~3/2 = 701.8424 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3384 ¢

Optimal ET sequence: 12f, 29, 41

Badness (Sintel): 1.19

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195

Mapping: [⟨1 0 15 25 32 37 -7 9], ⟨0 1 -8 -14 -18 -21 7 -3]]

Optimal tunings:

WE: ~2 = 1199.5242 ¢, ~3/2 = 702.0783 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3711 ¢

Optimal ET sequence: 12f, 29, 41

Badness (Sintel): 1.17

Schisicosiennic

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 154/153, 170/169, 196/195

Mapping: [⟨1 0 15 25 32 37 58], ⟨0 1 -8 -14 -18 -21 -34]]

Optimal tunings:

WE: ~2 = 1200.6122 ¢, ~3/2 = 703.0830 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6968 ¢

Optimal ET sequence: 12fg, 29g, 41, 70cd

Badness (Sintel): 1.11

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189

Mapping: [⟨1 0 15 25 32 37 58 9], ⟨0 1 -8 -14 -18 -21 -34 -3]]

Optimal tunings:

WE: ~2 = 1200.7981 ¢, ~3/2 = 703.2199 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7221 ¢

Optimal ET sequence: 12fg, 29g, 41, 70cd

Badness (Sintel): 1.09

Schisicosiennoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 85/84, 100/99, 105/104, 119/117, 221/220

Mapping: [⟨1 0 15 25 32 37 12], ⟨0 1 -8 -14 -18 -21 -5]]

Optimal tunings:

WE: ~2 = 1201.3146 ¢, ~3/2 = 703.4864 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6491 ¢

Optimal ET sequence: 12f, 29g, 41g

Badness (Sintel): 1.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152

Mapping: [⟨1 0 15 25 32 37 12 9], ⟨0 1 -8 -14 -18 -21 -5 -3]]

Optimal tunings:

WE: ~2 = 1201.3140 ¢, ~3/2 = 703.4860 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6578 ¢

Optimal ET sequence: 12f, 29g, 41g

Badness (Sintel): 1.02

Helenus

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 3125/3087

Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]

Optimal tunings:

WE: ~2 = 1199.7097 ¢, ~3/2 = 701.5554 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7370 ¢

Minimax tuning:

11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩

unchanged-interval (eigenmonzo) basis: 2.11/9

Optimal ET sequence: 12, 41e, 53, 118d

Badness (Sintel): 1.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 275/273, 847/845

Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]]

Optimal tunings:

WE: ~2 = 1199.7370 ¢, ~3/2 = 701.5937 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7570 ¢

Minimax tuning:

13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩

unchanged-interval (eigenmonzo) basis: 2.11/9

Optimal ET sequence: 12f, …, 41ef, 53, 118d

Badness (Sintel): 1.09

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 99/98, 120/119, 176/175, 275/273, 442/441

Mapping: [⟨1 0 15 25 51 56 -7], ⟨0 1 -8 -14 -30 -33 7]]

Optimal tunings:

WE: ~2 = 1199.2895 ¢, ~3/2 = 701.2643 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6967 ¢

Optimal ET sequence: 12f, 53, 65d, 118dg

Badness (Sintel): 1.21

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245

Mapping: [⟨1 0 15 25 51 56 -7 9], ⟨0 1 -8 -14 -30 -33 7 -3]]

Optimal tunings:

WE: ~2 = 1199.5280 ¢, ~3/2 = 701.4290 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7149 ¢

Optimal ET sequence: 12f, 53, 65d

Badness (Sintel): 1.18

Karadeniz

See also: Turkish maqam music temperaments § Karadeniz temperament

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3125/3087

Mapping: [⟨1 1 7 11 2], ⟨0 2 -16 -28 5]]

mapping generators: ~2, ~11/9

Optimal tunings:

WE: ~2 = 1199.7351 ¢, ~11/9 = 350.9167 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.9995 ¢

Optimal ET sequence: 24d, 41, 65d, 106, 147

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 640/637

Mapping: [⟨1 1 7 11 2 -8], ⟨0 2 -16 -28 5 40]]

Optimal tunings:

WE: ~2 = 1199.3042 ¢, ~11/9 = 350.7533 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.9686 ¢

Optimal ET sequence: 24d, 41, 65d, 106f

Badness (Sintel): 1.34

Hemigari

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 3125/3087

Mapping: [⟨1 0 15 25 9], ⟨0 2 -16 -28 -7]]

mapping generators: ~2, ~110/63

Optimal tunings:

WE: ~2 = 1200.7303 ¢, ~110/63 = 951.6605 ¢
CWE: ~2 = 1200.0000 ¢, ~110/63 = 951.0604 ¢

Optimal ET sequence: 24d, 29, 53, 82e, 135ee

Badness (Sintel): 1.68

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 225/224, 275/273

Mapping: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]

Optimal tunings:

WE: ~2 = 1200.8146 ¢, ~26/15 = 951.7273 ¢
CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.0574 ¢

Optimal ET sequence: 24d, 29, 53, 82e, 135eef

Badness (Sintel): 1.13

Sanjaab

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323, 3125/3087

Mapping: [⟨1 2 -1 -3 0], ⟨0 -3 24 42 25]]

mapping generators: ~2, ~11/10

Optimal tunings:

WE: ~2 = 1200.1997 ¢, ~11/10 = 166.0018 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9786 ¢

Optimal ET sequence: 29, 65d, 94

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 847/845, 1331/1323

Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]

Optimal tunings:

WE: ~2 = 1200.1224 ¢, ~11/10 = 165.9800 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9659 ¢

Optimal ET sequence: 29, 65d, 94

Badness (Sintel): 1.40

Pontiac

Main article: Pontiac

Pontiac tempers out the ragisma, rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^³A).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]

Optimal tunings:

WE: ~2 = 1200.0989 ¢, ~3/2 = 701.8145 ¢

error map: ⟨+0.099 -0.042 -0.138 -0.038]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7579 ¢

error map: ⟨0.000 -0.197 -0.377 -0.268]

Minimax tuning:

7-odd-limit: ~3/2 = [27/47 0 -1/47 1/47⟩

[[1 0 0 0⟩, [74/47 0 -1/47 1/47⟩, [113/47 0 8/47 -8/47⟩, [113/47 0 -39/47 39/47⟩]

unchanged-interval (eigenmonzo) basis: 2.7/5

9-odd-limit: ~3/2 = [1/2 1/5 -1/10⟩

[[1 0 0 0⟩, [3/2 1/5 -1/10 0⟩, [3 -8/5 4/5 0⟩, [-1/2 39/5 -39/10 0⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

7- and 9-odd-limit diamond monotone: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955]

Optimal ET sequence: 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd

Badness (Sintel): 0.358

Helenoid

Helenoid may be described as 53 & 118, and is closely related to the helenus temperament, differing only by the mapping of 7.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 4375/4374

Mapping: [⟨1 0 15 -59 51], ⟨0 1 -8 39 -30]]

Optimal tunings:

WE: ~2 = 1200.3277 ¢, ~3/2 = 701.9135 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7223 ¢

Minimax tuning:

11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69⟩

unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 53, 118, 289e, 407de

Badness (Sintel): 1.28

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 729/728

Mapping: [⟨1 0 15 -59 51 56], ⟨0 1 -8 39 -30 -33]]

Optimal tunings:

WE: ~2 = 1200.1780 ¢, ~3/2 = 701.8491 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7446 ¢

Minimax tuning:

13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72⟩

unchanged-interval (eigenmonzo) basis: 2.13/7

Optimal ET sequence: 53, 118, 171e

Badness (Sintel): 1.39

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 729/728

Mapping: [⟨1 0 15 -59 51 56 -91], ⟨0 1 -8 39 -30 -33 60]]

Optimal tunings:

WE: ~2 = 1200.1645 ¢, ~3/2 = 701.8385 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7425 ¢

Minimax tuning:

17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93⟩

unchanged-interval (eigenmonzo) basis: 2.17/13

Optimal ET sequence: 53, 118, 171e

Badness (Sintel): 1.47

Helena

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 3146/3125

Mapping: [⟨1 0 15 -59 51 -28], ⟨0 1 -8 39 -30 20]]

Optimal tunings:

WE: ~2 = 1200.5227 ¢, ~3/2 = 702.0456 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7418 ¢

Optimal ET sequence: 53, 118f, 171ef

Badness (Sintel): 1.50

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125

Mapping: [⟨1 0 15 -59 51 -28 -91], ⟨0 1 -8 39 -30 20 60]]

Optimal tunings:

WE: ~2 = 1200.4988 ¢, ~3/2 = 702.0218 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7332 ¢

Optimal ET sequence: 53, 118f, 171ef

Badness (Sintel): 1.56

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625

Mapping: [⟨1 0 15 -59 51 -28 -91 9], ⟨0 1 -8 39 -30 20 60 -3]]

Optimal tunings:

WE: ~2 = 1200.5185 ¢, ~3/2 = 702.0323 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7318 ¢

Optimal ET sequence: 53, 118f, 171ef

Badness (Sintel): 1.33

Ponta

Ponta tempers out 540/539 and may be described as 171 & 224. 224edo itself makes for an excellent tuning.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59 135], ⟨0 1 -8 39 -83]]

Optimal tunings:

WE: ~2 = 1199.9814 ¢, ~3/2 = 701.7725 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7834 ¢

Minimax tuning:

11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩

unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 53, 171, 224

Badness (Sintel): 1.61

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2200/2197

Mapping: [⟨1 0 15 -59 135 56], ⟨0 1 -8 39 -83 -33]]

Optimal tunings:

WE: ~2 = 1199.9601 ¢, ~3/2 = 701.7610 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7845 ¢

Minimax tuning:

13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩

unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 53, 171, 224

Badness (Sintel): 0.976

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197

Mapping: [⟨1 0 15 -59 135 56 -91], ⟨0 1 -8 39 -83 -33 60]]

Optimal tunings:

WE: ~2 = 1199.8850 ¢, ~3/2 = 701.7101 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7775 ¢

Minimax tuning:

17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143⟩

unchanged-interval (eigenmonzo) basis: 2.17/11

Optimal ET sequence: 53, 171, 224, 395e, 619eg

Badness (Sintel): 1.16

Pontic

Pontic temperament tempers out 441/440 and may be described as 118 & 171. 289edo may be recommended as a tuning.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59 -136], ⟨0 1 -8 39 88]]

Optimal tunings:

WE: ~2 = 1200.1259 ¢, ~3/2 = 701.7980 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7256 ¢

Minimax tuning:

11-odd-limit: ~3/2 = [6/11 0 0 0 1/88⟩

unchanged-interval (eigenmonzo) basis: 2.11

Optimal ET sequence: 53e, 118, 289, 407d

Badness (Sintel): 1.64

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 625/624, 729/728, 3584/3575

Mapping: [⟨1 0 15 -59 -136 56], ⟨0 1 -8 39 88 -33]]

Optimal tunings:

WE: ~2 = 1199.9254 ¢, ~3/2 = 701.6945 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7378 ¢

Minimax tuning:

13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩

unchanged-interval (eigenmonzo) basis: 2.13/11

Optimal ET sequence: 53e, 118, 171, 289f

Badness (Sintel): 1.87

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873

Mapping: [⟨1 0 15 -59 -136 56 -91], ⟨0 1 -8 39 88 -33 60]]

Optimal tunings:

WE: ~2 = 1199.9454 ¢, ~3/2 = 701.7085 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7401 ¢

Minimax tuning:

17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩

unchanged-interval (eigenmonzo) basis: 2.13/11

Optimal ET sequence: 53e, 118, 171, 289f

Badness (Sintel): 1.51

Pontoid

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59 -136 -215], ⟨0 1 -8 39 88 138]]

Optimal tunings:

WE: ~2 = 1200.0897 ¢, ~3/2 = 701.7874 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7356 ¢

Optimal ET sequence: 53ef, 118f, 171, 289

Badness (Sintel): 2.07

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768

Mapping: [⟨1 0 15 -59 -136 -215 -91], ⟨0 1 -8 39 88 138 60]]

Optimal tunings:

WE: ~2 = 1200.1045 ¢, ~3/2 = 701.7962 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7359 ¢

Optimal ET sequence: 53ef, 118f, 171, 289, 460e, 749defg

Badness (Sintel): 1.50

Bipont

Bipont tempers out the lehmerisma (3025/3024) and the kalisma (9801/9800). It may be described as 118 & 224. It has a period of half octave and a ploidacot signature of diploid monocot. 342edo may be recommended as a tuning.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 32805/32768

Mapping: [⟨2 0 30 -118 -85], ⟨0 1 -8 39 29]]

mapping generators: ~99/70, ~3

Optimal tunings:

WE: ~99/70 = 600.0500 ¢, ~3/2 = 701.8153 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7584 ¢

Optimal ET sequence: 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde

Badness (Sintel): 0.484

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 4096/4095

Mapping: [⟨2 0 30 -118 -85 112], ⟨0 1 -8 39 29 -33]]

Optimal tunings:

WE: ~99/70 = 599.9939 ¢, ~3/2 = 701.7657 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7728 ¢

Optimal ET sequence: 106, 118, 224, 566f, 790f

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873

Mapping: [⟨2 0 30 -118 -85 112 -182], ⟨0 1 -8 39 29 -33 60]]

Optimal tunings:

WE: ~99/70 = 599.9839 ¢, ~3/2 = 701.7463 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7649 ¢

Optimal ET sequence: 106g, 118, 224, 342, 566f

Badness (Sintel): 1.38

Counterbipont

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768

Mapping: [⟨2 0 30 -118 -85 -243], ⟨0 1 -8 39 29 79]]

Optimal tunings:

WE: ~99/70 = 600.0405 ¢, ~3/2 = 701.8160 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7697 ¢

Optimal ET sequence: 106f, 118f, 224, 342f, 566, 1356cf

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768

Mapping: [⟨2 0 30 -118 -85 -243 -182], ⟨0 1 -8 39 29 79 60]]

Optimal tunings:

WE: ~99/70 = 600.0336 ¢, ~3/2 = 701.8031 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7647 ¢

Optimal ET sequence: 106fg, 118f, 224, 342f, 566

Badness (Sintel): 1.29

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864

Mapping: [⟨2 0 30 -118 -85 -243 -182 -169], ⟨0 1 -8 39 29 79 60 56]]

Optimal tunings:

WE: ~99/70 = 600.0243 ¢, ~3/2 = 701.7891 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7613 ¢

Optimal ET sequence: 106fgh, 118f, 224, 342f, 566h, 908fgh

Badness (Sintel): 1.35

Quadrapont

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768

Mapping: [⟨4 0 60 -236 -170 -131], ⟨0 1 -8 39 29 23]]

mapping generators: ~208/175, ~3

Optimal tunings:

WE: ~208/175 = 300.0229 ¢, ~3/2 = 701.8097 ¢
CWE: ~208/175 = 300.0000 ¢, ~3/2 = 701.7578 ¢

Optimal ET sequence: 224, 460, 684, 2276cde, 2960cde

Badness (Sintel): 0.869

Grackle

Grackle tempers out [-44 26 0 1⟩ so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.

Subgroup: 2.3.5.7

Comma list: 126/125, 32805/32768

Mapping: [⟨1 0 15 44], ⟨0 1 -8 -26]]

mapping generators: ~2, ~3

Optimal tunings:

WE: ~2 = 1199.7974 ¢, ~3/2 = 701.1210 ¢

error map: ⟨-0.203 -1.037 +3.300 -1.618]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2465 ¢

error map: ⟨0.000 -0.709 +3.715 -1.234]

Minimax tuning:

7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 12, …, 65, 77, 166c

Badness (Sintel): 1.78

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 32805/32768

Mapping: [⟨1 0 15 44 70], ⟨0 1 -8 -26 -42]]

Optimal tunings:

WE: ~2 = 1199.7077 ¢, ~3/2 = 701.0017 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1804 ¢

Optimal ET sequence: 12, 65e, 77, 89, 166c

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 5445/5408

Mapping: [⟨1 0 15 44 70 75], ⟨0 1 -8 -26 -42 -45]]

Optimal tunings:

WE: ~2 = 1199.7782 ¢, ~3/2 = 701.0966 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2319 ¢

Optimal ET sequence: 12f, 65ef, 77, 166cf

Badness (Sintel): 1.56

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873

Mapping: [⟨1 0 15 44 70 75 -7], ⟨0 1 -8 -26 -42 -45 7]]

Optimal tunings:

WE: ~2 = 1199.5839 ¢, ~3/2 = 700.9632 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2137 ¢

Optimal ET sequence: 12f, 77, 89f, 166cf

Badness (Sintel): 1.52

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323

Mapping: [⟨1 0 15 44 70 75 -7 9], ⟨0 1 -8 -26 -42 -45 7 -3]]

Optimal tunings:

WE: ~2 = 1199.7146 ¢, ~3/2 = 701.0500 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2212 ¢

Optimal ET sequence: 12f, 77, 166cf

Badness (Sintel): 1.40

Grackloid

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 729/728, 1287/1280

Mapping: [⟨1 0 15 44 70 -47], ⟨0 1 -8 -26 -42 32]]

Optimal tunings:

WE: ~2 = 1200.0060 ¢, ~3/2 = 701.2202 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2167 ¢

Optimal ET sequence: 12, 77, 166c

Badness (Sintel): 2.00

Grack

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 896/891

Mapping: [⟨1 0 15 44 51], ⟨0 1 -8 -26 -30]]

Optimal tunings:

WE: ~2 = 1199.8388 ¢, ~3/2 = 701.3071 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4068 ¢

Optimal ET sequence: 12, 53d, 65, 77e

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 245/242, 832/825

Mapping: [⟨1 0 15 44 51 75], ⟨0 1 -8 -26 -30 -45]]

Optimal tunings:

WE: ~2 = 1199.7329 ¢, ~3/2 = 701.1918 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.3555 ¢

Optimal ET sequence: 12f, 53dff, 65f, 77e

Badness (Sintel): 1.84

Catahelenic

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 352/351

Mapping: [⟨1 0 15 44 51 56], ⟨0 1 -8 -26 -30 -33]]

Optimal tunings:

WE: ~2 = 1199.8928 ¢, ~3/2 = 701.4664 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.5327 ¢

Optimal ET sequence: 12f, …, 53d, 65

Badness (Sintel): 2.01

Quasipyth

Named by Xenllium in 2026, quasipyth tempers out [109 -67 0 -1⟩, the nanisma, as well as the catasyc comma, 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 390625/387072

Mapping: [⟨1 0 15 109], ⟨0 1 -8 -67]]

Optimal tunings:

WE: ~2 = 1200.2569 ¢, ~3/2 = 702.1149 ¢

error map: ⟨+0.2569 +0.4168 -1.4342 +0.2685]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9615 ¢

error map: ⟨0.0000 +0.0065 -2.0054 -0.2437]

Optimal ET sequence: 53, 147d, 200, 253, 306c, 559c

Badness (Sintel): 5.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 19712/19683, 78125/77616

Mapping: [⟨1 0 15 109 -117], ⟨0 1 -8 -67 76]]

Optimal tunings:

WE: ~2 = 1200.3283 ¢, ~3/2 = 702.1636 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9713 ¢

Optimal ET sequence: 53, 200, 253, 559ce

Badness (Sintel): 3.83

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 2200/2197, 19712/19683

Mapping: [⟨1 0 15 109 -117 -28], ⟨0 1 -8 -67 76 20]]

Optimal tunings:

WE: ~2 = 1200.3229 ¢, ~3/2 = 702.1603 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9714 ¢

Optimal ET sequence: 53, 200, 253, 559ce

Badness (Sintel): 2.13

Schism

See Archytas clan #Schism.

Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

Bischismic

Bischismic tempers out 3136/3125, the hemimean comma, as well as 321489/320000, the varunisma, and may be described as the 118 & 130 temperament. The octave is split in halves, so the ploidacot of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds 7/4 by a stack of two 10/9's plus a semi-octave period, and in the 11-limit, it simply finds 11/8 by a stack of three 10/9's. 248edo and 378edo make for excellent tunings in both cases.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 32805/32768

Mapping: [⟨2 0 30 69], ⟨0 1 -8 -20]]

mapping generators: ~567/400, ~3

Optimal tunings:

WE: ~567/400 = 600.0072 ¢, ~3/2 = 701.6005 ¢

error map: ⟨+0.014 -0.340 +0.982 -0.629]

CWE: ~567/400 = 600.0000 ¢, ~3/2 = 701.5915 ¢

error map: ⟨0.000 -0.364 +0.954 -0.656]

Minimax tuning:

7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 12, …, 106d, 118, 130, 248, 378

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 8019/8000

Mapping: [⟨2 0 30 69 102], ⟨0 1 -8 -20 -30]]

Optimal tunings:

WE: ~99/70 = 600.0165 ¢, ~3/2 = 701.6316 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.6110 ¢

Optimal ET sequence: 12, …, 106de, 118, 130, 248

Badness (Sintel): 0.931

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 3136/3125

Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]

Optimal tunings:

WE: ~99/70 = 599.9610 ¢, ~3/2 = 701.5445 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.5908 ¢

Optimal ET sequence: 12, 118, 130, 248, 378

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125

Mapping: [⟨2 0 30 69 102 -75 5], ⟨0 1 -8 -20 -30 26 1]]

Optimal tunings:

WE: ~99/70 = 600.0331 ¢, ~3/2 = 701.6387 ¢
CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.5994 ¢

Optimal ET sequence: 12, 118, 130, 248g

Badness (Sintel): 1.49

Bischis

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 3136/3125

Mapping: [⟨2 0 30 69 102 131], ⟨0 1 -8 -20 -30 -39]]

Optimal tunings:

WE: ~55/39 = 599.9766 ¢, ~3/2 = 701.5380 ¢
CWE: ~55/39 = 600.0000 ¢, ~3/2 = 701.5670 ¢

Optimal ET sequence: 12f, 106deff, 118f, 130

Badness (Sintel): 1.21

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125

Mapping: [⟨2 0 30 69 102 131 5], ⟨0 1 -8 -20 -30 -39 1]]

Optimal tunings:

WE: ~55/39 = 600.0997 ¢, ~3/2 = 701.7114 ¢
CWE: ~55/39 = 600.0000 ¢, ~3/2 = 701.5899 ¢

Optimal ET sequence: 12f, 106deff, 118f, 130, 248fg

Badness (Sintel): 1.37

Kleischismic

Kleischismic tempers out 1500625/1492992, the uniwiz comma, and may be described as the 94 & 118 temperament. The generator is a infrafifth, two of which plus a semi-octave period make the 3rd harmonic; its ploidacot is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of 9/5. Kleischismic then finds 7/4 by that minus a 36/35 quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for 16/11 and the quartertone stands in for 33/32 in the 11-limit. 212edo and 330edo in the 330e val may be recommended as tunings.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1500625/1492992

Mapping: [⟨2 1 22 -15], ⟨0 2 -16 19]]

mapping generators: ~1225/864, ~35/24

Optimal tunings:

WE: ~1225/864 = 600.1246 ¢, ~35/24 = 651.0550 ¢ (~36/35 = 50.9304 ¢)

error map: ⟨+0.249 +0.280 -0.453 -0.650]

CWE: ~1225/864 = 600.0000 ¢, ~35/24 = 650.9204 ¢ (~36/35 = 50.9204 ¢)

error map: ⟨0.000 -0.114 -1.041 -1.338]

Optimal ET sequence: 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd

Badness (Sintel): 2.80

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 9801/9800, 14641/14580

Mapping: [⟨2 1 22 -15 8], ⟨0 2 -16 19 -1]]

Optimal tunings:

WE: ~99/70 = 600.1645 ¢, ~35/24 = 651.0963 ¢ (~36/35 = 50.9319 ¢)
CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9184 ¢ (~36/35 = 50.9184 ¢)

Optimal ET sequence: 24, 94, 118, 212, 330e, 542dee, 872cddeee

Badness (Sintel): 1.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1575/1573

Mapping: [⟨2 1 22 -15 8 15], ⟨0 2 -16 19 -1 -7]]

Optimal tunings:

WE: ~99/70 = 600.0696 ¢, ~35/24 = 651.0136 ¢ (~36/35 = 50.9440 ¢)
CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9378 ¢ (~36/35 = 50.9378 ¢)

Optimal ET sequence: 24, 94, 118, 212f

Badness (Sintel): 1.56

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 385/384, 561/560

Mapping: [⟨2 1 22 -15 8 15 6], ⟨0 2 -16 19 -1 -7 2]]

Optimal tunings:

WE: ~99/70 = 600.1134 ¢, ~35/24 = 651.0646 ¢ (~36/35 = 50.9512 ¢)
CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9414 ¢ (~36/35 = 50.9414 ¢)

Optimal ET sequence: 24, 94, 118

Badness (Sintel): 1.30

Kleischis

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1573/1568, 14641/14580

Mapping: [⟨2 1 22 -15 8 -36], ⟨0 2 -16 19 -1 40]]

Optimal tunings:

WE: ~99/70 = 600.1909 ¢, ~35/24 = 651.1578 ¢ (~36/35 = 50.9670 ¢)
CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9541 ¢ (~36/35 = 50.9541 ¢)

Optimal ET sequence: 24f, 94, 118f, 212

Badness (Sintel): 1.55

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580

Mapping: [⟨2 1 22 -15 8 -36 6], ⟨0 2 -16 19 -1 40 2]]

Optimal tunings:

WE: ~99/70 = 600.2190 ¢, ~35/24 = 651.1578 ¢ (~36/35 = 50.9670 ¢)
CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9518 ¢ (~36/35 = 50.9518 ¢)

Optimal ET sequence: 24f, 94, 118f, 212g

Badness (Sintel): 1.26

Salsa

Salsa tempers out 245/243, the sensamagic comma, and may be described as the 41 & 65 temperament. It has a neutral third as a generator; its ploidacot is dicot. In fact it is related to hemififths, from which this less accurate temperament only differs by the mapping of 5.

Subgroup: 2.3.5.7

Comma list: 245/243, 32805/32768

Mapping: [⟨1 1 7 -1], ⟨0 2 -16 13]]

mapping generators: ~2, ~128/105

Optimal tunings:

WE: ~2 = 1200.7707 ¢, ~128/105 = 351.2748 ¢

error map: ⟨+0.771 +1.365 -1.315 -3.024]

CWE: ~2 = 1200.0000 ¢, ~128/105 = 351.0471 ¢

error map: ⟨0.000 +0.139 -3.068 -5.213]

Optimal ET sequence: 17, 24, 41, 106d, 147d, 188cd

Badness (Sintel): 2.03

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 245/242, 385/384

Mapping: [⟨1 1 7 -1 2], ⟨0 2 -16 13 5]]

Optimal tunings:

WE: ~2 = 1200.3891 ¢, ~11/9 = 351.1275 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.0141 ¢

Optimal ET sequence: 17, 24, 41, 106d

Badness (Sintel): 1.30

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 245/242

Mapping: [⟨1 1 7 -1 2 4], ⟨0 2 -16 13 5 -1]]

Optimal tunings:

WE: ~2 = 1199.9362 ¢, ~11/9 = 351.0061 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.0247 ¢

Optimal ET sequence: 17, 24, 41

Badness (Sintel): 1.27

Hemischis

Hemischis tempers out 6144/6125, the porwell comma, as well as 19683/19600, the cataharry comma, and may be described as the 53 & 130 temperament. Its ploidacot is alpha-dicot.

The S-expression-based comma list for 13-limit hemischis is {S12/S14, S13/S15 = S26, S27, S64, (S65)}. Tempering out 169/168 (S13), 225/224 (S15) or 625/624 (S25) leads to 53edo while tempering out 24192/24167 (S12/S13), 10985/10976 (S13/S14), 43904/43875 (S14/S15) or 2401/2400 (S49) leads to 130edo and implies S12, S13, S14, and S15 are tempered together.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 19683/19600

Mapping: [⟨1 0 15 -17], ⟨0 2 -16 25]]

mapping generators: ~2, ~140/81

Optimal tunings:

WE: ~2 = 1199.8579 ¢, ~140/81 = 951.6847 ¢

error map: ⟨-0.142 -0.586 +0.600 +0.708]

CWE: ~2 = 1200.0000 ¢, ~140/81 = 951.7966 ¢

error map: ⟨0.000 -0.362 +0.941 +1.088]

Optimal ET sequence: 24, 53, 130, 183, 313

Badness (Sintel): 1.16

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 8019/8000

Mapping: [⟨1 0 15 -17 51], ⟨0 2 -16 25 -60]]

Optimal tunings:

WE: ~2 = 1199.8482 ¢, ~140/81 = 950.6809 ¢
CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.8020 ¢

Optimal ET sequence: 53, 130, 183, 313, 809cd

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 4096/4095

Mapping: [⟨1 0 15 -17 51 14], ⟨0 2 -16 25 -60 -13]]

Optimal tunings:

WE: ~2 = 1199.9140 ¢, ~140/81 = 950.7324 ¢
CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.8010 ¢

Optimal ET sequence: 53, 130, 183, 313

Badness (Sintel): 0.860

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095

Mapping: [⟨1 0 15 -17 51 14 -49], ⟨0 2 -16 25 -60 -13 67]]

Optimal tunings:

WE: ~2 = 1199.9740 ¢, ~26/15 = 950.7894 ¢
CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8100 ¢

Optimal ET sequence: 53, 130, 183, 496d

Badness (Sintel): 1.07

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095

Mapping: [⟨1 0 15 -17 51 14 -49 9], ⟨0 2 -16 25 -60 -13 67 -6]]

Optimal tunings:

WE: ~2 = 1200.0464 ¢, ~26/15 = 950.8459 ¢
CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8091 ¢

Optimal ET sequence: 53, 130, 183, 313h

Badness (Sintel): 1.11

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095

Mapping: [⟨1 0 15 -17 51 14 -49 9 -24], ⟨0 2 -16 25 -60 -13 67 -6 36]]

Optimal tunings:

WE: ~2 = 1200.0215 ¢, ~26/15 = 950.8239 ¢
CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8069 ¢

Optimal ET sequence: 53, 130, 183, 313h

Badness (Sintel): 1.06

Music

HemischisMatic EP (2023) by Francium – Spotify | Bandcamp | YouTube – 4-piece extended play

Term

Term tempers out the landscape comma, mapping 63/50 to the 1/3-octave period. It can be described as 12 & 171, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three marvel commas. A septimal comma is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a kleisma, with three kleismas making a comma, so this temperament may be useful for modeling that. 171edo makes for an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 250047/250000

Mapping: [⟨3 0 45 94], ⟨0 1 -8 -18]]

mapping generators: ~63/50, ~3

Optimal tunings:

WE: ~63/50 = 400.0257 ¢, ~3/2 = 701.7873 ¢

error map: ⟨+0.077 -0.091 -0.072 +0.031]

CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.7383 ¢

error map: ⟨0.000 -0.217 -0.220 -0.115]

Minimax tuning:

7-odd-limit unchanged-interval (eigenmonzo) basis): 2.5/3
9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722

Badness (Sintel): 0.505

Terminal

Terminal tempers out 441/440 and 4375/4356, and may be described as 159 & 171. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 32805/32768

Mapping: [⟨3 0 45 94 134], ⟨0 1 -8 -18 -26]]

Optimal tunings:

WE: ~44/35 = 400.0464 ¢, ~3/2 = 701.9053 ¢
CWE: ~44/35 = 400.0000 ¢, ~3/2 = 701.8178 ¢

Optimal ET sequence: 12, …, 159, 330

Badness (Sintel): 1.97

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 13720/13689

Mapping: [⟨3 0 45 94 134 168], ⟨0 1 -8 -18 -26 -33]]

Optimal tunings:

WE: ~44/35 = 400.0449 ¢, ~3/2 = 701.8995 ¢
CWE: ~44/35 = 400.0000 ¢, ~3/2 = 701.8156 ¢

Optimal ET sequence: 12f, …, 159, 330

Badness (Sintel): 1.53

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619

Mapping: [⟨3 0 45 94 134 168 -2], ⟨0 1 -8 -18 -26 -33 3]]

Optimal tunings:

WE: ~34/27 = 400.0195 ¢, ~3/2 = 701.8439 ¢
CWE: ~34/27 = 400.0000 ¢, ~3/2 = 701.8081 ¢

Optimal ET sequence: 12f, 159, 171, 330

Badness (Sintel): 1.38

Terminator

Terminator tempers out 540/539, and may be described as 171 & 183.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 137781/137500

Mapping: [⟨3 0 45 94 -137], ⟨0 1 -8 -18 31]]

Optimal tunings:

WE: ~63/50 = 399.9677 ¢, ~3/2 = 701.6278 ¢
CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6846 ¢

Optimal ET sequence: 12e, 171, 183, 354, 537, 891de

Badness (Sintel): 2.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 31250/31213

Mapping: [⟨3 0 45 94 -137 -103], ⟨0 1 -8 -18 31 24]]

Optimal tunings:

WE: ~63/50 = 399.9731 ¢, ~3/2 = 701.6414 ¢
CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6881 ¢

Optimal ET sequence: 12e, 171, 183, 354, 891de

Badness (Sintel): 1.47

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095

Mapping: [⟨3 0 45 94 -137 -103 -2], ⟨0 1 -8 -18 31 24 3]]

Optimal tunings:

WE: ~63/50 = 399.9757 ¢, ~3/2 = 701.6458 ¢
CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6881 ¢

Optimal ET sequence: 12e, 171, 183, 354, 891de

Badness (Sintel): 1.04

Semiterm

The semiterm temperament tempers out 9801/9800 (kalisma) as well as 151263/151250 (odiheim comma), and may be described as 12 & 342. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 151263/151250

Mapping: [⟨6 0 90 188 287], ⟨0 1 -8 -18 -28]]

mapping generators: ~55/49, ~3

Optimal tunings:

WE: ~55/49 = 200.0134 ¢, ~3/2 = 701.7931 ¢
CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.7426 ¢

Optimal ET sequence: 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde

Badness (Sintel): 0.973

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375

Mapping: [⟨6 0 90 188 287 355], ⟨0 1 -8 -18 -28 -35]]

Optimal tunings:

WE: ~55/49 = 200.0083 ¢, ~3/2 = 701.7549 ¢
CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.7238 ¢

Optimal ET sequence: 12f, 330eff, 342f, 696f *

* optimal patent val: 354

Badness (Sintel): 1.85

Hemiterm

The hemiterm temperament tempers out 3025/3024 (lehmerisma), and may be described as 159 & 183. Its ploidacot is triploid alpha-dicot.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 32805/32768, 102487/102400

Mapping: [⟨3 0 45 94 8], ⟨0 2 -16 -36 1]]

mapping generators: ~63/50, ~693/400

Optimal tunings:

WE: ~63/50 = 400.0309 ¢, ~693/400 = 950.9458 ¢ (~12/11 = 150.8841 ¢)
CWE: ~63/50 = 400.0000 ¢, ~693/400 = 950.8707 ¢ (~12/11 = 150.8707 ¢)

Optimal ET sequence: 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce

Badness (Sintel): 0.684

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712

Mapping: [⟨3 0 45 94 8 42], ⟨0 2 -16 -36 1 -13]]

Optimal tunings:

WE: ~63/50 = 400.0541 ¢, ~26/15 = 951.0013 ¢ (~12/11 = 150.8932 ¢)
CWE: ~63/50 = 400.0000 ¢, ~26/15 = 950.8696 ¢ (~12/11 = 150.8696 ¢)

Optimal ET sequence: 24d, 159, 183, 342f

Badness (Sintel): 1.30

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264

Mapping: [⟨3 0 45 94 8 42 -2], ⟨0 2 -16 -36 1 -13 6]]

Optimal tunings:

WE: ~34/27 = 400.0373 ¢, ~26/15 = 950.9556 ¢ (~12/11 = 150.8809 ¢)
CWE: ~34/27 = 400.0000 ¢, ~26/15 = 950.8652 ¢ (~12/11 = 150.8652 ¢)

Optimal ET sequence: 24d, 159, 183, 342f, 525f

Badness (Sintel): 1.14

Altinex

Named by Aura in 2021, altinex is an alternative to hemiterm and may be described as 24 & 159. 159edo itself makes for a recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 367653125/362797056

Mapping: [⟨3 0 45 -32], ⟨0 2 -16 17]]

mapping generators: ~1536/1225, ~34300/19683

Optimal tunings:

WE: ~1536/1225 = 400.1360 ¢, ~34300/19683 = 951.2867 ¢

error map: ⟨+0.408 +0.618 -0.781 -1.304]

CWE: ~1536/1225 = 400.0000 ¢, ~34300/19683 = 950.9638 ¢

error map: ⟨0.000 -0.027 -1.735 -2.441]

Optimal ET sequence: 24, 135, 159, 612ccdd

Badness (Sintel): 10.7

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 14700/14641, 19712/19683

Mapping: [⟨3 0 45 -32 8], ⟨0 2 -16 17 1]]

Optimal tunings:

WE: ~44/35 = 400.1156 ¢, ~121/70 = 951.2377 ¢
CWE: ~44/35 = 400.0000 ¢, ~121/70 = 950.9634 ¢

Optimal ET sequence: 24, 135, 159

Badness (Sintel): 3.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384, 676/675, 19712/19683

Mapping: [⟨3 0 45 -32 8 42], ⟨0 2 -16 17 1 -13]]

Optimal tunings:

WE: ~44/35 = 400.1396 ¢, ~26/15 = 951.2799 ¢
CWE: ~44/35 = 400.0000 ¢, ~26/15 = 950.9462 ¢

Optimal ET sequence: 24, 135f, 159

Badness (Sintel): 2.27

Squirrel

Squirrel tempers out 686/675, the sengic comma, and may be described as 29 & 36. It has a ~11/10 generator, three of which give the fourth (4/3), and thirteen of which give 7/4 with octave reduction. Its ploidacot is omega-tricot.

