35th-octave temperaments: Difference between revisions
→Bromine: expand to 11-limit and 19-limit via 2730edo's consistency, also can't keep the "normal" matrix due to limitations of the sintel's finder so 152 and -7 for 19-limit in the mapping |
|||
| Line 4: | Line 4: | ||
Bromine is named after the 35th chemical element. | Bromine is named after the 35th chemical element. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|6 -37 13 8}} | [[Comma list]]: {{monzo|47 -7 -7 -7}}, {{monzo|6 -37 13 8}} | ||
Mapping: [{{val|35 0 -418 653}}, {{val|0 1 9 -10}}] | [[Mapping]]: [{{val|35 0 -418 653}}, {{val|0 1 9 -10}}] | ||
Mapping generators: ~38263752/37515625 = 1\35, ~3 | Mapping generators: ~38263752/37515625 = 1\35, ~3 | ||
Optimal tuning (CTE): ~3/2 = 701.973 | [[Optimal tuning]] ([[CTE]]): ~3/2 = 701.973 | ||
{{ | [[Support]]ing [[ET]]s: {{EDOs|665, 2065, 2730, 3395, 4060, 7455}} | ||
=== | === 11-limit === | ||
While 665edo still tunes bromine in the 11-limit, it is not recommended due to error on the 11th harmonic. [[2730edo]] or [[3395edo]] are better tunings. Alternately, considering bromine as a no-11s temperament keeps it within the realm of [[very high accuracy temperaments]], having [[TE error]] of less than 0.005 cents per octave. | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 151263/151250, 115091701760/115063885233, 45137758519296/45135986328125 | |||
Mapping: [{{val|35 0 -418 653 2451}}, {{val|0 1 9 -10 -42}}] | |||
Optimal tuning (CTE): ~3/2 = 701. | Mapping generators: ~1203125/1179648 = 1\35, ~3 | ||
Optimal tuning (CTE): ~3/2 = 701.975 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 123201/123200, 151263/151250, 1990656/1990625, 8683257856/8681088735 | |||
Mapping: [{{val|35 0 -418 653 2451 1239}}, {{val|0 1 9 -10 -42 -20}}] | |||
Mapping generators: ~1485/1456 = 1\35, ~3 | |||
Optimal tuning (CTE): ~3/2 = 701.975 | |||
Supporting ETs: 665, 3395, ... | Supporting ETs: 665, 3395, ... | ||
=== | === 17-limit === | ||
The period is mapped to ~51/50. | The period is mapped to ~51/50. | ||
Subgroup: 2.3.5.7.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 12376/12375, 123201/123200, 194481/194480, 1713660/1713481, 24635975/24634368 | ||
Mapping: [{{val|35 0 -418 653 2451 1239 -800}}, {{val|0 1 9 -10 -42 -20 17}}] | |||
Mapping generators: ~51/50 = 1\35, ~3 | |||
Optimal tuning (CTE): ~3/2 = 701. | Optimal tuning (CTE): ~3/2 = 701.97... | ||
Supporting ETs: {{EDOs|665, 2730, 3395, 6125, 7455, 10185}}... | Supporting ETs: {{EDOs|665, 2730, 3395, 6125, 7455, 10185}}... | ||
=== 19-limit === | |||
[[Subgroup]]: 2.3.5.7.11.13.17.19 | |||
[[Comma list]]: 12376/12375, 12636/12635, 13377/13376, 14080/14079, 14365/14364, 486400/486387 | |||
[[Mapping]]: [{{val|35 0 -418 653 2451 1239 -800 152}}, {{val|0 1 9 -10 -42 -20 17 -7}}] | |||
[[Mapping]] [[generator]]s: ~51/50 = 1\35, ~3 | |||
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.97... | |||
[[Support]]ing [[ET]]s: {{EDOs|665, 2730, 3395, 6125, 6790h, 7455eh}}... | |||
== Tritonopodismic (rank-3) == | == Tritonopodismic (rank-3) == | ||
Revision as of 12:32, 19 May 2023
35edo is a tuning system that's not remarkable in its own right from a regular temperament theory perspective, but some of its multiples are either zeta edos like 3395edo, have an exceptionally precise perfect fifth like 665edo, or have high consistency limits like 525edo. In addition, one step of 35edo is close to 50/49 in low limits and 51/50 in the higher limits.
