Fifive family: Difference between revisions

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

== 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.

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>

[[Subgroup]]: 2.3.5

[[Comma list]]: 9765625/9565938

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

[[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

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]].

[[Subgroup]]: 2.3.5.7

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

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

* [[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, 18bcd, 26, 34d, 60d }}

 

[[Badness]] (Sintel): 2.19

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 50/49, 99/98, 864/847

 

 

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, 18bcd, 26, 34d, 60d }}

 

Badness (Sintel): 1.35

 

Subgroup: 2.3.5.7.11.13

 

Comma list: 50/49, 78/77, 99/98, 144/143

 

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

 

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 }}

 

* WE: ~7/5 = 599.5348{{c}}, ~13/12 = 140.2961{{c}}

* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.2668{{c}}

 

 

Badness (Sintel): 0.946

 

[[Subgroup]]: 2.3.5.7

 

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

 

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

 

* [[WE]]: ~567/400 = 600.9312{{c}}, ~27/25 = 140.1261{{c}}

* [[CWE]]: ~567/400 = 600.0000{{c}}, ~27/25 = 139.9826{{c}}

 

 

 

Subgroup: 2.3.5.7.11

 

Comma list: 100/99, 385/384, 3969/3872

 

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

 

* 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: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}

 

* WE: ~55/39 = 600.4600{{c}}, ~12/11 = 139.9737{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 139.9092{{c}}

 

 

 

=== 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 }}

 

* 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}}

 

 

Badness (Sintel): 1.50

 

 

[[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

 

* [[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 }}

 

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 245/243, 385/384, 235298/234375

 

Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}

 

 

 

 

==== 13-limit ====

 

Comma list: 196/195, 245/243, 385/384, 20000/19773

 

 

* WE: ~25/21 = 299.9488{{c}}, ~13/12 = 140.7359{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7539{{c}}

 

 

Badness (Sintel): 2.78

 

Subgroup: 2.3.5.7.11

 

 

Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}

 

* 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 }}

 

* WE: ~25/21 = 300.0500{{c}}, ~13/12 = 140.7516{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7590{{c}}

 

 

 

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

 

Comma list: 121/120, 154/153, 170/169, 196/195, 245/243

 

 

* 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}}

 

 

 

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.

 

 

Comma list: 325/324, 20000/19773

 

Mapping: {{mapping| 2 2 3 6 | 0 5 7 6 }}

 

* 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

 

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}}

 

 

 

 

[[Category:Fifive family| ]] <!-- main article -->

[[Category:Fifive| ]] <!-- key article -->

[[Category:Catalogs of rank-2 temperaments]]