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No-fives subgroup temperaments: Difference between revisions

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** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]
== Magi ==
Subgroup: 2.3.7
[[Comma list]]: 537824/531441
[[Gencom]]: [2 243/196; 537824/531441]
[[Mapping|Sval mapping]]: [{{val|1 0 -1}}, {{val|0 5 12}}]
[[Tp tuning|POL2 generator]]: ~243/196 = 380.661
{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145, 186 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.4277 cents
=== 2.3.7.11 ===
Subgroup: 2.3.7.11
[[Comma list]]: 896/891, 26411/26244
[[Gencom]]: [2 96/77; 896/891 26411/26244]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6}}, {{val|0 5 12 -8}}]
[[Tp tuning|POL2 generator]]: ~96/77 = 380.768
{{Optimal ET sequence|legend=1| 19, 22, 41, 63, 104 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
==== Balthazar ====
Subgroup: 2.3.7.11.13
[[Comma list]]: 169/168, 896/891, 26411/26244
[[Gencom]]: [2 143/128; 169/168 896/891 26411/26244]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 1}}, {{val|0 10 24 -16 17}}]
[[Tp tuning|POL2 generator]]: ~143/128 = 190.407
{{Optimal ET sequence|legend=1| 19, 44, 63, 145f }}
[[Tp tuning #T2 tuning|RMS error]]: 0.6937 cents
==== Caspar ====
Subgroup: 2.3.7.11.13
[[Comma list]]: 144/143, 343/338, 729/728
[[Gencom]]: [2 26/21; 144/143 343/338 729/728]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2}}, {{val|0 5 12 -8 18}}]
[[Tp tuning|POL2 generator]]: ~26/21 = 380.531
{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.032 cents
==== Melchior ====
Subgroup: 2.3.7.11.13
[[Comma list]]: 352/351, 364/363, 26411/26244
[[Gencom]]: [2 96/77; 352/351 364/363 26411/26244]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 11}}, {{val|0 5 12 -8 -23}}]
[[Tp tuning|POL2 generator]]: ~96/77 = 380.766
{{Optimal ET sequence|legend=1| 19f, 22, 41, 63, 104 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3891 cents
=== Hogwarts ===
Subgroup: 2.3.7.29
[[Comma list]]: 784/783, 5887/5832
[[Gencom]]: [2 36/29; 784/783 5887/5832]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 2}}, {{val|0 5 12 9}}]
[[Tp tuning|POL2 generator]]: ~36/29 = 380.618
{{Optimal ET sequence|legend=1| 19, 22, 41, 145, 186j, 227j }}
==== Twenothology ====
Subgroup: 2.3.7.11.13.29
[[Comma list]]: 144/143, 232/231, 343/338, 729/728
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2 2}}, {{val|0 5 12 -8 18 9}}]
[[Tp tuning|POL2 generator]]: ~26/21 = 380.526
{{Optimal ET sequence|legend=1| 19, 22f, 41 }}


== Lee ==
== Lee ==
Line 458: Line 557:


[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
== Hearts ==
Subgroup: 2.3.7
[[Comma list]]: 34451725707/34359738368 (trila-quadzo comma)
[[Gencom]]: [2 567/512; 34451725707/34359738368]
[[Mapping|Sval mapping]]: [{{val|1 1 5}}, {{val|0 4 -15}}]
[[Tp tuning|POL2 generator]]: ~567/512 = 175.433
{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 89, 130, 171 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.0529 cents
Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquiquartififths]]
=== 2.3.7.11 ===
Subgroup: 2.3.7.11
[[Comma list]]: 243/242, 65536/65219
[[Gencom]]: [2 256/231; 243/242 65536/65219]
[[Mapping|Sval mapping]]: [{{val|1 1 5 2}}, {{val|0 4 -15 10}}]
[[Tp tuning|POL2 generator]]: ~256/231 = 175.369
{{Optimal ET sequence|legend=1| 7, 27de, 34, 41, 89, 130 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3224 cents
Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
==== 2.3.7.11.19 ====
Subgroup: 2.3.7.11.19
[[Comma list]]: 243/242, 513/512, 1083/1078
[[Gencom]]: [2 21/19; 243/242 513/512 1083/1078]
[[Mapping|Sval mapping]]: [{{val|1 1 5 2 6}}, {{val|0 4 -15 10 -12}}]
[[Tp tuning|POL2 generator]]: ~21/19 = 175.341
{{Optimal ET sequence|legend=1| 7, 27deh, 34, 41, 89, 130, 219 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3121 cents
Related temperaments: [[Tetracot family|monkey]], [[Schismatic family|sesquart]]
== Hectosaros leap week ==
Defined as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.
[[Subgroup]]: 2.3.7
[[Comma list]]: {{monzo|-50 -746 439}}
[[Mapping]]: [{{val|1 313 532}}, {{val|0 -439 -746}}]
[[Optimal tuning]] ([[CTE]]): ~{{monzo|17 343 143}} = 851.248
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.0164 cents
=== 2.3.7.13 subgroup ===
Subgroup: 2.3.7.13
Comma list: {{monzo|-42  -2 -5 16}}, {{monzo|10 -46 29 -5}}
Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
=== 2.3.7.13.17 subgroup ===
Subgroup: 2.3.7.13.17
Comma list: 39337984/39328497, {{monzo|0 -14  7  4 -3}}, {{monzo|-18 -24 14 -1  5}}
Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
Optimal tuning (CTE): ~6144/3757 = 851.248
{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
=== 2.3.7.13.17.19 subgroup ===
Subgroup: 2.3.7.13.17.19
Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256
Mapping: [{{val|1 313 532 208 58 432}}, {{val|0 -439 -746 -288 -76 -603}}]
Optimal tuning (CTE): ~6144/3757 = 851.248
{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}


