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This tempers out the fifive comma, 9765625/9565938 = |-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).


=Fifive=
== Fifive ==
Comma: 9765625/9565938
The head of this family is fifive, which splits the [[3/2|perfect fifth]] into five [[27/25]]'s, and [[5/4]] is found as seven generators minus a half-octave period. Its [[ploidacot]] is diploid pentacot, and it is a member of the [[diaschismic–gothmic equivalence continuum]] with equivalence number ''n'' = 5/2.


POTE generator: ~27/25 = 140.624
The name ''fifive'' was given by [[Petr Pařízek]] in 2011 for it splits the perfect fifth in five.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>


Map: [&lt;2 2 3|, &lt;0 5 7|]
[[Subgroup]]: 2.3.5


EDOs: 8, 26, 34, 94, 128
[[Comma list]]: 9765625/9565938


Badness: 0.2058
{{Mapping|legend=1| 2 2 3 | 0 5 7 }}
: mapping generators: ~78125/50421, ~27/25


=Crepuscular=
[[Optimal tuning]]s:
Commas: 50/49, 4375/4374
* [[WE]]: ~78125/50421 = 600.0168{{c}}, ~27/25 = 140.6276{{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 }}


POTE generator: ~27/25 = 140.349
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}


Map: [&lt;2 2 3 4|, &lt;0 5 7 7|]
[[Badness]] (Sintel): 4.83


Wedgie: &lt;&lt;10 14 14 -1 -6 -7||
=== Overview to extensions ===
The second comma in the comma list defines which 7-limit family member we are looking at. Crepuscular (26 & 34d) adds [[50/49]]. Fifives (26 & 34) adds [[875/864]]. Both use the same generators as fifive. The weak extension fourfives (60 & 68) adds [[245/243]]. All are considered below.


EDOs: 8d, 26, 34d, 60d
The fifive family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].


Badness: 0.0867
== Crepuscular ==
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 50/49, 4375/4374
Commas: 50/49, 99/98, 864/847


POTE generator: ~12/11 = 140.587
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}


Map: [&lt;2 2 3 4 6|, &lt;0 5 7 7 4|]
[[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 }}


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


Badness: 0.0408
[[Badness]] (Sintel): 2.19


==13-limit==
=== 11-limit ===
Commas: 50/49, 78/77, 99/98, 144/143
Subgroup: 2.3.5.7.11


POTE generator: ~12/11 = 140.554
Comma list: 50/49, 99/98, 864/847


Map: [&lt;2 2 3 4 6 6|, &lt;0 5 7 7 4 6|]
{{Mapping|legend=0| 2 2 3 4 6 | 0 5 7 7 4 }}


EDOs: 8d, 26, 34d, 60d
Optimal tunings:  
* WE: ~7/5 = 599.1577{{c}}, ~12/11 = 140.3893{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~12/11 = 140.3244{{c}}


Badness: 0.0244
{{Optimal ET sequence|legend=0| 8d, 18bcd, 26, 34d, 60d }}


=Fifives=
Badness (Sintel): 1.35
Commas: 875/864, 83349/81920


POTE generator: ~27/25 = 139.909
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 2 3 7|, &lt;0 5 7 -6|]
Comma list: 50/49, 78/77, 99/98, 144/143


EDOs: 8, 26, 34, 60, 94d
{{Mapping|legend=0| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}


Badness: 0.1306
Optimal tunings:  
* WE: ~7/5 = 599.2598{{c}}, ~13/12 = 140.3805{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.3237{{c}}


=Fourfives=
{{Optimal ET sequence|legend=0| 8d, 18bcdf, 26, 34d, 60d }}
Commas: 245/243, 235298/234375


POTE generator: ~27/25 = 140.754
Badness (Sintel): 1.01


Map: [&lt;4 4 6 7|, &lt;0 5 7 9|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 8d, 60, 68, 128
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143


Badness: 0.1141
{{Mapping|legend=0| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}


==11-limit==
Optimal tunings:
Commas: 245/243, 385/384, 235298/234375
* WE: ~7/5 = 599.5348{{c}}, ~13/12 = 140.2961{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.2668{{c}}


POTE generator: ~27/25 = 140.771
{{Optimal ET sequence|legend=0| 8d, 18bcdfg, 26, 34d, 60d }}


Map: [&lt;4 4 6 7 19|, &lt;0 5 7 9 -11|]
Badness (Sintel): 0.946


EDOs: 8de, 60, 68, 128, 196
== Fifives ==
[[Subgroup]]: 2.3.5.7


Badness: 0.1202
[[Comma list]]: 875/864, 83349/81920


==13-limit==
{{Mapping|legend=1| 2 2 3 7 | 0 5 7 -6 }}
Commas: 196/195, 245/243, 385/384, 20000/19773


POTE generator: ~13/12 = 140.760
[[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 }}


Map: [&lt;4 4 6 7 19 12|, &lt;0 5 7 9 -11 6|]
{{Optimal ET sequence|legend=1| 8, 26, 34, 60, 266bcccddd }}


EDOs: 8de, 60, 68, 128, 196f
[[Badness]] (Sintel): 3.30


Badness: 0.0674
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 3969/3872
 
{{Mapping|legend=0| 2 2 3 7 6 | 0 5 7 -6 4 }}
 
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 }}
 
Badness (Sintel): 2.65
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 144/143, 1352/1331
 
{{Mapping|legend=0| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}
 
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 }}
 
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|legend=0| 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 ET sequence|legend=0| 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|legend=1| 4 4 6 7 | 0 5 7 9 }}
: mapping generators: ~25/21, ~27/25
 
[[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, 196 }}
 
[[Badness]] (Sintel): 2.89
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 235298/234375
 
{{Mapping|legend=0| 4 4 6 7 19 | 0 5 7 9 -11 }}
 
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| 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|legend=0| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}
 
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| 60, 68, 128, 196f }}
 
Badness (Sintel): 2.78
 
=== Quadrafives ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 245/243, 1375/1372
 
{{Mapping|legend=0| 4 4 6 7 11 | 0 5 7 9 6 }}
 
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 }}
 
Badness (Sintel): 1.89
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 196/195, 245/243, 275/273
 
{{Mapping|legend=0| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}
 
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 }}
 
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|legend=0| 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 ET sequence|legend=0| 8d, …, 60e, 68 }}
 
Badness (Sintel): 1.26
 
== Subgroup extensions ==
=== Fifive (2.3.5.13) ===
As the [[~]][[27/25]] generator of fifive is so close to [[13/12]], one may temper out their difference, [[325/324]], to obtain this extension in the 2.3.5.13 subgroup. It is also not unreasonable to equate both [[17/12]] and [[24/17]] with the semi-octave period given its overall level of precision, tempering out [[289/288]] and leading to a 2.3.5.13.17-subgroup temperament.
 
Subgroup: 2.3.5.13
 
Comma list: 325/324, 20000/19773
 
{{Mapping|legend=2| 2 2 3 6 | 0 5 7 6 }}
 
Optimal tunings:
* WE: ~351/250 = 599.8593{{c}}, ~13/12 = 140.6362{{c}}
* CWE: ~351/250 = 600.0000{{c}}, ~13/12 = 140.6232{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=2| 2 2 3 6 7 | 0 5 7 6 5 }}
 
Optimal tunings:
* WE: ~17/12 = 599.9773{{c}}, ~13/12 = 140.6075{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.6057{{c}}
 
{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}
 
Badness (Sintel): 0.488
 
== References ==
 
[[Category:Fifive family| ]] <!-- main article -->
[[Category:Fifive| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Fifive_family"