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 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
@@ Line 24: / Line 24: @@
 [[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
+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]].
-Mapping: {{mapping| 2 2 3 6 7 | 0 5 7 6 5 }}
-Optimal tunings:
-* CTE: ~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
 == Crepuscular ==
@@ Line 76: / Line 51: @@
 Comma list: 50/49, 99/98, 864/847
-Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}
+{{Mapping|legend=0| 2 2 3 4 6 | 0 5 7 7 4 }}
 Optimal tunings:
@@ Line 91: / Line 66: @@
 Comma list: 50/49, 78/77, 99/98, 144/143
-Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}
+{{Mapping|legend=0| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}
 Optimal tunings:
@@ Line 106: / Line 81: @@
 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 }}
+{{Mapping|legend=0| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}
 Optimal tunings:
@@ Line 138: / Line 113: @@
 Comma list: 100/99, 385/384, 3969/3872
-Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}
+{{Mapping|legend=0| 2 2 3 7 6 | 0 5 7 -6 4 }}
 Optimal tunings:
@@ Line 153: / Line 128: @@
 Comma list: 100/99, 105/104, 144/143, 1352/1331
-Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}
+{{Mapping|legend=0| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}
 Optimal tunings:
@@ Line 168: / Line 143: @@
 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 }}
+{{Mapping|legend=0| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}
 Optimal tunings:
@@ Line 205: / Line 180: @@
 Comma list: 245/243, 385/384, 235298/234375
-Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}
+{{Mapping|legend=0| 4 4 6 7 19 | 0 5 7 9 -11 }}
 Optimal tunings:
@@ Line 220: / Line 195: @@
 Comma list: 196/195, 245/243, 385/384, 20000/19773
-Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}
+{{Mapping|legend=0| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}
 Optimal tunings:
@@ Line 235: / Line 210: @@
 Comma list: 121/120, 245/243, 1375/1372
-Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}
+{{Mapping|legend=0| 4 4 6 7 11 | 0 5 7 9 6 }}
 Optimal tunings:
@@ Line 250: / Line 225: @@
 Comma list: 121/120, 196/195, 245/243, 275/273
-Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}
+{{Mapping|legend=0| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}
 Optimal tunings:
@@ Line 265: / Line 240: @@
 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 }}
+{{Mapping|legend=0| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}
 Optimal tunings:
@@ Line 276: / Line 251: @@
 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 ==

Fifive family: Difference between revisions

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