Fifive family
The fifive family tempers out the fifive comma, 9765625/9565938 = [-1 -14 10⟩.
Temperaments discussed elsewhere include:
- Fourfives (+245/243) → Sensamagic clan
Considered below are crepuscular and fifives.
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
- See also: Jubilismic clan #Crepuscular
Subgroup: 2.3.5.7
Comma list: 50/49, 4375/4374
Mapping: [⟨2 2 3 4], ⟨0 5 7 7]]
Wedgie: ⟨⟨10 14 14 -1 -6 -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