Fifive family: Difference between revisions

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{{Mapping|legend=1| 2 2 3 | 0 5 7 }}
{{Mapping|legend=1| 2 2 3 | 0 5 7 }}
: mapping generators: ~78125/50421, ~27/25
: mapping generators: ~78125/50421, ~27/25


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~78125/50421 = 1\2, ~27/25 = 140.6349
* [[CTE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.6349{{c}}
* [[POTE]]: ~78125/50421 = 1\2, ~27/25 = 140.624
* [[POTE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.624{{c}}


{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}


[[Badness]]:
[[Badness]] (Sintel): 4.83
* Smith: 0.205812
* Dirichlet: 4.828


=== 2.3.5.13 subgroup ===
=== 2.3.5.13 subgroup ===
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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
: mapping generators: ~351/250, ~13/12


Optimal tunings:
Optimal tunings:
* CTE: ~351/250 = 1\2, ~13/12 = 140.5685
* CTE: ~351/250 = 600.0000{{c}}, ~13/12 = 140.5685{{c}}
* CWE: ~351/250 = 1\2, ~13/12 = 140.6232
* CWE: ~351/250 = 600.0000{{c}}, ~13/12 = 140.6232{{c}}


Optimal ET sequence: {{Optimal ET sequence| 8, 18bcf, 26, 34, 94, 128 }}
{{Optimal ET sequence|legend=0| 8, 18bcf, 26, 34, 94, 128 }}


Badness:
Badness (Sintel): 0.800
* Smith: 0.0240
* Dirichlet: 0.800


=== 2.3.5.13.17 subgroup ===
=== 2.3.5.13.17 subgroup ===
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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
: mapping generators: ~17/12, ~13/12


Optimal tunings:
Optimal tunings:
* CTE: ~17/12 = 1\2, ~13/12 = 140.5958
* CTE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.5958{{c}}
* CWE: ~17/12 = 1\2, ~13/12 = 140.6057
* CWE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.6057{{c}}


Optimal ET sequence: {{Optimal ET sequence| 8, 18bcfg, 26, 34, 94, 128 }}
{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}


Badness:
Badness (Sintel): 0.488
* Smith: 0.0110
* Dirichlet: 0.488


== Crepuscular ==
== Crepuscular ==
{{See also| Jubilismic clan #Crepuscular }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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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 = 1\2, ~27/25 = 140.349
[[Optimal tuning]] ([[POTE]]): ~7/5 = 600.000{{c}}, ~27/25 = 140.349{{c}}


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


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


=== 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 = 1\2, ~12/11 = 140.587
Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.587{{c}}


{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 26, 34d, 60d }}


Badness: 0.040758
Badness (Smith): 0.040758


=== 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 = 1\2, ~12/11 = 140.554
Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.554{{c}}


{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 26, 34d, 60d }}


Badness: 0.024368
Badness (Smith): 0.024368


=== 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 = 1\2, ~12/11 = 140.405
Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.405{{c}}


{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 26, 34d, 60d }}


Badness: 0.018567
Badness (Smith): 0.018567


== 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 = 1\2, ~27/25 = 139.909
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~27/25 = 139.909{{c}}


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


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


=== 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 = 1\2, ~12/11 = 139.884
Optimal tuning (POTE): ~63/44 = 600.000{{c}}, ~12/11 = 139.884{{c}}


{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness: 0.080306
Badness (Smith): 0.080306


=== 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 = 1\2, ~12/11 = 139.867
Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~12/11 = 139.867{{c}}


{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness: 0.044253
Badness (Smith): 0.044253


=== 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 tuning (POTE): ~17/12 = 1\2, ~12/11 = 139.868
Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~12/11 = 139.868{{c}}


{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness: 0.029429
Badness (Smith): 0.029429


== Fourfives ==
== Fourfives ==
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{{Mapping|legend=1| 4 4 6 7 | 0 5 7 9 }}
{{Mapping|legend=1| 4 4 6 7 | 0 5 7 9 }}
: mapping generators: ~25/21, ~27/25
: mapping generators: ~25/21, ~27/25


