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

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]]

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]]

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

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]]

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

Optimal ET sequence: 94, 282

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

Optimal ET sequence: 94, 282

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

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

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 tuning (CTE): ~161/150 = 1\10, ~11/8 = 553.918 (~36/35 = 46.082)

Optimal ET sequence: 30dh, 50, 80, 130

Badness: 0.016719