Fifive family: Difference between revisions

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


Considered below are crepuscular, fifives, and fourfives.  
== 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
[[Subgroup]]: 2.3.5


Line 9: Line 12:


{{Mapping|legend=1| 2 2 3 | 0 5 7 }}
{{Mapping|legend=1| 2 2 3 | 0 5 7 }}
: mapping generators: ~78125/50421, ~27/25
: mapping generators: ~78125/50421, ~27/25


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~78125/50421 = 1\2, ~27/25 = 140.6349
* [[WE]]: ~78125/50421 = 600.0168{{c}}, ~27/25 = 140.6276{{c}}
* [[POTE]]: ~78125/50421 = 1\2, ~27/25 = 140.624
: [[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 }}


[[Badness]]:
[[Badness]] (Sintel): 4.83
* Smith: 0.205812
* Dirichlet: 4.828


=== 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
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 | 0 5 7 6 }}
 
: mapping generators: ~351/250, ~13/12
 
Optimal tunings:
* CTE: ~351/250 = 1\2, ~13/12 = 140.5685
* CWE: ~351/250 = 1\2, ~13/12 = 140.6232
 
Optimal ET sequence: {{Optimal ET sequence| 8, 18bcf, 26, 34, 94, 128 }}
 
Badness:
* Smith: 0.0240
* Dirichlet: 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 = 1\2, ~13/12 = 140.5958
* CWE: ~17/12 = 1\2, ~13/12 = 140.6057
 
Optimal ET sequence: {{Optimal ET sequence| 8, 18bcfg, 26, 34, 94, 128 }}
 
Badness:
* Smith: 0.0110
* Dirichlet: 0.488


== Crepuscular ==
== Crepuscular ==
{{See also| Jubilismic clan #Crepuscular }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 69: Line 36:
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}


{{Multival|legend=1| 10 14 14 -1 -6 -7 }}
[[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 tuning]] ([[POTE]]): ~7/5 = 1\2, ~27/25 = 140.349
{{Optimal ET sequence|legend=1| 8d, 18bcd, 26, 34d, 60d }}


{{Optimal ET sequence|legend=1| 8d, 26, 34d, 60d }}
[[Badness]] (Sintel): 2.19
 
[[Badness]]: 0.086669


=== 11-limit ===
=== 11-limit ===
Line 82: Line 51:
Comma list: 50/49, 99/98, 864/847
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 tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.587
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=1| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcd, 26, 34d, 60d }}


Badness: 0.040758
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
Line 95: Line 66:
Comma list: 50/49, 78/77, 99/98, 144/143
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 tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.554
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=1| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcdf, 26, 34d, 60d }}


Badness: 0.024368
Badness (Sintel): 1.01


=== 17-limit ===
=== 17-limit ===
Line 108: Line 81:
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143
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 tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.405
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=1| 8d, 26, 34d, 60d }}
{{Optimal ET sequence|legend=0| 8d, 18bcdfg, 26, 34d, 60d }}


Badness: 0.018567
Badness (Sintel): 0.946


== Fifives ==
== Fifives ==
Line 123: Line 98:
{{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 = 1\2, ~27/25 = 139.909
[[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]]: 0.130589
[[Badness]] (Sintel): 3.30


=== 11-limit ===
=== 11-limit ===
Line 134: Line 113:
Comma list: 100/99, 385/384, 3969/3872
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 tuning (POTE): ~63/44 = 1\2, ~12/11 = 139.884
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=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness: 0.080306
Badness (Sintel): 2.65


=== 13-limit ===
=== 13-limit ===
Line 147: Line 128:
Comma list: 100/99, 105/104, 144/143, 1352/1331
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 tuning (POTE): ~55/39 = 1\2, ~12/11 = 139.867
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=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness: 0.044253
Badness (Sintel): 1.83


=== 17-limit ===
=== 17-limit ===
Line 160: Line 143:
Comma list: 100/99, 105/104, 144/143, 170/169, 221/220
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 tuning (POTE): ~17/12 = 1\2, ~12/11 = 139.868
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 ET sequence|legend=1| 8, 26, 34, 60 }}
Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~12/11 = 139.868{{c}}


Badness: 0.029429
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
 
Badness (Sintel): 1.50


== Fourfives ==
== Fourfives ==
Line 176: Line 163:


{{Mapping|legend=1| 4 4 6 7 | 0 5 7 9 }}
{{Mapping|legend=1| 4 4 6 7 | 0 5 7 9 }}
: mapping generators: ~25/21, ~27/25
: mapping generators: ~25/21, ~27/25


