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 '''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
[[Subgroup]]: 2.3.5


Line 12: 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 ===
Subgroup: 2.3.5.13
 
Comma list: 325/324, 20000/19773
 
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 }}
=== 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.


Badness:
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]].  
* Smith: 0.0110
* Dirichlet: 0.488


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


Line 72: Line 36:
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}


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


[[Badness]]: 0.086669
[[Badness]] (Sintel): 2.19


=== 11-limit ===
=== 11-limit ===
Line 85: Line 53:
Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}
Mapping: {{mapping| 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 98: Line 68:
Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}
Mapping: {{mapping| 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 111: Line 83:
Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}
Mapping: {{mapping| 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 124: 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 137: Line 115:
Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}
Mapping: {{mapping| 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 150: Line 130:
Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}
Mapping: {{mapping| 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 163: Line 145:
Mapping: {{mapping| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}
Mapping: {{mapping| 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 tuning (POTE): ~17/12 = 600.000{{c}}, ~12/11 = 139.868{{c}}


{{Optimal ET sequence|legend=1| 8, 26, 34, 60 }}
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}


Badness: 0.029429
Badness (Sintel): 1.50


== Fourfives ==
== Fourfives ==
Line 177: 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 193: Line 182:
Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}
Mapping: {{mapping| 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 206: Line 197:
Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}
Mapping: {{mapping| 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 219: Line 212:
Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}
Mapping: {{mapping| 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 232: Line 227:
Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}
Mapping: {{mapping| 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 245: Line 242:
Mapping: {{mapping| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}
Mapping: {{mapping| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.718
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: {{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=1| 8d, 60e, 68, 128e }}
{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}


Badness: 0.024796
Badness (Sintel): 0.488


== References ==
== References ==

Latest revision as of 04:21, 25 August 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

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

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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