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:

• CTE: ~2 = 1200.0000 ¢, ~3/2 = 701.7187 ¢
• POTE: ~2 = 1200.0000 ¢, ~3/2 = 701.7359 ¢

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 (Smith): 0.004259

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:

• Salsa (+245/243) → Sensamagic clan
• Guiron (+1029/1024) → Gamelismic 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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.085 ¢

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, 241c, 335cd, 576ccd

Badness (Smith): 0.021644

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.157 ¢

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: 41, 53, 94, 229c, 323c, 417cce

Badness (Smith): 0.027396

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.113 ¢

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): 0.020676

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.092 ¢

Optimal ET sequence: 41, 53, 94g

Badness (Smith): 0.023270

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.079 ¢

Optimal ET sequence: 41, 53, 94g

Badness (Smith): 0.018189

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.097 ¢

Optimal ET sequence: 41g, 53, 94, 241ce, 335cde

Badness (Smith): 0.023167

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 = 1200.000 ¢, ~3/2 = 702.0967 ¢
• POTE: ~2 = 1200.000 ¢, ~3/2 = 702.0978 ¢

Optimal ET sequence: 41g, 53, 94, 241ceh, 335cdehh

Badness (Smith): 0.017635

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 ¢
• POTE: ~2 = 1200.0000 ¢, ~3/2 = 702.0960 ¢

Optimal ET sequence: 41g, 53, 94

Badness (Smith): 0.015072

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.144 ¢

Optimal ET sequence: 41, 53g, 94

Badness (Smith): 0.022454

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.135 ¢

Optimal ET sequence: 41, 53g, 94

Badness (Smith): 0.017576

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.321 ¢

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, 217ce, 258ce

Badness (Smith): 0.023556

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.559 ¢

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, 152cdf, 193cdf, 234cdf

Badness (Smith): 0.020749

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.312 ¢

Optimal ET sequence: 12f, 29, 41

Badness (Smith): 0.023406

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.357 ¢

Optimal ET sequence: 12f, 29, 41

Badness (Smith): 0.019154

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.725 ¢

Optimal ET sequence: 12fg, 29g, 41, 70cd, 111cd

Badness (Smith): 0.021758

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.753 ¢

Optimal ET sequence: 12fg, 29g, 41, 70cd, 111cdh, 181ccddh

Badness (Smith): 0.017902

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.717 ¢

Optimal ET sequence: 12f, 29g, 41g, 70cdgg

Badness (Smith): 0.020895

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 702.716 ¢

Optimal ET sequence: 12f, 29g, 41g, 70cdgg

Badness (Smith): 0.016773

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.725 ¢

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, 171de

Badness (Smith): 0.035637

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.747 ¢

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, 171de

Badness (Smith): 0.026284

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.680 ¢

Optimal ET sequence: 12f, 41ef, 53, 65d, 118dg

Badness (Smith): 0.023732

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.705 ¢

Optimal ET sequence: 12f, 41ef, 53, 65d, 118dg

Badness (Smith): 0.019411

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 tuning (POTE): ~2 = 1200.000 ¢, ~110/63 = 951.082 ¢

Optimal ET sequence: 29, 53, 82e, 135e, 188ce

Badness (Smith): 0.050681

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 tuning (POTE): ~2 = 1200.000 ¢, ~26/15 = 951.082 ¢

Optimal ET sequence: 29, 53, 82e, 135ef, 188cef

Badness (Smith): 0.027464

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/9 = 350.994 ¢

Optimal ET sequence: 41, 106, 147

Badness (Smith): 0.041562

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/9 = 351.014 ¢

Optimal ET sequence: 41, 106, 147

Badness (Smith): 0.042564

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 165.974 ¢

Optimal ET sequence: 29, 65d, 94, 441cde, 535cde, 629cde

Badness (Smith): 0.058040

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 165.963 ¢

Optimal ET sequence: 29, 65d, 94

Badness (Smith): 0.033849

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.

