Hemifamity temperaments: Difference between revisions
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== Countriton == | == Countriton == | ||
: ''For the 5-limit version of this temperament, see [[High badness temperaments # | : ''For the 5-limit version of this temperament, see [[Schismic-Mercator equivalence continuum #Countritonic]] and [[High badness temperaments #Countritonic]].'' | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Revision as of 09:26, 6 September 2023
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, ketchup, and artoneutral. 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
- 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
- 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
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
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