Fifive family: Difference between revisions

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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.6349{{c}}
* [[WE]]: ~78125/50421 = 600.0168{{c}}, ~27/25 = 140.6276{{c}}
* [[POTE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.624{{c}}
: [[error map]]: {{val| +0.034 +1.217 -1.870 }}
* [[CWE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.6291{{c}}
: error map: {{val| 0.000 +1.191 -1.910 }}


{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}
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Mapping: {{mapping| 2 2 3 6 | 0 5 7 6 }}
Mapping: {{mapping| 2 2 3 6 | 0 5 7 6 }}
: mapping generators: ~351/250, ~13/12


Optimal tunings:
Optimal tunings:
* CTE: ~351/250 = 600.0000{{c}}, ~13/12 = 140.5685{{c}}
* WE: ~351/250 = 599.8593{{c}}, ~13/12 = 140.6362{{c}}
* CWE: ~351/250 = 600.0000{{c}}, ~13/12 = 140.6232{{c}}
* CWE: ~351/250 = 600.0000{{c}}, ~13/12 = 140.6232{{c}}


Line 44: Line 45:


Mapping: {{mapping| 2 2 3 6 7 | 0 5 7 6 5 }}
Mapping: {{mapping| 2 2 3 6 7 | 0 5 7 6 5 }}
: mapping generators: ~17/12, ~13/12


Optimal tunings:
Optimal tunings:
* CTE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.5958{{c}}
* CTE: ~17/12 = 599.9773{{c}}, ~13/12 = 140.6075{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.6057{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.6057{{c}}


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{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 600.000{{c}}, ~27/25 = 140.349{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.000{{c}}, ~27/25 = 140.349{{c}}
: [[error map]]: {{val| -0.787 -1.458 -5.696 +11.398 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~27/25 = 140.349{{c}}
: error map: {{val| 0.000 -0.876 -4.803 +12.685 }}


{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=1| 8d, 18bcd, 26, 34d, 60d }}


[[Badness]] (Smith): 0.086669
[[Badness]] (Sintel): 2.19


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}
Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}


Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.587{{c}}
Optimal tunings:
* WE: ~7/5 = 599.1577{{c}}, ~12/11 = 140.3893{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~12/11 = 140.3244{{c}}


{{Optimal ET sequence|legend=0| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcd, 26, 34d, 60d }}


Badness (Smith): 0.040758
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}
Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}


Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.554{{c}}
Optimal tunings:
* WE: ~7/5 = 599.2598{{c}}, ~13/12 = 140.3805{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.3237{{c}}


{{Optimal ET sequence|legend=0| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcdf, 26, 34d, 60d }}


Badness (Smith): 0.024368
Badness (Sintel): 1.01


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}
Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}


Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.405{{c}}
Optimal tunings:
* WE: ~7/5 = 599.5348{{c}}, ~13/12 = 140.2961{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.2668{{c}}


{{Optimal ET sequence|legend=0| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcdfg, 26, 34d, 60d }}


Badness (Smith): 0.018567
Badness (Sintel): 0.946


== Fifives ==
== Fifives ==
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{{Mapping|legend=1| 2 2 3 7 | 0 5 7 -6 }}
{{Mapping|legend=1| 2 2 3 7 | 0 5 7 -6 }}


[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~27/25 = 139.909{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.9312{{c}}, ~27/25 = 140.1261{{c}}
: [[error map]]: {{val| +1.862 +0.538 -2.637 -3.064 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~27/25 = 139.9826{{c}}
: error map: {{val| 0.000 -2.042 -6.435 -8.722 }}


{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=1| 8, 26, 34, 60, 266bcccddd }}


[[Badness]] (Smith): 0.130589
[[Badness]] (Sintel): 3.30


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}
Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}


Optimal tuning (POTE): ~63/44 = 600.000{{c}}, ~12/11 = 139.884{{c}}
Optimal tunings:
* WE: ~63/44 = 600.5091{{c}}, ~12/11 = 140.0024{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~12/11 = 139.9267{{c}}


{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness (Smith): 0.080306
Badness (Sintel): 2.65


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}
Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}


Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~12/11 = 139.867{{c}}
Optimal tunings:
* WE: ~55/39 = 600.4600{{c}}, ~12/11 = 139.9737{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 139.9092{{c}}


{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness (Smith): 0.044253
Badness (Sintel): 1.83


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}
Mapping: {{mapping| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}
Optimal tunings:
* WE: ~17/12 = 600.4903{{c}}, ~12/11 = 139.9825{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~12/11 = 139.9150{{c}}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~12/11 = 139.868{{c}}
Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~12/11 = 139.868{{c}}
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{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness (Smith): 0.029429
Badness (Sintel): 1.50


== Fourfives ==
== Fourfives ==
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: mapping generators: ~25/21, ~27/25
: mapping generators: ~25/21, ~27/25


[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.000{{c}}, ~27/25 = 140.754{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 300.0011{{c}}, ~27/25 = 140.7547{{c}}
: [[error map]]: {{val| +0.004 +1.823 -1.024 -2.026 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~27/25 = 140.7549{{c}}
: error map: {{val| 0.000 +1.819 -1.030 -2.032 }}


{{Optimal ET sequence|legend=1| 8d, 60, 68, 128 }}
{{Optimal ET sequence|legend=1| 8d, …, 60, 68, 128, 196 }}


[[Badness]] (Smith): 0.114143
[[Badness]] (Sintel): 2.89


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}
Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~27/25 = 140.771{{c}}
Optimal tunings:
* WE: ~25/21 = 299.9901{{c}}, ~27/25 = 140.7659{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/25 = 140.7693{{c}}


{{Optimal ET sequence|legend=0| 8de, 60, 68, 128, 196 }}
{{Optimal ET sequence|legend=0| 60, 68, 128, 196 }}


Badness (Smith): 0.120165
Badness (Sintel): 3.97


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}
Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.760{{c}}
Optimal tunings:
* WE: ~25/21 = 299.9488{{c}}, ~13/12 = 140.7359{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7539{{c}}


{{Optimal ET sequence|legend=0| 8de, 60, 68, 128, 196f }}
{{Optimal ET sequence|legend=0| 60, 68, 128, 196f }}


Badness (Smith): 0.067365
Badness (Sintel): 2.78


=== Quadrafives ===
=== Quadrafives ===
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Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}
Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~27/25 = 140.630{{c}}
Optimal tunings:
* WE: ~25/21 = 300.1673{{c}}, ~27/25 = 140.7084{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/25 = 140.7353{{c}}


{{Optimal ET sequence|legend=0| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68, 128e }}


Badness (Smith): 0.057268
Badness (Sintel): 1.89


==== 13-limit ====
==== 13-limit ====
Line 220: Line 252:
Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}
Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.728{{c}}
Optimal tunings:
* WE: ~25/21 = 300.0500{{c}}, ~13/12 = 140.7516{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7590{{c}}


{{Optimal ET sequence|legend=0| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68 }}


Badness (Smith): 0.036128
Badness (Sintel): 1.49


==== 17-limit ====
==== 17-limit ====
Line 232: Line 266:


Mapping: {{mapping| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}
Mapping: {{mapping| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}
Optimal tunings:
* WE: ~25/21 = 300.0593{{c}}, ~13/12 = 140.7457{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7520{{c}}


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


{{Optimal ET sequence|legend=0| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68 }}


Badness (Smith): 0.024796
Badness (Sintel): 1.26


== References ==
== References ==

Revision as of 03:48, 25 August 2026

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 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

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]]

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:

  • CTE: ~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

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

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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