Stearnsmic clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
- garistearn (moved to 94th-octave temps); - decistearn (moved to trisedodge family); fix formatting
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.994{{c}}, ~9/7 = 433.884{{c}}
* [[WE]]: ~343/243 = 599.994{{c}}, ~9/7 = 433.884{{c}}
: [[error map]]: {[val| -0.012 -0.309 +0.582 }}
: [[error map]]: {{val| -0.012 -0.309 +0.582 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.885{{c}}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.885{{c}}
: error map: {[val| 0.000 -0.300 +0.600 }}
: error map: {{val| 0.000 -0.300 +0.600 }}


{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d }}
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d }}
Line 33: Line 33:
Stearnsmic temperaments not listed include:  
Stearnsmic temperaments not listed include:  
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Harry]] (+2401/2400 or 19683/19600) → [[Gravity family #Harry|Gravity family]]
* [[Harry]] (+2401/2400 or 19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Octoid]]'' (+4375/4374 or 16875/16807) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* ''[[Octoid]]'' (+4375/4374 or 16875/16807) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Decistearn]]'' (+3136/3125) → [[Trisedodge family #Decistearn|Trisedodge family]]
* ''[[Garistearn]]'' (+33554432/33480783) → [[94th-octave temperaments #Garistearn|94th-octave temperaments]]


Considered below are pogo, supers, stearnscape, garistearn and decistearn.
Considered below are pogo, supers, stearnscape, garistearn and decistearn.
Line 52: Line 54:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}
* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}
: [[error map]]: {[val| -0.028 -0.298 +0.005 +0.598 }}
: [[error map]]: {{val| -0.028 -0.298 +0.005 +0.598 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.9010{{c}}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.9010{{c}}
: error map: {[val| 0.000 -0.252 +0.062 +0.679 }}
: error map: {{val| 0.000 -0.252 +0.062 +0.679 }}


{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
Line 101: Line 103:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.8062{{c}}, ~9/7 = 434.0778{{c}}
* [[WE]]: ~343/243 = 599.8062{{c}}, ~9/7 = 434.0778{{c}}
: [[error map]]: {[val| -0.388 +0.085 -0.199 +1.175 }}
: [[error map]]: {{val| -0.388 +0.085 -0.199 +1.175 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 434.2025{{c}}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 434.2025{{c}}
: error map: {[val| 0.000 +0.653 +0.335 +2.187 }}
: error map: {{val| 0.000 +0.653 +0.335 +2.187 }}


{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}
{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}
Line 166: Line 168:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2450/2187 = 199.9984{{c}}, ~567/500 = 216.9400{{c}}
* [[WE]]: ~2450/2187 = 199.9984{{c}}, ~567/500 = 216.9400{{c}}
: [[error map]]: {[val| -0.010 -0.320 +0.023 +0.564 }}
: [[error map]]: {{val| -0.010 -0.320 +0.023 +0.564 }}
* [[CWE]]: ~2450/2187 = 200.0000{{c}}, ~567/500 = 216.9406{{c}}
* [[CWE]]: ~2450/2187 = 200.0000{{c}}, ~567/500 = 216.9406{{c}}
: error map: {[val| 0.000 -0.312 +0.033 +0.580 }}
: error map: {{val| 0.000 -0.312 +0.033 +0.580 }}


{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
Line 218: Line 220:


Badness (Sintel): 0.782
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 interval|prime]] [[5/1|5]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 118098/117649, 33554432/33480783
{{Mapping|legend=1| 94 149 0 264 | 0 0 1 0 }}
: mapping generators: ~1029/1024, ~5
[[Optimal tuning]]s:
* [[WE]]: ~1029/1024 = 12.7638{{c}}, ~5/4 = 386.7156{{c}} (~5120/5103 = 3.8011{{c}})
: [[error map]]: {[val| -0.201 -0.146 -0.000 +0.822 }}
* [[CWE]]: ~1029/1024 = 12.7660{{c}}, ~5/4 = 386.6637{{c}} (~5120/5103 = 3.6850{{c}})
: error map: {[val| 0.000 +0.173 +0.350 +1.387 }}
{{Optimal ET sequence|legend=1| 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: {{mapping| 94 149 0 264 107 | 0 0 1 0 1 }}
Optimal tunings:
* WE: ~1029/1024 = 12.7637{{c}}, ~5/4 = 386.5270{{c}} (~5120/5103 = 3.6174{{c}})
* CWE: ~1029/1024 = 12.7660{{c}}, ~5/4 = 386.4545{{c}} (~5120/5103 = 3.4758{{c}})
{{Optimal ET sequence|legend=0| 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: {{mapping| 94 149 0 264 107 348 | 0 0 1 0 1 0 }}
Optimal tunings:
* WE: ~169/168 = 12.7628{{c}}, ~5/4 = 386.7181{{c}} (~352/351 = 3.8344{{c}})
* CWE: ~169/168 = 12.7660{{c}}, ~5/4 = 386.6570{{c}} (~352/351 = 3.6783{{c}})
{{Optimal ET sequence|legend=0| 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: {{mapping| 94 149 0 264 107 348 166 | 0 0 1 0 1 0 1 }}
Optimal tunings:
* WE: ~169/168 = 12.7628{{c}}, ~5/4 = 386.7471{{c}} (~352/351 = 3.8623{{c}})
* CWE: ~169/168 = 12.7660{{c}}, ~5/4 = 386.6751{{c}} (~352/351 = 3.6963{{c}})
{{Optimal ET sequence|legend=0| 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|legend=1| 10 2 14 5 | 0 3 2 5 }}
: mapping generators: ~15/14, ~135/98
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 119.9752{{c}}, ~135/98 = 553.8455{{c}}
: [[error map]]: {[val| -0.248 -0.468 +1.030 +0.277 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~135/98 = 553.8935{{c}}
: error map: {[val| 0.000 -0.275 +1.473 +0.642 }}
{{Optimal ET sequence|legend=1| 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: {{mapping| 10 2 14 5 30 | 0 3 2 5 1 }}
Optimal tunings:
* WE: ~15/14 = 119.9534{{c}}, ~11/8 = 553.8639{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9779{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 10 2 14 5 30 37 | 0 3 2 5 1 0 }}
Optimal tunings:
* WE: ~15/14 = 119.9741{{c}}, ~11/8 = 553.8275{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9051{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 10 2 14 5 30 37 27 | 0 3 2 5 1 0 3 }}
Optimal tunings:
* WE: ~15/14 = 119.9899{{c}}, ~11/8 = 553.8840{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9141{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 10 2 14 5 30 37 27 24 | 0 3 2 5 1 0 3 4 }}
Optimal tunings:
* WE: ~15/14 = 119.9949{{c}}, ~11/8 = 553.9253{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9402{{c}}
{{Optimal ET sequence|legend=0| 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, [[10edo|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: {{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{{c}}, ~11/8 = 553.9247{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9331{{c}}
{{Optimal ET sequence|legend=0| 30dh, 50, 80, 130 }}
Badness (Sintel): 1.02


[[Category:Stearnsmic clan| ]] <!-- main article -->
[[Category:Stearnsmic clan| ]] <!-- main article -->

Revision as of 07:38, 15 September 2026

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, 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: ⟨-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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