Amity family
The amity family tempers out the 5-limit amity comma, 1600000/1594323. The generator for the amity temperament is the acute minor third, which means the 6/5 just minor third raised by an 81/80 comma to 243/200, and from this it derives its name. If you are looking for a different kind of neutral third this could be the temperament for you.
Amity
In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)1/13, which gives a pure classical major third. Mos scales of 11, 18, 25, 32, 39, 46 or 53 notes are available.
Subgroup: 2.3.5
Comma list: 1600000/1594323
Mapping: [⟨1 3 6], ⟨0 -5 -13]]
- mapping generators: ~2, ~243/200
Optimal ET sequence: 7, 32c, 39, 46, 53, 152, 205, 258, 1085, 1343, 1601, 1859b, 2117bc
Badness: 0.021960
Overview to extensions
The second comma to extend the 5-limit amity include 4375/4374 for septimal amity, 225/224 for houborizic, 65625/65536 for paramity, 126/125 for accord, 245/243 for bamity, 2430/2401 for hamity, 1029/1024 for gamity, 10976/10935 for chromat, 703125/702464 for trinity, 2401/2400 for amicable, 2100875/2097152 for calamity, 420175/419904 for witcher, and 16875/16807 for familia.
Temperaments discussed elsewhere include:
- Chromat → Hemimage temperaments #Chromat (+10976/10935)
- Witcher → Wizmic microtemperaments #Witcher (+420175/419904)
The rest are considered below.
Septimal amity
Septimal amity can be described as the 46 & 53 temperament, which tempers out 4375/4374 and 5120/5103 in the 7-limit. 99edo is a good tuning, with generator 28\99.
Subgroup: 2.3.5.7
Comma list: 4375/4374, 5120/5103
Mapping: [⟨1 3 6 -2], ⟨0 -5 -13 17]]
Wedgie: ⟨⟨ 5 13 -17 9 -41 -76 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~128/105 = 339.446
- POTE: ~2 = 1\1, ~128/105 = 339.432
Optimal ET sequence: 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd
Badness: 0.023649
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4374, 5120/5103
Mapping: [⟨1 3 6 -2 21], ⟨0 -5 -13 17 -62]]
Optimal tunings:
- CTE: ~2 = 1\1, ~128/105 = 339.485
- POTE: ~2 = 1\1, ~128/105 = 339.464
Optimal ET sequence: 46e, 53, 99e, 152, 357d, 509dd, 661dd
Badness: 0.031506
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 540/539, 625/624, 729/728
Mapping: [⟨1 3 6 -2 21 17], ⟨0 -5 -13 17 -62 -47]]
Optimal tunings:
- CTE: ~2 = 1\1, ~128/105 = 339.508
- POTE: ~2 = 1\1, ~128/105 = 339.481
Optimal ET sequence: 46ef, 53, 99ef, 152f, 205
Badness: 0.028008
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 352/351, 375/374, 540/539, 729/728
Mapping: [⟨1 3 6 -2 21 17 -1], ⟨0 -5 -13 17 -62 -47 18]]
Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.496
Optimal ET sequence: 46ef, 53, 99ef, 152fg, 205gg
Badness: 0.026201
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455
Mapping: [⟨1 3 6 -2 21 17 -1 15], ⟨0 -5 -13 17 -62 -47 18 -38]]
Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.501
Optimal ET sequence: 46efh, 53, 99ef, 152fg, 205gg
Badness: 0.018782
Hitchcock
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175, 2200/2187
Mapping: [⟨1 3 6 -2 6], ⟨0 -5 -13 17 -9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 339.390
- POTE: ~2 = 1\1, ~11/9 = 339.390
Optimal ET sequence: 7, 25cdde, 32cd, 39, 46, 53, 99
Badness: 0.035187
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 176/175, 325/324
Mapping: [⟨1 3 6 -2 6 2], ⟨0 -5 -13 17 -9 6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 339.411
- POTE: ~2 = 1\1, ~11/9 = 339.419
Optimal ET sequence: 7, 25cddef, 32cd, 39, 46, 53, 99
Badness: 0.022448
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
Mapping: [⟨1 3 6 -2 6 2 -1], ⟨0 -5 -13 17 -9 6 18]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 339.366
- POTE: ~2 = 1\1, ~11/9 = 339.366
Optimal ET sequence: 7, 25cddefgg, 32cdg, 39, 46, 99
Badness: 0.019395
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
Mapping: [⟨1 3 6 -2 6 2 -1 0], ⟨0 -5 -13 17 -9 6 18 15]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 339.415
- POTE: ~2 = 1\1, ~11/9 = 339.407
Optimal ET sequence: 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h
Badness: 0.017513
Catamite
The catamite temperament (46 & 99ef) tempers out 441/440 (werckisma) and 896/891 (pentacircle) in the 11-limit; 196/195, 352/351 and 364/363 in the 13-limit. The word "catamite" itself is a term for male homosexual, but also a play on the words "cata-" (down) and "amity."
