Schismatic family

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{{c]}, ~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⟩,
• Schism adds [6 -2 0 -1⟩,
• Pontiac adds [-59 39 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.
• Hemischis adds [-34 25 0 -2⟩ and has a hemififth 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.
• Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.

Temperaments discussed elsewhere include:

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

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-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 {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{{c]}, ~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.

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-Exx#), 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{{c]}, ~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

The helenoid temperament (53 & 118) is closely related to the helenus temperament, but with the ragisma rather than the marvel comma tempered out.

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

The ponta temperament (53 & 171) tempers out the swetisma and the ragisma.

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

The pontic temperament (118 & 171) tempers out the werckisma and the ragisma.

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

The bipont temperament (118 & 224) has a period of half octave and tempers out the lehmerisma (3025/3024) and the kalisma (9801/9800).

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⟩. 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.

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{{c]}, ~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

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-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

Bischismic

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

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

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{c}}, ~11/9 = 351.0061 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.0247 ¢

Optimal ET sequence: 17, 24, 41

Badness (Sintel): 1.27

Hemischis

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

Its S-expression-based comma list is {S12/S14, S13/S15 = S26, S27, S64(, S65)}. Tempering out S13, S15 or S25 leads to 53edo (through Catakleismic) while tempering out S12/S13, S13/S14, S14/S15 or S49 (implying S12 = S13 = S14 = S15) leads to 130edo.

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

The terminal temperament (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

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

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 (12 & 342) has a period of 1/6 octave and tempers out 9801/9800 (kalisma) and 151263/151250 (odiheim comma).

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

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

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

The squirrel temperament (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

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

The tertiaschis temperament (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

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

The countertertiaschis temperament (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

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

The quadrant temperament (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

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

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

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

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

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

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

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

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

The quintilipyth temperament (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

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

The quintaschis temperament (12 & 289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.

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

The sextilifourths (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

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

Octant

The octant temperament (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.

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

The nonant temperament (36 & 135) has a period of 1/9 octave and tempers out the septimal ennealimma, [-11 -9 0 9⟩.

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

Septant

The septant temperament (224 & 301) has a period of 1/7 octave and tempers out the akjaysma, [47 -7 -7 -7⟩.

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

Septiquarschis

The septiquarschis temperament (89 & 94) splits septimal minor seventh (7/4) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).

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

Tridecafifths divides the perfect 3/2 into 13 quartertones.

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

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

The S-expression-based comma list of this temperament is {S16/S18, S19 , (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.

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