Stearnsmic clan: Difference between revisions

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The '''stearnsmic clan''' tempers out the stearnsma, the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.  
{{Technical data page}}
The '''stearnsmic clan''' tempers out the [[stearnsma]], the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.


== No-five stearnsmic ==
== No-five stearnsmic ==
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{{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}
{{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}


[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.888
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.994¢, ~9/7 = 433.884¢
* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 433.885¢


{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
[[Badness]] (Sintel): 0.382


=== Overview to extensions ===
=== Overview to extensions ===
The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|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]].  
The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|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]], as well as [[9801/9800]], so that [[11/10]] and [[9/7]] add up to the semioctave.  


Stearnsmic temperaments not listed include:  
Stearnsmic temperaments not listed include:  
Line 27: Line 32:
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]


Considered below are pogo, supers, stearnscape, garistearn and decistearn.  
Considered below are pogo, supers, stearnscape, garistearn and decistearn.


== Pogo ==
== Pogo ==
{{See also| Schismatic family #Pogo }}
{{See also| Schismatic family }}


The pogo temperament (94 & 130) tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma.  
The pogo temperament (94 & 130) tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It is also the temperament that reconciles the difference between [[tertiaschis]] and [[countertertiaschis]], using its semi-octave period.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 39: Line 44:


{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
: mapping generators: ~343/243, ~9/7


{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~343/243 = 599.986¢, ~9/7 = 433.890¢
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 433.901¢


{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}


[[Badness]]: 0.079635
[[Badness]] (Sintel): 2.015


=== 11-limit ===
=== 11-limit ===
Line 55: Line 61:
Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}
Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~9/7 = 433.911{{c}}


{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}


Badness: 0.031857
Badness (Smith): 0.031857


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}
Mapping: {{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 tuning (POTE): ~99/70 = 600.000{{c}}, ~9/7 = 433.911{{c}}


{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}


Badness: 0.017514
Badness (Smith): 0.017514


== Supers ==
== Supers ==
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{{Mapping|legend=1| 2 1 -12 2 | 0 3 23 5 }}
{{Mapping|legend=1| 2 1 -12 2 | 0 3 23 5 }}


{{Multival|legend=1| 6 46 10 59 -1 -106 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~343/243 = 599.806¢, ~9/7 = 434.078¢
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 434.218
* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 434.203¢


{{Optimal ET sequence|legend=1| 58, 94, 152 }}
{{Optimal ET sequence|legend=1| 58, 94, 152 }}


[[Badness]]: 0.092748
[[Badness]] (Sintel): 2.347


=== 11-limit ===
=== 11-limit ===
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Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.217
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.217


{{Optimal ET sequence|legend=1| 58, 94, 152 }}
{{Optimal ET sequence|legend=0| 58, 94, 152 }}


Badness: 0.028240
Badness: 0.028240
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Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.221
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.221


{{Optimal ET sequence|legend=1| 58, 94, 152f }}
{{Optimal ET sequence|legend=0| 58, 94, 152f }}


Badness: 0.021645
Badness: 0.021645
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Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.181
Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.181


{{Optimal ET sequence|legend=1| 58, 94, 152f }}
{{Optimal ET sequence|legend=0| 58, 94, 152f }}


Badness: 0.021316
Badness: 0.021316
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: mapping generators: ~2450/2187, ~567/500
: mapping generators: ~2450/2187, ~567/500


{{Multival|legend=1| 36 66 60 21 -6 -46 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2450/2187 = 199.998¢, ~567/500 = 216.940¢
[[Optimal tuning]] ([[CTE]]): ~2450/2187 = 1\6, ~567/500 = 216.9394 (~245/243 = 16.9394)
* [[CWE]]: ~2450/2187 = 200.000¢, ~567/500 = 216.941¢


{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}


[[Badness]]: 0.090467
[[Badness]] (Sintel): 2.289


=== 11-limit ===
=== 11-limit ===
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Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)
Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)


{{Optimal ET sequence|legend=1| 72, 210e, 282, 354 }}
{{Optimal ET sequence|legend=0| 72, 210e, 282, 354 }}


Badness: 0.032096
Badness: 0.032096
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Optimal tuning (CTE): ~55/49 = 1\6, ~312/275 = 216.9332 (~105/104 = 16.9332)
Optimal tuning (CTE): ~55/49 = 1\6, ~312/275 = 216.9332 (~105/104 = 16.9332)


{{Optimal ET sequence|legend=1| 72, 210ef, 282, 354 }}
{{Optimal ET sequence|legend=0| 72, 210ef, 282, 354 }}


