Xenharmonic Wiki

Stearnsmic clan

Revision as of 06:32, 15 September 2026 by FloraC (talk | contribs) (Switch to Sintel's badness, WE & CWE tunings)
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: {[val| -0.012 -0.309 +0.582 }}
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.885 ¢
error map: {[val| 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, stearnscape, garistearn and decistearn.

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: {[val| -0.028 -0.298 +0.005 +0.598 }}
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.9010 ¢
error map: {[val| 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: {[val| -0.388 +0.085 -0.199 +1.175 }}
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 434.2025 ¢
error map: {[val| 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: {[val| -0.010 -0.320 +0.023 +0.564 }}
  • CWE: ~2450/2187 = 200.0000 ¢, ~567/500 = 216.9406 ¢
error map: {[val| 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

Garistearn

The garistearn temperament (94 & 282) tempers out the same commas as 94edo in the 2.3.7 subgroup while adding an independent generator for prime 5.

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

Optimal tunings:

  • WE: ~1029/1024 = 12.7638 ¢, ~5/4 = 386.7156 ¢ (~5120/5103 = 3.8011 ¢)
error map: {[val| -0.201 -0.146 -0.000 +0.822 }}
  • CWE: ~1029/1024 = 12.7660 ¢, ~5/4 = 386.6637 ¢ (~5120/5103 = 3.6850 ¢)
error map: {[val| 0.000 +0.173 +0.350 +1.387 }}

Optimal ET sequence: 94, 188, 282, 658d, 940dd

Badness (Sintel): 7.77

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:

  • WE: ~1029/1024 = 12.7637 ¢, ~5/4 = 386.5270 ¢ (~5120/5103 = 3.6174 ¢)
  • CWE: ~1029/1024 = 12.7660 ¢, ~5/4 = 386.4545 ¢ (~5120/5103 = 3.4758 ¢)

Optimal ET sequence: 94, 188e, 282, 376, 658de

Badness (Sintel): 2.72

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:

  • WE: ~169/168 = 12.7628 ¢, ~5/4 = 386.7181 ¢ (~352/351 = 3.8344 ¢)
  • CWE: ~169/168 = 12.7660 ¢, ~5/4 = 386.6570 ¢ (~352/351 = 3.6783 ¢)

Optimal ET sequence: 94, 188e, 282, 658deff, 940ddeefff

Badness (Sintel): 1.90

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:

  • WE: ~169/168 = 12.7628 ¢, ~5/4 = 386.7471 ¢ (~352/351 = 3.8623 ¢)
  • CWE: ~169/168 = 12.7660 ¢, ~5/4 = 386.6751 ¢ (~352/351 = 3.6963 ¢)

Optimal ET sequence: 94, 188eg, 282, 658deff, 940ddeefffg

Badness (Sintel): 1.41

Decistearn

See also: Trisedodge family

The decistearn temperament has a period of 1/10-octave and tempers out the hemimean comma, 3136/3125 as well as the linus comma. It may be described as 80 & 130.

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 tunings:

  • WE: ~15/14 = 119.9752 ¢, ~135/98 = 553.8455 ¢
error map: {[val| -0.248 -0.468 +1.030 +0.277 }}
  • CWE: ~15/14 = 120.0000 ¢, ~135/98 = 553.8935 ¢
error map: {[val| 0.000 -0.275 +1.473 +0.642 }}

Optimal ET sequence: 50, 80, 130, 470cd, 600cd, 730cd

Badness (Sintel): 2.42

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 = 119.9534 ¢, ~11/8 = 553.8639 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~11/8 = 553.9779 ¢

Optimal ET sequence: 50, 80, 130, 210e, 340ce, 470cdee, 810ccddeee

Badness (Sintel): 1.27

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 = 119.9741 ¢, ~11/8 = 553.8275 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~11/8 = 553.9051 ¢

Optimal ET sequence: 50, 80, 130

Badness (Sintel): 1.11

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 = 119.9899 ¢, ~11/8 = 553.8840 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~11/8 = 553.9141 ¢

Optimal ET sequence: 50, 80, 130

Badness (Sintel): 1.23

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 = 119.9949 ¢, ~11/8 = 553.9253 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~11/8 = 553.9402 ¢

Optimal ET sequence: 50, 80, 130

Badness (Sintel): 1.11

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 = 119.9973 ¢, ~11/8 = 553.9247 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~11/8 = 553.9331 ¢

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

Badness (Sintel): 1.02

Retrieved from "https://en.xen.wiki/index.php?title=Stearnsmic_clan&oldid=237177"