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This 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).
 
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.  


== Fifive ==
== Fifive ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 9765625/9565938
 
{{Mapping|legend=1| 2 2 3 | 0 5 7 }}


[[Comma]]: 9765625/9565938
: mapping generators: ~78125/50421, ~27/25


[[Mapping]]: [{{val|2 2 3}}, {{val|0 5 7}}]
[[Optimal tuning]]s:
* [[CTE]]: ~78125/50421 = 1\2, ~27/25 = 140.6349
* [[POTE]]: ~78125/50421 = 1\2, ~27/25 = 140.624


[[POTE generator]]: ~27/25 = 140.624
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}


{{Val list|legend=1| 8, 26, 34, 94, 128 }}
[[Badness]]:
* Smith: 0.205812
* Dirichlet: 4.828


[[Badness]]: 0.205812
=== 2.3.5.13 subgroup ===
Subgroup: 2.3.5.13
 
Comma list: 325/324, 20000/19773
 
Mapping: {{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: {{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: {{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: {{Optimal ET sequence| 8, 18bcfg, 26, 34, 94, 128 }}
 
Badness:
* Smith: 0.0110
* Dirichlet: 0.488


== Crepuscular ==
== Crepuscular ==
{{see also|Jubilismic clan #Crepuscular}}
{{See also| Jubilismic clan #Crepuscular }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 4375/4374
[[Comma list]]: 50/49, 4375/4374


[[Mapping]]: [{{val|2 2 3 4}}, {{val|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


[[POTE generator]]: ~27/25 = 140.349
{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}
 
{{Val list|legend=1| 8d, 26, 34d, 60d }}


[[Badness]]: 0.086669
[[Badness]]: 0.086669
Line 36: Line 83:
Comma list: 50/49, 99/98, 864/847
Comma list: 50/49, 99/98, 864/847


Mapping: [{{val|2 2 3 4 6}}, {{val|0 5 7 7 4}}]
Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}


POTE generator: ~12/11 = 140.587
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.587


Optimal GPV sequence: {{Val list| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}


Badness: 0.040758
Badness: 0.040758
Line 49: Line 96:
Comma list: 50/49, 78/77, 99/98, 144/143
Comma list: 50/49, 78/77, 99/98, 144/143


Mapping: [{{val|2 2 3 4 6 6}}, {{val|0 5 7 7 4 6}}]
Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}


POTE generator: ~12/11 = 140.554
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.554


Optimal GPV sequence: {{Val list| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}


Badness: 0.024368
Badness: 0.024368
Line 62: Line 109:
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143


Mapping: [{{val|2 2 3 4 6 6 7}}, {{val|0 5 7 7 4 6 5}}]
Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}


POTE generator: ~12/11 = 140.405
Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.405


Optimal GPV sequence: {{Val list| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}


Badness: 0.018567
Badness: 0.018567


== Fifives ==
== Fifives ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 83349/81920
[[Comma list]]: 875/864, 83349/81920


[[Mapping]]: [{{val|2 2 3 7}}, {{val|0 5 7 -6}}]
{{Mapping|legend=1| 2 2 3 7 | 0 5 7 -6 }}


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


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


[[Badness]]: 0.130589
[[Badness]]: 0.130589
Line 88: Line 135:
Comma list: 100/99, 385/384, 3969/3872
Comma list: 100/99, 385/384, 3969/3872


Mapping: [{{val|2 2 3 7 6}}, {{val|0 5 7 -6 4}}]
Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}


POTE generator: ~12/11 = 139.884
Optimal tuning (POTE): ~63/44 = 1\2, ~12/11 = 139.884


Optimal GPV sequence: {{Val list| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}


Badness: 0.080306
Badness: 0.080306
Line 101: Line 148:
Comma list: 100/99, 105/104, 144/143, 1352/1331
Comma list: 100/99, 105/104, 144/143, 1352/1331


Mapping: [{{val|2 2 3 7 6 6}}, {{val|0 5 7 -6 4 6}}]
Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}


POTE generator: ~12/11 = 139.867
Optimal tuning (POTE): ~55/39 = 1\2, ~12/11 = 139.867


