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Fifive family: Difference between revisions

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


[[Comma]]: 9765625/9565938
Mapping: {{mapping| 2 2 3 6 | 0 5 7 6 }}


[[Mapping]]: [{{val|2 2 3}}, {{val|0 5 7}}]
: mapping generators: ~351/250, ~13/12


[[POTE generator]]: ~27/25 = 140.624
Optimal tunings:
* CTE: ~351/250 = 1\2, ~13/12 = 140.5685
* CWE: ~351/250 = 1\2, ~13/12 = 140.6232


{{Val list|legend=1| 8, 26, 34, 94, 128 }}
Optimal ET sequence: {{Optimal ET sequence| 8, 18bcf, 26, 34, 94, 128 }}


[[Badness]]: 0.205812
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
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143
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 ET sequence|legend=1| 8d, 26, 34d, 60d }}
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


=== 11-limit ===
=== 11-limit ===
Line 75: 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: ~27/25 = 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 88: 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: ~27/25 = 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
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 144/143, 170/169, 221/220
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 ET sequence|legend=1| 8, 26, 34, 60 }}
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| 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 114: 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


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


[[Category:Theory]]
=== 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| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Fifive]]
Retrieved from "https://en.xen.wiki/w/Fifive_family"