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 of temperaments tempers out the stearnsma (monzo: [1 10 0 -6⟩, ratio: 118098/117649), a no-5's comma of about 6.59 cents.
No-5 stearnsmic
This temperament is generated by a very slightly flat ~9/7 major third, three of which minus a semi-octave period give the perfect fifth. Its ploidacot is diploid alpha-tricot.
Subgroup: 2.3.7
Comma list: 118098/117649
Subgroup-val mapping: [⟨2 1 2], ⟨0 3 5]]
Gencom mapping: [⟨2 1 0 2], ⟨0 3 0 5]]
- mapping generators: ~343/243, ~9/7
- WE: ~343/243 = 599.994 ¢, ~9/7 = 433.884 ¢
- error map: ⟨-0.012 -0.309 +0.582]
- CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.885 ¢
- error map: ⟨0.000 -0.300 +0.600]
Optimal ET sequence: 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d
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 (94 & 130) adds 32805/32768, supers (58 & 94) adds 5120/5103, echidna (22 & 58) adds 1728/1715, and hedgehog (14c & 22) adds 50/49. Those are the strong extensions.
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.
All of them extend naturally 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 semi-octave period.
Stearnsmic temperaments not listed include:
- Hedgehog (+50/49 or 245/243) → Porcupine family
- Echidna (+1728/1715 or 2048/2025) → Diaschismic family
- Wizard (+225/224) → Marvel temperaments
- Harry (+2401/2400 or 19683/19600) → Gravity family
- Septisuperfourth (+6144/6125) → Escapade family
- Octoid (+4375/4374 or 16875/16807) → Ragismic microtemperaments
- Decistearn (+3136/3125) → Trisedodge family
- Garistearn (+33554432/33480783) → 94th-octave temperaments
Considered below are pogo, supers, and stearnscape.
Pogo
The pogo temperament tempers out the schisma, whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It may be described as 94 & 130, and is 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]]
- WE: ~343/243 = 599.9860 ¢, ~9/7 = 433.8905 ¢
- error map: ⟨-0.028 -0.298 +0.005 +0.598]
- CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.9010 ¢
- error map: ⟨0.000 -0.252 +0.062 +0.679]
Optimal ET sequence: 36, 94, 130, 224, 354
Badness (Sintel): 2.02
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 tunings:
- WE: ~99/70 = 599.9700 ¢, ~9/7 = 433.8898 ¢
- CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9125 ¢
Optimal ET sequence: 36, 58ce, 94, 130, 224, 354, 578
Badness (Sintel): 1.05
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 tunings:
- WE: ~99/70 = 599.9670 ¢, ~9/7 = 433.8867 ¢
- CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9112 ¢
Optimal ET sequence: 36, 94, 130, 224, 354, 578
Badness (Sintel): 0.724
Supers
Supers tempers out 5120/5103, the aberschisma, and may be described as the 58 & 94 temperament.
Subgroup: 2.3.5.7
Comma list: 5120/5103, 118098/117649
Mapping: [⟨2 1 -12 2], ⟨0 3 23 5]]
- WE: ~343/243 = 599.8062 ¢, ~9/7 = 434.0778 ¢
- error map: ⟨-0.388 +0.085 -0.199 +1.175]
- CWE: ~343/243 = 600.000 ¢, ~9/7 = 434.2025 ¢
- error map: ⟨0.000 +0.653 +0.335 +2.187]
Optimal ET sequence: 36c, 58, 94, 152
Badness (Sintel): 2.35
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 tunings:
- WE: ~99/70 = 599.8051 ¢, ~9/7 = 434.0755 ¢
- CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.2002 ¢
Optimal ET sequence: 36ce, 58, 94, 152
Badness (Sintel): 0.934
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 tunings:
- WE: ~99/70 = 599.7318 ¢, ~9/7 = 434.0271 ¢
- CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.1985 ¢
Optimal ET sequence: 36ce, 58, 94, 152f
Badness (Sintel): 0.894
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 tunings:
- WE: ~99/70 = 599.8786 ¢, ~9/7 = 434.0934 ¢
- CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.1726 ¢
Optimal ET sequence: 36ce, 58, 94, 152f
Badness (Sintel): 1.09
Stearnscape
Stearnscape tempers out 250047/250000, the landscape comma, with a period of 1/6 octave, and may be described as the 72 & 282 temperament.
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
- WE: ~2450/2187 = 199.9984 ¢, ~567/500 = 216.9400 ¢
- error map: ⟨-0.010 -0.320 +0.023 +0.564]
- CWE: ~2450/2187 = 200.0000 ¢, ~567/500 = 216.9406 ¢
- error map: ⟨0.000 -0.312 +0.033 +0.580]
Optimal ET sequence: 72, 210, 282, 354
Badness (Sintel): 2.29
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 tunings:
- WE: ~55/49 = 199.9835 ¢, ~567/500 = 216.9321 ¢ (~100/99 = 16.9487 ¢)
- CWE: ~55/49 = 200.0000 ¢, ~567/500 = 216.9381 ¢ (~100/99 = 16.9381 ¢)
Optimal ET sequence: 72, 210e, 282, 354
Badness (Sintel): 1.06
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 tunings:
- WE: ~55/49 = 199.9810 ¢, ~312/275 = 216.9360 ¢ (~105/104 = 16.9550 ¢)
- CWE: ~55/49 = 200.0000 ¢, ~312/275 = 216.9474 ¢ (~105/104 = 16.9474 ¢)
Optimal ET sequence: 72, 210ef, 282, 354
Badness (Sintel): 1.06
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 tunings:
- WE: ~55/49 = 199.9814 ¢, ~17/15 = 216.9376 ¢ (~105/104 = 16.9563 ¢)
- CWE: ~55/49 = 200.0000 ¢, ~17/15 = 216.9487 ¢ (~105/104 = 16.9345 ¢)
Optimal ET sequence: 72, 210efg, 282, 354
Badness (Sintel): 0.782