Xenharmonic Wiki

Fifive family

Revision as of 04:05, 25 August 2026 by FloraC (talk | contribs) (Re-organize)
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]

Fifive

Subgroup: 2.3.5

Comma list: 9765625/9565938

Mapping: [⟨2 2 3], ⟨0 5 7]]

mapping generators: ~78125/50421, ~27/25

Optimal tunings:

  • WE: ~78125/50421 = 600.0168 ¢, ~27/25 = 140.6276 ¢
error map: ⟨+0.034 +1.217 -1.870]
  • CWE: ~78125/50421 = 600.0000 ¢, ~27/25 = 140.6291 ¢
error map: ⟨0.000 +1.191 -1.910]

Optimal ET sequence: 8, 18bc, 26, 34, 94, 128

Badness (Sintel): 4.83

Overview to extensions

The second comma in the comma list defines which 7-limit family member we are looking at. Crepuscular (26 & 34d) adds 50/49. Fifives (26 & 34) adds 875/864. Both use the same generators as fifive. The weak extension fourfives (60 & 68) adds 245/243. All are considered below.

The fifive family boasts a very remarkable extension to the 2.3.5.13 subgroup, which has further extensions with higher primes. These are listed at the bottom of this page, in #Subgroup extensions.

Crepuscular

Subgroup: 2.3.5.7

Comma list: 50/49, 4375/4374

Mapping: [⟨2 2 3 4], ⟨0 5 7 7]]

Optimal tunings:

  • WE: ~7/5 = 600.000 ¢, ~27/25 = 140.349 ¢
error map: ⟨-0.787 -1.458 -5.696 +11.398]
  • CWE: ~7/5 = 600.000 ¢, ~27/25 = 140.349 ¢
error map: ⟨0.000 -0.876 -4.803 +12.685]

Optimal ET sequence: 8d, 18bcd, 26, 34d, 60d

Badness (Sintel): 2.19

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 tunings:

  • WE: ~7/5 = 599.1577 ¢, ~12/11 = 140.3893 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~12/11 = 140.3244 ¢

Optimal ET sequence: 8d, 18bcd, 26, 34d, 60d

Badness (Sintel): 1.35

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 tunings:

  • WE: ~7/5 = 599.2598 ¢, ~13/12 = 140.3805 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~13/12 = 140.3237 ¢

Optimal ET sequence: 8d, 18bcdf, 26, 34d, 60d

Badness (Sintel): 1.01

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 tunings:

  • WE: ~7/5 = 599.5348 ¢, ~13/12 = 140.2961 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~13/12 = 140.2668 ¢

Optimal ET sequence: 8d, 18bcdfg, 26, 34d, 60d

Badness (Sintel): 0.946

Fifives

Subgroup: 2.3.5.7

Comma list: 875/864, 83349/81920

Mapping: [⟨2 2 3 7], ⟨0 5 7 -6]]

Optimal tunings:

  • WE: ~567/400 = 600.9312 ¢, ~27/25 = 140.1261 ¢
error map: ⟨+1.862 +0.538 -2.637 -3.064]
  • CWE: ~567/400 = 600.0000 ¢, ~27/25 = 139.9826 ¢
error map: ⟨0.000 -2.042 -6.435 -8.722]

Optimal ET sequence: 8, 26, 34, 60, 266bcccddd

Badness (Sintel): 3.30

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 tunings:

  • WE: ~63/44 = 600.5091 ¢, ~12/11 = 140.0024 ¢
  • CWE: ~63/44 = 600.0000 ¢, ~12/11 = 139.9267 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 2.65

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 tunings:

  • WE: ~55/39 = 600.4600 ¢, ~12/11 = 139.9737 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~12/11 = 139.9092 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 1.83

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 tunings:

  • WE: ~17/12 = 600.4903 ¢, ~12/11 = 139.9825 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~12/11 = 139.9150 ¢

Optimal tuning (POTE): ~17/12 = 600.000 ¢, ~12/11 = 139.868 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 1.50

Fourfives

See also: Sensamagic clan

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 tunings:

  • WE: ~25/21 = 300.0011 ¢, ~27/25 = 140.7547 ¢
error map: ⟨+0.004 +1.823 -1.024 -2.026]
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7549 ¢
error map: ⟨0.000 +1.819 -1.030 -2.032]

Optimal ET sequence: 8d, …, 60, 68, 128, 196

Badness (Sintel): 2.89

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 tunings:

  • WE: ~25/21 = 299.9901 ¢, ~27/25 = 140.7659 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7693 ¢

Optimal ET sequence: 60, 68, 128, 196

Badness (Sintel): 3.97

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 tunings:

  • WE: ~25/21 = 299.9488 ¢, ~13/12 = 140.7359 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7539 ¢

Optimal ET sequence: 60, 68, 128, 196f

Badness (Sintel): 2.78

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 tunings:

  • WE: ~25/21 = 300.1673 ¢, ~27/25 = 140.7084 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7353 ¢

Optimal ET sequence: 8d, …, 60e, 68, 128e

Badness (Sintel): 1.89

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 tunings:

  • WE: ~25/21 = 300.0500 ¢, ~13/12 = 140.7516 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7590 ¢

Optimal ET sequence: 8d, …, 60e, 68

Badness (Sintel): 1.49

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 tunings:

  • WE: ~25/21 = 300.0593 ¢, ~13/12 = 140.7457 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7520 ¢

Optimal tuning (POTE): ~25/21 = 300.000 ¢, ~13/12 = 140.718 ¢

Optimal ET sequence: 8d, …, 60e, 68

Badness (Sintel): 1.26

Subgroup extensions

Fifive (2.3.5.13)

Subgroup: 2.3.5.13

Comma list: 325/324, 20000/19773

Mapping: [⟨2 2 3 6], ⟨0 5 7 6]]

Optimal tunings:

  • WE: ~351/250 = 599.8593 ¢, ~13/12 = 140.6362 ¢
  • CWE: ~351/250 = 600.0000 ¢, ~13/12 = 140.6232 ¢

Optimal ET sequence: 8, 18bcf, 26, 34, 94, 128

Badness (Sintel): 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]]

Optimal tunings:

  • WE: ~17/12 = 599.9773 ¢, ~13/12 = 140.6075 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~13/12 = 140.6057 ¢

Optimal ET sequence: 8, 18bcfg, 26, 34, 94, 128

Badness (Sintel): 0.488

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
Retrieved from "https://en.xen.wiki/index.php?title=Fifive_family&oldid=236249"