Fifive family
- 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 fifive family of temperaments tempers out the fifive comma (monzo: [-1 -14 10⟩, ratio: 9765625/9565938).
The name fifive was given by Petr Pařízek in 2011 for it splits the perfect fifth in five.[1]
Considered below are crepuscular, fifives, and fourfives.
Fifive
Subgroup: 2.3.5
Comma list: 9765625/9565938
Mapping: [⟨2 2 3], ⟨0 5 7]]
- mapping generators: ~78125/50421, ~27/25
Optimal ET sequence: 8, 18bc, 26, 34, 94, 128
- Smith: 0.205812
- Dirichlet: 4.828
2.3.5.13 subgroup
Subgroup: 2.3.5.13
Comma list: 325/324, 20000/19773
Mapping: [⟨2 2 3 6], ⟨0 5 7 6]]
- mapping generators: ~351/250, ~13/12
Optimal tunings:
- CTE: ~351/250 = 1\2, ~13/12 = 140.5685
- CWE: ~351/250 = 1\2, ~13/12 = 140.6232
Optimal ET sequence: 8, 18bcf, 26, 34, 94, 128
Badness:
- Smith: 0.0240
- Dirichlet: 0.800
2.3.5.13.17 subgroup
Subgroup: 2.3.5.13.17
Comma list: 170/169, 289/288, 325/324
Mapping: [⟨2 2 3 6 7], ⟨0 5 7 6 5]]
- mapping generators: ~17/12, ~13/12
Optimal tunings:
- CTE: ~17/12 = 1\2, ~13/12 = 140.5958
- CWE: ~17/12 = 1\2, ~13/12 = 140.6057
Optimal ET sequence: 8, 18bcfg, 26, 34, 94, 128
Badness:
- Smith: 0.0110
- Dirichlet: 0.488
Crepuscular
Subgroup: 2.3.5.7
Comma list: 50/49, 4375/4374
Mapping: [⟨2 2 3 4], ⟨0 5 7 7]]
Optimal tuning (POTE): ~7/5 = 1\2, ~27/25 = 140.349
Optimal ET sequence: 8d, 26, 34d, 60d
Badness: 0.086669
11-limit
Subgroup: 2.3.5.7.11
Comma list: 50/49, 99/98, 864/847
Mapping: [⟨2 2 3 4 6], ⟨0 5 7 7 4]]
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.587
Optimal ET sequence: 8d, 26, 34d, 60d
Badness: 0.040758
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 78/77, 99/98, 144/143
Mapping: [⟨2 2 3 4 6 6], ⟨0 5 7 7 4 6]]
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.554
Optimal ET sequence: 8d, 26, 34d, 60d
Badness: 0.024368
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143
Mapping: [⟨2 2 3 4 6 6 7], ⟨0 5 7 7 4 6 5]]
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.405
Optimal ET sequence: 8d, 26, 34d, 60d
Badness: 0.018567
Fifives
Subgroup: 2.3.5.7
Comma list: 875/864, 83349/81920
Mapping: [⟨2 2 3 7], ⟨0 5 7 -6]]
Optimal tuning (POTE): ~567/400 = 1\2, ~27/25 = 139.909
Optimal ET sequence: 8, 26, 34, 60
Badness: 0.130589
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 385/384, 3969/3872
Mapping: [⟨2 2 3 7 6], ⟨0 5 7 -6 4]]
Optimal tuning (POTE): ~63/44 = 1\2, ~12/11 = 139.884
Optimal ET sequence: 8, 26, 34, 60
Badness: 0.080306
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 144/143, 1352/1331
Mapping: [⟨2 2 3 7 6 6], ⟨0 5 7 -6 4 6]]
Optimal tuning (POTE): ~55/39 = 1\2, ~12/11 = 139.867
Optimal ET sequence: 8, 26, 34, 60
Badness: 0.044253
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 144/143, 170/169, 221/220
Mapping: [⟨2 2 3 7 6 6 7], ⟨0 5 7 -6 4 6 5]]
Optimal tuning (POTE): ~17/12 = 1\2, ~12/11 = 139.868
Optimal ET sequence: 8, 26, 34, 60
Badness: 0.029429
Fourfives
Subgroup: 2.3.5.7
Comma list: 245/243, 235298/234375
Mapping: [⟨4 4 6 7], ⟨0 5 7 9]]
- mapping generators: ~25/21, ~27/25
Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.754
Optimal ET sequence: 8d, 60, 68, 128
Badness: 0.114143
11-limit
Subgroup: 2.3.5.7.11
Comma list: 245/243, 385/384, 235298/234375
Mapping: [⟨4 4 6 7 19], ⟨0 5 7 9 -11]]
Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.771
Optimal ET sequence: 8de, 60, 68, 128, 196
Badness: 0.120165
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 245/243, 385/384, 20000/19773
Mapping: [⟨4 4 6 7 19 12], ⟨0 5 7 9 -11 6]]
Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.760
Optimal ET sequence: 8de, 60, 68, 128, 196f
Badness: 0.067365
Quadrafives
Subgroup: 2.3.5.7.11
Comma list: 121/120, 245/243, 1375/1372
Mapping: [⟨4 4 6 7 11], ⟨0 5 7 9 6]]
Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.630
Optimal ET sequence: 8d, 60e, 68, 128e
Badness: 0.057268
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 196/195, 245/243, 275/273
Mapping: [⟨4 4 6 7 11 12], ⟨0 5 7 9 6 6]]
Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.728
Optimal ET sequence: 8d, 60e, 68, 128e
Badness: 0.036128
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 121/120, 154/153, 170/169, 196/195, 245/243
Mapping: [⟨4 4 6 7 11 12 14], ⟨0 5 7 9 6 6 5]]
Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.718
Optimal ET sequence: 8d, 60e, 68, 128e
Badness: 0.024796