Badness: 0.0258
Badness: 0.0258
Line 180: Line 186:
Optimal tuning (CTE): ~55/49 = 1\6, ~17/15 = 216.9345 (~105/104 = 16.9345)
Optimal tuning (CTE): ~55/49 = 1\6, ~17/15 = 216.9345 (~105/104 = 16.9345)


{{Optimal ET sequence|legend=1| 72, 210efg, 282, 354 }}
{{Optimal ET sequence|legend=0| 72, 210efg, 282, 354 }}


Badness: 0.0154
Badness: 0.0154
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{{Mapping|legend=1| 94 149 0 264 | 0 0 1 0 }}
{{Mapping|legend=1| 94 149 0 264 | 0 0 1 0 }}


{{Multival|legend=1| 0 94 0 149 0 -264 }}
: mapping generators: ~1029/1024, ~5


[[Optimal tuning]] ([[POTE]]): ~1029/1024 = 1\94, ~5/4 = 386.7805
[[Optimal tuning]]s:
* [[CTE]]: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.3137¢
* [[CWE]]: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6637¢


{{Optimal ET sequence|legend=1| 94, 282, 658d, 940dd }}
{{Optimal ET sequence|legend=1| 94, 282, 658d, 940dd }}


[[Badness]]: 0.307043
[[Badness]] (Sintel): 7.770


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 94 149 0 264 107 | 0 0 1 0 1 }}
Mapping: {{mapping| 94 149 0 264 107 | 0 0 1 0 1 }}


Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.5968
Optimal tunings:
* CTE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
* CWE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.4545¢


{{Optimal ET sequence|legend=1| 94, 282, 376, 658de }}
{{Optimal ET sequence|legend=0| 94, 282, 376, 658de }}


Badness: 0.082255
Badness (Sintel): 2.719


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 94 149 0 264 107 348 | 0 0 1 0 1 0 }}
Mapping: {{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 tunings:
* CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
* CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6570¢


{{Optimal ET sequence|legend=1| 94, 282 }}
{{Optimal ET sequence|legend=0| 94, 282 }}


Badness: 0.045941
Badness (Sintel): 1.898


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 94 149 0 264 107 348 166 | 0 0 1 0 1 0 1 }}
Mapping: {{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 tunings:
* CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 385.9792¢
* CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6751¢


{{Optimal ET sequence|legend=1| 94, 282 }}
{{Optimal ET sequence|legend=0| 94, 282 }}


Badness: 0.027608
Badness (Sintel): 1.406


== Decistearn ==
== Decistearn ==
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: mapping generators: ~15/14, ~135/98
: mapping generators: ~15/14, ~135/98


{{Multival|legend=1| 30 20 50 -38 -5 60 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~15/14 = 119.975¢, ~135/98 = 553.845¢
[[Optimal tuning]] ([[CTE]]): ~15/14 = 1\10, ~135/98 = 553.810 (~36/35 = 46.190)
* [[CWE]]: ~15/14 = 120.000¢, ~135/98 = 553.893¢


{{Optimal ET sequence|legend=1| 50, 80, 130, 470cd, 600cd, 730cd, 860ccd }}
{{Optimal ET sequence|legend=1| 50, 80, 130, 470cd, 600cd, 730cd, 860ccd }}


[[Badness]]: 0.095543
[[Badness]] (Sintel): 2.418


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 10 2 14 5 30 | 0 3 2 5 1 }}
Mapping: {{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 tunings:
* WE: ~15/14 = 119.953¢, ~11/8 = 553.864¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.978¢


{{Optimal ET sequence|legend=1| 50, 80, 130, 210e, 340ce }}
{{Optimal ET sequence|legend=0| 50, 80, 130, 210e, 340ce }}


Badness: 0.038556
Badness (Sintel): 1.275


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 10 2 14 5 30 37 | 0 3 2 5 1 0 }}
Mapping: {{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 tunings:
* WE: ~15/14 = 119.974¢, ~11/8 = 553.827¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.905¢


{{Optimal ET sequence|legend=1| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}


Badness: 0.026895
Badness (Sintel): 1.111


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 10 2 14 5 30 37 27 | 0 3 2 5 1 0 3 }}
Mapping: {{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 tunings:
* WE: ~15/14 = 119.990¢, ~11/8 = 553.884¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.914¢


{{Optimal ET sequence|legend=1| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}


Badness: 0.024234
Badness (Sintel): 1.235


=== 19-limit ===
=== 19-limit ===
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Mapping: {{mapping| 10 2 14 5 30 37 27 24 | 0 3 2 5 1 0 3 4 }}
Mapping: {{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 tunings:
* WE: ~15/14 = 119.995¢, ~11/8 = 553.925¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.940¢