Subgroup: 2.3.5.7

Comma list: 686/675, 32805/32768

Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]

Optimal tunings:

WE: ~2 = 1200.7408 ¢, ~160/147 = 166.2424 ¢

error map: ⟨+0.741 +0.799 +2.763 -6.934]

CWE: ~2 = 1200.0000 ¢, ~160/147 = 166.1597 ¢

error map: ⟨0.000 -0.434 +1.518 -8.750]

Optimal ET sequence: 29, 36, 65

Badness (Sintel): 4.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 686/675, 896/891

Mapping: [⟨1 2 -1 1 0], ⟨0 -3 24 13 25]]

Optimal tunings:

WE: ~2 = 1200.6379 ¢, ~11/10 = 166.1853 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.1157 ¢

Optimal ET sequence: 29, 36, 65

Badness (Sintel): 2.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 245/242, 896/891

Mapping: [⟨1 2 -1 1 0 3], ⟨0 -3 24 13 25 5]]

Optimal tunings:

WE: ~2 = 1201.1361 ¢, ~11/10 = 166.2110 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0833 ¢

Optimal ET sequence: 29, 65f, 94df

Badness (Sintel): 1.81

Tertiaschis

Named by Xenllium in 2021, tertiaschis may be described as 94 & 159. It has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with squirrel, but tempers out 1071875/1062882 for prime 7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1071875/1062882

Mapping: [⟨1 2 -1 10], ⟨0 -3 24 -52]]

Optimal tunings:

WE: ~2 = 1200.3627 ¢, ~192/175 = 166.0691 ¢

error map: ⟨+0.363 +0.563 -1.019 -0.790]

CWE: ~2 = 1200.0000 ¢, ~192/175 = 166.0172 ¢

error map: ⟨0.000 -0.007 -1.901 -1.720]

Optimal ET sequence: 65, 94, 159, 253, 412cd

Badness (Sintel): 5.36

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 19712/19683

Mapping: [⟨1 2 -1 10 0], ⟨0 -3 24 -52 25]]

Optimal tunings:

WE: ~2 = 1200.3379 ¢, ~11/10 = 166.0638 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0167 ¢

Optimal ET sequence: 65, 94, 159, 253, 412cd, 665ccde

Badness (Sintel): 2.07

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 10985/10976

Mapping: [⟨1 2 -1 10 0 12], ⟨0 -3 24 -52 25 -60]]

Optimal tunings:

WE: ~2 = 1200.3467 ¢, ~11/10 = 166.0635 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0142 ¢

Optimal ET sequence: 65f, 94, 159, 253, 412cdf, 665ccdef

Badness (Sintel): 1.52

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

Mapping: [⟨1 2 -1 10 0 12 -2], ⟨0 -3 24 -52 25 -60 44]]

Optimal tunings:

WE: ~2 = 1200.3019 ¢, ~11/10 = 166.0535 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0114 ¢

Optimal ET sequence: 65f, 94, 159, 253

Badness (Sintel): 1.35

Countertertiaschis

Named by Flora Canou in 2021, Countertertiaschis may be described as 159 & 224. It has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with squirrel, but tempers out 244140625/243045684 for prime 7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 244140625/243045684

Mapping: [⟨1 2 -1 -12], ⟨0 -3 24 107]]

Optimal tunings:

WE: ~2 = 1200.1265 ¢, ~625/567 = 166.0797 ¢

error map: ⟨+0.127 +0.059 -0.529 +0.178]

CWE: ~2 = 1200.0000 ¢, ~625/567 = 166.0632 ¢

error map: ⟨0.000 -0.145 -0.797 -0.065]

Optimal ET sequence: 65d, 159, 224, 383, 607

Badness (Sintel): 4.76

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 32805/32768

Mapping: [⟨1 2 -1 -12 0], ⟨0 -3 24 107 25]]

Optimal tunings:

WE: ~2 = 1200.0804 ¢, ~11/10 = 166.0739 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0634 ¢

Optimal ET sequence: 65d, 159, 224, 383, 607

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976

Mapping: [⟨1 2 -1 -12 0 -10], ⟨0 -3 24 107 25 99]]

Optimal tunings:

WE: ~2 = 1200.0805 ¢, ~11/10 = 166.0740 ¢
CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0635 ¢

Optimal ET sequence: 65d, 159, 224, 383, 607

Badness (Sintel): 1.01

Quadrant

Named by Xenllium in 2021, quadrant tempers out 390625/388962, the dimcomp comma, and maps 25/21 to the 1/4-octave period. It may be described as the 12 & 212 temperament; its ploidacot is tetraploid monocot. Just as term equates the syntonic~Pythagorean comma with three marvel commas, quadrant equates the syntonic~Pythagorean comma with four. A septimal comma is then found as a stack of five marvel commas.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 390625/388962

Mapping: [⟨4 0 60 119], ⟨0 1 -8 -17]]

mapping generators: ~25/21, ~3

Optimal tunings:

WE: ~2 = 300.0255 ¢, ~3/2 = 701.8831 ¢

error map: ⟨+0.102 +0.030 -0.664 +0.462]

CWE: ~2 = 300.0000 ¢, ~3/2 = 701.8180 ¢

error map: ⟨0.000 -0.137 -0.858 +0.268]

Optimal ET sequence: 12, …, 200, 212, 224, 436, 660

Badness (Sintel): 2.79

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 32805/32768

Mapping: [⟨4 0 60 119 185], ⟨0 1 -8 -17 -27]]

Optimal tunings:

WE: ~25/21 = 300.0244 ¢, ~3/2 = 701.8759 ¢
CWE: ~25/21 = 300.0000 ¢, ~3/2 = 701.8145 ¢

Optimal ET sequence: 12, …, 212, 224, 436, 660

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647

Mapping: [⟨4 0 60 119 185 224], ⟨0 1 -8 -17 -27 -33]]

Optimal tunings:

WE: ~25/21 = 300.0234 ¢, ~3/2 = 701.8707 ¢
CWE: ~25/21 = 300.0000 ¢, ~3/2 = 701.8123 ¢

Optimal ET sequence: 12f, …, 212, 224, 436, 660

Badness (Sintel): 1.13

Sesquiquartififths

Sesquiquartififths tempers out 2401/2400, the breedsma, and may be described as the 41 & 171 temperament. It splits the fifth into four; its ploidacot is thus tetracot.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 32805/32768

Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]

mapping generators: ~2, ~448/405

Optimal tunings:

WE: ~2 = 1200.0846 ¢, ~448/405 = 175.4460 ¢

error map: ⟨+0.085 -0.086 +0.007 -0.093]

CWE: ~2 = 1200.0000 ¢, ~448/405 = 175.4320 ¢

error map: ⟨0.000 -0.227 -0.137 -0.306]

Minimax tuning:

7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd

Badness (Sintel): 0.285

Sesquart

Sesquart is the main 11- and 13-limit extension of sesquiquartififths of practical interest, as it identifies the neutral third with 11/9, which is realized in 41edo, 89edo, 130edo, and 171edo also makes for a possible tuning.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 16384/16335

Mapping: [⟨1 1 7 5 2], ⟨0 4 -32 -15 10]]

Optimal tunings:

WE: ~2 = 1199.8171 ¢, ~256/231 = 175.3793 ¢
CWE: ~2 = 1200.0000 ¢, ~256/231 = 175.4081 ¢

Optimal ET sequence: 41, 89, 130, 301e, 431e

Badness (Sintel): 0.969

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 3584/3575

Mapping: [⟨1 1 7 5 2 -2], ⟨0 4 -32 -15 10 39]]

Optimal tunings:

WE: ~2 = 1199.8352 ¢, ~72/65 = 175.3852 ¢
CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.4095 ¢

Optimal ET sequence: 41, 89, 130, 301e, 431e

Badness (Sintel): 0.925

Heartia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 256/255, 273/272, 364/363, 441/440

Mapping: [⟨1 1 7 5 2 -2 0], ⟨0 4 -32 -15 10 39 28]]

Optimal tunings:

WE: ~2 = 1199.6422 ¢, ~72/65 = 175.3338 ¢
CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3857 ¢

Optimal ET sequence: 41, 89, 130g

Badness (Sintel): 1.45

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440

Mapping: [⟨1 1 7 5 2 -2 0 6], ⟨0 4 -32 -15 10 39 28 -12]]

Optimal tunings:

WE: ~2 = 1199.7499 ¢, ~21/19 = 175.3432 ¢
CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.3797 ¢

Optimal ET sequence: 41, 89, 130g

Badness (Sintel): 1.40

Sesquartia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575

Mapping: [⟨1 1 7 5 2 -2 -6], ⟨0 4 -32 -15 10 39 69]]

Optimal tunings:

WE: ~2 = 1199.8902 ¢, ~72/65 = 175.4077 ¢
CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.4234 ¢

Optimal ET sequence: 41, 130, 171

Badness (Sintel): 1.18

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: [⟨1 1 7 5 2 -2 -6 6], ⟨0 4 -32 -15 10 39 69 -12]]

Optimal tunings:

WE: ~2 = 1199.9864 ¢, ~21/19 = 175.4169 ¢
CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.4189 ¢

Optimal ET sequence: 41, 130, 171

Badness (Sintel): 1.24

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: [⟨1 1 7 5 2 -2 -6 6 -6], ⟨0 4 -32 -15 10 39 69 -12 72]]

Optimal tunings:

WE: ~2 = 1199.9606 ¢, ~21/19 = 175.4067 ¢
CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.4123 ¢

Optimal ET sequence: 41i, 130, 171

Badness (Sintel): 1.36

Hearty

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625

Mapping: [⟨1 1 7 5 2 -2 13], ⟨0 4 -32 -15 10 39 -61]]

Optimal tunings:

WE: ~2 = 1199.9458 ¢, ~72/65 = 175.3689 ¢
CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3770 ¢

Optimal ET sequence: 41g, 89, 130

Badness (Sintel): 1.56

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455

Mapping: [⟨1 1 7 5 2 -2 13 6], ⟨0 4 -32 -15 10 39 -61 -12]]

Optimal tunings:

WE: ~2 = 1200.0114 ¢, ~72/65 = 175.3783 ¢
CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3765 ¢

Optimal ET sequence: 41g, 89, 130

Badness (Sintel): 1.39

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440

Mapping: [⟨1 1 7 5 2 -2 13 6 13], ⟨0 4 -32 -15 10 39 -61 -12 -58]]

Optimal tunings:

WE: ~2 = 1200.0122 ¢, ~72/65 = 175.3782 ¢
CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3763 ¢

Optimal ET sequence: 41g, 89, 130

Badness (Sintel): 1.37

Bisesqui

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 32805/32768

Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]

mapping generators: ~99/70, ~448/405

Optimal tunings:

WE: ~99/70 = 600.0429 ¢, ~448/405 = 175.4474 ¢
CWE: ~99/70 = 600.0000 ¢, ~448/405 = 175.4334 ¢

Optimal ET sequence: 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee

Badness (Sintel): 0.561

Tsaharuk

Main article: Tsaharuk

Tsaharuk tempers out 420175/419904, the wizma, and may be described as the 77 & 94 temperament. It is generated by a slightly flat neutral second of ~13/12, five of which make the perfect fifth, so its ploidacot is pentacot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 420175/419904

Mapping: [⟨1 1 7 0], ⟨0 5 -40 24]]

mapping generators: ~2, ~243/224

Optimal tunings:

WE: ~2 = 1200.1039 ¢, ~243/224 = 140.3620 ¢

error map: ⟨+0.104 -0.041 -0.067 -0.137]

CWE: ~2 = 1200.0000 ¢, ~243/224 = 140.3496 ¢

error map: ⟨0.000 -0.207 -0.296 -0.436]

Optimal ET sequence: 17, 77, 94, 171

Badness (Sintel): 0.777

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 19712/19683

Mapping: [⟨1 1 7 0 1], ⟨0 5 -40 24 21]]

Optimal tunings:

WE: ~2 = 1200.3103 ¢, ~88/81 = 140.4011 ¢
CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.3649 ¢

Optimal ET sequence: 17, 77, 94, 171e, 265e

Badness (Sintel): 2.10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1331/1323

Mapping: [⟨1 1 7 0 1 3], ⟨0 5 -40 24 21 6]]

Optimal tunings:

WE: ~2 = 1200.1840 ¢, ~13/12 = 140.3840 ¢
CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.3627 ¢

Optimal ET sequence: 17, 77, 94, 171e

Badness (Sintel): 1.57

Quanharuk

Quanharuk tempers out 16875/16807, the mirkwai comma, and may be described as the 41 & 183 temperament. The generator is a slightly flat major third of ~56/45, five of which make the 3rd harmonic, so the ploidacot of this temperament is alpha-pentacot. 224edo makes for a recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 32805/32768

Mapping: [⟨1 0 15 12], ⟨0 5 -40 -29]]

mapping generators: ~2, ~56/45

Optimal tunings:

WE: ~2 = 1200.0032 ¢, ~56/45 = 380.3557 ¢

error map: ⟨+0.003 -0.177 -0.493 +0.898]

CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3546 ¢

error map: ⟨0.000 -0.182 -0.498 +0.890]

Optimal ET sequence: 41, 142, 183, 224

Badness (Sintel): 1.82

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 32805/32768

Mapping: [⟨1 0 15 12 -7], ⟨0 5 -40 -29 33]]

Optimal tunings:

WE: ~2 = 1199.9709 ¢, ~56/45 = 380.3423 ¢
CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3517 ¢

Optimal ET sequence: 41, 142, 183, 224, 631d, 855d

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 4096/4095

Mapping: [⟨1 0 15 12 -7 -15], ⟨0 5 -40 -29 33 59]]

Optimal tunings:

WE: ~2 = 1199.9663 ¢, ~56/45 = 380.3403 ¢
CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3509 ¢

Optimal ET sequence: 41, 142, 183, 224, 631d, 855d

Badness (Sintel): 0.884

Quintilipyth

Named by Xenllium in 2021, quintilipyth (formerly quintilischis) slices the perfect fourth into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit. It may be described as the 12 & 253 temperament, and its ploidacot is omega-pentacot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 9765625/9680832

Mapping: [⟨1 2 -1 -4], ⟨0 -5 40 82]]

mapping generators: ~2, ~625/588

Optimal tunings:

WE: ~2 = 1200.1138 ¢, ~625/588 = 99.6347 ¢

error map: ⟨+0.114 +0.099 -1.041 +0.761]

CWE: ~2 = 1200.0000 ¢, ~625/588 = 99.6265 ¢

error map: ⟨0.000 -0.087 -1.255 +0.544]

Optimal ET sequence: 12, …, 253, 265

Badness (Sintel): 6.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 32805/32768

Mapping: [⟨1 2 -1 -4 -7], ⟨0 -5 40 82 126]]

Optimal tunings:

WE: ~2 = 1200.1503 ¢, ~35/33 = 99.6287 ¢
CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6176 ¢

Optimal ET sequence: 12, …, 253, 265, 518c

Badness (Sintel): 3.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647

Mapping: [⟨1 2 -1 -4 -7 -9], ⟨0 -5 40 82 126 153]]

Optimal tunings:

WE: ~2 = 1200.1774 ¢, ~35/33 = 99.6267 ¢
CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6134 ¢

Optimal ET sequence: 12f, …, 241cdef, 253

Badness (Sintel): 2.86

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619

Mapping: [⟨1 2 -1 -4 -7 -9 5], ⟨0 -5 40 82 126 153 -11]]

Optimal tunings:

WE: ~2 = 1200.1745 ¢, ~18/17 = 99.6265 ¢
CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6131 ¢

