Fifive family: Difference between revisions

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 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 ==
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 [[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 }}
-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 }}
+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 ==
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 Badness (Sintel): 1.26
+== Subgroup extensions ==
+=== Fifive (2.3.5.13) ===
+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 ==