{{Optimal ET sequence|legend=1| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}


Badness: 0.018232
Badness (Sintel): 1.109


=== 23-limit ===
=== 23-limit ===
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Mapping: {{mapping| 10 2 14 5 30 37 27 24 36 | 0 3 2 5 1 0 3 4 2 }}
Mapping: {{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 tunings:
* WE: ~15/14 = 119.997¢, ~11/8 = 553.925¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.933¢


{{Optimal ET sequence|legend=1| 30dh, 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 30dh, 50, 80, 130 }}


Badness: 0.016719
Badness (Sintel): 1.017


[[Category:Temperament clans]]
[[Category:Temperament clans]]

Latest revision as of 05:57, 7 March 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 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

Subgroup-val mapping: [⟨2 1 2], ⟨0 3 5]]

mapping generators: ~343/243, ~9/7

Gencom mapping: [⟨2 1 0 2], ⟨0 3 0 5]]

Optimal tunings:

  • WE: ~343/243 = 599.994¢, ~9/7 = 433.884¢
  • CWE: ~343/243 = 600.000¢, ~9/7 = 433.885¢

Optimal ET sequence: 14, 22, 36, 94, 130, 224, 354, 484, 838

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 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, as well as 9801/9800, so that 11/10 and 9/7 add up to the semioctave.

Stearnsmic temperaments not listed include:

Considered below are pogo, supers, stearnscape, garistearn and decistearn.

Pogo

See also: Schismatic family

The pogo temperament (94 & 130) tempers out the schisma, whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It is also 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]]

mapping generators: ~343/243, ~9/7

Optimal tunings:

  • WE: ~343/243 = 599.986¢, ~9/7 = 433.890¢
  • CWE: ~343/243 = 600.000¢, ~9/7 = 433.901¢

Optimal ET sequence: 36, 94, 130, 224, 354

Badness (Sintel): 2.015

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 = 600.000 ¢, ~9/7 = 433.911 ¢

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness (Smith): 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 = 600.000 ¢, ~9/7 = 433.911 ¢

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness (Smith): 0.017514

Supers

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 = 599.806¢, ~9/7 = 434.078¢
  • CWE: ~343/243 = 600.000¢, ~9/7 = 434.203¢

Optimal ET sequence: 58, 94, 152

Badness (Sintel): 2.347

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 tunings:

  • WE: ~2450/2187 = 199.998¢, ~567/500 = 216.940¢
  • CWE: ~2450/2187 = 200.000¢, ~567/500 = 216.941¢

Optimal ET sequence: 72, 210, 282, 354

Badness (Sintel): 2.289

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]]

mapping generators: ~1029/1024, ~5

Optimal tunings:

  • CTE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.3137¢
  • CWE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6637¢

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

Badness (Sintel): 7.770

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:

  • CTE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
  • CWE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.4545¢

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

Badness (Sintel): 2.719

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:

  • CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
  • CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6570¢

Optimal ET sequence: 94, 282

Badness (Sintel): 1.898

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:

  • CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 385.9792¢
  • CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6751¢

Optimal ET sequence: 94, 282

Badness (Sintel): 1.406

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 tunings:

  • WE: ~15/14 = 119.975¢, ~135/98 = 553.845¢
  • CWE: ~15/14 = 120.000¢, ~135/98 = 553.893¢

Optimal ET sequence: 50, 80, 130, 470cd, 600cd, 730cd, 860ccd

Badness (Sintel): 2.418

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 = 119.953¢, ~11/8 = 553.864¢
  • CWE: ~15/14 = 120.000¢, ~11/8 = 553.978¢

Optimal ET sequence: 50, 80, 130, 210e, 340ce

Badness (Sintel): 1.275

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 = 119.974¢, ~11/8 = 553.827¢
  • CWE: ~15/14 = 120.000¢, ~11/8 = 553.905¢

Optimal ET sequence: 50, 80, 130

Badness (Sintel): 1.111

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 = 119.990¢, ~11/8 = 553.884¢
  • CWE: ~15/14 = 120.000¢, ~11/8 = 553.914¢

Optimal ET sequence: 50, 80, 130

Badness (Sintel): 1.235

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 = 119.995¢, ~11/8 = 553.925¢
  • CWE: ~15/14 = 120.000¢, ~11/8 = 553.940¢

Optimal ET sequence: 50, 80, 130

Badness (Sintel): 1.109

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 = 119.997¢, ~11/8 = 553.925¢
  • CWE: ~15/14 = 120.000¢, ~11/8 = 553.933¢

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

Badness (Sintel): 1.017

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