Subgroup: 2.3.5.7.11
Comma list: 441/440, 896/891, 4375/4374
Mapping: [⟨1 3 6 -2 -7], ⟨0 -5 -13 17 37]]
Optimal tunings:
- CTE: ~2 = 1\1, ~128/105 = 339.314
- POTE: ~2 = 1\1, ~128/105 = 339.340
Optimal ET sequence: 46, 99e, 145
Badness: 0.040976
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 364/363, 4375/4374
Mapping: [⟨1 3 6 -2 -7 -11], ⟨0 -5 -13 17 37 52]]
Optimal tunings:
- CTE: ~2 = 1\1, ~128/105 = 339.277
- POTE: ~2 = 1\1, ~128/105 = 339.313
Optimal ET sequence: 46, 99ef, 145, 191c, 336cef, 527bccef
Badness: 0.034215
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
Mapping: [⟨1 3 6 -2 -7 -11 -1], ⟨0 -5 -13 17 37 52 18]]
Optimal tunings:
- CTE: ~2 = 1\1, ~17/14 = 339.272
- POTE: ~2 = 1\1, ~17/14 = 339.313
Optimal ET sequence: 46, 99ef, 145, 191c
Badness: 0.021193
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
Mapping: [⟨1 3 6 -2 -7 -11 -1 -13], ⟨0 -5 -13 17 37 52 18 61]]
Optimal tunings:
- CTE: ~2 = 1\1, ~17/14 = 339.282
- POTE: ~2 = 1\1, ~17/14 = 339.325
Optimal ET sequence: 46, 99ef, 145, 191c, 336cefg
Badness: 0.018864
Hemiamity
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 5120/5103
Mapping: [⟨2 1 -1 13 13], ⟨0 5 13 -17 -14]]
- mapping generators: ~99/70, ~64/55
Optimal tunings:
- CTE: ~99/70 = 1\2, ~64/55 = 260.566
- POTE: ~99/70 = 1\2, ~64/55 = 260.561
Optimal ET sequence: 14cde, 32cde, 46, 106, 152, 350
Badness: 0.031307
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 1716/1715, 3025/3024
Mapping: [⟨2 1 -1 13 13 20], ⟨0 5 13 -17 -14 -29]]
Optimal tunings:
- CTE: ~99/70 = 1\2, ~64/55 = 260.607
- POTE: ~99/70 = 1\2, ~64/55 = 260.583
Optimal ET sequence: 46, 106f, 152f, 198
Badness: 0.025784
Accord
Subgroup: 2.3.5.7
Comma list: 126/125, 100352/98415
Mapping: [⟨1 3 6 11], ⟨0 -5 -13 -29]]
Wedgie: ⟨⟨ 5 13 29 9 32 31 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~243/200 = 339.154
- POTE: ~2 = 1\1, ~243/200 = 338.993
Optimal ET sequence: 7d, 25cddd, 32cdd, 39d, 46
Badness: 0.095612
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 126/125, 896/891
Mapping: [⟨1 3 6 11 6], ⟨0 -5 -13 -29 -9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 339.136
- POTE: ~2 = 1\1, ~11/9 = 339.047
Optimal ET sequence: 7d, 25cddde, 32cdd, 39d, 46
Badness: 0.042468
Houborizic
The houborizic temperament (53 & 60) tempers out the marvel comma, 225/224. It is so named because it is closely related to the houboriz tuning (generator: 339.774971 cents).