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.757 ¢

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, 1934c, 2105c, 2276cd, 2447cd, 2618cd, 2789cd, 2960cd, 3131bcd

Badness (Smith): 0.014133

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.722 ¢

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 (Smith): 0.038863

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.745 ¢

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 (Smith): 0.033677

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.742 ¢

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, 289ef, 460eef

Badness (Smith): 0.028891

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.740 ¢

Optimal ET sequence: 53, 118f, 171ef

Badness (Smith): 0.036281

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.730 ¢

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

Badness (Smith): 0.030688

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.729 ¢

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

Badness (Smith): 0.021892

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.783 ¢

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, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee

Badness (Smith): 0.048692

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.784 ¢

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 (Smith): 0.023616

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.777 ¢

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 (Smith): 0.022853

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.724 ¢

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, 696d

Badness (Smith): 0.049573

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.738 ¢

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, 460ef

Badness (Smith): 0.045308

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.740 ¢

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, 460ef

Badness (Smith): 0.029618

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.735 ¢

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

Badness (Smith): 0.050188

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.735 ¢

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

Badness (Smith): 0.029383

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 tuning (POTE): ~99/70 = 600.000 ¢, ~3/2 = 701.757 ¢

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

Badness (Smith): 0.014629

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 tuning (POTE): ~99/70 = 600.000 ¢, ~3/2 = 701.773 ¢

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

Badness (Smith): 0.030172

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 tuning (POTE): ~99/70 = 600.000 ¢, ~3/2 = 701.765 ¢

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

Badness (Smith): 0.027051

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 tuning (POTE): ~99/70 = 600.000 ¢, ~3/2 = 701.769 ¢

Optimal ET sequence: 106f, 118f, 224, 342f, 566, 1356cf, 1922cff

Badness (Smith): 0.025547

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 tuning (POTE): ~99/70 = 600.000 ¢, ~3/2 = 701.764 ¢

Optimal ET sequence: 106fg, 118f, 224, 342f, 566, 908fg, 1474cffgg

Badness (Smith): 0.025251

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 tuning (POTE): ~99/70 = 600.000 ¢, ~3/2 = 701.761 ¢

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

Badness (Smith): 0.022267

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 tuning (POTE): ~208/175 = 300.000 ¢, ~3/2 = 701.756 ¢

Optimal ET sequence: 224, 460, 684, 2276cde, 2960cde, 3644bccddee

Badness (Smith): 0.021025

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.239 ¢

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, 53d, 65, 77, 166c, 243c

Badness (Smith): 0.070407

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.172 ¢

Optimal ET sequence: 12, 53dee, 65e, 77, 89, 166c, 255c

Badness (Smith): 0.048887

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.226 ¢

Optimal ET sequence: 12f, 53deeff, 65ef, 77, 166cf, 243cf

Badness (Smith): 0.037859

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.206 ¢

Optimal ET sequence: 12f, 53deeff, 65ef, 77, 89f, 166cf

Badness (Smith): 0.029864

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.217 ¢

Optimal ET sequence: 12f, 53deeff, 65ef, 77, 166cf

Badness (Smith): 0.023096

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.217 ¢

Optimal ET sequence: 12, 53deef, 65e, 77, 166c

Badness (Smith): 0.048511

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.401 ¢

Optimal ET sequence: 12, 53d, 65, 77e, 142de

Badness (Smith): 0.055908

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.348 ¢

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

Badness (Smith): 0.044458

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.529 ¢

Optimal ET sequence: 12f, 53df, 65

Badness (Smith): 0.048524

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 tuning (CTE): ~567/400 = 600.0000 ¢, ~3/2 = 701.5899 ¢

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 (Smith): 0.054744

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 tuning (CTE): ~99/70 = 600.0000 ¢, ~3/2 = 701.6077 ¢

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

Badness (Smith): 0.028160

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 tuning (CTE): ~99/70 = 600.0000 ¢, ~3/2 = 701.5949 ¢

Optimal ET sequence: 12, 106def, 118, 130, 248, 378

Badness (Smith): 0.028722

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 tuning (CTE): ~99/70 = 600.0000 ¢, ~3/2 = 701.5959 ¢

Optimal ET sequence: 12, 106def, 118, 130, 248g

Badness (Smith): 0.029340

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 tuning (CTE): ~55/39 = 600.0000 ¢, ~3/2 = 701.5708 ¢

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

Badness (Smith): 0.029321

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 tuning (CTE): ~55/39 = 600.0000 ¢, ~3/2 = 701.5717 ¢

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

Badness (Smith): 0.026894

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 tuning (POTE): ~1225/864 = 600.000 ¢, ~35/24 = 650.920 ¢ (~36/35 = 50.920 ¢)

