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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).


== Fifive ==
== Fifive ==
Subgroup: 2.3.5
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.  


[[Comma]]: 9765625/9565938
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>


[[Mapping]]: [{{val|2 2 3}}, {{val|0 5 7}}]
[[Subgroup]]: 2.3.5


[[POTE generator]]: ~27/25 = 140.624
[[Comma list]]: 9765625/9565938


{{Val list|legend=1| 8, 26, 34, 94, 128 }}
{{Mapping|legend=1| 2 2 3 | 0 5 7 }}
: mapping generators: ~78125/50421, ~27/25


[[Badness]]: 0.205812
[[Optimal tuning]]s:
* [[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 }}
 
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}
 
[[Badness]] (Sintel): 4.83
 
=== 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.
 
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]].  


== Crepuscular ==
== 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]]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 }}


[[POTE generator]]: ~27/25 = 140.349
{{Optimal ET sequence|legend=1| 8d, 18bcd, 26, 34d, 60d }}


{{Val list|legend=1| 8d, 26, 34d, 60d }}
[[Badness]] (Sintel): 2.19
 
[[Badness]]: 0.086669


=== 11-limit ===
=== 11-limit ===
Line 36: Line 51:
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 tunings:  
* WE: ~7/5 = 599.1577{{c}}, ~12/11 = 140.3893{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~12/11 = 140.3244{{c}}


Optimal GPV sequence: {{Val list| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcd, 26, 34d, 60d }}


Badness: 0.040758
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
Line 49: Line 66:
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 }}
 
Optimal tunings:
* WE: ~7/5 = 599.2598{{c}}, ~13/12 = 140.3805{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.3237{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 18bcdf, 26, 34d, 60d }}
 
Badness (Sintel): 1.01
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~12/11 = 140.554
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143


Optimal GPV sequence: {{Val list| 8d, 26, 34d, 60d }}
Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}


Badness: 0.024368
Optimal tunings:
* WE: ~7/5 = 599.5348{{c}}, ~13/12 = 140.2961{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.2668{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 18bcdfg, 26, 34d, 60d }}
 
Badness (Sintel): 0.946


== 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]]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 }}


{{Val list|legend=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=1| 8, 26, 34, 60, 266bcccddd }}


Badness: 0.130589
[[Badness]] (Sintel): 3.30


=== 11-limit ===
=== 11-limit ===
Line 75: Line 113:
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 tunings:  
* WE: ~63/44 = 600.5091{{c}}, ~12/11 = 140.0024{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~12/11 = 139.9267{{c}}


Optimal GPV sequence: {{Val list| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness: 0.080306
Badness (Sintel): 2.65


=== 13-limit ===
=== 13-limit ===
Line 88: Line 128:
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 }}
 
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


POTE generator: ~27/25 = 139.867
Mapping: {{mapping| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}


Optimal GPV sequence: {{Val list| 8, 26, 34, 60 }}
Optimal tunings:  
* WE: ~17/12 = 600.4903{{c}}, ~12/11 = 139.9825{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~12/11 = 139.9150{{c}}


Badness: 0.044253
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 ==
== 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 }}
: mapping generators: ~25/21, ~27/25


[[POTE generator]]: ~27/25 = 140.754
[[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 }}


{{Val list| 8d, 60, 68, 128 }}
{{Optimal ET sequence|legend=1| 8d, …, 60, 68, 128, 196 }}


[[Badness]]: 0.114143
[[Badness]] (Sintel): 2.89


=== 11-limit ===
=== 11-limit ===
Line 114: Line 180:
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 tunings:  
* WE: ~25/21 = 299.9901{{c}}, ~27/25 = 140.7659{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/25 = 140.7693{{c}}


Optimal GPV sequence: {{Val list| 8de, 60, 68, 128, 196 }}
{{Optimal ET sequence|legend=0| 60, 68, 128, 196 }}


Badness: 0.120165
Badness (Sintel): 3.97


=== 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 }}
 
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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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}}


POTE generator: ~13/12 = 140.760
{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}


Optimal GPV sequence: {{Val list| 8de, 60, 68, 128, 196f }}
Badness (Sintel): 0.488


Badness: 0.067365
== References ==


[[Category:Theory]]
[[Category:Fifive family| ]] <!-- main article -->
[[Category:Fifive| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Fifive]]
Retrieved from "https://en.xen.wiki/w/Fifive_family"