Optimal GPV sequence: {{Val list| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}


Badness: 0.044253
Badness: 0.044253
Line 114: Line 161:
Comma list: 100/99, 105/104, 144/143, 170/169, 221/220
Comma list: 100/99, 105/104, 144/143, 170/169, 221/220


Mapping: [{{val|2 2 3 7 6 6 7}}, {{val|0 5 7 -6 4 6 5}}]
Mapping: {{mapping| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}


POTE generator: ~12/11 = 139.868
Optimal tuning (POTE): ~17/12 = 1\2, ~12/11 = 139.868


Optimal GPV sequence: {{Val list| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}


Badness: 0.029429
Badness: 0.029429


== Fourfives ==
== Fourfives ==
Subgroup: 2.3.5.7
{{See also| Sensamagic clan }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 235298/234375
[[Comma list]]: 245/243, 235298/234375


[[Mapping]]: [{{val|4 4 6 7}}, {{val|0 5 7 9}}]
{{Mapping|legend=1| 4 4 6 7 | 0 5 7 9 }}


[[POTE generator]]: ~27/25 = 140.754
: mapping generators: ~25/21, ~27/25


{{Val list|legend=1| 8d, 60, 68, 128 }}
[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~27/25 = 140.754
 
{{Optimal ET sequence|legend=1| 8d, 60, 68, 128 }}


[[Badness]]: 0.114143
[[Badness]]: 0.114143
Line 140: Line 191:
Comma list: 245/243, 385/384, 235298/234375
Comma list: 245/243, 385/384, 235298/234375


Mapping: [{{val|4 4 6 7 19}}, {{val|0 5 7 9 -11}}]
Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}


POTE generator: ~27/25 = 140.771
Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.771


Optimal GPV sequence: {{Val list| 8de, 60, 68, 128, 196 }}
{{Optimal ET sequence|legend=1| 8de, 60, 68, 128, 196 }}


Badness: 0.120165
Badness: 0.120165
Line 153: Line 204:
Comma list: 196/195, 245/243, 385/384, 20000/19773
Comma list: 196/195, 245/243, 385/384, 20000/19773


Mapping: [{{val|4 4 6 7 19 12}}, {{val|0 5 7 9 -11 6}}]
Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}


POTE generator: ~13/12 = 140.760
Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.760


Optimal GPV sequence: {{Val list| 8de, 60, 68, 128, 196f }}
{{Optimal ET sequence|legend=1| 8de, 60, 68, 128, 196f }}


Badness: 0.067365
Badness: 0.067365
Line 166: Line 217:
Comma list: 121/120, 245/243, 1375/1372
Comma list: 121/120, 245/243, 1375/1372


Mapping: [{{val|4 4 6 7 11}}, {{val|0 5 7 9 6}}]
Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}


POTE generator: ~27/25 = 140.630
Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.630


Optimal GPV sequence: {{Val list| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=1| 8d, 60e, 68, 128e }}


Badness: 0.057268
Badness: 0.057268
Line 179: Line 230:
Comma list: 121/120, 196/195, 245/243, 275/273
Comma list: 121/120, 196/195, 245/243, 275/273


Mapping: [{{val|4 4 6 7 11 12}}, {{val|0 5 7 9 6 6}}]
Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}


POTE generator: ~13/12 = 140.728
Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.728


Optimal GPV sequence: {{Val list| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=1| 8d, 60e, 68, 128e }}


Badness: 0.036128
Badness: 0.036128
Line 192: Line 243:
Comma list: 121/120, 154/153, 170/169, 196/195, 245/243
Comma list: 121/120, 154/153, 170/169, 196/195, 245/243


Mapping: [{{val|4 4 6 7 11 12 14}}, {{val|0 5 7 9 6 6 5}}]
Mapping: {{mapping| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}


POTE generator: ~13/12 = 140.718
Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.718


Optimal GPV sequence: {{Val list| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=1| 8d, 60e, 68, 128e }}


Badness: 0.024796
Badness: 0.024796
== Notes ==


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

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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