Optimal ET sequence: 24, 70c, 94, 118, 212, 330, 542d, 872cd

Badness (Smith): 0.110583

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 tuning (POTE): ~99/70 = 600.000 ¢, ~35/24 = 650.918 ¢ (~36/35 = 50.918 ¢)

Optimal ET sequence: 24, 70c, 94, 118, 212, 330e, 542de

Badness (Smith): 0.036749

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 tuning (POTE): ~99/70 = 600.000 ¢, ~35/24 = 650.938 ¢ (~36/35 = 50.938 ¢)

Optimal ET sequence: 24, 70c, 94, 118, 212f

Badness (Smith): 0.037640

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 tuning (POTE): ~99/70 = 600.000 ¢, ~35/24 = 650.942 ¢ (~36/35 = 50.942 ¢)

Optimal ET sequence: 24, 70c, 94, 118, 212fg

Badness (Smith): 0.025615

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 tuning (POTE): ~99/70 = 600.000 ¢, ~35/24 = 650.951 ¢ (~36/35 = 50.951 ¢)

Optimal ET sequence: 24f, 70cf, 94, 118f, 212

Badness (Smith): 0.037607

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 tuning (POTE): ~99/70 = 600.000 ¢, ~35/24 = 650.948 ¢ (~36/35 = 50.948 ¢)

Optimal ET sequence: 24f, 70cf, 94, 118f, 212g

Badness (Smith): 0.024734

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 tuning (POTE): ~2 = 1200.000 ¢, ~140/81 = 950.797 ¢

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

Badness (Smith): 0.045817

2.3.5.7.13.19.23 subgroup

Subgroup: 2.3.5.7.13.19.23

Comma list: 351/350, 456/455, 513/512, 576/575, 676/675

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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~26/15 = 950.783 ¢

Optimal ET sequence: 24i, 53, 130

Badness (Sintel): 0.583

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 tuning (POTE): ~2 = 1200.000 ¢, ~140/81 = 950.801 ¢

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

Badness (Smith): 0.036289

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 tuning (POTE): ~2 = 1200.000 ¢, ~26/15 = 950.801 ¢

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

Badness (Smith): 0.020816

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 tuning (POTE): ~2 = 1200.000 ¢, ~26/15 = 950.810 ¢

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

Badness (Smith): 0.021073

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 tuning (POTE): ~2 = 1200.000 ¢, ~26/15 = 950.809 ¢

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

Badness (Smith): 0.018262

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 tuning (POTE): ~2 = 1200.000 ¢, ~26/15 = 950.807 ¢

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

Badness (Smith): 0.014819

Music

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

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 tuning (POTE): ~2 = 1200.000 ¢, ~160/147 = 166.140 ¢

Optimal ET sequence: 29, 36, 65

Badness (Smith): 0.174705

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 166.097 ¢

Optimal ET sequence: 29, 36, 65

Badness (Smith): 0.068310

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 166.054 ¢

Optimal ET sequence: 29, 36, 65f, 94df, 159df

Badness (Smith): 0.043750

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 tuning (POTE): ~2 = 1200.000 ¢, ~192/175 = 166.019 ¢

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

Badness (Smith): 0.211859

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 166.017 ¢

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

Badness (Smith): 0.061336

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 166.016 ¢

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

Badness (Smith): 0.036700

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 166.012 ¢

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

Badness (Smith): 0.026504

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 tuning (POTE): ~2 = 1200.000 ¢, ~625/567 = 166.0621 ¢

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

Badness (Smith): 0.188043

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 166.0628 ¢

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

Badness (Smith): 0.048943

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 tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 166.0628 ¢

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

Badness (Smith): 0.024506

Pogo

See also: Stearnsmic clan § Pogo

The pogo temperament (94 & 130) splits the period in two to address the difference between #Tertiaschis and #Countertertiaschis. The schismic tempering of the fifth is just about right for tempering out the stearnsma.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 118098/117649

Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]

mapping generators: ~343/243, ~9/7

Optimal tuning (POTE): ~343/243 = 600.000 ¢, ~9/7 = 433.901 ¢

Optimal ET sequence: 36, 94, 130, 224, 354

Badness (Smith): 0.079635

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 32805/32768

Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~9/7 = 433.911 ¢

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness (Smith): 0.031857

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 4096/4095

Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~9/7 = 433.911 ¢

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness (Smith): 0.017514

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 tuning (POTE): ~63/50 = 400.000 ¢, ~3/2 = 701.742 ¢

