Hemifamity 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. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. 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, hemidromeda, mystery, quanic, septiquarter, countriton, artoneutral and ketchup. 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
Hemidromeda
The name hemidromeda comes from "hemi-" (Ancient Greek for "one half") and "Andromeda", because the generator is 1/2 of the andromeda fourth (~4/3, about 497.6 cents).
Subgroup: 2.3.5.7
Comma list: 5120/5103, 52734375/52706752
Mapping: [⟨1 0 38 48], ⟨0 2 -45 -57]]
- Mapping generator: ~2, ~12500/7203
Wedgie: ⟨⟨2 -45 -57 -76 -96 -6]]
Optimal tuning (CTE): ~2 = 1\1, ~7203/6250 = 248.581
Optimal ET sequence: 29, 82cd, 111, 140, 531, 671, 811b, 951b
Badness: 0.115803
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1331/1323, 1375/1372, 5120/5103
Mapping: [⟨1 0 38 48 32], ⟨0 2 -45 -57 -36]]
Optimal tuning (CTE): ~2 = 1\1, ~405/352 = 248.589
Optimal ET sequence: 29, 82cd, 111, 140, 251, 391e
Badness: 0.060808
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 676/675, 847/845, 1331/1323
Mapping: [⟨1 0 38 48 32 37], ⟨0 2 -45 -57 -36 -42]]
Optimal tuning (CTE): ~2 = 1\1, ~15/13 = 248.588
Optimal ET sequence: 29, 82cdf, 111, 140, 391e, 531e
Badness: 0.028632
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
Mapping: [⟨1 0 38 48 32 37 58], ⟨0 2 -45 -57 -36 -42 -68]]
Optimal tuning (CTE): ~2 = 1\1, ~15/13 = 248.591
Optimal ET sequence: 29g, 82cdfg, 111, 140, 251, 391e
Badness: 0.019054
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
Mapping: [⟨1 0 38 48 32 37 58 32], ⟨0 2 -45 -57 -36 -42 -68 -35]]
Optimal tuning (CTE): ~2 = 1\1, ~15/13 = 248.587
Optimal ET sequence: 29g, 82cdfgh, 111, 140, 391ehh, 531ehh
Badness: 0.016609
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459
Mapping: [⟨1 0 38 48 32 37 58 32 18], ⟨0 2 -45 -57 -36 -42 -68 -35 -17]]
Optimal tuning (CTE): ~2 = 1\1, ~15/13 = 248.588
Optimal ET sequence: 29g, 82cdfgh, 111, 140, 391ehhi, 531ehhii
Badness: 0.015361
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
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: 0.157120
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: 0.045920
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: 0.026257
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: 0.022749
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: 0.019585
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: 0.016332
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