Aberschismic temperaments

The hemifamity temperaments temper out the hemifamity comma, [10 -6 1 -1⟩ = 5120/5103, dividing 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 circle 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.

Belonging to it and considered below are undecental, leapday, mystery, quanic, septiquarter, countriton and ketchup. Other hemifamity temperaments are:

• Dominant → Meantone family

+36/35, the 12 & 17c temperament, generated by the fifth with 5/4 mapped to the M3.

• Garibaldi → Schismatic family

+225/224, the 41 & 53 temperament, generated by the fifth with 5/4 mapped to the d4.

• Kwai → Mirkwai clan

+16875/16807, the 41 & 70 temperament, generated by the fifth with 5/4 mapped to the 4A7 aka m3 + 3 Pyth. commas.

• Diaschismic → Diaschismic family

+126/125, the 46 & 58 temperament, generated by the fifth and using a semioctave period.

• Hemififths → Breedsmic temperaments

+2401/2400, the 41 & 58 temperament, fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma.

• Rodan → Gamelismic clan

+245/243, the 41 & 46 temperament, fifth sliced into three as does slendric.

• Trimot → Tricot family

+2430/2401, the 53 & 70 temperament, twelfth sliced into three as does tricot.

• Monkey → Tetracot family

+875/864, the 34 & 41 temperament, fifth sliced into four as does tetracot.

• Buzzard → Vulture family

+1728/1715, the 53 & 58 temperament, twelfth sliced into four as does vulture.

• Misty → Misty family

+3136/3125, the 12 & 99 temperament, generated by the fifth and using a 1/3-octave period.

• Supers → Stearnsmic clan

+118098/117649, the 36c & 58 temperament, fifth sliced into three and using a semioctave period.

• Undim → Undim family

+390625/388962, the 12 & 140 temperament, generated by the fifth and using a 1/4-octave period.

• Quinticosiennic → Quintaleap family

+395136/390625, the 12 & 145 temperament, fourth sliced into five.

• Quintakwai → Quindromeda family

+9765625/9680832, the 12 & 181 temperament, fourth sliced into five.

• Amity → Amity family

+4375/4374, the 46 & 53 temperament, eleventh sliced into five.

• Countercata → Kleismic family

+15625/15552, the 34 & 53 temperament, twelfth sliced into six as does hanson.

• Warrior → Sensipent family

+78732/78125, the 46 & 65d temperament, 6th harmonic sliced into seven as does sensi.

• Alphaquarter → Escapade family

+29360128/29296875, the 65d & 87 temperament, fourth sliced into nine as does escapade.

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 High badness 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 High badness temperaments #Countriton.

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