Fifive family: Difference between revisions

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The '''fifive family''' tempers out the [[fifive comma]], 9765625/9565938 = {{monzo| -1 -14 10 }}.
{{Technical data page}}
The '''fifive family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[fifive comma]] ({{monzo|legend=1| -1 -14 10 }}, [[ratio]]: 9765625/9565938).


Temperaments discussed elsewhere include:
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>
* ''[[Fourfives]]'' (+245/243) → [[Sensamagic clan #Fourfives|Sensamagic clan]]


Considered below are crepuscular and fifives.  
Considered below are crepuscular, fifives, and fourfives.  


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


[[Optimal tuning]] ([[CWE]]): ~78125/50421 = 1\2, ~27/25 = 140.623
[[Optimal tuning]]s:
* [[CTE]]: ~78125/50421 = 1\2, ~27/25 = 140.6349
* [[POTE]]: ~78125/50421 = 1\2, ~27/25 = 140.624


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


[[Badness]]: 0.205812
[[Badness]]:
 
* Smith: 0.205812
Dirichlet badness: 4.828
* Dirichlet: 4.828


=== 2.3.5.13 subgroup ===
=== 2.3.5.13 subgroup ===
Subgroup: 2.3.5.13


[[Subgroup]]: 2.3.5.13
Comma list: 325/324, 20000/19773
 
[[Comma list]]: 9765625/9565938, 325/324


{{Mapping|legend=1| 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 tuning]] ([[CWE]]): ~351/250 = 1\2, ~13/12 = 140.623
Optimal tunings:
* CTE: ~351/250 = 1\2, ~13/12 = 140.5685
* CWE: ~351/250 = 1\2, ~13/12 = 140.6232


[[Patent val]] EDO tunings: 8, 26, 34, 60, 68, 94, 128
Optimal ET sequence: {{Optimal ET sequence| 8, 18bcf, 26, 34, 94, 128 }}


Dirichlet badness: 0.800
Badness:
* Smith: 0.0240
* Dirichlet: 0.800


=== 2.3.5.13.17 subgroup ===
=== 2.3.5.13.17 subgroup ===
Subgroup: 2.3.5.13.17


[[Subgroup]]: 2.3.5.13.17
Comma list: 170/169, 289/288, 325/324
 
[[Comma list]]: 9765625/9565938, 325/324, 289/288


{{Mapping|legend=1| 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 tuning]] ([[CWE]]): ~17/12 = 1\2, ~13/12 = 140.606
Optimal tunings:
* CTE: ~17/12 = 1\2, ~13/12 = 140.5958
* CWE: ~17/12 = 1\2, ~13/12 = 140.6057


[[Patent val]] EDO tunings: 8, 26, 34, 60, 68, 94, 128
Optimal ET sequence: {{Optimal ET sequence| 8, 18bcfg, 26, 34, 94, 128 }}


Dirichlet badness: 0.488  
Badness:
* Smith: 0.0110
* Dirichlet: 0.488


== Crepuscular ==
== Crepuscular ==
Line 63: Line 71:


{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}
{{Multival|legend=1| 10 14 14 -1 -6 -7 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~27/25 = 140.349
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~27/25 = 140.349
Line 162: Line 168:


Badness: 0.029429
Badness: 0.029429
== Fourfives ==
{{See also| Sensamagic clan }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 245/243, 235298/234375
{{Mapping|legend=1| 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|legend=1| 8d, 60, 68, 128 }}
[[Badness]]: 0.114143
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 245/243, 385/384, 235298/234375
Mapping: {{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|legend=1| 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: {{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|legend=1| 8de, 60, 68, 128, 196f }}
Badness: 0.067365
=== Quadrafives ===
Subgroup: 2.3.5.7.11
Comma list: 121/120, 245/243, 1375/1372
Mapping: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 8d, 60e, 68, 128e }}
Badness: 0.024796
== Notes ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Fifive family| ]] <!-- main article -->
[[Category:Fifive family| ]] <!-- main article -->
[[Category:Fifive| ]] <!-- key article -->
[[Category:Fifive| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:27, 24 June 2025

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 = 1\2, ~27/25 = 140.6349
  • POTE: ~78125/50421 = 1\2, ~27/25 = 140.624

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

Badness:

  • 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]]

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

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 = 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

Notes

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