Stearnsmic clan

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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

Optimal tunings:

  • 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:

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]]

Optimal tunings:

  • 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]]

Optimal tunings:

  • 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

Optimal tunings:

  • 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

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