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No-threes subgroup temperaments: Difference between revisions

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Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Trader has a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Insect has a ~55/32 generator, three of which give the ~5/1.
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Badness (Sintel): 1.29
Badness (Sintel): 1.29
=== French decimal ===
French decimal is conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.
[[Subgroup]]: 2.5.7
[[Comma list]]: {{monzo| 372 -159 -1 }}
{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
: mapping generators: ~2, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
: error map: {{val| 0.000 +0.046 +0.019 }}
{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
[[Badness]] (Sintel): 148
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}
Optimal tunings:
* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}
Badness (Sintel): 52.2
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
Optimal tunings:
* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}
Badness (Sintel): 10.5


=== Bastille ===
=== Bastille ===
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{{See also| Chromatic pairs #Sephiroth }}
{{See also| Chromatic pairs #Sephiroth }}


Sephiroth is the no-7 [[restriction]] of [[muggles]].  
Sephiroth is the common [[restriction]] of [[muggles]] and [[submerged]].  


[[Subgroup]]: 2.5.11
[[Subgroup]]: 2.5.11
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[[Badness]] (Sintel): 0.386
[[Badness]] (Sintel): 0.386
=== Arsenic ===
Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.
[[Subgroup]]: 2.11.13.17.19
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432
{{Mapping|legend=2| 33 0 8 249 26 | 0 1 1 -1 1 }}
: mapping generators: ~247/242, ~11
[[Optimal tuning]]s:
* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}
* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}
{{Optimal ET sequence|legend=1| 33, 165, 198, 231, 495h }}
[[Badness]] (Sintel): 2.30


=== Berylic ===
=== Berylic ===
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