Aberschismic temperaments

This is a collection of rank-2 temperaments tempering out the hemifamity comma, [10 -6 1 -1⟩ = 5120/5103. These temperaments divide an exact or approximate septimal quartertone, 36/35 into two equal steps, each representing 81/80~64/63, the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same chain of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify 10/7 by the augmented fourth and 50/49 by the Pythagorean comma.

Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.

Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Rodan has the fifth sliced into three as does slendric. Trimot has the twelfth sliced into three as does tricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.

Temperaments considered below are undecental, leapday, mystery, quanic, septiquarter, countriton, ketchup, and artoneutral. Discussed elsewhere are:

• Dominant (+36/35) → Meantone family
• Garibaldi (+225/224) → Schismatic family
• Kwai (+16875/16807) → Mirkwai clan
• Diaschismic (+126/125) → Diaschismic family
• Hemififths (+2401/2400) → Breedsmic temperaments
• Rodan (+245/243) → Gamelismic clan
• Trimot (+2430/2401) → Tricot family
• Monkey (+875/864) → Tetracot family
• Buzzard (+1728/1715) → Vulture family
• Misty (+3136/3125) → Misty family
• Supers (+118098/117649) → Stearnsmic clan
• Undim (+390625/388962) → Undim family
• Quinticosiennic (+395136/390625) → Quintaleap family
• Quintakwai (+9765625/9680832) → Quindromeda family
• Amity (+4375/4374) → Amity family
• Countercata (+15625/15552) → Kleismic family
• Warrior (+78732/78125) → Sensipent family
• Alphaquarter (+29360128/29296875) → Escapade family

Undecental

Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple diminished seventh (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas. 58\99 is an almost perfect generator, just as the name suggests. Another interesting choice is the argent fifth, 2^{(2 - sqrt (2))}.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 235298/234375

Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]

Wedgie: ⟨⟨ 1 -37 -43 -61 -71 4 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.039

Optimal ET sequence: 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd

Badness: 0.094603

Leapday

Main article: Leapday

For the 5-limit version of this temperament, see High badness temperaments #Leapday.

Leapday tempers out the leapday comma, [31 -21 1⟩, in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 & 46 temperament, which tempers out the hemifamity and 686/675 (senga). The alternative extension polypyth (46 & 121) tempers out the same 5-limit comma as the leapday, but with the porwell (6144/6125) rather than the hemifamity tempered out.

Subgroup: 2.3.5.7

Comma list: 686/675, 5120/5103

Mapping: [⟨1 0 -31 -21], ⟨0 1 21 15]]

Wedgie: ⟨⟨ 1 21 15 31 21 -24 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.263

Optimal ET sequence: 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd

Badness: 0.096123

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 686/675

Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.250

Optimal ET sequence: 17c, 29, 46, 167de, 213de, 259cdde

Badness: 0.038624

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.214

Optimal ET sequence: 17c, 29, 46, 121def, 167def, 213deff

Badness: 0.024732

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.229

Optimal ET sequence: 17cg, 29g, 46, 121defg, 167defg, 213deffg

Badness: 0.017863

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 9], ⟨0 1 21 15 11 8 24 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.135

Optimal ET sequence: 17cg, 29g, 46, 75dfgh, 121defgh

Badness: 0.017356

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 9 -5], ⟨0 1 21 15 11 8 24 -3 6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.141

Optimal ET sequence: 17cg, 29g, 46, 75dfgh, 121defgh

Badness: 0.014065

Leapling

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37], ⟨0 1 21 15 11 8 24 26]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.123

Optimal ET sequence: 17cgh, 29g, 46h, 75dfg, 121defghh

Badness: 0.019065

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37 -5], ⟨0 1 21 15 11 8 24 26 6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.114

Optimal ET sequence: 17cgh, 29g, 46h, 75dfg, 121defghh

Badness: 0.016067

Mystery

Main article: Mystery

For the 5-limit version of this temperament, see 29th-octave temperaments #Mystery.

Mystery has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. 145edo or 232edo are good candidates for tunings.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 50421/50000

Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]

Wedgie: ⟨⟨ 0 29 29 46 46 -14 ]]

Optimal tuning (POTE): ~50/49 = 1\29, ~5/4 = 388.646

Optimal ET sequence: 29, 58, 87, 145

Badness: 0.103734

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3388/3375

Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]

Optimal tuning (POTE): ~45/44 = 1\29, ~5/4 = 388.460

Optimal ET sequence: 29, 58, 87, 145

Badness: 0.034291

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 676/675

Mapping: [⟨29 46 0 14 33 40], ⟨0 0 1 1 1 1]]

Optimal tuning (POTE): ~45/44 = 1\29, ~5/4 = 388.354

Optimal ET sequence: 29, 58, 87, 145, 232, 377cef

Badness: 0.018591

Quanic

Subgroup: 2.3.5.7

Comma list: 5120/5103, 5832000/5764801

Mapping: [⟨1 1 -4 0], ⟨0 5 54 24]]

Optimal tuning (POTE): ~2 = 1\1, ~160/147 = 140.493

Optimal ET sequence: 94, 111, 205

Badness: 0.179475

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1331/1323, 5120/5103

Mapping: [⟨1 1 -4 0 1], ⟨0 5 54 24 21]]

Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.489

Optimal ET sequence: 94, 111, 205

Badness: 0.058678

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1331/1323

Mapping: [⟨1 1 -4 0 1 3], ⟨0 5 54 24 21 6]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.496

