Stearnsmic clan
The stearnsmic clan tempers out the stearnsma, the no-fives comma [1 10 0 -6⟩ = 118098/117649.
No-five stearnsmic
Subgroup: 2.3.7
Comma list: 118098/117649
Sval mapping: [⟨2 1 2], ⟨0 3 5]]
- mapping generators: ~343/243, ~9/7
Gencom mapping: [⟨2 1 0 2], ⟨0 3 0 5]]
Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 433.888
Optimal ET sequence: 14, 22, 36, 94, 130, 224, 354, 484, 838
Overview to extensions
The second comma in the comma list determines how we extend it to include the harmonic 5. Pogo adds 32805/32768, supers 5120/5103, echidna 1728/1715, and hedgehog 50/49. Those are strong extensions. The others are weak. Wizard adds 225/224. Harry adds 2401/2400. Those split the generator in two. Septisuperfourth adds 6144/6125 and splits the generator in three. Stearnscape adds 250047/250000 and splits the period in three. Octoid adds 4375/4374 and splits the period in four. Decistearn adds 3136/3125 splits the period in five. They all have neat extensions to the 11-limit via tempering out both 540/539 and 4000/3993.
Stearnsmic temperaments not listed include:
- Hedgehog (+50/49 or 245/243) → Porcupine family
- Wizard (+225/224) → Marvel temperaments
- Echidna (+1728/1715 or 2048/2025) → Diaschismic family
- Harry (+2401/2400 or 19683/19600) → Gravity family
- Octoid (+4375/4374 or 16875/16807) → Ragismic microtemperaments
- Septisuperfourth (+6144/6125) → Escapade family
Considered below are pogo, supers, stearnscape, garistearn and decistearn.
Pogo
- See also: Schismatic family #Pogo
The pogo temperament (94 & 130) tempers out the schisma, whose amount of tempering of the fifth is just about right for the stearnsma.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 118098/117649
Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]
Wedgie: ⟨⟨6 -48 10 -90 -1 158]]
Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 433.901
Optimal ET sequence: 36, 94, 130, 224, 354
Badness: 0.079635
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 32805/32768
Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal ET sequence: 36, 94, 130, 224, 354, 578
Badness: 0.031857
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal ET sequence: 36, 94, 130, 224, 354, 578
Badness: 0.017514
Supers
Subgroup: 2.3.5.7
Comma list: 5120/5103, 118098/117649
Mapping: [⟨2 1 -12 2], ⟨0 3 23 5]]
Wedgie: ⟨⟨6 46 10 59 -1 -106]]
Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 434.218
Optimal ET sequence: 58, 94, 152
Badness: 0.092748
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 5120/5103
Mapping: [⟨2 1 -12 2 -9], ⟨0 3 23 5 22]]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.217
Optimal ET sequence: 58, 94, 152
Badness: 0.028240
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 540/539, 729/728, 1575/1573
Mapping: [⟨2 1 -12 2 -9 -2], ⟨0 3 23 5 22 13]]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.221
Optimal ET sequence: 58, 94, 152f
Badness: 0.021645
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 289/288, 352/351, 442/441, 561/560
Mapping: [⟨2 1 -12 2 -9 -2 6], ⟨0 3 23 5 22 13 3]]
Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.181
Optimal ET sequence: 58, 94, 152f
Badness: 0.021316
Stearnscape
Subgroup: 2.3.5.7
Comma list: 118098/117649, 250047/250000
Mapping: [⟨6 3 2 6], ⟨0 6 11 10]]
- mapping generators: ~2450/2187, ~567/500
Wedgie: ⟨⟨36 66 60 21 -6 -46]]
Optimal tuning (CTE): ~2450/2187 = 1\6, ~567/500 = 216.9394 (~245/243 = 16.9394)
Optimal ET sequence: 72, 210, 282, 354
Badness: 0.090467
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 137781/137500
Mapping: [⟨6 3 2 6 11], ⟨0 6 11 10 9]]
Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)
Optimal ET sequence: 72, 210e, 282, 354
Badness: 0.032096
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1575/1573, 34398/34375
Mapping: [⟨6 3 2 6 11 -6], ⟨0 6 11 10 9 26]]
Optimal tuning (CTE): ~55/49 = 1\6, ~312/275 = 216.9332 (~105/104 = 16.9332)
Optimal ET sequence: 72, 210ef, 282, 354
Badness: 0.0258
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573
Mapping: [⟨6 3 2 6 11 -6 5], ⟨0 6 11 10 9 26 18]]
Optimal tuning (CTE): ~55/49 = 1\6, ~17/15 = 216.9345 (~105/104 = 16.9345)
Optimal ET sequence: 72, 210efg, 282, 354
Badness: 0.0154
Garistearn
The garistearn temperament (94 & 282) has a period of 1/94-octave and tempers out 118098/117649 and the garischisma, 33554432/33480783.
Subgroup: 2.3.5.7
Comma list: 118098/117649, 33554432/33480783
Mapping: [⟨94 149 0 264], ⟨0 0 1 0]]
Wedgie: ⟨⟨0 94 0 149 0 -264]]
Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.7805
Optimal ET sequence: 94, 282, 658d, 940dd
Badness: 0.307043
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 33554432/33480783
Mapping: [⟨94 149 0 264 107], ⟨0 0 1 0 1]]
Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.5968
Optimal ET sequence: 94, 282, 376, 658de
Badness: 0.082255
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1575/1573, 28672/28561
Mapping: [⟨94 149 0 264 107 348], ⟨0 0 1 0 1 0]]
Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8141
Badness: 0.045941
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 1156/1155, 1575/1573, 2880/2873
Mapping: [⟨94 149 0 264 107 348 166], ⟨0 0 1 0 1 0 1]]
Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8420
Badness: 0.027608
Decistearn
- See also: Trisedodge family
The decistearn temperament (80 & 130) has a period of 1/10-octave and tempers out the hemimean comma, 3136/3125 as well as the linus comma.
Subgroup: 2.3.5.7
Comma list: 3136/3125, 118098/117649
Mapping: [⟨10 2 14 5], ⟨0 3 2 5]]
- mapping generators: ~15/14, ~135/98
Wedgie: ⟨⟨30 20 50 -38 -5 60]]
Optimal tuning (CTE): ~15/14 = 1\10, ~135/98 = 553.810 (~36/35 = 46.190)
Optimal ET sequence: 50, 80, 130, 470cd, 600cd, 730cd, 860ccd
Badness: 0.095543
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 4000/3993
Mapping: [⟨10 2 14 5 30], ⟨0 3 2 5 1]]
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)
Optimal ET sequence: 50, 80, 130, 210e, 340ce
Badness: 0.038556
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 540/539, 3136/3125
Mapping: [⟨10 2 14 5 30 37], ⟨0 3 2 5 1 0]]
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)
Optimal ET sequence: 50, 80, 130
Badness: 0.026895
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 289/288, 351/350, 540/539, 1632/1625
Mapping: [⟨10 2 14 5 30 37 27], ⟨0 3 2 5 1 0 3]]
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.862 (~36/35 = 46.138)
Optimal ET sequence: 50, 80, 130
Badness: 0.024234
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475
Mapping: [⟨10 2 14 5 30 37 27 24], ⟨0 3 2 5 1 0 3 4]]
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.913 (~36/35 = 46.087)
Optimal ET sequence: 50, 80, 130
Badness: 0.018232