Optimal ET sequence: 12f, 241cdef, 253

Badness (Sintel): 2.34

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971

Mapping: [⟨1 2 -1 -4 -7 -9 5 4], ⟨0 -5 40 82 126 153 -11 3]]

Optimal tunings:

WE: ~2 = 1200.0713 ¢, ~18/17 = 99.6208 ¢
CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6152 ¢

Optimal ET sequence: 12f, 253, 265

Badness (Sintel): 2.32

Quintaschis

Named by Xenllium in 2021, quintaschis slices the perfect fourth into five semitones and tempers out 49009212/48828125 in the 7-limit. It may be described as the 12 & 289 temperament, and its ploidacot is omega-pentacot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 49009212/48828125

Mapping: [⟨1 2 -1 -5], ⟨0 -5 40 94]]

Optimal tunings:

WE: ~2 = 1200.0536 ¢, ~200/189 = 99.6684 ¢

error map: ⟨+0.054 -0.190 +0.370 -0.262]

CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6645 ¢

error map: ⟨0.000 -0.277 +0.266 -0.363]

Optimal ET sequence: 12, …, 289, 301, 590, 891, 1192

Badness (Sintel): 3.36

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 1953125/1951488

Mapping: [⟨1 2 -1 -5 -8], ⟨0 -5 40 94 138]]

Optimal tunings:

WE: ~2 = 1200.0988 ¢, ~35/33 = 99.6613 ¢
CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6540 ¢

Optimal ET sequence: 12, …, 277d, 289

Badness (Sintel): 3.69

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 32805/32768, 109512/109375

Mapping: [⟨1 2 -1 -5 -8 -11], ⟨0 -5 40 94 138 177]]

Optimal tunings:

WE: ~2 = 1200.0625 ¢, ~35/33 = 99.6630 ¢
CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6583 ¢

Optimal ET sequence: 12f, …, 277dff, 289

Badness (Sintel): 3.07

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768

Mapping: [⟨1 2 -1 -5 -8 -11 5], ⟨0 -5 40 94 138 177 -11]]

Optimal tunings:

WE: ~2 = 1200.1286 ¢, ~18/17 = 99.6668 ¢
CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6568 ¢

Optimal ET sequence: 12f, 277dff, 289

Badness (Sintel): 2.58

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859

Mapping: [⟨1 2 -1 -5 -8 -11 5 4], ⟨0 -5 40 94 138 177 -11 3]]

Optimal tunings:

WE: ~2 = 1200.0289 ¢, ~18/17 = 99.6609 ¢
CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6586 ¢

Optimal ET sequence: 12f, 289

Badness (Sintel): 2.56

Quintahelenic

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 8019/8000, 151263/151250

Mapping: [⟨1 2 -1 -5 -9], ⟨0 -5 40 94 150]]

Optimal tunings:

WE: ~2 = 1200.0195 ¢, ~200/189 = 99.6723 ¢
CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6709 ¢

Optimal ET sequence: 12, …, 289e, 301, 915

Badness (Sintel): 2.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000

Mapping: [⟨1 2 -1 -5 -9 -11], ⟨0 -5 40 94 150 177]]

Optimal tunings:

WE: ~2 = 1200.0442 ¢, ~200/189 = 99.6709 ¢
CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6675 ¢

Optimal ET sequence: 12f, …, 289e, 301

Badness (Sintel): 2.30

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750

Mapping: [⟨1 2 -1 -5 -9 -11 5], ⟨0 -5 40 94 150 177 -11]]

Optimal tunings:

WE: ~2 = 1200.1227 ¢, ~200/189 = 99.6753 ¢
CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6658 ¢

Optimal ET sequence: 12f, 289e, 301

Badness (Sintel): 2.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700

Mapping: [⟨1 2 -1 -5 -9 -11 5 4], ⟨0 -5 40 94 150 177 -11 3]]

Optimal tunings:

WE: ~2 = 1200.0230 ¢, ~200/189 = 99.6694 ¢
CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6676 ¢

Optimal ET sequence: 12f, 301

Badness (Sintel): 2.24

Quintahelenoid

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436

Mapping: [⟨1 2 -1 -5 -9 14], ⟨0 -5 40 94 150 -124]]

Optimal tunings:

WE: ~2 = 1199.9919 ¢, ~200/189 = 99.6712 ¢
CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6718 ¢

Optimal ET sequence: 12, 301, 614, 915

Badness (Sintel): 2.73

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157

Mapping: [⟨1 2 -1 -5 -9 14 5], ⟨0 -5 40 94 150 -124 -11]]

Optimal tunings:

WE: ~2 = 1200.0469 ¢, ~18/17 = 99.6749 ¢
CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6710 ¢

Optimal ET sequence: 12, 301

Badness (Sintel): 2.44

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137

Mapping: [⟨1 2 -1 -5 -9 14 5 4], ⟨0 -5 40 94 150 -124 -11 3]]

Optimal tunings:

WE: ~2 = 1199.9925 ¢, ~18/17 = 99.6710 ¢
CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6716 ¢

Optimal ET sequence: 12, 301

Badness (Sintel): 2.41

Sextilifourths

Named by Xenllium in 2021, sextilifourths (also known as sextilischis, formerly sextilififths) slices the perfect fourth into six small semitones, which serves as both 21/20 and 22/21. It may be described as 130 & 159, and its ploidacot is omega-hexacot. 289edo gives a highly recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 235298/234375

Mapping: [⟨1 2 -1 -1], ⟨0 -6 48 55]]

mapping generators: ~2, ~21/20

Optimal tunings:

WE: ~2 = 1200.0987 ¢, ~21/20 = 83.0599 ¢

error map: ⟨+0.099 -0.117 +0.462 -0.630]

CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0543 ¢

error map: ⟨0.000 -0.281 +0.295 -0.837]

Optimal ET sequence: 29, 72cd, 101, 130, 289, 419

Badness (Sintel): 2.75

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 235298/234375

Mapping: [⟨1 2 -1 -1 0], ⟨0 -6 48 55 50]]

Optimal tunings:

WE: ~2 = 1200.0424 ¢, ~21/20 = 83.0520 ¢
CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0497 ¢

Optimal ET sequence: 29, 72cde, 101e, 130, 289

Badness (Sintel): 1.50

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 10985/10976

Mapping: [⟨1 2 -1 -1 0 1], ⟨0 -6 48 55 50 39]]

Optimal tunings:

WE: ~2 = 1200.1056 ¢, ~21/20 = 83.0566 ¢
CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0508 ¢

Optimal ET sequence: 29, 72cdef, 101e, 130, 289

Badness (Sintel): 1.04

Septant

Named by Xenllium in 2021, septant notably tempers out the akjaysma ([47 -7 -7 -7⟩) and may be described as the 224 & 301 temperament. It has a period of 1/7 octave, and its ploidacot is heptaploid monocot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 516560652/514714375

Mapping: [⟨7 0 105 -56], ⟨0 1 -8 7]]

mapping generators: ~8575/7776, ~3

Optimal tunings:

WE: ~8575/7776 = 171.4303 ¢, ~3/2 = 701.7091 ¢

error map: ⟨+0.012 -0.234 +0.096 +0.265]

CWE: ~8575/7776 = 171.4286 ¢, ~3/2 = 701.7022 ¢

error map: ⟨0.000 -0.253 +0.069 +0.232]

Optimal ET sequence: 77, 147, 224, 301, 525, 826, 1351

Badness (Sintel): 2.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 24057/24010, 32805/32768

Mapping: [⟨7 0 105 -56 -120], ⟨0 1 -8 7 13]]

Optimal tunings:

WE: ~495/448 = 171.4334 ¢, ~3/2 = 701.7387 ¢
CWE: ~495/448 = 171.4286 ¢, ~3/2 = 701.7198 ¢

Optimal ET sequence: 77, 147, 224, 301, 525

Badness (Sintel): 1.46

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024

Mapping: [⟨7 0 105 -56 -120 37], ⟨0 1 -8 7 13 -1]]

Optimal tunings:

WE: ~495/448 = 171.4282 ¢, ~3/2 = 701.7229 ¢
CWE: ~495/448 = 171.4286 ¢, ~3/2 = 701.7242 ¢

Optimal ET sequence: 77, 147, 224, 525, 1274f

Badness (Sintel): 1.02

Octant

Octant may be described as the 224 & 248 temperament. It has a period of 1/8 octave, and its ploidacot is octaploid monocot. In this temperament, 12/11, 35/27, and 99/70 are mapped to 1\8, 3\8, and 4\8 respectively.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 2259436291848/2251875390625

Mapping: [⟨8 0 120 -117], ⟨0 1 -8 11]]

mapping generators: ~42875/39366, ~3

Optimal tunings:

WE: ~42875/39366 = 150.0048 ¢, ~3/2 = 701.7356 ¢

error map: ⟨+0.039 -0.181 +0.071 +0.127]

CWE: ~42875/39366 = 150.0000 ¢, ~3/2 = 701.7134 ¢

error map: ⟨0.000 -0.242 -0.021 +0.022]

Optimal ET sequence: 24, …, 224, 472, 696, 1168

Badness (Sintel): 3.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 46656/46585

Mapping: [⟨8 0 120 -117 15], ⟨0 1 -8 11 1]]

Optimal tunings:

WE: ~12/11 = 150.0010 ¢, ~3/2 = 701.7177 ¢
CWE: ~12/11 = 150.0000 ¢, ~3/2 = 701.7131 ¢

Optimal ET sequence: 24, …, 224, 472, 696, 1168

Badness (Sintel): 1.48

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655

Mapping: [⟨8 0 120 -117 15 93], ⟨0 1 -8 11 1 -5]]

Optimal tunings:

WE: ~12/11 = 149.9957 ¢, ~3/2 = 701.7046 ¢
CWE: ~12/11 = 150.0000 ¢, ~3/2 = 701.7247 ¢

Optimal ET sequence: 24, 224, 472, 696

Badness (Sintel): 1.26

Nonant

Named by Xenllium in 2023, nonant tempers out the septimal ennealimma ([-11 -9 0 9⟩) and may be described as the 36 & 171 temperament. It has a period of 1/9 octave, and its ploidacot is enneaploid monocot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 40353607/40310784

Mapping: [⟨9 0 135 11], ⟨0 1 -8 1]]

mapping generators: ~2592/2401, ~3

Optimal tunings:

WE: ~2592/2401 = 133.3442 ¢, ~3/2 = 701.8000 ¢

error map: ⟨+0.098 -0.057 -0.027 -0.141]

CWE: ~2592/2401 = 133.3333 ¢, ~3/2 = 701.7384 ¢

error map: ⟨0.000 -0.217 -0.221 -0.421]

Optimal ET sequence: 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd

Badness (Sintel): 1.77

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 42875/42592

Mapping: [⟨9 0 135 11 131], ⟨0 1 -8 1 -7]]

Optimal tunings:

WE: ~242/225 = 133.3308 ¢, ~3/2 = 701.8205 ¢
CWE: ~242/225 = 133.3333 ¢, ~3/2 = 701.8351 ¢

Optimal ET sequence: 36, 135, 171

Badness (Sintel): 4.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 16807/16731

Mapping: [⟨9 0 135 11 131 -38], ⟨0 1 -8 1 -7 5]]

Optimal tunings:

WE: ~242/225 = 133.3180 ¢, ~3/2 = 701.6956 ¢
CWE: ~242/225 = 133.3333 ¢, ~3/2 = 701.7800 ¢

Optimal ET sequence: 36, 99cf, 135, 171

Badness (Sintel): 3.15

Septiquarschis

Named by Xenllium in 2021, septiquarschis tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma), and may be described as the 89 & 94 temperament. It splits septimal minor seventh (7/4) into four generators. Note that in the data below, the generator is the octave complement so that seven of them minus five octaves make a perfect fifth; its ploidacot is thus epsilon-heptacot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 829440/823543

Mapping: [⟨1 -4 47 6], ⟨0 7 56 -4]]

mapping generators: ~2, ~256/147

Optimal tunings:

WE: ~2 = 1199.8855 ¢, ~256/147 = 957.2944 ¢

error map: ⟨-0.114 -0.436 -0.182 +1.310]

CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3867 ¢

error map: ⟨0.000 -0.248 +0.032 +1.627]

Optimal ET sequence: 89, 94, 183, 460d, 643d

Badness (Sintel): 4.73

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 15488/15435, 32805/32768

Mapping: [⟨1 -4 47 6 25], ⟨0 7 56 -4 -27]]

Optimal tunings:

WE: ~2 = 1199.9430 ¢, ~256/147 = 957.3390 ¢
CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3849 ¢

Optimal ET sequence: 89, 94, 183, 460d

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1573/1568, 4096/4095

Mapping: [⟨1 -4 47 6 25 -33], ⟨0 7 56 -4 -27 46]]

Optimal tunings:

WE: ~2 = 1200.0058 ¢, ~256/147 = 957.3946 ¢
CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3900 ¢

Optimal ET sequence: 89, 94, 183, 277, 460d

Badness (Sintel): 1.46

Tridecafifths

Named by Eliora in 2023, tridecafifths may be described as the 89 & 200 temperament. It divides the perfect fifth into thirteen quartertones, so its ploidacot is 13-cot. 289edo gives a highly recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 32805/32768, [-14 -1 -9 13⟩

Mapping: [⟨1 1 7 6], ⟨0 13 -104 -71]]

mapping generators: ~2, ~1323/1280

Optimal tunings:

WE: ~2 = 1200.1431 ¢, ~1323/1280 = 53.9838 ¢

error map: ⟨+0.143 -0.023 +0.375 -0.816]

CWE: ~2 = 1200.0000 ¢, ~1323/1280 = 53.9764 ¢

error map: ⟨0.000 -0.261 -0.221 -0.421]

Optimal ET sequence: 89, 200, 289

Badness (Sintel): 10.9

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 55296000/55240493

Mapping: [⟨1 1 7 6 4], ⟨0 13 -104 -71 -12]]

Optimal tunings:

WE: ~2 = 1200.0311 ¢, ~33/32 = 53.9766 ¢
CWE: ~2 = 1200.0000 ¢, ~33/32 = 53.9750 ¢

Optimal ET sequence: 89, 200, 289

Badness (Sintel): 4.23

Subgroup extensions

Maqamschismic (2.3.5.11)

Proposed by Eufalesio in 2026, maqamschismic is equivalent to the no-7 cassandra. The 2.3.5.11.13 subgroup adds 352/351 to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning maqamat (especially the Turkish version), as outlined by Ozan Yarman. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 helenus.

Subgroup: 2.3.5.11

Comma list: 2200/2187, 4125/4096

Subgroup-val mapping: [⟨1 0 15 -33], ⟨0 1 -8 23]]

Optimal tunings:

WE: ~2 = 1200.5458 ¢ ~3/2 = 702.4021 ¢
CWE: 2 = 1200.0000 ¢, ~3/2 = 702.0906 ¢

Optimal ET sequence: 12e, …, 41, 53, 94, 147e, 241ce, 335ce

Badness (Sintel): 1.34

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 325/324, 352/351, 4125/4096

Subgroup-val mapping: [⟨1 0 15 -33 -28], ⟨0 1 -8 23 20]]

Optimal tunings:

WE: ~2 = 1200.4565 ¢ ~3/2 = 702.3057 ¢
CWE: 2 = 1200.0000 ¢, ~3/2 = 702.0485 ¢

Optimal ET sequence: 12e, …, 41, 53, 94, 147e

Badness (Sintel): 0.862

Tridecaschismic (2.3.5.13)

Proposed by Eufalesio in 2026, tridecaschismic adds the marveltwin comma to the comma list, or equivalently, the tridecapyth comma. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 restriction of 13-limit cassandra.

Subgroup: 2.3.5.13

Comma list: 325/324, 32805/32768

Subgroup-val mapping: [⟨1 0 15 -28], ⟨0 1 -8 20]]

Optimal tunings:

WE: ~2 = 1200.3326 ¢ ~3/2 = 702.1092 ¢
CWE: 2 = 1200.0000 ¢, ~3/2 = 701.9189 ¢

Optimal ET sequence: 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff

Badness (Sintel): 0.582

2.3.5.13.19 subgroup

Subgroup: 2.3.5.13.19

Comma list: 325/324, 361/360, 513/512

Subgroup-val mapping: [⟨1 0 15 -28 9], ⟨0 1 -8 20 -3]]

Optimal tunings:

WE: ~2 = 1200.4236 ¢, ~3/2 = 702.1510 ¢
CWE: 2 = 1200.0000 ¢, ~3/2 = 701.9064 ¢

Optimal ET sequence: 12, …, 41, 53

Badness (Sintel): 0.354

Photia (2.3.5.17)

See also: No-elevens subgroup temperaments § Garibaldia

Subgroup: 2.3.5.17

Comma list: 256/255, 1458/1445

Subgroup-val mapping: [⟨1 0 15 -7], ⟨0 1 -8 7]]

Gencom mapping: [⟨1 0 15 0 0 0 -7], ⟨0 1 -8 0 0 0 7]]

mapping generators: ~2, ~3

Optimal tunings:

WE: ~2 = 1199.5471 ¢, ~3/2 = 701.2262 ¢

error map: ⟨-0.453 -1.182 +0.706 +3.628]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4976 ¢

error map: ⟨0.000 -0.457 +1.705 +5.528]

Optimal ET sequence: 12, 41, 53, 65, 207g, 272gg

Badness (Sintel): 0.479

2.3.5.17.19 subgroup

Subgroup: 2.3.5.17.19

Comma list: 171/170, 256/255, 324/323

Subgroup-val mapping: [⟨1 0 15 -7 9], ⟨0 1 -8 7 -3]]

Gencom mapping: [⟨1 0 15 0 0 0 -7 9], ⟨0 1 -8 0 0 0 7 -3]]

Optimal tunings:

WE: ~2 = 1199.7225 ¢, ~3/2 = 701.3077 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4754 ¢

Optimal ET sequence: 12, 41, 53, 65, 142g

Badness (Sintel): 0.332

Nestoria (2.3.5.19)

See also: No-elevens subgroup temperaments #Garibaldia and #Pontia

Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like 19/10 and 19/15.

