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

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. For strong extensions, the generator can be taken to be ~11/10, and three ~11/10's then make a ~4/3 as a result of tempering out 4000/3993, making 10/9 and 12/11 equidistant from 11/10.

Temperaments discussed elsewhere are:

• Hedgehog (+50/49 or 245/243) → Porcupine 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

The rest are considered below.

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

Echidna

See also: Chords of echidna

Echidna adds 1728/1715, the orwellisma, and 2048/2025, the diaschisma, to the commas. It may be described as the 22 & 58 temperament. 58edo or 80edo make for good tunings, or their vals can be added to ⟨138 219 321 388] (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about 6 cents of error within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note 2mos much more. Better yet, it relates three important 11-limit edos: 22edo is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with hedgehog; 58edo is the smallest tuning that is distinctly consistent in the 11-odd-limit, and 80edo is the third smallest distinctly consistent in the 11-odd-limit.

Like most diaschismic extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out 176/175, tempering out 351/350 and 352/351 which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with srutal archagall, leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, the last two of which are consistent in the 17-odd-limit.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 2048/2025

Mapping: [⟨2 1 9 2], ⟨0 3 -6 5]]

mapping generators: ~45/32, ~9/7

Optimal tunings:

• WE: ~45/32 = 599.3056 ¢, ~9/7 = 434.3524 ¢

error map: ⟨-1.389 +0.408 +1.322 +1.547]

• CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8327 ¢

error map: ⟨0.000 +2.543 +4.690 +5.338]

Optimal ET sequence: 22, 58, 80, 138cd, 218cd

Badness (Sintel): 1.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 896/891

Mapping: [⟨2 1 9 2 12], ⟨0 3 -6 5 -7]]

Optimal tunings:

• WE: ~45/32 = 599.3085 ¢, ~9/7 = 434.3511 ¢
• CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8647 ¢

Minimax tuning:

• 11-odd-limit: ~9/7 = [5/12 0 0 1/12 -1/12⟩

[[1 0 0 0 0⟩, [7/4 0 0 1/4 -1/4⟩, [2 0 0 -1/2 1/2⟩, [37/12 0 0 5/12 -5/12⟩, [37/12 0 0 -7/12 7/12⟩]

unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 22, 58, 80, 138cde, 218cde

Badness (Sintel): 0.859

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 364/363, 540/539

Mapping: [⟨2 1 9 2 12 19], ⟨0 3 -6 5 -7 -16]]

Optimal tunings:

• WE: ~45/32 = 599.3397 ¢, ~9/7 = 434.2772 ¢
• CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.7864 ¢

Optimal ET sequence: 22, 36f, 58, 80, 138cde

Badness (Sintel): 0.978

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

Mapping: [⟨2 1 9 2 12 19 6], ⟨0 3 -6 5 -7 -16 3]]

Optimal tunings:

• WE: ~45/32 = 599.4645 ¢, ~9/7 = 434.4282 ¢
• CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8340 ¢

Optimal ET sequence: 22, 36f, 58, 80, 138cde

Badness (Sintel): 1.03

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