Subgroup: 2.3.5.7
Comma list: 225/224, 1250000/1240029
Mapping: [⟨1 3 6 13], ⟨0 -5 -13 -36]]
Wedgie: ⟨⟨ 5 13 36 9 43 47 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~243/200 = 339.711
- POTE: ~2 = 1\1, ~243/200 = 339.763
Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c
Badness: 0.066638
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 1250000/1240029
Mapping: [⟨1 3 6 13 -9], ⟨0 -5 -13 -36 44]]
Optimal tunings:
- CTE: ~2 = 1\1, ~243/200 = 339.751
- POTE: ~2 = 1\1, ~243/200 = 339.763
Optimal ET sequence: 53, 113, 166, 551ccee
Badness: 0.067891
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 385/384, 2200/2197
Mapping: [⟨1 3 6 13 -9 2], ⟨0 -5 -13 -36 44 6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~39/32 = 339.754
- POTE: ~2 = 1\1, ~39/32 = 339.764
Optimal ET sequence: 53, 113, 166
Badness: 0.032996
Houbor
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 2200/2187
Mapping: [⟨1 3 6 13 6], ⟨0 -5 -13 -36 -9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 339.680
- POTE: ~2 = 1\1, ~11/9 = 339.814
Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53
Badness: 0.045232
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 225/224, 275/273, 325/324
Mapping: [⟨1 3 6 13 6 2], ⟨0 -5 -13 -36 -9 6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 339.685
- POTE: ~2 = 1\1, ~11/9 = 339.784
Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53
Badness: 0.031331
Paramity
The paramity temperament (53 & 311) tempers out the horwell comma (65625/65536) and garischisma (33554432/33480783).
Subgroup: 2.3.5.7
Comma list: 65625/65536, 1600000/1594323
Mapping: [⟨1 3 6 -17], ⟨0 -5 -13 70]]
Optimal tunings:
- CTE: ~2 = 1\1, ~243/200 = 339.554
- POTE: ~2 = 1\1, ~243/200 = 339.553
Optimal ET sequence: 53, 205d, 258, 311, 675, 986
Badness: 0.113655
11-limit
Subgroup: 2.3.5.7.11
Comma list: 6250/6237, 19712/19683, 41503/41472
Mapping: [⟨1 3 6 -17 36], ⟨0 -5 -13 70 -115]]
Optimal tunings:
- CTE: ~2 = 1\1, ~243/200 = 339.554
- POTE: ~2 = 1\1, ~243/200 = 339.554
Optimal ET sequence: 53, 205de, 258, 311, 675, 986
Badness: 0.064853
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683
Mapping: [⟨1 3 6 -17 36 17], ⟨0 -5 -13 70 -115 -47]]
Optimal tunings:
- CTE: ~2 = 1\1, ~243/200 = 339.554
- POTE: ~2 = 1\1, ~243/200 = 339.554
Optimal ET sequence: 53, 205de, 258, 311, 675, 986, 1661cf
Badness: 0.030347
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430
Mapping: [⟨1 3 6 -17 36 17 -31], ⟨0 -5 -13 70 -115 -47 124]]
Optimal tunings:
- CTE: ~2 = 1\1, ~243/200 = 339.555
- POTE: ~2 = 1\1, ~243/200 = 339.555
Optimal ET sequence: 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf
Badness: 0.024118
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197
Mapping: [⟨1 3 6 -17 36 17 -31 15], ⟨0 -5 -13 70 -115 -47 124 -38]]
Optimal tunings:
- CTE: ~2 = 1\1, ~208/171 = 339.555
- POTE: ~2 = 1\1, ~208/171 = 339.555
Optimal ET sequence: 53, 205deg, 258g, 311, 675, 986, 1661cfh
Badness: 0.017420
Bamity
Bamity has a period of half octave and tempers out the sensamagic comma, 245/243. The name bamity is a contraction of bi- and amity.
Subgroup: 2.3.5.7
Comma list: 245/243, 64827/64000
Mapping: [⟨2 1 -1 3], ⟨0 5 13 6]]
- mapping generators: ~343/240, ~7/6
Wedgie: ⟨⟨ 10 26 12 18 -9 -45 ]]
Optimal tunings:
- CTE: ~343/240 = 1\2, ~7/6 = 260.563
- POTE: ~343/240 = 1\2, ~7/6 = 260.402
Optimal ET sequence: 14c, 32c, 46, 106d, 152d
Badness: 0.083601
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 245/243, 441/440
Mapping: [⟨2 1 -1 3 3], ⟨0 5 13 6 9]]
Optimal tunings:
- CTE: ~99/70 = 1\2, ~7/6 = 260.653
- POTE: ~99/70 = 1\2, ~7/6 = 260.393
Optimal ET sequence: 14c, 32c, 46, 152de, 198, 244dee
Badness: 0.035504
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 245/243, 441/440
Mapping: [⟨2 1 -1 3 3 0], ⟨0 5 13 6 9 17]]
Optimal tunings:
- CTE: ~55/39 = 1\2, ~7/6 = 260.811
- POTE: ~55/39 = 1\2, ~7/6 = 260.618
Optimal ET sequence: 14cf, 32cf, 46
Badness: 0.030885
Hamity
Hamity has a generator of about 430 cents which represents 9/7. It is also generated by half of acute minor "tenth" (acute minor third of 243/200 plus an octave), and its name is a contraction of half and amity.
