Fifive family: Difference between revisions
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The '''fifive family''' tempers out the [[fifive comma]] | The '''fifive family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[fifive comma]] ({{monzo|legend=1| -1 -14 10 }}, [[ratio]]: 9765625/9565938). | ||
The name ''fifive'' was given by [[Petr Pařízek]] in 2011 for it splits the [[3/2|perfect fifth]] in five.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref> | |||
Considered below are crepuscular, fifives, and fourfives. | Considered below are crepuscular, fifives, and fourfives. | ||
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Badness: 0.024796 | Badness: 0.024796 | ||
== Notes == | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
Revision as of 11:56, 1 April 2025
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]]
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
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