Subgroup: 2.3.5.19

Comma list: 361/360, 513/512

Subgroup-val mapping: [⟨1 0 15 9], ⟨0 1 -8 -3]]

Gencom mapping: [⟨1 0 15 0 0 0 0 9], ⟨0 1 -8 0 0 0 0 -3]]

mapping generators: ~2, ~3

Optimal tunings:

WE: ~2 = 1200.2250 ¢, ~3/2 = 701.8776 ¢

error map: ⟨+0.225 +0.148 +0.240 -1.796]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7307 ¢

error map: ⟨0.000 -0.224 -0.159 -2.705]

Optimal ET sequence: 12, 29, 41, 53, 118, 171, 460hh, 631hh

Badness (Sintel): 0.126

Taylor (2.3.5.13)

This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.

Subgroup: 2.3.5.13

Comma list: 676/675, 32805/32768

Subgroup-val mapping: [⟨1 0 15 14], ⟨0 2 -16 -13]]

Gencom mapping: [⟨1 0 15 0 0 14], ⟨0 2 -16 0 0 -13]]

mapping generators: ~2, ~26/15

Optimal tunings:

WE: ~2 = 1200.1497 ¢, ~26/15 = 950.9740 ¢

error map: ⟨+0.150 -0.007 +0.348 -1.094]

CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8493 ¢

error map: ⟨0.000 -0.256 +0.098 -1.568]

Optimal ET sequence: 24, 53, 130, 183, 236, 525f, 761ff

Badness (Sintel): 0.334

Dakota (2.3.5.13.19)

Subgroup: 2.3.5.13.19

Comma list: 361/360, 513/512, 676/675

Subgroup-val mapping: [⟨1 0 15 14 9], ⟨0 2 -16 -13 -6]]

Optimal tunings:

WE: ~2 = 1200.2611 ¢, ~26/15 = 951.0703 ¢
CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8532 ¢

Optimal ET sequence: 24, 29, 53, 130, 183, 236h, 289h

Badness (Sintel): 0.262

2.3.5.13.19.37 subgroup

Subgroup: 2.3.5.13.19.37

Comma list: 361/360, 481/480, 513/512, 676/675

Subgroup-val mapping: [⟨1 0 15 14 9 6], ⟨0 2 -16 -13 -6 -1]]

Optimal tunings:

WE: ~2 = 1200.2987 ¢, ~26/15 = 951.1060 ¢
CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8595 ¢

Optimal ET sequence: 24, 29, 53, 183, 236h, 289hl, 631fhhll

Badness (Sintel): 0.223

Quintilischis (2.3.5.17)

For full 17- and 19-limit extensions, see #Quintilipyth or #Quintaschis.

Subgroup: 2.3.5.17

Comma list: 32805/32768, 1419857/1417176

Subgroup-val mapping: [⟨1 2 -1 5], ⟨0 -5 40 -11]]

Gencom mapping: [⟨1 2 -1 0 0 0 5], ⟨0 -5 40 0 0 0 -11]]

mapping generators: ~2, ~18/17

Optimal tunings:

WE: ~2 = 1200.1370 ¢, ~18/17 = 99.6602 ¢

error map: ⟨+0.137 +0.018 -0.042 -0.533]

CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6499 ¢

error map: ⟨0.000 -0.205 -0.317 -1.104]

Optimal ET sequence: 12, …, 253, 265, 277, 289, 566g, 855g

Badness (Sintel): 1.34

2.3.5.17.19 subgroup

Subgroup: 2.3.5.17.19

Comma list: 4624/4617, 6144/6137, 6885/6859

Subgroup-val mapping: [⟨1 2 -1 5 4], ⟨0 -5 40 -11 3]]

Gencom mapping: [⟨1 2 -1 0 0 0 5 4], ⟨0 -5 40 0 0 0 -11 3]]

Optimal tunings:

WE: ~2 = 1200.0350 ¢, ~18/17 = 99.6550 ¢
CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6520 ¢