Optimal ET sequence: 94, 111, 205

Badness: 0.032481

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 715/714, 847/845

Mapping: [⟨1 1 -4 0 1 3 -2], ⟨0 5 54 24 21 6 52]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.497

Optimal ET sequence: 94, 111, 205

Badness: 0.021112

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

Mapping: [⟨1 1 -4 0 1 3 -2 -5], ⟨0 5 54 24 21 6 52 79]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.496

Optimal ET sequence: 94, 111, 205

Badness: 0.017273

Septiquarter

Subgroup: 2.3.5.7

Comma list: 5120/5103, 420175/419904

Mapping: [⟨1 3 10 2], ⟨0 -7 -38 4]]

Wedgie: ⟨⟨ 7 38 -4 44 -26 -116 ]]

Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.453

Optimal ET sequence: 94, 99, 292, 391, 881bd, 1272bcd

Badness: 0.053760

Semiseptiquarter

Subgroup: 2.3.5.7.11

Comma list: 5120/5103, 9801/9800, 14641/14580

Mapping: [⟨2 6 20 4 15], ⟨0 -7 -38 4 -20]]

Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4511

Optimal ET sequence: 94, 198, 292, 490

Badness: 0.064160

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 14641/14580

Mapping: [⟨2 6 20 4 15 7], ⟨0 -7 -38 4 -20 1]]

Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4448

Optimal ET sequence: 94, 198, 490f

Badness: 0.034834

Countriton

For the 5-limit version of this temperament, see Schismic-Mercator equivalence continuum #Countritonic and High badness temperaments #Countritonic.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 7558272/7503125

Mapping: [⟨1 6 19 -7], ⟨0 -9 -34 20]]

Wedgie: ⟨⟨ 9 34 -20 33 -57 -142 ]]

Optimal tuning (POTE): ~2 = 1\1, ~1728/1225 = 588.582

Optimal ET sequence: 53, 157, 210

Badness: 0.131191

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103, 41503/41472

Mapping: [⟨1 6 19 -7 27], ⟨0 -9 -34 20 -48]]

Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.545

Optimal ET sequence: 53, 104c, 157

Badness: 0.084782

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 2197/2187

Mapping: [⟨1 6 19 -7 27 14], ⟨0 -9 -34 20 -48 -21]]

Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.544

Optimal ET sequence: 53, 104c, 157

Badness: 0.042321

Ketchup

Subgroup: 2.3.5.7

Comma list: 5120/5103, 1071875/1062882

Mapping: [⟨2 3 4 6], ⟨0 4 15 -9]]

Wedgie: ⟨⟨ 8 30 -18 29 -51 -126 ]]

Optimal tuning (POTE): ~1225/864 = 1\2, ~64/63 = 25.719

Optimal ET sequence: 46, 94, 140

Badness: 0.084538

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 2200/2187

Mapping: [⟨2 3 4 6 7], ⟨0 4 15 -9 -2]]

Optimal tuning (POTE): ~99/70 = 1\2, ~64/63 = 25.693

Optimal ET sequence: 46, 94, 140

Badness: 0.039555

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1331/1323

Mapping: [⟨2 3 4 6 7 8], ⟨0 4 15 -9 -2 -14]]

Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 25.697

Optimal ET sequence: 46, 94, 140

Badness: 0.024824

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 385/384, 561/560

Mapping: [⟨2 3 4 6 7 8 8], ⟨0 4 15 -9 -2 -14 4]]

Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.701

Optimal ET sequence: 46, 94, 140

Badness: 0.016591

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560

Mapping: [⟨2 3 4 6 7 8 8 9], ⟨0 4 15 -9 -2 -14 4 -12]]

Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.660

Optimal ET sequence: 46, 94, 140h, 234eh

Badness: 0.018170

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384

Mapping: [⟨2 3 4 6 7 8 8 9 9], ⟨0 4 15 -9 -2 -14 4 -12 1]]

Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.661

Optimal ET sequence: 46, 94, 140h, 234ehi

Badness: 0.014033

Artoneutral

Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the 87 & 94 temperament. 181edo is a recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 3828125/3779136

Mapping: [⟨1 8 18 -20], ⟨0 -9 -22 32]]

mapping generators: ~2, ~105/64

Optimal tuning (CTE): ~2 = 1\1, ~105/64 = 855.2452

Optimal ET sequence: 87, 94, 181

Badness: 0.157

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 4000/3993

Mapping: [⟨1 8 18 -20 17], ⟨0 -9 -22 32 -19]]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2397

Optimal ET sequence: 87, 181

Badness: 0.0459

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1575/1573

Mapping: [⟨1 8 18 -20 17 -2], ⟨0 -9 -22 32 -19 8]]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2369

Optimal ET sequence: 87, 181

Badness: 0.0263

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 385/384, 595/594

Mapping: [⟨1 8 18 -20 17 -2 44], ⟨0 -9 -22 32 -19 8 -56]]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2495

Optimal ET sequence: 87, 94, 181

Badness: 0.0227

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594

Mapping: [⟨1 8 18 -20 17 -2 44 52], ⟨0 -9 -22 32 -19 8 -56 -67]]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2534

Optimal ET sequence: 87, 94, 181

Badness: 0.0196

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483

Mapping: [⟨1 8 18 -20 17 -2 44 52 48], ⟨0 -9 -22 32 -19 8 -56 -67 -61]]

Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2576

Optimal ET sequence: 87, 94, 181

Badness: 0.0163