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~27/25 = 140.754
[[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]]: 0.114143
[[Badness]] (Sintel): 2.89


=== 11-limit ===
=== 11-limit ===
Line 190: Line 180:
Comma list: 245/243, 385/384, 235298/234375
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 tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.771
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=1| 8de, 60, 68, 128, 196 }}
{{Optimal ET sequence|legend=0| 60, 68, 128, 196 }}


Badness: 0.120165
Badness (Sintel): 3.97


==== 13-limit ====
==== 13-limit ====
Line 203: Line 195:
Comma list: 196/195, 245/243, 385/384, 20000/19773
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 tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.760
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=1| 8de, 60, 68, 128, 196f }}
{{Optimal ET sequence|legend=0| 60, 68, 128, 196f }}


Badness: 0.067365
Badness (Sintel): 2.78


=== Quadrafives ===
=== Quadrafives ===
Line 216: Line 210:
Comma list: 121/120, 245/243, 1375/1372
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 tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.630
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=1| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68, 128e }}


Badness: 0.057268
Badness (Sintel): 1.89


==== 13-limit ====
==== 13-limit ====
Line 229: Line 225:
Comma list: 121/120, 196/195, 245/243, 275/273
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 tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.728
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=1| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68 }}


Badness: 0.036128
Badness (Sintel): 1.49


==== 17-limit ====
==== 17-limit ====
Line 242: Line 240:
Comma list: 121/120, 154/153, 170/169, 196/195, 245/243
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:
* 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 ET sequence|legend=0| 8d, …, 60e, 68 }}
 
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 tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.718
{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}


{{Optimal ET sequence|legend=1| 8d, 60e, 68, 128e }}
Badness (Sintel): 0.488


Badness: 0.024796
== References ==


[[Category:Temperament families]]
[[Category:Fifive family| ]] <!-- main article -->
[[Category:Fifive family| ]] <!-- main article -->
[[Category:Fifive| ]] <!-- key article -->
[[Category:Fifive| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 17:44, 30 September 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The fifive family of temperaments tempers out the fifive comma (monzo: [-1 -14 10⟩, ratio: 9765625/9565938).

Fifive

The head of this family is fifive, which splits the 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.[1]

Subgroup: 2.3.5

Comma list: 9765625/9565938

Mapping: [⟨2 2 3], ⟨0 5 7]]

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

Optimal tunings:

  • WE: ~78125/50421 = 600.0168 ¢, ~27/25 = 140.6276 ¢
error map: ⟨+0.034 +1.217 -1.870]
  • CWE: ~78125/50421 = 600.0000 ¢, ~27/25 = 140.6291 ¢
error map: ⟨0.000 +1.191 -1.910]

Optimal ET sequence: 8, 18bc, 26, 34, 94, 128

Badness (Sintel): 4.83

Overview to extensions

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.

Crepuscular

Subgroup: 2.3.5.7

Comma list: 50/49, 4375/4374

Mapping: [⟨2 2 3 4], ⟨0 5 7 7]]

Optimal tunings:

  • WE: ~7/5 = 600.000 ¢, ~27/25 = 140.349 ¢
error map: ⟨-0.787 -1.458 -5.696 +11.398]
  • CWE: ~7/5 = 600.000 ¢, ~27/25 = 140.349 ¢
error map: ⟨0.000 -0.876 -4.803 +12.685]

Optimal ET sequence: 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

Mapping: [⟨2 2 3 4 6], ⟨0 5 7 7 4]]

Optimal tunings:

  • WE: ~7/5 = 599.1577 ¢, ~12/11 = 140.3893 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~12/11 = 140.3244 ¢

Optimal ET sequence: 8d, 18bcd, 26, 34d, 60d

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 2 3 4 6 6], ⟨0 5 7 7 4 6]]

Optimal tunings:

  • WE: ~7/5 = 599.2598 ¢, ~13/12 = 140.3805 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~13/12 = 140.3237 ¢

Optimal ET sequence: 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: [⟨2 2 3 4 6 6 7], ⟨0 5 7 7 4 6 5]]

Optimal tunings:

  • WE: ~7/5 = 599.5348 ¢, ~13/12 = 140.2961 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~13/12 = 140.2668 ¢