Optimal ET sequence: 12, …, 253, 265, 277, 289

Badness (Sintel): 1.17

@@ Line 23: / Line 23: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7187{{c}}
+* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
-* [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}}
+: [[error map]]: {{val| +0.075 -0.100 -0.027 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}
+: error map: {{val| 0.000 -0.224 -0.160 }}
 [[Tuning ranges]]:
-* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
+* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
 * 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)
 {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}
-[[Badness]] (Smith): 0.004259
+[[Badness]] (Sintel): 0.0999
 === Overview to extensions ===
-The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.
-* [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]],
-* [[#Grackle|Grackle]] adds {{monzo| -44 26 0 1 }},
-* [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]],
-* [[#Pontiac|Pontiac]] adds {{monzo| -59 39 0 -1 }}.
-Those all have a fifth as generator.
-* [[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period.
+[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
-* [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemififth generator.
-* [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth.
+Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Septant|septant]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.
-* [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3 octave period.
-* [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
 Temperaments discussed elsewhere include:
 * ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
 * ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]
+Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.
 The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].
@@ Line 57: / Line 54: @@
 {{Main| Garibaldi }}
-Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
+Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
 [[Subgroup]]: 2.3.5.7
@@ Line 65: / Line 62: @@
 {{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.085{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
+: [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}
+: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}
 [[Minimax tuning]]:
@@ Line 79: / Line 80: @@
 * 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]
-{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576ccd }}
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94 }}
-[[Badness]] (Smith): 0.021644
+[[Badness]] (Sintel): 0.548
 === Cassandra ===
-Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.
+Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.
 Subgroup: 2.3.5.7.11
@@ Line 92: / Line 93: @@
 Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.157{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}}
 Minimax tuning:
@@ Line 102: / Line 105: @@
 * 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
-{{Optimal ET sequence|legend=0| 41, 53, 94, 229c, 323c, 417cce }}
+{{Optimal ET sequence|legend=0| 12e, 41, 53, 94, 229c }}
-Badness (Smith): 0.027396
+Badness (Sintel): 0.906
 ==== 13-limit ====
@@ Line 113: / Line 116: @@
 Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.113{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}}
 Minimax tuning:
@@ Line 126: / Line 131: @@
 {{Optimal ET sequence|legend=0| 41, 53, 94, 429ccdeef, 523ccdeef }}
-Badness (Sintel): 0.020676
+Badness (Sintel): 0.854
 ===== Cassie =====
@@ Line 135: / Line 140: @@
 Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.092{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}}
-{{Optimal ET sequence|legend=0| 41, 53, 94g }}
+{{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }}
-Badness (Smith): 0.023270
+Badness (Sintel): 1.19
 ====== 19-limit ======
@@ Line 148: / Line 155: @@
 Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.079{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}}
-{{Optimal ET sequence|legend=0| 41, 53, 94g }}
+{{Optimal ET sequence|legend=0| 12e, 41, 53 }}
-Badness (Smith): 0.018189
+Badness (Sintel): 1.11
 ===== Cassandric =====
@@ Line 161: / Line 170: @@
 Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.097{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}}
-{{Optimal ET sequence|legend=0| 41g, 53, 94, 241ce, 335cde }}
+{{Optimal ET sequence|legend=0| 41g, 53, 94 }}
-Badness (Smith): 0.023167
+Badness (Sintel): 1.18
 ====== 19-limit ======
@@ Line 176: / Line 187: @@
 Optimal tunings:
 * WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.0967{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.0978{{c}}
-{{Optimal ET sequence|legend=1| 41g, 53, 94, 241ceh, 335cdehh }}
+{{Optimal ET sequence|legend=1| 41g, 53, 94 }}
-Badness (Smith): 0.017635
+Badness (Sintel): 1.07
 ====== 23-limit ======
@@ Line 193: / Line 203: @@
 * WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}}
-* POTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0960{{c}}
 {{Optimal ET sequence|legend=0| 41g, 53, 94 }}
-Badness (Smith): 0.015072
+Badness (Sintel): 1.08
 ===== Cassander =====
@@ Line 206: / Line 215: @@
 Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.144{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}
 {{Optimal ET sequence|legend=0| 41, 53g, 94 }}
-Badness (Smith): 0.022454
+Badness (Sintel): 1.14
 ====== 19-limit ======
@@ Line 219: / Line 230: @@
 Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.135{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}}
 {{Optimal ET sequence|legend=0| 41, 53g, 94 }}
-Badness (Smith): 0.017576
+Badness (Sintel): 1.07
 === Andromeda ===
@@ Line 232: / Line 245: @@
 Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.321{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}}
 Minimax tuning:
@@ Line 242: / Line 257: @@
 * 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
-{{Optimal ET sequence|legend=0| 12, 29, 41, 217ce, 258ce }}
+{{Optimal ET sequence|legend=0| 12, 29, 41 }}
-Badness (Smith): 0.023556
+Badness (Sintel): 0.779
 ==== 13-limit ====
@@ Line 253: / Line 268: @@
 Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.559{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}}
 Minimax tuning:
@@ Line 264: / Line 281: @@
 * 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]
-{{Optimal ET sequence|legend=0| 12f, 29, 41, 152cdf, 193cdf, 234cdf }}
+{{Optimal ET sequence|legend=0| 12f, 29, 41 }}
-Badness (Smith): 0.020749
+Badness (Sintel): 0.857
 ===== 17-limit =====
@@ Line 275: / Line 292: @@
 Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.312{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}}
 {{Optimal ET sequence|legend=0| 12f, 29, 41 }}
-Badness (Smith): 0.023406
+Badness (Sintel): 1.19
 ====== 19-limit ======
@@ Line 288: / Line 307: @@
 Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.357{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}}
 {{Optimal ET sequence|legend=0| 12f, 29, 41 }}
-Badness (Smith): 0.019154
+Badness (Sintel): 1.17
 ===== Schisicosiennic =====
@@ Line 301: / Line 322: @@
 Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.725{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}}
-{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd, 111cd }}
+{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
-Badness (Smith): 0.021758
+Badness (Sintel): 1.11
 ====== 19-limit ======
@@ Line 314: / Line 337: @@
 Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.753{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}}
-{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd, 111cdh, 181ccddh }}
+{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
-Badness (Smith): 0.017902
+Badness (Sintel): 1.09
 ===== Schisicosiennoid =====
@@ Line 327: / Line 352: @@
 Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.717{{c}}
+Optimal tunings:
+* WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}}
-{{Optimal ET sequence|legend=0| 12f, 29g, 41g, 70cdgg }}
+{{Optimal ET sequence|legend=0| 12f, 29g, 41g }}
-Badness (Smith): 0.020895
+Badness (Sintel): 1.06
 ====== 19-limit ======
@@ Line 340: / Line 367: @@
 Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.716{{c}}
+Optimal tunings:
+* WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}}
-{{Optimal ET sequence|legend=1| 12f, 29g, 41g, 70cdgg }}
+{{Optimal ET sequence|legend=1| 12f, 29g, 41g }}
-Badness (Smith): 0.016773
+Badness (Sintel): 1.02
 === Helenus ===
@@ Line 353: / Line 382: @@
 Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.725{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}}
 Minimax tuning:
@@ Line 359: / Line 390: @@
 : unchanged-interval (eigenmonzo) basis: 2.11/9
-{{Optimal ET sequence|legend=0| 12, 41e, 53, 118d, 171de }}
+{{Optimal ET sequence|legend=0| 12, 41e, 53, 118d }}
-Badness (Smith): 0.035637
+Badness (Sintel): 1.18
 ==== 13-limit ====
@@ Line 370: / Line 401: @@
 Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.747{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}}
 Minimax tuning:
@@ Line 376: / Line 409: @@
 : unchanged-interval (eigenmonzo) basis: 2.11/9
-{{Optimal ET sequence|legend=0| 12f, 41ef, 53, 118d, 171de }}
+{{Optimal ET sequence|legend=0| 12f, …, 41ef, 53, 118d }}
-Badness (Smith): 0.026284
+Badness (Sintel): 1.09
 ==== 17-limit ====
@@ Line 387: / Line 420: @@
 Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.680{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}}
-{{Optimal ET sequence|legend=0| 12f, 41ef, 53, 65d, 118dg }}
+{{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }}
-Badness (Smith): 0.023732
+Badness (Sintel): 1.21
 ==== 19-limit ====
@@ Line 400: / Line 435: @@
 Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.705{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}}
-{{Optimal ET sequence|legend=1| 12f, 41ef, 53, 65d, 118dg }}
+{{Optimal ET sequence|legend=0| 12f, 53, 65d }}
-Badness (Smith): 0.019411
+Badness (Sintel): 1.18
+=== Karadeniz ===
+{{See also| Turkish maqam music temperaments #Karadeniz temperament }}
-=== Hemigari ===
 Subgroup: 2.3.5.7.11
-Comma list: 121/120, 225/224, 3125/3087
+Comma list: 225/224, 243/242, 3125/3087
-Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
+Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
-: mapping generators: ~2, ~110/63
+: mapping generators: ~2, ~11/9
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~110/63 = 951.082{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}
-{{Optimal ET sequence|legend=0| 29, 53, 82e, 135e, 188ce }}
+{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}
-Badness (Smith): 0.050681
+Badness (Sintel): 1.37
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 169/168, 225/224, 275/273
+Comma list: 225/224, 243/242, 325/324, 640/637
-Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
+Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 951.082{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}
-{{Optimal ET sequence|legend=0| 29, 53, 82e, 135ef, 188cef }}
+{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}
-Badness (Smith): 0.027464
+Badness (Sintel): 1.34
-=== Karadeniz ===
-{{See also| Turkish maqam music temperaments #Karadeniz temperament }}
+=== Hemigari ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 243/242, 3125/3087
+Comma list: 121/120, 225/224, 3125/3087
-Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
+Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
-: mapping generators: ~2, ~11/9
+: mapping generators: ~2, ~110/63
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 350.994{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
-{{Optimal ET sequence|legend=0| 41, 106, 147 }}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
-Badness (Smith): 0.041562
+Badness (Sintel): 1.68
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 243/242, 325/324, 640/637
+Comma list: 121/120, 169/168, 225/224, 275/273
-Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
+Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/9 = 351.014{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
-{{Optimal ET sequence|legend=0| 41, 106, 147 }}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
-Badness (Smith): 0.042564
+Badness (Sintel): 1.13
 === Sanjaab ===
@@ Line 470: / Line 515: @@
 : mapping generators: ~2, ~11/10
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 165.974{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}}
-{{Optimal ET sequence|legend=1| 29, 65d, 94, 441cde, 535cde, 629cde }}
+{{Optimal ET sequence|legend=0| 29, 65d, 94 }}
-Badness (Smith): 0.058040
+Badness (Sintel): 1.92
 ==== 13-limit ====
@@ Line 483: / Line 530: @@
 Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 165.963{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}
 {{Optimal ET sequence|legend=0| 29, 65d, 94 }}
-Badness (Smith): 0.033849
+Badness (Sintel): 1.40
-== Schism ==
+== Pontiac ==
-See [[Archytas clan #Schism]].
+{{Main| Pontiac }}
-Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
-== Pontiac ==
-{{Main| Pontiac }}
-Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^<sup>3</sup>A).
+Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).
 [[Subgroup]]: 2.3.5.7
@@ Line 505: / Line 549: @@
 {{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.757{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
+: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
+: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}
 [[Minimax tuning]]:
@@ Line 519: / Line 567: @@
 * 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955]
-{{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, 1934c, 2105c, 2276cd, 2447cd, 2618cd, 2789cd, 2960cd, 3131bcd }}
+{{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd }}
-[[Badness]] (Smith): 0.014133
+[[Badness]] (Sintel): 0.358
 === Helenoid ===
-The helenoid temperament ({{nowrap| 53 & 118 }}) is closely related to the helenus temperament, but with the ragisma rather than the [[225/224|marvel comma]] tempered out.
+Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.
 Subgroup: 2.3.5.7.11
@@ Line 532: / Line 580: @@
 Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.722{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}}
 Minimax tuning:
@@ Line 540: / Line 590: @@
 {{Optimal ET sequence|legend=0| 53, 118, 289e, 407de }}
-Badness (Smith): 0.038863
+Badness (Sintel): 1.28
 ==== 13-limit ====
@@ Line 549: / Line 599: @@
 Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.745{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}}
 Minimax tuning:
@@ Line 557: / Line 609: @@
 {{Optimal ET sequence|legend=0| 53, 118, 171e }}
-Badness (Smith): 0.033677
+Badness (Sintel): 1.39
 ===== 17-limit =====
@@ Line 566: / Line 618: @@
 Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.742{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}}
 Minimax tuning:
@@ Line 572: / Line 626: @@
 : unchanged-interval (eigenmonzo) basis: 2.17/13
-{{Optimal ET sequence|legend=0| 53, 118, 171e, 289ef, 460eef }}
+{{Optimal ET sequence|legend=0| 53, 118, 171e }}
-Badness (Smith): 0.028891
+Badness (Sintel): 1.47
 ==== Helena ====
@@ Line 583: / Line 637: @@
 Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.740{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}}
 {{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
-Badness (Smith): 0.036281
+Badness (Sintel): 1.50
 ===== 17-limit =====
@@ Line 596: / Line 652: @@
 Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.730{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}}
-{{Optimal ET sequence|legend=0| 53, 118f, 171ef, 289eff }}
+{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
-Badness (Smith): 0.030688
+Badness (Sintel): 1.56
 ===== 19-limit =====
@@ Line 609: / Line 667: @@
 Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.729{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}}
-{{Optimal ET sequence|legend=0| 53, 118f, 171ef, 289effh }}
+{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
-Badness (Smith): 0.021892
+Badness (Sintel): 1.33
 === Ponta ===
-The ponta temperament ({{nowrap| 53 & 171 }}) tempers out the [[540/539|swetisma]] and the ragisma.
+Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.
 Subgroup: 2.3.5.7.11
@@ Line 624: / Line 684: @@
 Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.783{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}}
 Minimax tuning:
@@ Line 630: / Line 692: @@
 : unchanged-interval (eigenmonzo) basis: 2.11/7
-{{Optimal ET sequence|legend=0| 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee }}
+{{Optimal ET sequence|legend=0| 53, 171, 224 }}
-Badness (Smith): 0.048692
+Badness (Sintel): 1.61
 ==== 13-limit ====
@@ Line 641: / Line 703: @@
 Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.784{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}}
 Minimax tuning:
@@ Line 649: / Line 713: @@
 {{Optimal ET sequence|legend=0| 53, 171, 224 }}
-Badness (Smith): 0.023616
+Badness (Sintel): 0.976
 ==== 17-limit ====
@@ Line 658: / Line 722: @@
 Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.777{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}}
 Minimax tuning:
@@ Line 666: / Line 732: @@
 {{Optimal ET sequence|legend=0| 53, 171, 224, 395e, 619eg }}
-Badness (Smith): 0.022853
+Badness (Sintel): 1.16
 === Pontic ===
-The pontic temperament ({{nowrap| 118 & 171 }}) tempers out the [[441/440|werckisma]] and the ragisma.
+Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 677: / Line 743: @@
 Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.724{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}}
 Minimax tuning:
@@ Line 683: / Line 751: @@
 : unchanged-interval (eigenmonzo) basis: 2.11
-{{Optimal ET sequence|legend=0| 53e, 118, 289, 407d, 696d }}
+{{Optimal ET sequence|legend=0| 53e, 118, 289, 407d }}
-Badness (Smith): 0.049573
+Badness (Sintel): 1.64
 ==== 13-limit ====
@@ Line 694: / Line 762: @@
 Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.738{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}}
 Minimax tuning:
@@ Line 700: / Line 770: @@
 : unchanged-interval (eigenmonzo) basis: 2.13/11
-{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f, 460ef }}
+{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}
-Badness (Smith): 0.045308
+Badness (Sintel): 1.87
 ===== 17-limit =====
@@ Line 711: / Line 781: @@
 Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.740{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}}
 Minimax tuning:
@@ Line 717: / Line 789: @@
 : unchanged-interval (eigenmonzo) basis: 2.13/11
-{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f, 460ef }}
+{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}
-Badness (Smith): 0.029618
+Badness (Sintel): 1.51
 ==== Pontoid ====
@@ Line 728: / Line 800: @@
 Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.735{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}}
-{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749def }}
+{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }}
-Badness (Smith): 0.050188
+Badness (Sintel): 2.07
 ===== 17-limit =====
@@ Line 741: / Line 815: @@
 Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.735{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1045{{c}}, ~3/2 = 701.7962{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}}
 {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749defg }}
-Badness (Smith): 0.029383
+Badness (Sintel): 1.50
 === Bipont ===
-The bipont temperament ({{nowrap| 118 & 224 }}) has a period of half octave and tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]].
+Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 757: / Line 833: @@
 : mapping generators: ~99/70, ~3
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~3/2 = 701.757{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}}
 {{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }}
-Badness (Smith): 0.014629
+Badness (Sintel): 0.484
 ==== 13-limit ====
@@ Line 770: / Line 848: @@
 Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~3/2 = 701.773{{c}}
+Optimal tunings:
+* WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}}
 {{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }}
-Badness (Smith): 0.030172
+Badness (Sintel): 1.25
 ===== 17-limit =====
@@ Line 783: / Line 863: @@
 Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~3/2 = 701.765{{c}}
+Optimal tunings:
+* WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}}
 {{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }}
-Badness (Smith): 0.027051
+Badness (Sintel): 1.38
 ==== Counterbipont ====
@@ Line 796: / Line 878: @@
 Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~3/2 = 701.769{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0405{{c}}, ~3/2 = 701.8160{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7697{{c}}
-{{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf, 1922cff }}
+{{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }}
-Badness (Smith): 0.025547
+Badness (Sintel): 1.06
 ===== 17-limit =====
@@ Line 809: / Line 893: @@
 Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~3/2 = 701.764{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}}
-{{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566, 908fg, 1474cffgg }}
+{{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }}
-Badness (Smith): 0.025251
+Badness (Sintel): 1.29
 ===== 19-limit =====
@@ Line 822: / Line 908: @@
 Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~3/2 = 701.761{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.0243{{c}}, ~3/2 = 701.7891{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7613{{c}}
 {{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }}
-Badness (Smith): 0.022267
+Badness (Sintel): 1.35
 ==== Quadrapont ====
@@ Line 836: / Line 924: @@
 : mapping generators: ~208/175, ~3
-Optimal tuning (POTE): ~208/175 = 300.000{{c}}, ~3/2 = 701.756{{c}}
+Optimal tunings:
+* WE: ~208/175 = 300.0229{{c}}, ~3/2 = 701.8097{{c}}
+* CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}}
-{{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde, 3644bccddee }}
+{{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }}
-Badness (Smith): 0.021025
+Badness (Sintel): 0.869
 == Grackle ==
-Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
+Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
 [[Subgroup]]: 2.3.5.7
@@ Line 852: / Line 942: @@
 : mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.239{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
+: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}
+: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}
 [[Minimax tuning]]:
@@ Line 858: / Line 952: @@
 * [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 12, 53d, 65, 77, 166c, 243c }}
+{{Optimal ET sequence|legend=1| 12, …, 65, 77, 166c }}
-[[Badness]] (Smith): 0.070407
+[[Badness]] (Sintel): 1.78
 === 11-limit ===
@@ Line 869: / Line 963: @@
 Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.172{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}}
-{{Optimal ET sequence|legend=0| 12, 53dee, 65e, 77, 89, 166c, 255c }}
+{{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }}
-Badness (Smith): 0.048887
+Badness (Sintel): 1.62
 ==== 13-limit ====
@@ Line 882: / Line 978: @@
 Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.226{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}}
-{{Optimal ET sequence|legend=0| 12f, 53deeff, 65ef, 77, 166cf, 243cf }}
+{{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }}
-Badness (Smith): 0.037859
+Badness (Sintel): 1.56
 ===== 17-limit =====
@@ Line 895: / Line 993: @@
 Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.206{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}}
-{{Optimal ET sequence|legend=0| 12f, 53deeff, 65ef, 77, 89f, 166cf }}
+{{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }}
-Badness (Smith): 0.029864
+Badness (Sintel): 1.52
 ===== 19-limit =====
@@ Line 908: / Line 1,008: @@
 Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.217{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}}
-{{Optimal ET sequence|legend=0| 12f, 53deeff, 65ef, 77, 166cf }}
+{{Optimal ET sequence|legend=0| 12f, 77, 166cf }}
-Badness (Smith): 0.023096
+Badness (Sintel): 1.40
 ==== Grackloid ====
@@ Line 921: / Line 1,023: @@
 Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.217{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}}
-{{Optimal ET sequence|legend=0| 12, 53deef, 65e, 77, 166c }}
+{{Optimal ET sequence|legend=0| 12, 77, 166c }}
-Badness (Smith): 0.048511
+Badness (Sintel): 2.00
 === Grack ===
@@ Line 934: / Line 1,038: @@
 Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.401{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}}
-{{Optimal ET sequence|legend=0| 12, 53d, 65, 77e, 142de }}
+{{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }}
-Badness (Smith): 0.055908
+Badness (Sintel): 1.85
 ==== 13-limit ====
@@ Line 947: / Line 1,053: @@
 Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.348{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}}
 {{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }}
-Badness (Smith): 0.044458
+Badness (Sintel): 1.84
 ==== Catahelenic ====
@@ Line 960: / Line 1,068: @@
 Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.529{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}}
+{{Optimal ET sequence|legend=0| 12f, …, 53d, 65 }}
-{{Optimal ET sequence|legend=0| 12f, 53df, 65 }}
+Badness (Sintel): 2.01
-Badness (Smith): 0.048524
+== Quasipyth ==
+Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.
-== Bischismic ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 3136/3125, 32805/32768
+[[Comma list]]: 32805/32768, 390625/387072
-{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
+{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
-: mapping generators: ~567/400, ~3
-[[Optimal tuning]] ([[CTE]]): ~567/400 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
+: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
+: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
-[[Minimax tuning]]:
+{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
-* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 12, 106d, 118, 130, 248, 378 }}
-[[Badness]] (Smith): 0.054744
+[[Badness]] (Sintel): 5.04
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 3136/3125, 8019/8000
+Comma list: 385/384, 19712/19683, 78125/77616
-Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
+Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
-Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 701.6077{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
-{{Optimal ET sequence|legend=0| 12, 106de, 118, 130, 248 }}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
-Badness (Smith): 0.028160
+Badness (Sintel): 3.83
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 441/440, 729/728, 1001/1000, 3136/3125
+Comma list: 325/324, 385/384, 2200/2197, 19712/19683
-Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
+Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
-Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 701.5949{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
-{{Optimal ET sequence|legend=0| 12, 106def, 118, 130, 248, 378 }}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
-Badness (Smith): 0.028722
+Badness (Sintel): 2.13
-===== 17-limit =====
+== Schism ==
-Subgroup: 2.3.5.7.11.13.17
+See [[Archytas clan #Schism]].
-Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
+Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
-Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
+== Bischismic ==
+Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.
-Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 701.5959{{c}}
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=0| 12, 106def, 118, 130, 248g }}
+[[Comma list]]: 3136/3125, 32805/32768
-Badness (Smith): 0.029340
+{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
+: mapping generators: ~567/400, ~3
-==== Bischis ====
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13
+* [[WE]]: ~567/400 = 600.0072{{c}}, ~3/2 = 701.6005{{c}}
+: [[error map]]: {{val| +0.014 -0.340 +0.982 -0.629 }}
+* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}}
+: error map: {{val| 0.000 -0.364 +0.954 -0.656 }}
-Comma list: 351/350, 364/363, 441/440, 3136/3125
+[[Minimax tuning]]:
+* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
+{{Optimal ET sequence|legend=1| 12, …, 106d, 118, 130, 248, 378 }}
-Optimal tuning (CTE): ~55/39 = 600.0000{{c}}, ~3/2 = 701.5708{{c}}
+[[Badness]] (Sintel): 1.39
-{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness (Smith): 0.029321
+Comma list: 441/440, 3136/3125, 8019/8000
-===== 17-limit =====
+Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
+Optimal tunings:
+* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}
-Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
+{{Optimal ET sequence|legend=0| 12, …, 106de, 118, 130, 248 }}
-Optimal tuning (CTE): ~55/39 = 600.0000{{c}}, ~3/2 = 701.5717{{c}}
+Badness (Sintel): 0.931
-{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness (Smith): 0.026894
+Comma list: 441/440, 729/728, 1001/1000, 3136/3125
-== Kleischismic ==
+Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 1500625/1492992
+Optimal tunings:
+* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}
-{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
+{{Optimal ET sequence|legend=0| 12, 118, 130, 248, 378 }}
-: mapping generators: ~1225/864, ~35/24
-[[Optimal tuning]] ([[POTE]]): ~1225/864 = 600.000{{c}}, ~35/24 = 650.920{{c}} (~36/35 = 50.920{{c}})
+Badness (Sintel): 1.19
-{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]] (Smith): 0.110583
+Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
-=== 11-limit ===
+Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 9801/9800, 14641/14580
+Optimal tunings:
+* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
-Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.918{{c}} (~36/35 = 50.918{{c}})
+{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}
-{{Optimal ET sequence|legend=0| 24, 70c, 94, 118, 212, 330e, 542de }}
+Badness (Sintel): 1.49
-Badness (Smith): 0.036749
+==== Bischis ====
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 385/384, 729/728, 1575/1573
+Comma list: 351/350, 364/363, 441/440, 3136/3125
-Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
+Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.938{{c}} (~36/35 = 50.938{{c}})
+Optimal tunings:
+* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}}
-{{Optimal ET sequence|legend=0| 24, 70c, 94, 118, 212f }}
+{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}
-Badness (Smith): 0.037640
+Badness (Sintel): 1.21
 ===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
+Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
-Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
+Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.942{{c}} (~36/35 = 50.942{{c}})
+Optimal tunings:
+* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}
-{{Optimal ET sequence|legend=0| 24, 70c, 94, 118, 212fg }}
+{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
-Badness (Smith): 0.025615
+Badness (Sintel): 1.37
-==== Kleischis ====
+== Kleischismic ==
-Subgroup: 2.3.5.7.11.13
+Kleischismic tempers out 1500625/1492992, the [[uniwiz comma]], and may be described as the {{nowrap| 94 & 118 }} temperament. The generator is a infrafifth, two of which plus a semi-octave period make the [[3/1|3rd]] [[harmonic]]; its [[ploidacot]] is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of [[9/5]]. Kleischismic then finds [[7/4]] by that minus a [[36/35]] quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for [[16/11]] and the quartertone stands in for [[33/32]] in the [[11-limit]]. [[212edo]] and [[330edo]] in the 330e val may be recommended as tunings.
+[[Subgroup]]: 2.3.5.7
-Comma list: 325/324, 385/384, 1573/1568, 14641/14580
+[[Comma list]]: 32805/32768, 1500625/1492992
-Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
+{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
+: mapping generators: ~1225/864, ~35/24
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.951{{c}} (~36/35 = 50.951{{c}})
+[[Optimal tuning]]s:
+* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}})
+: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }}
+* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}})
+: error map: {{val| 0.000 -0.114 -1.041 -1.338 }}
-{{Optimal ET sequence|legend=0| 24f, 70cf, 94, 118f, 212 }}
+{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}
-Badness (Smith): 0.037607
+[[Badness]] (Sintel): 2.80
-===== 17-limit =====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
-Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
+Comma list: 385/384, 9801/9800, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
+Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.948{{c}} (~36/35 = 50.948{{c}})
+Optimal tunings:
+* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})
-{{Optimal ET sequence|legend=0| 24f, 70cf, 94, 118f, 212g }}
+{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}
-Badness (Smith): 0.024734
+Badness (Sintel): 1.21
-== Salsa ==
+==== 13-limit ====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 245/243, 32805/32768
+Comma list: 352/351, 385/384, 729/728, 1575/1573
-{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
+Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
-: mapping generators: ~2, ~128/105
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}
+* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})
-: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}
-: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}
-{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
+{{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}
-[[Badness]] (Sintel): 2.03
+Badness (Sintel): 1.56
-=== 11-limit ===
+===== 17-limit =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 245/242, 385/384
+Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
-Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
+Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
 Optimal tunings:
-* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
+* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})
-{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}
+{{Optimal ET sequence|legend=0| 24, 94, 118 }}
 Badness (Sintel): 1.30
-=== 13-limit ===
+==== Kleischis ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 144/143, 243/242, 245/242
+Comma list: 325/324, 385/384, 1573/1568, 14641/14580
-Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
+Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
 Optimal tunings:
-* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}
+* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})
-{{Optimal ET sequence|legend=0| 17, 24, 41 }}
+{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212 }}
-Badness (Sintel): 1.27
+Badness (Sintel): 1.55
-== Hemischis ==
+===== 17-limit =====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 6144/6125, 19683/19600
+Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
-{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
+Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
-: mapping generators: ~2, ~140/81
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~140/81 = 950.797{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}})
-{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
+{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}
-[[Badness]] (Smith): 0.045817
+Badness (Sintel): 1.26
-=== 11-limit ===
+== Salsa ==
-Subgroup: 2.3.5.7.11
+Salsa tempers out 245/243, the [[sensamagic comma]], and may be described as the {{nowrap| 41 & 65 }} temperament. It has a neutral third as a generator; its [[ploidacot]] is dicot. In fact it is related to [[hemififths]], from which this less accurate temperament only differs by the mapping of [[5/1|5]].
-Comma list: 540/539, 5632/5625, 8019/8000
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
+[[Comma list]]: 245/243, 32805/32768
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~140/81 = 950.801{{c}}
+{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
+: mapping generators: ~2, ~128/105
-{{Optimal ET sequence|legend=0| 24e, 53, 130, 183, 313 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}
+: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}
+: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}
-Badness (Smith): 0.036289
+{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
-=== 13-limit ===
+[[Badness]] (Sintel): 2.03
-Its [[S-expression]]-based comma list is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]](, [[4225/4224|S65]])}. Tempering out [[169/168|S13]], [[225/224|S15]] or [[625/624|S25]] leads to [[53edo]] (through [[Catakleismic]]) while tempering out [[24192/24167|S12/S13]], [[10985/10976|S13/S14]], [[43904/43875|S14/S15]] or [[2401/2400|S49]] (implying S12 = S13 = S14 = S15) leads to [[130edo]].
-Subgroup: 2.3.5.7.11.13
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Comma list: 351/350, 540/539, 676/675, 4096/4095
+Comma list: 243/242, 245/242, 385/384
-Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
+Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.801{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
-{{Optimal ET sequence|legend=0| 24e, 53, 130, 183, 313 }}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}
-Badness (Smith): 0.020816
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}
-=== 17-limit ===
+Badness (Sintel): 1.30
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
+Comma list: 105/104, 144/143, 243/242, 245/242
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.810{{c}}
+Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
-{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
+Optimal tunings:
+* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
-Badness (Smith): 0.021073
+{{Optimal ET sequence|legend=0| 17, 24, 41 }}
-=== 19-limit ===
+Badness (Sintel): 1.27
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
+== Hemischis ==
+Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
+The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together.
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.809{{c}}
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
+[[Comma list]]: 6144/6125, 19683/19600
-Badness (Smith): 0.018262
+{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
+: mapping generators: ~2, ~140/81
-=== 23-limit ===
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13.17.19.23
+* [[WE]]: ~2 = 1199.8579{{c}}, ~140/81 = 951.6847{{c}}
+: [[error map]]: {{val| -0.142 -0.586 +0.600 +0.708 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}}
+: error map: {{val| 0.000 -0.362 +0.941 +1.088 }}
-Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
+{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
+[[Badness]] (Sintel): 1.16
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.807{{c}}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
+Comma list: 540/539, 5632/5625, 8019/8000
-Badness (Smith): 0.014819
+Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
-; Music
+Optimal tunings:
-* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play
+* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}
-== Squirrel ==
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }}
-The squirrel temperament ({{nowrap| 29 & 36 }}) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
-[[Subgroup]]: 2.3.5.7
+Badness (Sintel): 1.20
-[[Comma list]]: 686/675, 32805/32768
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
+Comma list: 351/350, 540/539, 676/675, 4096/4095
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~160/147 = 166.140{{c}}
+Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
-{{Optimal ET sequence|legend=1| 29, 36, 65 }}
+Optimal tunings:
+* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}
-[[Badness]] (Smith): 0.174705
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313 }}
-=== 11-limit ===
+Badness (Sintel): 0.860
-Subgroup: 2.3.5.7.11
-Comma list: 245/242, 686/675, 896/891
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
+Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.097{{c}}
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
-{{Optimal ET sequence|legend=0| 29, 36, 65 }}
+Optimal tunings:
+* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}
-Badness (Smith): 0.068310
+{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
-=== 13-limit ===
+Badness (Sintel): 1.07
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 169/168, 245/242, 896/891
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
+Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.054{{c}}
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
-{{Optimal ET sequence|legend=0| 29, 36, 65f, 94df, 159df }}
+Optimal tunings:
+* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}
-Badness (Smith): 0.043750
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-== Tertiaschis ==
+Badness (Sintel): 1.11
-The tertiaschis temperament ({{nowrap| 94 & 159 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 1071875/1062882 for prime 7.
-[[Subgroup]]: 2.3.5.7
+=== 23-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23
-[[Comma list]]: 32805/32768, 1071875/1062882
+Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
-{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~192/175 = 166.019{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}
-{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-[[Badness]] (Smith): 0.211859
+Badness (Sintel): 1.06
-=== 11-limit ===
+; Music
-Subgroup: 2.3.5.7.11
+* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play
-Comma list: 385/384, 4000/3993, 19712/19683
+== Term ==
+Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
-Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.017{{c}}
+[[Comma list]]: 32805/32768, 250047/250000
-{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd }}
+{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+: mapping generators: ~63/50, ~3
-Badness (Smith): 0.061336
+[[Optimal tuning]]s:
+* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
+: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
+* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
+: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}
-=== 13-limit ===
+[[Minimax tuning]]:
-Subgroup: 2.3.5.7.11.13
+* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Comma list: 325/324, 385/384, 1575/1573, 10985/10976
+{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
-Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
+[[Badness]] (Sintel): 0.505
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.016{{c}}
+=== Terminal ===
+Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
-{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
+Subgroup: 2.3.5.7.11
-Badness (Smith): 0.036700
+Comma list: 441/440, 4375/4356, 32805/32768
-=== 17-limit ===
+Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
+Optimal tunings:
+* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
-Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
+{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.012{{c}}
+Badness (Sintel): 1.97
-{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness (Smith): 0.026504
+Comma list: 364/363, 441/440, 625/624, 13720/13689
-== Countertertiaschis ==
+Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
-The countertertiaschis temperament ({{nowrap| 159 & 224 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.
-[[Subgroup]]: 2.3.5.7
+Optimal tunings:
+* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
-[[Comma list]]: 32805/32768, 244140625/243045684
+{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
-{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
+Badness (Sintel): 1.53
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~625/567 = 166.0621{{c}}
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
+Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
+Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
+Optimal tunings:
+* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
+* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
+{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
+Badness (Sintel): 1.38
-[[Badness]] (Smith): 0.188043
+=== Terminator ===
+Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4000/3993, 32805/32768
+Comma list: 540/539, 32805/32768, 137781/137500
-Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
+Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}}
+Optimal tunings:
+* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
-Badness (Smith): 0.048943
+Badness (Sintel): 2.21
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
+Comma list: 540/539, 729/728, 4096/4095, 31250/31213
-Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
+Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}}
+Optimal tunings:
+* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-Badness (Smith): 0.024506
+Badness (Sintel): 1.47
-== Term ==
+==== 17-limit ====
-Term tempers out the [[landscape comma]], mapping ~63/50 to the 1/3-octave period. It can be described as {{nowrap|12 & 171}}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
-[[Comma list]]: 32805/32768, 250047/250000
+Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
-{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+Optimal tunings:
-: mapping generators: ~63/50, ~3
+* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~3/2 = 701.742{{c}}
-[[Minimax tuning]]:
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
+Badness (Sintel): 1.04
-[[Badness]] (Smith): 0.019950
+=== Semiterm ===
+The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.
-=== Terminal ===
-The terminal temperament ({{nowrap| 12 & 159 }}) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4375/4356, 32805/32768
+Comma list: 9801/9800, 32805/32768, 151263/151250
-Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
+Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+: mapping generators: ~55/49, ~3
-Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~3/2 = 701.824{{c}}
+Optimal tunings:
+* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
-{{Optimal ET sequence|legend=0| 12, 147de, 159, 330 }}
+{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
-Badness (Smith): 0.059502
+Badness (Sintel): 0.973
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 625/624, 13720/13689
+Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
-Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
+Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
-Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~3/2 = 701.821{{c}}
+Optimal tunings:
+* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
-{{Optimal ET sequence|legend=0| 12f, 147def, 159, 330 }}
+{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
-Badness (Smith): 0.037082
+<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
-==== 17-limit ====
+Badness (Sintel): 1.85
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
+=== Hemiterm ===
+The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.
-Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~34/27 = 400.000{{c}}, ~3/2 = 701.810{{c}}
+Comma list: 3025/3024, 32805/32768, 102487/102400
-{{Optimal ET sequence|legend=0| 12f, 147def, 159, 171, 330 }}
+Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+: mapping generators: ~63/50, ~693/400
-Badness (Smith): 0.027073
+Optimal tunings:
+* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
-=== Terminator ===
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 32805/32768, 137781/137500
-Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~3/2 = 701.685{{c}}
-{{Optimal ET sequence|legend=0| 12e, 159e, 171, 183, 354, 537, 891de }}
-Badness (Smith): 0.066968
+Badness (Sintel): 0.684
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 31250/31213
+Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
-Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
+Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~3/2 = 701.689{{c}}
+Optimal tunings:
+* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})
-{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
-Badness (Smith): 0.035487
+Badness (Sintel): 1.30
 ==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
+Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
-Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
+Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~3/2 = 701.688{{c}}
+Optimal tunings:
+* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
+* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})
-{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
-Badness (Smith): 0.020434
+Badness (Sintel): 1.14
-=== Semiterm ===
+== Altinex ==
-The semiterm temperament ({{nowrap| 12 & 342 }}) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
+Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7
-Comma list: 9801/9800, 32805/32768, 151263/151250
+[[Comma list]]: 32805/32768, 367653125/362797056
-Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
-: mapping generators: ~55/49, ~3
+: mapping generators: ~1536/1225, ~34300/19683
-Optimal tuning (POTE): ~55/49 = 200.0000{{c}}, ~3/2 = 701.7460{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
+: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
+* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
+: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}
-{{Optimal ET sequence|legend=0| 12, 330e, 342, 1380, 1722, 2064, 2406c }}
+{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
-Badness (Smith): 0.029438
+[[Badness]] (Sintel): 10.7
-==== 13-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
+Comma list: 385/384, 14700/14641, 19712/19683
-Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
+Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
-Optimal tuning (POTE): ~55/49 = 200.0000{{c}}, ~3/2 = 701.7256{{c}}
+Optimal tunings:
+* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}
-{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
+{{Optimal ET sequence|legend=0| 24, 135, 159 }}
-<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
+Badness (Sintel): 3.35
-Badness (Smith): 0.044657
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== Hemiterm ===
+Comma list: 364/363, 385/384, 676/675, 19712/19683
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 32805/32768, 102487/102400
+Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
-Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+Optimal tunings:
-: mapping generators: ~63/50, ~693/400
+* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~693/400 = 950.872{{c}} (~12/11 = 150.872{{c}})
+{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
+Badness (Sintel): 2.27
-Badness (Smith): 0.020687
+== Squirrel ==
+Squirrel tempers out 686/675, the [[sengic comma]], and may be described as {{nowrap| 29 & 36 }}. It has a [[~]][[11/10]] generator, three of which give the fourth ([[4/3]]), and thirteen of which give [[7/4]] with octave reduction. Its [[ploidacot]] is omega-tricot.
-==== 13-limit ====
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
+[[Comma list]]: 686/675, 32805/32768
-Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
+{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~26/15 = 950.873{{c}} (~12/11 = 150.873{{c}})
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
+: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}
+: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
+{{Optimal ET sequence|legend=1| 29, 36, 65 }}
-Badness (Smith): 0.031362
+[[Badness]] (Sintel): 4.42
-==== 17-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
-Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
+Comma list: 245/242, 686/675, 896/891
-Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
+Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
-Optimal tuning (POTE): ~34/27 = 400.000{{c}}, ~26/15 = 950.867{{c}} (~12/11 = 150.867{{c}})
+Optimal tunings:
+* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f, 867ff }}
+{{Optimal ET sequence|legend=0| 29, 36, 65 }}
-Badness (Smith): 0.022316
+Badness (Sintel): 2.26
-== Altinex ==
+=== 13-limit ===
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 32805/32768, 367653125/362797056
+Comma list: 91/90, 169/168, 245/242, 896/891
-{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
+Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
-: mapping generators: ~1536/1225, ~34300/19683
-[[Optimal tuning]] ([[CTE]]): ~1536/1225 = 400.000{{c}}, ~34300/19683 = 950.9654{{c}}
+Optimal tunings:
+* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
-{{Optimal ET sequence|legend=1| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}
+{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
-[[Badness]] (Smith): 0.422026
+Badness (Sintel): 1.81
-=== 11-limit ===
+== Tertiaschis ==
-Subgroup: 2.3.5.7.11
+Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7.
-Comma list: 385/384, 14700/14641, 19712/19683
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
+[[Comma list]]: 32805/32768, 1071875/1062882
-Optimal tuning (CTE): ~44/35 = 400.000{{c}}, ~121/70 = 950.9658{{c}}
+{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
-{{Optimal ET sequence|legend=0| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
+: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
+: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}
-Badness (Smith): 0.101224
+{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
-=== 13-limit ===
+[[Badness]] (Sintel): 5.36
-Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 385/384, 676/675, 19712/19683
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
+Comma list: 385/384, 4000/3993, 19712/19683
-Optimal tuning (CTE): ~44/35 = 400.000{{c}}, ~26/15 = 950.9360{{c}}
+Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
-{{Optimal ET sequence|legend=0| 24, …, 111cf, 135f, 159 }}
+Optimal tunings:
+* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
-Badness (Smith): 0.054894
+{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
-== Sesquiquartififths ==
+Badness (Sintel): 2.07
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2401/2400, 32805/32768
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+Comma list: 325/324, 385/384, 1575/1573, 10985/10976
-: napping generators: ~2, ~448/405
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~448/405 = 175.434{{c}}
+Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
-[[Minimax tuning]]:
+Optimal tunings:
-* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
+* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
-{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
+{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
-[[Badness]] (Smith): 0.011244
+Badness (Sintel): 1.52
-=== Sesquart ===
+=== 17-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 441/440, 16384/16335
+Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
-Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
+Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~256/231 = 175.406{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
-Badness (Smith): 0.029306
+Badness (Sintel): 1.35
-==== 13-limit ====
+== Countertertiaschis ==
-Subgroup: 2.3.5.7.11.13
+Named by [[Flora Canou]] in 2021, Countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.
-Comma list: 243/242, 364/363, 441/440, 3584/3575
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
+[[Comma list]]: 32805/32768, 244140625/243045684
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.409{{c}}
+{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
+: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
+: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}
-Badness (Smith): 0.022396
+{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
-===== Sesquartia =====
+[[Badness]] (Sintel): 4.76
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
+Comma list: 3025/3024, 4000/3993, 32805/32768
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.424{{c}}
+Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
-{{Optimal ET sequence|legend=0| 41, 89g, 130, 171, 301e }}
+Optimal tunings:
+* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
-Badness (Smith): 0.023126
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-====== 19-limit ======
+Badness (Sintel): 1.62
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
+Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.419{{c}}
+Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
-{{Optimal ET sequence|legend=0| 41, 89g, 130, 171, 301eh }}
+Optimal tunings:
+* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
-Badness (Smith): 0.020466
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-====== 23-limit ======
+Badness (Sintel): 1.01
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
+== Quadrant ==
+Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.412{{c}}
+[[Comma list]]: 32805/32768, 390625/388962
-{{Optimal ET sequence|legend=0| 41i, 89gi, 130, 171, 301eh }}
+{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+: mapping generators: ~25/21, ~3
-Badness (Smith): 0.019043
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
+: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
+* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
+: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}
-===== Heartia =====
+{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
+[[Badness]] (Sintel): 2.79
-Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.386{{c}}
+Comma list: 1375/1372, 6250/6237, 32805/32768
-{{Optimal ET sequence|legend=0| 41, 89, 130g }}
+Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
-Badness (Smith): 0.028443
+Optimal tunings:
+* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
-====== 19-limit ======
+{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
+Badness (Sintel): 1.51
-Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.380{{c}}
+Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
-{{Optimal ET sequence|legend=0| 41, 89, 130g }}
+Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
-Badness (Smith): 0.023059
+Optimal tunings:
+* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
-===== Hearty =====
+{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
+Badness (Sintel): 1.13
-Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
+== Sesquiquartififths ==
+Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.377{{c}}
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefgg }}
+[[Comma list]]: 2401/2400, 32805/32768
-Badness (Smith): 0.030680
+{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+: mapping generators: ~2, ~448/405
-====== 19-limit ======
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13.17.19
+* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}}
+: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}}
+: error map: {{val| 0.000 -0.227 -0.137 -0.306 }}
-Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
+[[Minimax tuning]]:
+* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
+{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.377{{c}}
+[[Badness]] (Sintel): 0.285
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefggh }}
+=== Sesquart ===
+Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning.
-Badness (Smith): 0.022816
+Subgroup: 2.3.5.7.11
-====== 23-limit ======
+Comma list: 243/242, 441/440, 16384/16335
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
+Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
+Optimal tunings:
+* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.376{{c}}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefggh }}
+Badness (Sintel): 0.969
-Badness (Smith): 0.019121
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== Bisesqui ===
+Comma list: 243/242, 364/363, 441/440, 3584/3575
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 9801/9800, 32805/32768
+Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
-Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+Optimal tunings:
-: mapping generators: ~99/70, ~448/405
+* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~448/405 = 175.435{{c}}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d }}
+Badness (Sintel): 0.925
-Badness (Smith): 0.016968
+===== Heartia =====
+Subgroup: 2.3.5.7.11.13.17
-== Quintilipyth ==
+Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
-The quintilipyth temperament ({{nowrap| 12 & 253 }}, formerly ''quintilischis'') slices the pythagorean fourth ([[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.
-[[Subgroup]]: 2.3.5.7
+Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
-[[Comma list]]: 32805/32768, 9765625/9680832
+Optimal tunings:
+* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
-{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~625/588 = 99.625{{c}}
+Badness (Sintel): 1.45
-{{Optimal ET sequence|legend=1| 12, 253, 265 }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]] (Smith): 0.253966
+Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
-=== 11-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 4375/4356, 32805/32768
+Optimal tunings:
+* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.616{{c}}
+Badness (Sintel): 1.40
-{{Optimal ET sequence|legend=0| 12, 253, 265, 518c, 783cc }}
+===== Sesquartia =====
+Subgroup: 2.3.5.7.11.13.17
-Badness (Smith): 0.113044
+Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