Subgroup: 2.3.5.7
Comma list: 2430/2401, 4000/3969
Mapping: [⟨1 8 19 15], ⟨0 -10 -26 -19]]
- mapping generators: ~2, ~14/9
Wedgie: ⟨⟨ 10 26 19 18 2 -29 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 430.168
- POTE: ~2 = 1\1, ~9/7 = 430.219
Optimal ET sequence: 14c, 39d, 53
Badness: 0.073956
11-limit
Subgroup: 2.3.5.7.11
Comma list: 99/98, 121/120, 2200/2187
Mapping: [⟨1 8 19 15 15], ⟨0 -10 -26 -19 -18]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 430.220
- POTE: ~2 = 1\1, ~9/7 = 430.192
Optimal ET sequence: 14c, 39d, 53
Badness: 0.042947
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 121/120, 275/273, 572/567
Mapping: [⟨1 8 19 15 15 30], ⟨0 -10 -26 -19 -18 -41]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 430.233
- POTE: ~2 = 1\1, ~9/7 = 430.216
Optimal ET sequence: 14cf, 39df, 53
Badness: 0.029753
Gamity
The gamity temperament (46 & 113) tempers out the gamelisma, 1029/1024. It splits the interval of grave major sixth (~400/243, an octave minus acute minor third) in three.
Subgroup: 2.3.5.7
Comma list: 1029/1024, 1071875/1062882
Mapping: [⟨1 13 32 -1], ⟨0 -15 -39 5]]
- mapping generators: ~2, ~320/189
Wedgie: ⟨⟨ 15 39 -5 27 -50 -121 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~189/160 = 286.816
- POTE: ~2 = 1\1, ~189/160 = 286.787
Optimal ET sequence: 46, 113, 159, 205d, 364d
Badness: 0.125733
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 1071875/1062882
Mapping: [⟨1 13 32 -1 -11], ⟨0 -15 -39 5 19]]
Optimal tunings:
- CTE: ~2 = 1\1, ~33/28 = 286.813
- POTE: ~2 = 1\1, ~33/28 = 286.797
Optimal ET sequence: 46, 113, 159, 205d, 364d
Badness: 0.051111
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 364/363, 385/384, 10985/10976
Mapping: [⟨1 13 32 -1 -11 -10], ⟨0 -15 -39 5 19 18]]
Optimal tunings:
- CTE: ~2 = 1\1, ~13/11 = 286.803
- POTE: ~2 = 1\1, ~13/11 = 286.789
Optimal ET sequence: 46, 113, 159, 364df, 523ddff
Badness: 0.030297
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757
Mapping: [⟨1 13 32 -1 -11 -10 -2], ⟨0 -15 -39 5 19 18 8]]
Optimal tunings:
- CTE: ~2 = 1\1, ~13/11 = 286.804
- POTE: ~2 = 1\1, ~13/11 = 286.795
Optimal ET sequence: 46, 113, 159, 364df, 523ddff
Badness: 0.022036
Trinity
The trinity temperament (152 & 159) tempers out the meter, 703125/702464. It splits the acute minor tenth (~243/100, an octave plus acute minor third) in three. It was so named for the following reason – 133\311 (133 steps of 311edo) is a possible generator, which is placed around 3\7 (1.1¢ flat), three of which makes acute minor third of ~243/200 with octave reduction.