Bromine
Bromine is named after the 35th chemical element.
Subgroup: 2.3.5.7
Comma list: [47 -7 -7 -7⟩, [6 -37 13 8⟩
Mapping: [⟨35 0 -418 653], ⟨0 1 9 -10]]
Mapping generators: ~38263752/37515625 = 1\35, ~3
Optimal tuning (CTE): ~3/2 = 701.973
Supporting ETs: 665, 2065, 2730, 3395, 4060, 7455
11-limit
While 665edo still tunes bromine in the 11-limit, it is not recommended due to error on the 11th harmonic. 2730edo or 3395edo are better tunings. Alternately, considering bromine as a no-11s temperament keeps it within the realm of very high accuracy temperaments, having TE error of less than 0.005 cents per octave.
Subgroup: 2.3.5.7.11
Comma list: 151263/151250, 115091701760/115063885233, 45137758519296/45135986328125
Mapping: [⟨35 0 -418 653 2451], ⟨0 1 9 -10 -42]]
Mapping generators: ~1203125/1179648 = 1\35, ~3
Optimal tuning (CTE): ~3/2 = 701.975
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 123201/123200, 151263/151250, 1990656/1990625, 8683257856/8681088735
Mapping: [⟨35 0 -418 653 2451 1239], ⟨0 1 9 -10 -42 -20]]
Mapping generators: ~1485/1456 = 1\35, ~3
Optimal tuning (CTE): ~3/2 = 701.975
Supporting ETs: 665, 3395, ...
17-limit
The period is mapped to ~51/50.
Subgroup: 2.3.5.7.11.13.17
Comma list: 12376/12375, 123201/123200, 194481/194480, 1713660/1713481, 24635975/24634368
Mapping: [⟨35 0 -418 653 2451 1239 -800], ⟨0 1 9 -10 -42 -20 17]]
Mapping generators: ~51/50 = 1\35, ~3
Optimal tuning (CTE): ~3/2 = 701.97...
Supporting ETs: 665, 2730, 3395, 6125, 7455, 10185...
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 12376/12375, 12636/12635, 13377/13376, 14080/14079, 14365/14364, 486400/486387
Mapping: [⟨35 0 -418 653 2451 1239 -800 152], ⟨0 1 9 -10 -42 -20 17 -7]]
Mapping generators: ~51/50 = 1\35, ~3
Optimal tuning (CTE): ~3/2 = 701.97...
Supporting ETs: 665, 2730, 3395, 6125, 6790h, 7455eh...
Tritonopodismic (rank-3)
Tritonopodismic tempers out tritonopod comma, the comma which sets 7/5 to 17\35, "one leg", and 10/7 to 18\35, the "other leg". Also it sets 50/49 to be equal to 1\35.
Subgroup:2.3.5.7
Comma list: [17 0 35 -35⟩
Mapping: [⟨35 0 0 17], ⟨0 1 0 0], ⟨0 0 1 1]]
Mapping generators: ~50/49 = 1\35, ~3, ~5
Optimal tuning (CTE): ~3/2 = 701.955, ~5/4 = 386.174
Optimal ET sequence: 35, 70, 140, 210, 525, 665
Tritonopod (rank-2)
Subgroup:2.5.7
Comma list: [17 35 -35⟩
Mapping: [⟨35 0 17], ⟨0 1 1]]
Mapping generators: ~50/49, ~5
Optimal tuning (CTE): ~5/4 = 386.174