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Fifive family: Difference between revisions

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


Considered below are crepuscular, fifives, and fourfives.  
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>


== Fifive ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 15: Line 15:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.6349{{c}}
* [[WE]]: ~78125/50421 = 600.0168{{c}}, ~27/25 = 140.6276{{c}}
* [[POTE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.624{{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 }}
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}
Line 22: Line 24:
[[Badness]] (Sintel): 4.83
[[Badness]] (Sintel): 4.83


=== 2.3.5.13 subgroup ===
=== Overview to extensions ===
Subgroup: 2.3.5.13
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.  
 
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 = 600.0000{{c}}, ~13/12 = 140.5685{{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 }}
: mapping generators: ~17/12, ~13/12
 
Optimal tunings:
* CTE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.6057{{c}}


{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}
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 (Sintel): 0.488


== Crepuscular ==
== Crepuscular ==
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{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 600.000{{c}}, ~27/25 = 140.349{{c}}
[[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 }}


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


[[Badness]] (Smith): 0.086669
[[Badness]] (Sintel): 2.19


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}
Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}


Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.587{{c}}
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 ET sequence|legend=0| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcd, 26, 34d, 60d }}


Badness (Smith): 0.040758
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}
Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}


Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.554{{c}}
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, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcdf, 26, 34d, 60d }}


Badness (Smith): 0.024368
Badness (Sintel): 1.01


=== 17-limit ===
=== 17-limit ===
Line 100: Line 83:
Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}
Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}


Optimal tuning (POTE): ~7/5 = 600.000{{c}}, ~12/11 = 140.405{{c}}
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, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcdfg, 26, 34d, 60d }}


Badness (Smith): 0.018567
Badness (Sintel): 0.946


== Fifives ==
== Fifives ==
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{{Mapping|legend=1| 2 2 3 7 | 0 5 7 -6 }}
{{Mapping|legend=1| 2 2 3 7 | 0 5 7 -6 }}


[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~27/25 = 139.909{{c}}
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}
Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}


Optimal tuning (POTE): ~63/44 = 600.000{{c}}, ~12/11 = 139.884{{c}}
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 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness (Smith): 0.080306
Badness (Sintel): 2.65


=== 13-limit ===
=== 13-limit ===
Line 139: Line 130:
Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}
Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}


Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~12/11 = 139.867{{c}}
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 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness (Smith): 0.044253
Badness (Sintel): 1.83


=== 17-limit ===
=== 17-limit ===
Line 151: Line 144:


Mapping: {{mapping| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}
Mapping: {{mapping| 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 tuning (POTE): ~17/12 = 600.000{{c}}, ~12/11 = 139.868{{c}}
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{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness (Smith): 0.029429
Badness (Sintel): 1.50


== Fourfives ==
== Fourfives ==
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: mapping generators: ~25/21, ~27/25
: mapping generators: ~25/21, ~27/25


[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.000{{c}}, ~27/25 = 140.754{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 8d, …, 60, 68, 128, 196 }}


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


=== 11-limit ===
=== 11-limit ===
Line 181: Line 182:
Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}
Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~27/25 = 140.771{{c}}
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| 8de, 60, 68, 128, 196 }}
{{Optimal ET sequence|legend=0| 60, 68, 128, 196 }}


Badness (Smith): 0.120165
Badness (Sintel): 3.97


==== 13-limit ====
==== 13-limit ====
Line 194: Line 197:
Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}
Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.760{{c}}
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| 8de, 60, 68, 128, 196f }}
{{Optimal ET sequence|legend=0| 60, 68, 128, 196f }}


Badness (Smith): 0.067365
Badness (Sintel): 2.78


=== Quadrafives ===
=== Quadrafives ===
Line 207: Line 212:
Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}
Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~27/25 = 140.630{{c}}
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 }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68, 128e }}


Badness (Smith): 0.057268
Badness (Sintel): 1.89


==== 13-limit ====
==== 13-limit ====
Line 220: Line 227:
Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}
Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}


Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.728{{c}}
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, 128e }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68 }}


Badness (Smith): 0.036128
Badness (Sintel): 1.49


==== 17-limit ====
==== 17-limit ====
Line 232: Line 241:


Mapping: {{mapping| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}
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 tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.718{{c}}


{{Optimal ET sequence|legend=0| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68 }}


Badness (Smith): 0.024796
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}}
 
{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}
 
Badness (Sintel): 0.488


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Fifive_family"