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, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722

Badness (Smith): 0.019950

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 tuning (POTE): ~44/35 = 400.000 ¢, ~3/2 = 701.824 ¢

Optimal ET sequence: 12, 147de, 159, 330

Badness (Smith): 0.059502

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 tuning (POTE): ~44/35 = 400.000 ¢, ~3/2 = 701.821 ¢

Optimal ET sequence: 12f, 147def, 159, 330

Badness (Smith): 0.037082

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 tuning (POTE): ~34/27 = 400.000 ¢, ~3/2 = 701.810 ¢

Optimal ET sequence: 12f, 147def, 159, 171, 330

Badness (Smith): 0.027073

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 tuning (POTE): ~63/50 = 400.000 ¢, ~3/2 = 701.685 ¢

Optimal ET sequence: 12e, 159e, 171, 183, 354, 537, 891de

Badness (Smith): 0.066968

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 tuning (POTE): ~63/50 = 400.000 ¢, ~3/2 = 701.689 ¢

Optimal ET sequence: 171, 183, 354, 891de, 1245dee, 1599ddee

Badness (Smith): 0.035487

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 tuning (POTE): ~63/50 = 400.000 ¢, ~3/2 = 701.688 ¢

Optimal ET sequence: 171, 183, 354, 891de, 1245dee, 1599ddee

Badness (Smith): 0.020434

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 tuning (POTE): ~55/49 = 200.0000 ¢, ~3/2 = 701.7460 ¢

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

Badness (Smith): 0.029438

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 tuning (POTE): ~55/49 = 200.0000 ¢, ~3/2 = 701.7256 ¢

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

* optimal patent val: 354

Badness (Smith): 0.044657

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 tuning (POTE): ~63/50 = 400.000 ¢, ~693/400 = 950.872 ¢ (~12/11 = 150.872 ¢)

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

Badness (Smith): 0.020687

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 tuning (POTE): ~63/50 = 400.000 ¢, ~26/15 = 950.873 ¢ (~12/11 = 150.873 ¢)

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

Badness (Smith): 0.031362

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 tuning (POTE): ~34/27 = 400.000 ¢, ~26/15 = 950.867 ¢ (~12/11 = 150.867 ¢)

Optimal ET sequence: 24d, 159, 183, 342f, 525f, 867ff

Badness (Smith): 0.022316

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 tuning (CTE): ~1536/1225 = 400.000 ¢, ~34300/19683 = 950.9654 ¢

Optimal ET sequence: 24, …, 111c, 135, 159, 612ccdd, 771ccdd

Badness (Smith): 0.422026

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 tuning (CTE): ~44/35 = 400.000 ¢, ~121/70 = 950.9658 ¢

Optimal ET sequence: 24, …, 111c, 135, 159, 612ccdd, 771ccdd

Badness (Smith): 0.101224

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 tuning (CTE): ~44/35 = 400.000 ¢, ~26/15 = 950.9360 ¢

Optimal ET sequence: 24, …, 111cf, 135f, 159

Badness (Smith): 0.054894

Sesquiquartififths

Subgroup: 2.3.5.7

Comma list: 2401/2400, 32805/32768

Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]

napping generators: ~2, ~448/405

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~448/405 = 175.434 ¢

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 (Smith): 0.011244

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 tuning (POTE): ~2 = 1200.000 ¢, ~256/231 = 175.406 ¢

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

Badness (Smith): 0.029306

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 tuning (POTE): ~2 = 1200.000 ¢, ~72/65 = 175.409 ¢

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

Badness (Smith): 0.022396

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 tuning (POTE): ~2 = 1200.000 ¢, ~72/65 = 175.424 ¢

Optimal ET sequence: 41, 89g, 130, 171, 301e

Badness (Smith): 0.023126

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 tuning (POTE): ~2 = 1200.000 ¢, ~21/19 = 175.419 ¢

Optimal ET sequence: 41, 89g, 130, 171, 301eh

Badness (Smith): 0.020466

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 tuning (POTE): ~2 = 1200.000 ¢, ~21/19 = 175.412 ¢

Optimal ET sequence: 41i, 89gi, 130, 171, 301eh

Badness (Smith): 0.019043

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 tuning (POTE): ~2 = 1200.000 ¢, ~72/65 = 175.386 ¢