Optimal ET sequence: 8d, 18bcdfg, 26, 34d, 60d

Badness (Sintel): 0.946

Fifives

Subgroup: 2.3.5.7

Comma list: 875/864, 83349/81920

Mapping: [⟨2 2 3 7], ⟨0 5 7 -6]]

Optimal tunings:

  • WE: ~567/400 = 600.9312 ¢, ~27/25 = 140.1261 ¢
error map: ⟨+1.862 +0.538 -2.637 -3.064]
  • CWE: ~567/400 = 600.0000 ¢, ~27/25 = 139.9826 ¢
error map: ⟨0.000 -2.042 -6.435 -8.722]

Optimal ET sequence: 8, 26, 34, 60, 266bcccddd

Badness (Sintel): 3.30

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 2 3 7 6], ⟨0 5 7 -6 4]]

Optimal tunings:

  • WE: ~63/44 = 600.5091 ¢, ~12/11 = 140.0024 ¢
  • CWE: ~63/44 = 600.0000 ¢, ~12/11 = 139.9267 ¢

Optimal ET sequence: 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: [⟨2 2 3 7 6 6], ⟨0 5 7 -6 4 6]]

Optimal tunings:

  • WE: ~55/39 = 600.4600 ¢, ~12/11 = 139.9737 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~12/11 = 139.9092 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 1.83

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 170/169, 221/220

Mapping: [⟨2 2 3 7 6 6 7], ⟨0 5 7 -6 4 6 5]]

Optimal tunings:

  • WE: ~17/12 = 600.4903 ¢, ~12/11 = 139.9825 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~12/11 = 139.9150 ¢

Optimal tuning (POTE): ~17/12 = 600.000 ¢, ~12/11 = 139.868 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 1.50

Fourfives

See also: Sensamagic clan

Subgroup: 2.3.5.7

Comma list: 245/243, 235298/234375

Mapping: [⟨4 4 6 7], ⟨0 5 7 9]]

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

Optimal tunings:

  • WE: ~25/21 = 300.0011 ¢, ~27/25 = 140.7547 ¢
error map: ⟨+0.004 +1.823 -1.024 -2.026]
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7549 ¢
error map: ⟨0.000 +1.819 -1.030 -2.032]

Optimal ET sequence: 8d, …, 60, 68, 128, 196

Badness (Sintel): 2.89

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨4 4 6 7 19], ⟨0 5 7 9 -11]]

Optimal tunings:

  • WE: ~25/21 = 299.9901 ¢, ~27/25 = 140.7659 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7693 ¢

Optimal ET sequence: 60, 68, 128, 196

Badness (Sintel): 3.97

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨4 4 6 7 19 12], ⟨0 5 7 9 -11 6]]

Optimal tunings:

  • WE: ~25/21 = 299.9488 ¢, ~13/12 = 140.7359 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7539 ¢

Optimal ET sequence: 60, 68, 128, 196f

Badness (Sintel): 2.78

Quadrafives

Subgroup: 2.3.5.7.11

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

Mapping: [⟨4 4 6 7 11], ⟨0 5 7 9 6]]

Optimal tunings:

  • WE: ~25/21 = 300.1673 ¢, ~27/25 = 140.7084 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7353 ¢

Optimal ET sequence: 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: [⟨4 4 6 7 11 12], ⟨0 5 7 9 6 6]]

Optimal tunings:

  • WE: ~25/21 = 300.0500 ¢, ~13/12 = 140.7516 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7590 ¢

Optimal ET sequence: 8d, …, 60e, 68

Badness (Sintel): 1.49

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨4 4 6 7 11 12 14], ⟨0 5 7 9 6 6 5]]

Optimal tunings:

  • WE: ~25/21 = 300.0593 ¢, ~13/12 = 140.7457 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7520 ¢

Optimal tuning (POTE): ~25/21 = 300.000 ¢, ~13/12 = 140.718 ¢

Optimal ET sequence: 8d, …, 60e, 68

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

Subgroup-val mapping: [⟨2 2 3 6], ⟨0 5 7 6]]

Optimal tunings:

  • WE: ~351/250 = 599.8593 ¢, ~13/12 = 140.6362 ¢
  • CWE: ~351/250 = 600.0000 ¢, ~13/12 = 140.6232 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨2 2 3 6 7], ⟨0 5 7 6 5]]

Optimal tunings:

  • WE: ~17/12 = 599.9773 ¢, ~13/12 = 140.6075 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~13/12 = 140.6057 ¢

Optimal ET sequence: 8, 18bcfg, 26, 34, 94, 128

Badness (Sintel): 0.488

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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