Subgroup: 2.3.5.7
Comma list: 703125/702464, 1600000/1594323
Mapping: [⟨1 8 19 46], ⟨0 -15 -39 -101]]
Wedgie: ⟨⟨ 15 39 101 27 118 125 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~168/125 = 513.180
- POTE: ~2 = 1\1, ~168/125 = 513.178
Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085
Badness: 0.119453
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993, 19712/19683
Mapping: [⟨1 8 19 46 18], ⟨0 -15 -39 -101 -34]]
Optimal tunings:
- CTE: ~2 = 1\1, ~121/90 = 513.181
- POTE: ~2 = 1\1, ~121/90 = 513.177
Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085e, 1396e
Badness: 0.031296
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689
Mapping: [⟨1 8 19 46 18 64], ⟨0 -15 -39 -101 -34 -141]]
Optimal tunings:
- CTE: ~2 = 1\1, ~35/26 = 513.184
- POTE: ~2 = 1\1, ~35/26 = 513.182
Optimal ET sequence: 152f, 159, 311
Badness: 0.026418
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619
Mapping: [⟨1 8 19 46 18 64 -22], ⟨0 -15 -39 -101 -34 -141 61]]
Optimal tunings:
- CTE: ~2 = 1\1, ~35/26 = 513.185
- POTE: ~2 = 1\1, ~35/26 = 513.186
Optimal ET sequence: 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg
Badness: 0.025588
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573
Mapping: [⟨1 8 19 46 18 64 -22 53], ⟨0 -15 -39 -101 -34 -141 61 -114]]
Optimal tunings:
- CTE: ~2 = 1\1, ~35/26 = 513.184
- POTE: ~2 = 1\1, ~35/26 = 513.185
Optimal ET sequence: 152f, 159, 311
Badness: 0.018412
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104
Mapping: [⟨1 8 19 46 18 64 -22 53 49], ⟨0 -15 -39 -101 -34 -141 61 -114 -104]]
Optimal tunings:
- CTE: ~2 = 1\1, ~35/26 = 513.184
- POTE: ~2 = 1\1, ~35/26 = 513.185
Optimal ET sequence: 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi
Badness: 0.014343
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044
Mapping: [⟨1 8 19 46 18 64 -22 53 49 72], ⟨0 -15 -39 -101 -34 -141 61 -114 -104 -157]]
Optimal tunings:
- CTE: ~2 = 1\1, ~35/26 = 513.185
- POTE: ~2 = 1\1, ~35/26 = 513.186
Optimal ET sequence: 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij
Badness: 0.012038
Amicable
The amicable temperament (99 & 311) tempers out the amity comma and the canousma in addition to the breedsma, and is closely associated with the canou temperament.
While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1600000/1594323
Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]
Wedgie: ⟨⟨ 20 52 31 36 -7 -74 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.8831
- POTE: ~2 = 1\1, ~21/20 = 84.880
Optimal ET sequence: 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b
Badness: 0.045473
Amical
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 131072/130977, 1600000/1594323
Mapping: [⟨1 3 6 5 -8], ⟨0 -20 -52 -31 162]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.8843
- POTE: ~2 = 1\1, ~21/20 = 84.8843
Optimal ET sequence: 99, 212e, 311, 721, 1032, 1343, 2375bc
Badness: 0.100668
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206
Mapping: [⟨1 3 6 5 -8 -5], ⟨0 -20 -52 -31 162 123]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.8837
- POTE: ~2 = 1\1, ~21/20 = 84.8838
Optimal ET sequence: 99, 212ef, 311, 721, 1032
Badness: 0.049893
Amorous
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 6250/6237, 19712/19683
Mapping: [⟨1 3 6 5 14], ⟨0 -20 -52 -31 -149]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.8883
- POTE: ~2 = 1\1, ~21/20 = 84.8896
Optimal ET sequence: 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde
Badness: 0.048924
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647
Mapping: [⟨1 3 6 5 14 17], ⟨0 -20 -52 -31 -149 -188]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.8895
- POTE: ~2 = 1\1, ~21/20 = 84.8910
Optimal ET sequence: 99ef, 212, 311, 1145c, 1456cd, 1767cd
Badness: 0.034681
Pseudoamical
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 1600000/1594323
Mapping: [⟨1 3 6 5 -1], ⟨0 -20 -52 -31 63]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.9005
- POTE: ~2 = 1\1, ~21/20 = 84.9091
