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.

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

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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]].

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 ==

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

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

Optimal tunings:

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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:

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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 }}

Optimal tunings:

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

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

Optimal tunings:

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Comma list: 100/99, 105/104, 144/143, 1352/1331

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

Optimal tunings:

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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 }}

Optimal tunings:

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

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

Optimal tunings:

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

~~Mapping:~~ {{~~mapping~~| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}

Optimal tunings:

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Comma list: 121/120, 245/243, 1375/1372

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

Optimal tunings:

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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:

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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:

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== 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:

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Comma list: 170/169, 289/288, 325/324

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

Optimal tunings:

@@ Line 2: / Line 2: @@
 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.
+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 25: / Line 27: @@
 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]].
+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 ==
@@ Line 49: / 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 64: / 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 79: / 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 111: / 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 126: / 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 141: / 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 178: / 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 193: / 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 208: / 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 223: / 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 238: / 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 252: / Line 254: @@
 == 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 }}
+{{Mapping|legend=2| 2 2 3 6 | 0 5 7 6 }}
 Optimal tunings:
@@ Line 271: / Line 275: @@
 Comma list: 170/169, 289/288, 325/324
-Mapping: {{mapping| 2 2 3 6 7 | 0 5 7 6 5 }}
+{{Mapping|legend=2| 2 2 3 6 7 | 0 5 7 6 5 }}
 Optimal tunings:

Fifive family: Difference between revisions

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