-=== 13-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
+Optimal tunings:
+* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.612{{c}}
+Badness (Sintel): 1.18
-{{Optimal ET sequence|legend=0| 12f, 253, 518c, 771cc }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness (Smith): 0.069127
+Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
-=== 17-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
+Optimal tunings:
+* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.612{{c}}
+Badness (Sintel): 1.24
-{{Optimal ET sequence|legend=0| 12f, 253, 518c, 771cc }}
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-Badness (Smith): 0.045992
+Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
-=== 19-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
+Optimal tunings:
+* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
+{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.615{{c}}
+Badness (Sintel): 1.36
-{{Optimal ET sequence|legend=0| 12f, 253, 265, 518ch }}
+===== Hearty =====
+Subgroup: 2.3.5.7.11.13.17
-Badness (Smith): 0.038155
+Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
-== Quintaschis ==
+Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
-The quintaschis temperament ({{nowrap| 12 & 289 }}) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
-[[Subgroup]]: 2.3.5.7
+Optimal tunings:
+* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}
-[[Comma list]]: 32805/32768, 49009212/48828125
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
+Badness (Sintel): 1.56
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~200/189 = 99.664{{c}}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
+Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
-[[Badness]] (Smith): 0.132890
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
-=== 11-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11
+* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}
-Comma list: 441/440, 32805/32768, 1953125/1951488
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
+Badness (Sintel): 1.39
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.653{{c}}
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
+Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
-Badness (Smith): 0.111477
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
-==== 13-limit ====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13
+* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
-Comma list: 364/363, 441/440, 32805/32768, 109512/109375
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
+Badness (Sintel): 1.37
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.658{{c}}
+=== Bisesqui ===
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
+Comma list: 2401/2400, 9801/9800, 32805/32768
-Badness (Smith): 0.074218
+Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+: mapping generators: ~99/70, ~448/405
-==== 17-limit ====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17
+* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}
-Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
+{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
+Badness (Sintel): 0.561
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.656{{c}}
+== Tsaharuk ==
+{{Main| Tsaharuk }}
-{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
+Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.
-Badness (Smith): 0.050571
+[[Subgroup]]: 2.3.5.7
-==== 19-limit ====
+[[Comma list]]: 32805/32768, 420175/419904
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
+{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+: mapping generators: ~2, ~243/224
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.659{{c}}
+: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}
+: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}
-{{Optimal ET sequence|legend=0| 12f, 289 }}
+{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
-Badness (Smith): 0.042120
+[[Badness]] (Sintel): 0.777
-=== Quintahelenic ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 5632/5625, 8019/8000, 151263/151250
+Comma list: 385/384, 1331/1323, 19712/19683
-Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
+Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.671{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
-{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
-Badness (Smith): 0.082225
+Badness (Sintel): 2.10
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
+Comma list: 352/351, 385/384, 729/728, 1331/1323
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
+Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.661{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}
-{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
-Badness (Smith): 0.055570
+Badness (Sintel): 1.57
-===== 17-limit =====
+== Quanharuk ==
-Subgroup: 2.3.5.7.11.13.17
+Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd]] [[harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.
-Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
+[[Comma list]]: 16875/16807, 32805/32768
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.665{{c}}
+{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+: mapping generators: ~2, ~56/45
-{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
+: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}
+: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}
+{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}
-Badness (Smith): 0.040412
+[[Badness]] (Sintel): 1.82
-===== 19-limit =====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11
-Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
+Comma list: 540/539, 1375/1372, 32805/32768
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
+Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.668{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}
-{{Optimal ET sequence|legend=0| 12f, 301 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Smith): 0.036840
+Badness (Sintel): 1.04
-==== Quintahelenoid ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
+Comma list: 540/539, 729/728, 1375/1372, 4096/4095
-Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
+Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.672{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}
-{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Smith): 0.066108
+Badness (Sintel): 0.884
-===== 17-limit =====
+== Quintilipyth ==
-Subgroup: 2.3.5.7.11.13.17
+Named by [[Xenllium]] in 2021, quintilipyth (formerly ''quintilischis'') slices the [[4/3|perfect fourth]] into five semitones and tempers out the [[compass comma]] (9765625/9680832) in the [[7-limit]]. It may be described as the {{nowrap| 12 & 253 }} temperament, and its [[ploidacot]] is omega-pentacot.
-Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
+[[Comma list]]: 32805/32768, 9765625/9680832
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.671{{c}}
+{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+: mapping generators: ~2, ~625/588
-{{Optimal ET sequence|legend=0| 12, 301 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1138{{c}}, ~625/588 = 99.6347{{c}}
+: [[error map]]: {{val| +0.114 +0.099 -1.041 +0.761 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6265{{c}}
+: error map: {{val| 0.000 -0.087 -1.255 +0.544 }}
-Badness (Smith): 0.047908
+{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}
-===== 19-limit =====
+[[Badness]] (Sintel): 6.43
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
+Comma list: 1375/1372, 4375/4356, 32805/32768
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.672{{c}}
+Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
-{{Optimal ET sequence|legend=0| 12, 301 }}
+Optimal tunings:
+* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}
-Badness (Smith): 0.039542
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
-== Sextilifourths ==
+Badness (Sintel): 3.74
-The sextilifourths ({{nowrap| 130 & 159 }}, also known as ''sextilischis'', formerly ''sextilififths'') temperament slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
-[[Subgroup]]: 2.3.5.7
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 32805/32768, 235298/234375
+Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
-{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
-: mapping generators: ~2, ~21/2
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 83.053{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
-{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
+{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}
-[[Badness]] (Smith): 0.108794
+Badness (Sintel): 2.86
-=== 11-limit ===
+=== 17-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 441/440, 4000/3993, 235298/234375
+Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
-Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.049{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}
-{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}
-Badness (Smith): 0.045457
+Badness (Sintel): 2.34
-=== 13-limit ===
+=== 19-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 364/363, 441/440, 676/675, 10985/10976
+Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
-Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.049{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}
-{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
-Badness (Smith): 0.025276
+Badness (Sintel): 2.32
-== Septiquarschis ==
+== Quintaschis ==
-The septiquarschis temperament ({{nowrap| 89 & 94 }}) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
+Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 829440/823543
+[[Comma list]]: 32805/32768, 49009212/48828125
-{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}
+{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~147/128 = 242.614{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
+: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
+: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}
-{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d, 1103dd }}
+{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
-[[Badness]] (Smith): 0.187047
+[[Badness]] (Sintel): 3.36
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 15488/15435, 32805/32768
+Comma list: 441/440, 32805/32768, 1953125/1951488
-Mapping: {{mapping| 1 3 -9 2 -2 | 0 -7 -56 4 27 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~147/128 = 242.616{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}
-{{Optimal ET sequence|legend=0| 89, 94, 183, 460d, 643d, 826dd }}
+{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
-Badness (Smith): 0.052002
+Badness (Sintel): 3.69
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1573/1568, 4096/4095
+Comma list: 364/363, 441/440, 32805/32768, 109512/109375
-Mapping: {{mapping| 1 3 -9 2 -2 13 | 0 -7 -56 4 27 -46 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~147/128 = 242.610{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}
-{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
+{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
-Badness (Smith): 0.035315
+Badness (Sintel): 3.07
-== Tsaharuk ==
+==== 17-limit ====
-{{Main| Tsaharuk }}
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
-[[Comma list]]: 32805/32768, 420175/419904
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
-{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+Optimal tunings:
-: mapping generators: ~2, ~243/224
+* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~243/224 = 140.350{{c}}
+{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
-{{Optimal ET sequence|legend=1| 17, 60c, 77, 94, 171 }}
+Badness (Sintel): 2.58
-[[Badness]] (Smith): 0.030697
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 11-limit ===
+Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1331/1323, 19712/19683
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
-Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
+Optimal tunings:
+* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~88/81 = 140.365{{c}}
+{{Optimal ET sequence|legend=0| 12f, 289 }}
-{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 265e, 436ee }}
+Badness (Sintel): 2.56
-Badness (Smith): 0.063499
+=== Quintahelenic ===
+Subgroup: 2.3.5.7.11
-=== 13-limit ===
+Comma list: 5632/5625, 8019/8000, 151263/151250
-Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 385/384, 729/728, 1331/1323
+Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
-Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
+Optimal tunings:
+* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.363{{c}}
+{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
-{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 436ee }}
+Badness (Sintel): 2.72
-Badness (Smith): 0.037886
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-== Quanharuk ==
+Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 16875/16807, 32805/32768
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
-{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+Optimal tunings:
-: mapping generators: ~2, ~56/45
+* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~56/45 = 380.355{{c}}
+{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
-{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd }}
+Badness (Sintel): 2.30
-[[Badness]] (Smith): 0.071950
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-=== 11-limit ===
+Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 32805/32768
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
-Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
+Optimal tunings:
+* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 380.352{{c}}
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d, 1079d }}
+Badness (Sintel): 2.06
-Badness (Smith): 0.031549
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 13-limit ===
+Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1375/1372, 4096/4095
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
-Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
+Optimal tunings:
+* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 380.351{{c}}
+{{Optimal ET sequence|legend=0| 12f, 301 }}
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
+Badness (Sintel): 2.24
-Badness (Smith): 0.021392
+==== Quintahelenoid ====
+Subgroup: 2.3.5.7.11.13
-== Quadrant ==
+Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
-The ''quadrant'' temperament ({{nowrap| 12 & 224 }}) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
-[[Subgroup]]: 2.3.5.7
+Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
-[[Comma list]]: 32805/32768, 390625/388962
+Optimal tunings:
+* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
-{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
-: mapping generators: ~25/21, ~3
-[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8234{{c}}
+Badness (Sintel): 2.73
-{{Optimal ET sequence|legend=1| 212, 224, 436, 660, 1096c }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]] (Smith): 0.110242
+Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
-=== 11-limit ===
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 6250/6237, 32805/32768
+Optimal tunings:
+* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
-Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8176{{c}}
+Badness (Sintel): 2.44
-{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness (Smith): 0.045738
+Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
-=== 13-limit ===
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
+Optimal tunings:
+* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
-Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8158{{c}}
+Badness (Sintel): 2.41
-{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}
+== Sextilifourths ==
+Named by [[Xenllium]] in 2021, sextilifourths (also known as ''sextilischis'', formerly ''sextilififths'') slices the [[4/3|perfect fourth]] into six small semitones, which serves as both [[21/20]] and [[22/21]]. It may be described as {{nowrap| 130 & 159 }}, and its [[ploidacot]] is omega-hexacot. [[289edo]] gives a highly recommendable tuning.
-Badness (Smith): 0.027243
-== Septant ==
-The septant temperament ({{nowrap| 224 & 301 }}) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }}.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 516560652/514714375
+[[Comma list]]: 32805/32768, 235298/234375
-{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
+{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
-: mapping generators: ~8575/7776, ~3
+: mapping generators: ~2, ~21/20
-[[Optimal tuning]] ([[POTE]]): ~8575/7776 = 171.429{{c}}, ~3/2 = 701.702{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}
+: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}
+: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}
-{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
+{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
-[[Badness]] (Smith): 0.111142
+[[Badness]] (Sintel): 2.75
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 24057/24010, 32805/32768
+Comma list: 441/440, 4000/3993, 235298/234375
-Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
+Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
-Optimal tuning (POTE): ~495/448 = 171.429{{c}}, ~3/2 = 701.719{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
-{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
+{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
-Badness (Smith): 0.044122
+Badness (Sintel): 1.50
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
+Comma list: 364/363, 441/440, 676/675, 10985/10976
-Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
+Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
-Optimal tuning (POTE): ~495/448 = 171.429{{c}}, ~3/2 = 701.724{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}
-{{Optimal ET sequence|legend=0| 77, 147, 224, 525 }}
+{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
-Badness (Smith): 0.024706
+Badness (Sintel): 1.04
-== Octant ==
+== Septant ==
-The octant temperament ({{nowrap| 224 & 472 }}) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.
+Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 2259436291848/2251875390625
+[[Comma list]]: 32805/32768, 516560652/514714375
-{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
+{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
-: mapping generators: ~42875/39366, ~3
+: mapping generators: ~8575/7776, ~3
-[[Optimal tuning]] ([[POTE]]): ~42875/39366 = 150.000{{c}}, ~3/2 = 701.713{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
+: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
+* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
+: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
-{{Optimal ET sequence|legend=1| 24, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
-[[Badness]] (Smith): 0.157186
+[[Badness]] (Sintel): 2.81
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 9801/9800, 32805/32768, 46656/46585
+Comma list: 3025/3024, 24057/24010, 32805/32768
-Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
+Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~3/2 = 701.713{{c}}
+Optimal tunings:
+* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
-{{Optimal ET sequence|legend=0| 24, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
-Badness (Smith): 0.044778
+Badness (Sintel): 1.46
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
+Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
-Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
+Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~3/2 = 701.725{{c}}
+Optimal tunings:
+* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
-{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
-Badness (Smith): 0.030425
+Badness (Sintel): 1.02
-== Nonant ==
+== Octant ==
-The ''nonant'' temperament ({{nowrap| 36 & 135 }}) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}.
+Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 40353607/40310784
+[[Comma list]]: 32805/32768, 2259436291848/2251875390625
-{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
+{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
-: mapping generators: ~2592/2401, ~3
+: mapping generators: ~42875/39366, ~3
-[[Optimal tuning]] ([[CTE]]): ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7232{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
+: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
+* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
+: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
-{{Optimal ET sequence|legend=1| 36, 99c, 135, 171 }}
+{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
-[[Badness]] (Smith): 0.069896
+[[Badness]] (Sintel): 3.98
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 32805/32768, 42875/42592
+Comma list: 9801/9800, 32805/32768, 46656/46585
-Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
+Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
-Optimal tuning (CTE): ~242/225 = 133.3333{{c}}, ~3/2 = 701.8398{{c}}
+Optimal tunings:
+* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
-{{Optimal ET sequence|legend=0| 36, 99c, 135, 171, 477ce, 648cee }}
+{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
-Badness (Smith): 0.126910
+Badness (Sintel): 1.48
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 16807/16731
+Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
-Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
+Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
-Optimal tuning (CTE): ~242/225 = 133.3333{{c}}, ~3/2 = 701.7998{{c}}
+Optimal tunings:
+* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
-{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
+{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
-Badness (Smith): 0.076195
+Badness (Sintel): 1.26
-== Tridecafifths ==
+== Nonant ==
-Tridecafifths divides the perfect 3/2 into 13 quartertones.
+Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
+[[Comma list]]: 32805/32768, 40353607/40310784
-{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
+{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
-: mapping generators: ~2, ~1323/1280
+: mapping generators: ~2592/2401, ~3
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9741{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
-{{Optimal ET sequence|legend=1| 89, 200, 289 }}
+: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
+* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
-[[Badness]] (Smith): 0.432580
+: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
-=== 11-limit ===
+{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
-Subgroup: 2.3.5.7.11
+[[Badness]] (Sintel): 1.77
-Comma list: 441/440, 32805/32768, 55296000/55240493
+=== 11-limit ===
-Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
+Subgroup: 2.3.5.7.11
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 53.9744{{c}}
+Comma list: 540/539, 32805/32768, 42875/42592
-{{Optimal ET sequence|legend=0| 89, 200, 289 }}
+Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
-Badness (Smith): 0.127820
+Optimal tunings:
+* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
-== Subgroup extensions ==
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
-=== Photia (2.3.5.17) ===
-{{See also| No-elevens subgroup temperaments #Garibaldia }}
+{{Optimal ET sequence|legend=0| 36, 135, 171 }}
-[[Subgroup]]: 2.3.5.17
+Badness (Sintel): 4.20
-[[Comma list]]: 256/255, 1458/1445
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
+Comma list: 540/539, 729/728, 4096/4095, 16807/16731
-{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
-: mapping generators: ~2, ~3
+Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.491{{c}}
+Optimal tunings:
+* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
-{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
-[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
+{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
-==== 2.3.5.17.19 subgroup ====
+Badness (Sintel): 3.15
-Subgroup: 2.3.5.17.19
+== Septiquarschis ==
-Comma list: 171/170, 256/255, 324/323
+Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.
-Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 32805/32768, 829440/823543
+{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
+: mapping generators: ~2, ~256/147
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
+: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
+: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}
+{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
+[[Badness]] (Sintel): 4.73
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 540/539, 15488/15435, 32805/32768
+Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}
+Optimal tunings:
+* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}
+{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}
+Badness (Sintel): 1.72
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 540/539, 729/728, 1573/1568, 4096/4095
+Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
+Optimal tunings:
+* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}
+{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
+Badness (Sintel): 1.46
+== Tridecafifths ==
+Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
+{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
+: mapping generators: ~2, ~1323/1280
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
+: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
+: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}
+{{Optimal ET sequence|legend=1| 89, 200, 289 }}
+[[Badness]] (Sintel): 10.9
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 441/440, 32805/32768, 55296000/55240493
+Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
+Optimal tunings:
+* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}
+{{Optimal ET sequence|legend=0| 89, 200, 289 }}
+Badness (Sintel): 4.23
+== Subgroup extensions ==
+=== Maqamschismic (2.3.5.11) ===
+Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].
+Subgroup: 2.3.5.11
+Comma list: 2200/2187, 4125/4096
+Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}
+Optimal tunings:
+* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}
+{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}
+Badness (Sintel): 1.34
+==== 2.3.5.11.13 subgroup ====
+Subgroup: 2.3.5.11.13
+Comma list: 325/324, 352/351, 4125/4096
+Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}
+Optimal tunings:
+* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}
+{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e }}
+Badness (Sintel): 0.862
+=== Tridecaschismic (2.3.5.13) ===
+Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].
+Subgroup: 2.3.5.13
+Comma list: 325/324, 32805/32768
+Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}
+Optimal tunings:
+* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}
+Badness (Sintel): 0.582
+==== 2.3.5.13.19 subgroup ====
+Subgroup: 2.3.5.13.19
+Comma list: 325/324, 361/360, 513/512
+Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
+Optimal tunings:
+* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}
+Badness (Sintel): 0.354
+=== Photia (2.3.5.17) ===
+{{See also| No-elevens subgroup temperaments #Garibaldia }}
+[[Subgroup]]: 2.3.5.17
+[[Comma list]]: 256/255, 1458/1445
+{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
+{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
+: mapping generators: ~2, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
+: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
+: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}
+{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}
+[[Badness]] (Sintel): 0.479
+==== 2.3.5.17.19 subgroup ====
+Subgroup: 2.3.5.17.19
+Comma list: 171/170, 256/255, 324/323
+Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
 Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.470{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}
-{{Optimal ET sequence|legend=0| 12, 41, 53, 65 }}
+{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}
-RMS error: 0.5374 cents
+Badness (Sintel): 0.332
 === Nestoria (2.3.5.19) ===
 : ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
-The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20]]'')}. Strangely, despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is actually sharper than the fifth in optimal schismic. This is likely due to its optimization considering intervals like 19/10 and 19/15.
+Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]].
 [[Subgroup]]: 2.3.5.19
@@ Line 2,404: / Line 2,898: @@
 : mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.746{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
+: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
+: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}
-{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
+[[Badness]] (Sintel): 0.126
 === Taylor (2.3.5.13) ===
@@ Line 2,422: / Line 2,920: @@
 : mapping generators: ~2, ~26/15
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~26/15 = 950.855{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}
+: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}
+: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}
-{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
+{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
+[[Badness]] (Sintel): 0.334
 ==== Dakota (2.3.5.13.19) ====
@@ Line 2,435: / Line 2,937: @@
 Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8199{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}
 {{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}
-Badness (Smith): 0.00575
+Badness (Sintel): 0.262
 ===== 2.3.5.13.19.37 subgroup =====
@@ Line 2,448: / Line 2,952: @@
 Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8187{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}
 {{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
-Badness (Smith): 0.00357
+Badness (Sintel): 0.223
 === Quintilischis (2.3.5.17) ===
@@ Line 2,466: / Line 2,972: @@
 : mapping generators: ~2, ~18/17
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~18/17 = 99.649{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}
+: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}
+: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}
-{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
+{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
+[[Badness]] (Sintel): 1.34
 ==== 2.3.5.17.19 subgroup ====
@@ Line 2,481: / Line 2,991: @@
 Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.652{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}
-{{Optimal ET sequence|legend=0| 12, 253, 265, 277, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}
-RMS error: 0.1636 cents
+Badness (Sintel): 1.17
 [[Category:Temperament families]]
 [[Category:Schismatic family| ]] <!-- main article -->
 [[Category:Rank 2]]