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No-threes subgroup temperaments: Difference between revisions

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=== Frostburn ===
=== Frostburn ===
{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
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: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~28/25


[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.3479{{c}}, ~28/25 = 204.3389{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}
: [[error map]]: {{val| +0.346 -2.630 +2.189 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 204.2027{{c}}
: error map: {{val| 0.000 -3.125 +1.755 }}


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
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Comma list: 245/242, 625/616
Comma list: 245/242, 625/616


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~28/25


Optimal tuning (TE): ~2 = 1200.6817{{c}}, ~28/25 = 205.0745{{c}}
Optimal tunings:
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}


{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
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=== Mabilic ===
=== Mabilic ===
{{Main| Mabilic and trismegistus}}
{{Main| Mabilic and trismegistus }}
{{See also| Chromatic pairs #Mabilic }}
{{See also| Chromatic pairs #Mabilic }}


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: mapping generators: ~2, ~175/128
: mapping generators: ~2, ~175/128


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~175/128 = 527.236{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}
: [[error map]]: {{val| +1.254 -1.698 -1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}
: error map: {{val| 0.000 -4.701 -4.846 }}


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents
[[Badness]] (Sintel): 1.70


=== Rainy ===
=== Rainy ===
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{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~256/245


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~256/245 = 77.205{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
: error map: {{val| 0.000 -0.267 -0.454 }}


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
[[Badness]] (Sintel): 0.156


=== French decimal ===
=== French decimal ===
Conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.
French decimal is conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
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[[Comma list]]: {{monzo| 372 -159 -1 }}
[[Comma list]]: {{monzo| 372 -159 -1 }}


{{Mapping|legend=2| 1 2 54 | 0 1 -159 }}
{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
: mapping generators: ~2, ~5


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~5/4 = 386.360{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
: error map: {{val| 0.000 +0.046 +0.019 }}


{{Optimal ET sequence|legend=1| 205, 264, 469, 733, 997, 1261, 1525, 1789 }}
{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}


[[Badness]] (Sintel): 148.6
[[Badness]] (Sintel): 148


==== 2.5.7.11 subgroup ====
==== 2.5.7.11 subgroup ====
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Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}


Subgroup-val mapping: {{mapping| 1 2 54 -177 | 0 1 -159 -539 }}
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~5/4 = 386.361{{c}}
Optimal tunings:
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


{{Optimal ET sequence|legend=0| 264, 733, … }}
{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}


Badness (Sintel): 52.150
Badness (Sintel): 52.2


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
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Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


Subgroup-val mapping: {{mapping| 1 2 54 -177 52 | 0 1 -159 -539 173 }}
Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~5/4 = 386.361{{c}}
Optimal tunings:
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}


{{Optimal ET sequence|legend=0| 1525, 1789, … }}
{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}


Badness (Sintel): 10.518
Badness (Sintel): 10.5


=== Bastille ===
=== Bastille ===
{{Main| Bastille }}
{{Main| Bastille }}


Bastille is described as the 2.5.7-subgroup 1407 & 1789 temperament, and named after an [[Wikipedia: Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].
Bastille is described as the 2.5.7-subgroup 1407 & 1789 temperament, and named after an [[Wikipedia: Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins #Double bastille|double bastille]].


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
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[[Comma list]]: {{monzo| 1426 -596 -15 }}
[[Comma list]]: {{monzo| 1426 -596 -15 }}


{{Mapping|legend=2| 1 -4 254 | 0 -15 596 }}
{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~{{monzo| 381 0 -159 -4 }} = 694.243{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
: error map: {{val| 0.000 +0.041 +0.018 }}


{{Optimal ET sequence|legend=1| 382, 1025, 1407, 1789, 3196, … }}
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}


[[Badness]] (Sintel): 7224.3
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>


=== Augment ===
=== Augment ===
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Augment is related to [[augmented]].  
Augment is related to [[augmented]].  


[[Subgroup]]: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma list]]: 56/55, 128/125
Comma list: 56/55, 128/125


{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}


{{Mapping|legend=3| 3 0 7 9 11 | 0 0 0 -1 -1 }}
Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}
: mapping generators: ~5/4, ~8/7
: mapping generators: ~5/4, ~7


[[Optimal tuning]] ([[POTE]]): ~5/4 = 400.000{{c}}, ~8/7 = 228.275{{c}}
Optimal tunings:
* [[WE]]: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}


{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents
Badness (Sintel): 0.196


=== Ostara ===
=== Ostara ===
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