Optimal ET sequence: 14ce, …, 85cee, 99, 212
Badness: 0.085837
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1375/1372, 19773/19712
Mapping: [⟨1 3 6 5 -1 2], ⟨0 -20 -52 -31 63 24]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.9049
- POTE: ~2 = 1\1, ~21/20 = 84.9127
Optimal ET sequence: 14ce, …, 85ceef, 99, 113, 212
Badness: 0.047025
Pseudoamorous
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 980000/970299
Mapping: [⟨1 3 6 5 7], ⟨0 -20 -52 -31 -50]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.9022
- POTE: ~2 = 1\1, ~21/20 = 84.8917
Optimal ET sequence: 14c, …, 85ce, 99e, 113, 212e
Badness: 0.056583
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 1875/1859
Mapping: [⟨1 3 6 5 7 10], ⟨0 -20 -52 -31 -50 -89]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 84.9153
- POTE: ~2 = 1\1, ~21/20 = 84.9164
Optimal ET sequence: 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef
Badness: 0.042826
Floral
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 14641/14580
Mapping: [⟨2 6 12 10 13], ⟨0 -20 -52 -31 -43]]
Optimal tunings:
- CTE: ~99/70 = 1\2, ~21/20 = 84.8781
- POTE: ~99/70 = 1\2, ~21/20 = 84.8788
Optimal ET sequence: 14c, …, 170bccde, 184c, 198, 212, 410
Badness: 0.065110
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580
Mapping: [⟨2 6 12 10 13 19], ⟨0 -20 -52 -31 -43 -82]]
Optimal tunings:
- CTE: ~99/70 = 1\2, ~21/20 = 84.8759
- POTE: ~99/70 = 1\2, ~21/20 = 84.8750
Optimal ET sequence: 14c, …, 184cff, 198, 410
Badness: 0.037013
Humorous
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 1600000/1594323
Mapping: [⟨1 3 6 5 3], ⟨0 -40 -104 -62 13]]
Optimal tunings:
- CTE: ~2 = 1\1, ~4096/3993 = 42.4414
- POTE: ~2 = 1\1, ~4096/3993 = 42.4391
Optimal ET sequence: 85c, 113, 198, 311, 1131, 1442, 1753be
Badness: 0.058249
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024
Mapping: [⟨1 3 6 5 3 6], ⟨0 -40 -104 -62 13 -65]]
Optimal tunings:
- CTE: ~2 = 1\1, ~40/39 = 42.4425
- POTE: ~2 = 1\1, ~40/39 = 42.4391
Optimal ET sequence: 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff
Badness: 0.028267
Calamity
The calamity temperament (46 & 311) tempers out the rainy comma, 2100875/2097152, splitting the interval of two octaves plus an acute minor third into five.
Subgroup: 2.3.5.7
Comma list: 1600000/1594323, 2100875/2097152
Mapping: [⟨1 13 32 -15], ⟨0 -25 -65 39]]
Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.909
Optimal ET sequence: 46, 219c, 265, 311
Badness: 0.198130
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 12005/11979, 131072/130977
Mapping: [⟨1 13 32 -15 -18], ⟨0 -25 -65 39 47]]
Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.908
Optimal ET sequence: 46, 219c, 265, 311, 979, 1290
Badness: 0.060408
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979
Mapping: [⟨1 13 32 -15 -18 -31], ⟨0 -25 -65 39 47 76]]
Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.907
Optimal ET sequence: 46, 265, 311, 668, 979, 1290
Badness: 0.033617
Familia
The familia temperament (113 & 152) tempers out the mirkwai comma, 16875/16807. It splits the interval of acute minor tenth (~243/100) in five.
Subgroup: 2.3.5.7
Comma list: 16875/16807, 1600000/1594323
Mapping: [⟨1 8 19 20], ⟨0 -25 -65 -67]]
Wedgie: ⟨⟨ 25 65 67 45 36 -27 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11907/10000 = 307.915
- POTE: ~2 = 1\1, ~11907/10000 = 307.941
Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd
Badness: 0.144551
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 1600000/1594323
Mapping: [⟨1 8 19 20 5], ⟨0 -25 -65 -67 -6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3200/2673 = 307.915
- POTE: ~2 = 1\1, ~3200/2673 = 307.906
Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1099cdee
Badness: 0.051740
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1375/1372, 2205/2197
Mapping: [⟨1 8 19 20 5 25], ⟨0 -25 -65 -67 -6 -83]]
Optimal tunings:
- CTE: ~2 = 1\1, ~143/120 = 307.922
- POTE: ~2 = 1\1, ~143/120 = 307.913
Optimal ET sequence: 39df, 74cdf, 113, 152f, 265
Badness: 0.038473