Stearnsmic clan
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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
Subgroup-val mapping: [⟨2 1 2], ⟨0 3 5]]
- mapping generators: ~343/243, ~9/7
Gencom mapping: [⟨2 1 0 2], ⟨0 3 0 5]]
Optimal ET sequence: 14, 22, 36, 94, 130, 224, 354, 484, 838
Badness (Sintel): 0.382
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, as well as 9801/9800, so that 11/10 and 9/7 add up to the semioctave.
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
The pogo temperament (94 & 130) tempers out the schisma, whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It is also the temperament that reconciles the difference between tertiaschis and countertertiaschis, using its semi-octave period.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 118098/117649
Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]
- mapping generators: ~343/243, ~9/7
Optimal ET sequence: 36, 94, 130, 224, 354
Badness (Sintel): 2.015
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 = 600.000 ¢, ~9/7 = 433.911 ¢
Optimal ET sequence: 36, 94, 130, 224, 354, 578
Badness (Smith): 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 = 600.000 ¢, ~9/7 = 433.911 ¢
Optimal ET sequence: 36, 94, 130, 224, 354, 578
Badness (Smith): 0.017514
Supers
Subgroup: 2.3.5.7
Comma list: 5120/5103, 118098/117649
Mapping: [⟨2 1 -12 2], ⟨0 3 23 5]]
Optimal ET sequence: 58, 94, 152
Badness (Sintel): 2.347
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
Optimal ET sequence: 72, 210, 282, 354
Badness (Sintel): 2.289
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]]
- mapping generators: ~1029/1024, ~5
- CTE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.3137¢
- CWE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6637¢
Optimal ET sequence: 94, 282, 658d, 940dd
Badness (Sintel): 7.770
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 tunings:
- CTE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
- CWE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.4545¢
Optimal ET sequence: 94, 282, 376, 658de
Badness (Sintel): 2.719
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 tunings:
- CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
- CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6570¢
Badness (Sintel): 1.898
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 tunings:
- CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 385.9792¢
- CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6751¢
Badness (Sintel): 1.406
Decistearn
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
Optimal ET sequence: 50, 80, 130, 470cd, 600cd, 730cd, 860ccd
Badness (Sintel): 2.418
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 tunings:
- WE: ~15/14 = 119.953¢, ~11/8 = 553.864¢
- CWE: ~15/14 = 120.000¢, ~11/8 = 553.978¢
Optimal ET sequence: 50, 80, 130, 210e, 340ce
Badness (Sintel): 1.275
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 tunings:
- WE: ~15/14 = 119.974¢, ~11/8 = 553.827¢
- CWE: ~15/14 = 120.000¢, ~11/8 = 553.905¢
Optimal ET sequence: 50, 80, 130
Badness (Sintel): 1.111
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 tunings:
- WE: ~15/14 = 119.990¢, ~11/8 = 553.884¢
- CWE: ~15/14 = 120.000¢, ~11/8 = 553.914¢
Optimal ET sequence: 50, 80, 130
Badness (Sintel): 1.235
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 tunings:
- WE: ~15/14 = 119.995¢, ~11/8 = 553.925¢
- CWE: ~15/14 = 120.000¢, ~11/8 = 553.940¢
Optimal ET sequence: 50, 80, 130
Badness (Sintel): 1.109
23-limit
By equating 16/13 with 69/56 and 85/69 (enabling 56:69:85 chords), we find 23/16 at 2 generators underneath 6 periods (which is the 5edo fifth, 6\10).
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475, 897/896
Mapping: [⟨10 2 14 5 30 37 27 24 36], ⟨0 3 2 5 1 0 3 4 2]]
Optimal tunings:
- WE: ~15/14 = 119.997¢, ~11/8 = 553.925¢
- CWE: ~15/14 = 120.000¢, ~11/8 = 553.933¢
Optimal ET sequence: 30dh, 50, 80, 130
Badness (Sintel): 1.017