Optimal ET sequence: 41, 89, 130g

Badness (Smith): 0.028443

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 tuning (POTE): ~2 = 1200.000 ¢, ~21/19 = 175.380 ¢

Optimal ET sequence: 41, 89, 130g

Badness (Smith): 0.023059

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 tuning (POTE): ~2 = 1200.000 ¢, ~72/65 = 175.377 ¢

Optimal ET sequence: 41g, 89, 130, 609ceefgg

Badness (Smith): 0.030680

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 tuning (POTE): ~2 = 1200.000 ¢, ~21/19 = 175.377 ¢

Optimal ET sequence: 41g, 89, 130, 609ceefggh

Badness (Smith): 0.022816

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 tuning (POTE): ~2 = 1200.000 ¢, ~21/19 = 175.376 ¢

Optimal ET sequence: 41g, 89, 130, 609ceefggh

Badness (Smith): 0.019121

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 tuning (POTE): ~99/70 = 600.000 ¢, ~448/405 = 175.435 ¢

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

Badness (Smith): 0.016968

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]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~625/588 = 99.625 ¢

Optimal ET sequence: 12, 253, 265

Badness (Smith): 0.253966

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 tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 99.616 ¢

Optimal ET sequence: 12, 253, 265, 518c, 783cc

Badness (Smith): 0.113044

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 tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 99.612 ¢

Optimal ET sequence: 12f, 253, 518c, 771cc

Badness (Smith): 0.069127

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.612 ¢

Optimal ET sequence: 12f, 253, 518c, 771cc

Badness (Smith): 0.045992

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.615 ¢

Optimal ET sequence: 12f, 253, 265, 518ch

Badness (Smith): 0.038155

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 tuning (POTE): ~2 = 1200.000 ¢, ~200/189 = 99.664 ¢

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

Badness (Smith): 0.132890

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 tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 99.653 ¢

Optimal ET sequence: 12, …, 277d, 289

Badness (Smith): 0.111477

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 tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 99.658 ¢

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

Badness (Smith): 0.074218

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.656 ¢

Optimal ET sequence: 12f, 277dff, 289

Badness (Smith): 0.050571

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.659 ¢

Optimal ET sequence: 12f, 289

Badness (Smith): 0.042120

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 tuning (POTE): ~2 = 1200.000 ¢, ~200/189 = 99.671 ¢

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

Badness (Smith): 0.082225

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 tuning (POTE): ~2 = 1200.000 ¢, ~200/189 = 99.661 ¢

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

Badness (Smith): 0.055570

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.665 ¢

Optimal ET sequence: 12f, 289e, 301

Badness (Smith): 0.040412

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.668 ¢

Optimal ET sequence: 12f, 301

Badness (Smith): 0.036840

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 tuning (POTE): ~2 = 1200.000 ¢, ~200/189 = 99.672 ¢

Optimal ET sequence: 12, 301, 614, 915

Badness (Smith): 0.066108

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.671 ¢

Optimal ET sequence: 12, 301

Badness (Smith): 0.047908

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.672 ¢

Optimal ET sequence: 12, 301

Badness (Smith): 0.039542

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, ~21/2

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.053 ¢

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

Badness (Smith): 0.108794

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 tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.049 ¢

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

Badness (Smith): 0.045457

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 tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 83.049 ¢

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

Badness (Smith): 0.025276

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 3 -9 2], ⟨0 -7 -56 4]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~147/128 = 242.614 ¢

Optimal ET sequence: 89, 94, 183, 460d, 643d, 1103dd

Badness (Smith): 0.187047

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 -9 2 -2], ⟨0 -7 -56 4 27]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~147/128 = 242.616 ¢

Optimal ET sequence: 89, 94, 183, 460d, 643d, 826dd

Badness (Smith): 0.052002

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 -9 2 -2 13], ⟨0 -7 -56 4 27 -46]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~147/128 = 242.610 ¢

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

Badness (Smith): 0.035315

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 tuning (POTE): ~2 = 1200.000 ¢, ~243/224 = 140.350 ¢

Optimal ET sequence: 17, 60c, 77, 94, 171

Badness (Smith): 0.030697

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 tuning (POTE): ~2 = 1200.000 ¢, ~88/81 = 140.365 ¢

Optimal ET sequence: 17, 60ce, 77, 94, 171e, 265e, 436ee

Badness (Smith): 0.063499

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 tuning (POTE): ~2 = 1200.000 ¢, ~13/12 = 140.363 ¢

