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Stearnsmic clan: Difference between revisions

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The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|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 extension]]s.  
The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|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 extension]]s.  


Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Bisemidim adds [[126/125]]. Septisuperfourth adds [[6144/6125]]. Those split 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.  
Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Bisemidim adds [[126/125]]. Septisuperfourth adds [[6144/6125]]. Those split the generator in three. Stearnscape adds [[250047/250000]] and splits the period in three as well as the generator in two. 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.  
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.  
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Badness (Sintel): 1.91
Badness (Sintel): 1.91
== Bisemidim ==
Bisemidim tempers out [[126/125]] and may be described as the {{nowrap| 50 & 58 }} temperament. It splits the ~9/7 generator into three ~[[49/45]]'s, so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 118098/117649
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
[[Badness]] (Sintel): 2.47
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 126/125, 540/539, 1344/1331
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
Badness (Sintel): 1.36
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 196/195, 364/363
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
Badness (Sintel): 0.987


== Stearnscape ==
== Stearnscape ==
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Badness (Sintel): 0.782
Badness (Sintel): 0.782
== Bisemidim ==
Bisemidim tempers out [[126/125]] and may be described as the {{nowrap| 50 & 58 }} temperament. It splits the ~9/7 generator into three ~[[49/45]]'s, so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 118098/117649
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
[[Badness]] (Sintel): 2.47
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 126/125, 540/539, 1344/1331
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
Badness (Sintel): 1.36
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 196/195, 364/363
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
Badness (Sintel): 0.987


== Octoid ==
== Octoid ==
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