== Ennea ==
== Ennea ==
Line 554: Line 752:


[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
== Heartland (rank 3) ==
Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, [[3971/3969]]. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).
Subgroup: 2.3.7.11.19
[[Comma list]]: 243/242, 1083/1078
[[Gencom]]: [2 21/19 7; 243/242 1083/1078]
[[Mapping|Sval mapping]]: [{{val|1 1 0 2 1}}, {{val|0 4 0 10 3}}, {{val| 0 0 1 0 1 }}]
[[Tp tuning|POL2 generator]]: ~21/19 = 175.2713, ~7 = 3369.3784
{{Optimal ET sequence|legend=1| 7, 14, 27e, 34dh, 41, 89, 130 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents


== Reversed meantone ==
== Reversed meantone ==
Line 599: Line 814:


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
== Magi ==
Subgroup: 2.3.7.11
[[Comma list]]: 896/891, 26411/26244
[[Gencom]]: [2 96/77; 896/891, 26411/26244]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6}}, {{val|0 5 12 -8}}]
[[Tp tuning|POL2 generator]]: ~96/77 = 380.768
{{Optimal ET sequence|legend=1| 19, 22, 41, 63, 104 }}
=== Balthazar ===
Subgroup: 2.3.7.11.13
[[Comma list]]: 169/168, 896/891, 26411/26244
[[Gencom]]: [2 143/128; 169/168, 896/891, 26411/26244]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 1}}, {{val|0 10 24 -16 17}}]
[[Tp tuning|POL2 generator]]: ~143/128 = 190.407
{{Optimal ET sequence|legend=1| 19, 44, 63, 145f }}
=== Caspar ===
Subgroup: 2.3.7.11.13
[[Comma list]]: 144/143, 343/338, 729/728
[[Gencom]]: [2 26/21; 144/143, 343/338, 729/728]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 -2}}, {{val|0 5 12 -8 18}}]
[[Tp tuning|POL2 generator]]: ~26/21 = 380.531
{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
=== Melchior ===
Subgroup: 2.3.7.11.13
[[Comma list]]: 352/351, 364/363, 26411/26244
[[Gencom]]: [2 96/77; 352/351, 364/363, 26411/26244]
[[Mapping|Sval mapping]]: [{{val|1 0 -1 6 11}}, {{val|0 5 12 -8 -23}}]
[[Tp tuning|POL2 generator]]: ~96/77 = 380.766
{{Optimal ET sequence|legend=1| 19f, 22, 41, 63, 104 }}
== Hogwarts ==
Subgroup: 2.3.7.29
[[Comma list]]: 537824/531441, 784/783
[[Gencom]]: [2 36/29; 537824/531441, 784/783]
{{Optimal ET sequence|legend=1| 16dj, 19, 22, 41, 145, 186j, 227j }}
=== Twenothology ===
Subgroup: 2.3.7.11.13.29
[[Comma list]]: 537824/531441, 896/891, 144/143, 784/783
{{Optimal ET sequence|legend=1| 19, 22f, 41 }}
== Hectosaros leap week ==
Defined as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.
[[Subgroup]]: 2.3.7
[[Comma list]]: {{monzo|-50 -746 439}}
[[Mapping]]: [{{val|1 313 532}}, {{val|0 -439 -746}}]
[[Optimal tuning]] ([[CTE]]): ~{{monzo|17 343 143}} = 851.248
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
=== 2.3.7.13 subgroup ===
Subgroup: 2.3.7.13
Comma list: {{monzo|-42  -2 -5 16}}, {{monzo|10 -46 29 -5}}
Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
{{Optimal ET sequence|legend=1| 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689 }}
=== 2.3.7.13.17 subgroup ===
Subgroup: 2.3.7.13.17
Comma list: 39337984/39328497, {{monzo|0 -14  7  4 -3}}, {{monzo|-18 -24 14 -1  5}}
Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
Optimal tuning (CTE): ~6144/3757 = 851.248
{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}
=== 2.3.7.13.17.19 subgroup ===
Subgroup: 2.3.7.13.17.19
Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256
Mapping: [{{val|1 313 532 208 58 432}}, {{val|0 -439 -746 -288 -76 -603}}]
Optimal tuning (CTE): ~6144/3757 = 851.248
{{Optimal ET sequence|legend=1| 320, 1483bd, 1803, 2123 }}


== Sematology ==
== Sematology ==
Line 730: Line 831:


=== 2.3.7.37 subgroup ===
=== 2.3.7.37 subgroup ===
[[Subgroup]]: 2.3.7.37
[[Subgroup]]: 2.3.7.37


Line 758: Line 858:
{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}
{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}


=== 2.3.5.7.37 subgroup ===
==== 2.3.5.7.37 subgroup ====
 
[[Subgroup]]: 2.3.5.7.37
[[Subgroup]]: 2.3.5.7.37


Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"