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~27/25 = 140.754
[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.000{{c}}, ~27/25 = 140.754{{c}}


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


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


=== 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 = 1\4, ~27/25 = 140.771
Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~27/25 = 140.771{{c}}


{{Optimal ET sequence|legend=1| 8de, 60, 68, 128, 196 }}
{{Optimal ET sequence|legend=0| 8de, 60, 68, 128, 196 }}


Badness: 0.120165
Badness (Smith): 0.120165


==== 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 = 1\4, ~13/12 = 140.760
Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.760{{c}}


{{Optimal ET sequence|legend=1| 8de, 60, 68, 128, 196f }}
{{Optimal ET sequence|legend=0| 8de, 60, 68, 128, 196f }}


Badness: 0.067365
Badness (Smith): 0.067365


=== 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 = 1\4, ~27/25 = 140.630
Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~27/25 = 140.630{{c}}


{{Optimal ET sequence|legend=1| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, 60e, 68, 128e }}


Badness: 0.057268
Badness (Smith): 0.057268


==== 13-limit ====
==== 13-limit ====
Line 232: Line 220:
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 = 1\4, ~13/12 = 140.728
Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.728{{c}}


{{Optimal ET sequence|legend=1| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, 60e, 68, 128e }}


Badness: 0.036128
Badness (Smith): 0.036128


==== 17-limit ====
==== 17-limit ====
Line 245: Line 233:
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 tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.718
Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.718{{c}}


{{Optimal ET sequence|legend=1| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, 60e, 68, 128e }}


Badness: 0.024796
Badness (Smith): 0.024796


== References ==
== References ==

Revision as of 03:29, 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:

  • CTE: ~78125/50421 = 600.0000 ¢, ~27/25 = 140.6349 ¢
  • POTE: ~78125/50421 = 600.0000 ¢, ~27/25 = 140.624 ¢

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]]

mapping generators: ~351/250, ~13/12

Optimal tunings:

  • CTE: ~351/250 = 600.0000 ¢, ~13/12 = 140.5685 ¢
  • 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]]

mapping generators: ~17/12, ~13/12

Optimal tunings:

  • CTE: ~17/12 = 600.0000 ¢, ~13/12 = 140.5958 ¢
  • 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 tuning (POTE): ~7/5 = 600.000 ¢, ~27/25 = 140.349 ¢

Optimal ET sequence: 8d, 26, 34d, 60d

Badness (Smith): 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 = 600.000 ¢, ~12/11 = 140.587 ¢

Optimal ET sequence: 8d, 26, 34d, 60d

Badness (Smith): 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 = 600.000 ¢, ~12/11 = 140.554 ¢

Optimal ET sequence: 8d, 26, 34d, 60d

Badness (Smith): 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 = 600.000 ¢, ~12/11 = 140.405 ¢

Optimal ET sequence: 8d, 26, 34d, 60d

Badness (Smith): 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 = 600.000 ¢, ~27/25 = 139.909 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Smith): 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 = 600.000 ¢, ~12/11 = 139.884 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Smith): 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 = 600.000 ¢, ~12/11 = 139.867 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Smith): 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 = 600.000 ¢, ~12/11 = 139.868 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Smith): 0.029429

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 tuning (POTE): ~25/21 = 300.000 ¢, ~27/25 = 140.754 ¢

Optimal ET sequence: 8d, 60, 68, 128

Badness (Smith): 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 = 300.000 ¢, ~27/25 = 140.771 ¢

Optimal ET sequence: 8de, 60, 68, 128, 196

Badness (Smith): 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 = 300.000 ¢, ~13/12 = 140.760 ¢

Optimal ET sequence: 8de, 60, 68, 128, 196f

Badness (Smith): 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 = 300.000 ¢, ~27/25 = 140.630 ¢

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

Badness (Smith): 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 = 300.000 ¢, ~13/12 = 140.728 ¢

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

Badness (Smith): 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 = 300.000 ¢, ~13/12 = 140.718 ¢

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

Badness (Smith): 0.024796

References

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