Optimal ET sequence: 17, 60ce, 77, 94, 171e, 436ee

Badness (Smith): 0.037886

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 tuning (POTE): ~2 = 1200.000 ¢, ~56/45 = 380.355 ¢

Optimal ET sequence: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd

Badness (Smith): 0.071950

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 tuning (POTE): ~2 = 1200.000 ¢, ~56/45 = 380.352 ¢

Optimal ET sequence: 41, 142, 183, 224, 631d, 855d, 1079d

Badness (Smith): 0.031549

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 tuning (POTE): ~2 = 1200.000 ¢, ~56/45 = 380.351 ¢

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

Badness (Smith): 0.021392

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 tuning (POTE): ~25/21 = 300.0000 ¢, ~3/2 = 701.8234 ¢

Optimal ET sequence: 212, 224, 436, 660, 1096c

Badness (Smith): 0.110242

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 tuning (POTE): ~25/21 = 300.0000 ¢, ~3/2 = 701.8176 ¢

Optimal ET sequence: 212, 224, 436, 660

Badness (Smith): 0.045738

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 tuning (POTE): ~25/21 = 300.0000 ¢, ~3/2 = 701.8158 ¢

Optimal ET sequence: 212, 224, 436, 660

Badness (Smith): 0.027243

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 tuning (POTE): ~8575/7776 = 171.429 ¢, ~3/2 = 701.702 ¢

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

Badness (Smith): 0.111142

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 tuning (POTE): ~495/448 = 171.429 ¢, ~3/2 = 701.719 ¢

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

Badness (Smith): 0.044122

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 tuning (POTE): ~495/448 = 171.429 ¢, ~3/2 = 701.724 ¢

Optimal ET sequence: 77, 147, 224, 525

Badness (Smith): 0.024706

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 tuning (POTE): ~42875/39366 = 150.000 ¢, ~3/2 = 701.713 ¢

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

Badness (Smith): 0.157186

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 tuning (POTE): ~12/11 = 150.000 ¢, ~3/2 = 701.713 ¢

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

Badness (Smith): 0.044778

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 tuning (POTE): ~12/11 = 150.000 ¢, ~3/2 = 701.725 ¢

Optimal ET sequence: 24, 224, 472, 696

Badness (Smith): 0.030425

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 tuning (CTE): ~2592/2401 = 133.3333 ¢, ~3/2 = 701.7232 ¢

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

Badness (Smith): 0.069896

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 tuning (CTE): ~242/225 = 133.3333 ¢, ~3/2 = 701.8398 ¢

Optimal ET sequence: 36, 99c, 135, 171, 477ce, 648cee

Badness (Smith): 0.126910

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 tuning (CTE): ~242/225 = 133.3333 ¢, ~3/2 = 701.7998 ¢

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

Badness (Smith): 0.076195

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 tuning (CTE): ~2 = 1200.0000 ¢, ~1323/1280 = 53.9741 ¢

Optimal ET sequence: 89, 200, 289

Badness (Smith): 0.432580

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 tuning (CTE): ~2 = 1200.0000 ¢, ~33/32 = 53.9744 ¢

Optimal ET sequence: 89, 200, 289

Badness (Smith): 0.127820

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.491 ¢

Optimal ET sequence: 12, 41, 53, 65

RMS error: 0.4842 cents

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.470 ¢

Optimal ET sequence: 12, 41, 53, 65

RMS error: 0.5374 cents

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 tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 701.746 ¢

Optimal ET sequence: 12, 29, 41, 53, 118, 171

RMS error: 0.1763 cents

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 tuning (POTE): ~2 = 1200.000 ¢, ~26/15 = 950.855 ¢

Optimal ET sequence: 24, 53, 130, 183, 236

RMS error: 0.1485 cents

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 tuning (CTE): ~2 = 1200.0000 ¢, ~26/15 = 950.8199 ¢

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

Badness (Smith): 0.00575

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 tuning (CTE): ~2 = 1200.0000 ¢, ~26/15 = 950.8187 ¢

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

Badness (Smith): 0.00357

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.649 ¢

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

RMS error: 0.0719 cents

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 tuning (POTE): ~2 = 1200.000 ¢, ~18/17 = 99.652 ¢

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

RMS error: 0.1636 cents