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No-threes subgroup temperaments: Difference between revisions

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== 2.5.7 temperaments ==
== 2.5.7 temperaments ==
=== Llywelyn aka shoe ===
{{See also| Chromatic pairs #Shoe }}
{{See also| Llywelyn clan #Llywelyn aka shoe }}
[[Subgroup]]: 2.5.7
[[Comma list]]: 4194304/4117715
[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
Mapping generators: 2, ~8/7
[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
[[Gencom]]: [2 8/7; 4194304/4117715]
[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Comma list: 176/175, 1310720/1294139
Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
Gencom: [2 8/7; 176/175 1310720/1294139]
Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
Optimal tuning (POTE): ~8/7 = 227.114
{{Optimal ET sequence|legend=1| 16, 21, 37 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637, 847/845
Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
Optimal tuning (POTE): ~8/7 = 227.108
{{Optimal ET sequence|legend=1| 16, 21, 37 }}
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
Comma list: 176/175, 221/200, 640/637, 833/832
Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
Optimal tuning (POTE): ~8/7 = 227.242
{{Optimal ET sequence|legend=1| 16, 21, 37 }}


=== Didacus ===
=== Didacus ===
Line 124: Line 59:
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents


=== Mercy ===
=== Bastille ===
{{See also| Quince clan #Mercy }}
{{Main|Bastille}}
Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Pure Bastille|pure bastille]].
 
Subgroup: 2.5.7
 
Comma list: {{Monzo|1426 -596 -15}}
 
Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
 
Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
 
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
 
=== Frostburn ===
{{See also| Magic family #Quadrimage }}
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 78125/76832
 
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
 
: Sval mapping generators: ~2, ~28/25
 
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
 
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
 
==== 2.5.7.11 ====
Subgroup: 2.5.7.11
 
Comma list: 245/242, 625/616
 
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
 
: Sval mapping generators: ~2, ~28/25
 
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
 
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}


Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
=== Llywelyn aka shoe ===
{{See also| Chromatic pairs #Shoe }}
{{See also| Llywelyn clan #Llywelyn aka shoe }}


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 823543/819200
[[Comma list]]: 4194304/4117715


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]


[[Gencom]]: [2 2744/2560; 823543/819200]
Mapping generators: 2, ~8/7


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]


Optimal tuning ([[POTE]]): ~343/320 = 116.291
[[Gencom]]: [2 8/7; 4194304/4117715]


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910


==== 2.5.7.13 ====
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
Comma list: 176/175, 1310720/1294139


[[Gencom]]: [2 14/13; 343/338 640/637]
Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
Gencom: [2 8/7; 176/175 1310720/1294139]


Optimal tuning ([[POTE]]): ~14/13 = 116.094
Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]


{{Optimal ET sequence|legend=1| 10, 21, 31}}
Optimal tuning (POTE): ~8/7 = 227.114


==== 2.5.7.13.17 ====
{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
Comma list: 176/175, 640/637, 847/845


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]


Optimal tuning ([[POTE]]): ~14/13 = 115.769
Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]


{{Optimal ET sequence|legend=1| 10, 21, 31}}
Optimal tuning (POTE): ~8/7 = 227.108


==== 2.5.7.13.17.19 ====
{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
Comma list: 176/175, 221/200, 640/637, 833/832


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]


Optimal tuning ([[POTE]]): ~14/13 = 115.716
Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
Optimal tuning (POTE): ~8/7 = 227.242


=== Frostburn ===
{{Optimal ET sequence|legend=1| 16, 21, 37 }}
{{See also| Magic family #Quadrimage }}


[[Subgroup]]: 2.5.7
=== 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.


[[Comma list]]: 78125/76832
Subgroup: 2.5.7


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
Comma basis: {{monzo|372 -159 -1}}


: Sval mapping generators: ~2, ~28/25
Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
Optimal tuning (CTE): ~5/4 = 386.360


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...


==== 2.5.7.11 ====
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
 
Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
 
Optimal tuning (CTE): ~5/4 = 386.361
 
{{Optimal ET sequence|legend=1|264, 733}}, ...
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


: Sval mapping generators: ~2, ~28/25
Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Optimal tuning (CTE): ~5/4 = 386.361


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
{{Optimal ET sequence|legend=1|1525, 1789}}, ...


=== Ostara ===
=== Ostara ===
Line 275: Line 260:
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...


=== French decimal ===
=== Mercy ===
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.
{{See also| Quince clan #Mercy }}
 
Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 823543/819200
 
[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
 
[[Gencom]]: [2 2744/2560; 823543/819200]
 
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
 
Optimal tuning ([[POTE]]): ~343/320 = 116.291
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}


Subgroup: 2.5.7
==== 2.5.7.13 ====
[[Subgroup]]: 2.5.7.13


Comma basis: {{monzo|372 -159 -1}}
[[Comma list]]: 343/338, 640/637


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]


Optimal tuning (CTE): ~5/4 = 386.360
[[Gencom]]: [2 14/13; 343/338 640/637]


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]


==== 2.5.7.11 subgroup ====
Optimal tuning ([[POTE]]): ~14/13 = 116.094
Subgroup: 2.5.7.11


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
{{Optimal ET sequence|legend=1| 10, 21, 31}}


Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
==== 2.5.7.13.17 ====
[[Subgroup]]: 2.5.7.13.17


Optimal tuning (CTE): ~5/4 = 386.361
[[Comma list]]: 170/169, 224/221, 640/637


{{Optimal ET sequence|legend=1|264, 733}}, ...
[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]


==== 2.5.7.11.13 subgroup ====
[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
Subgroup: 2.5.7.11.13


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
Optimal tuning ([[POTE]]): ~14/13 = 115.769


Optimal tuning (CTE): ~5/4 = 386.361
{{Optimal ET sequence|legend=1| 10, 21, 31}}


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
==== 2.5.7.13.17.19 ====
[[Subgroup]]: 2.5.7.13.17.19


=== Bastille ===
[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
{{Main|Bastille}}
Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Pure Bastille|pure bastille]].


Subgroup: 2.5.7
[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]


Comma list: {{Monzo|1426 -596 -15}}
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
Optimal tuning ([[POTE]]): ~14/13 = 115.716


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}


=== Pakkanen (rank 3) ===
=== Pakkanen (rank 3) ===
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