35th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|35}} | |||
[[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. | [[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. | ||
== Insanobromismic == | |||
{{Main|Insanobromisma}} | |||
This is the comma that identifies [[51/50]] with 1 step of 35edo. | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: {{monzo|36 -35 70 0 0 0 -35}} | |||
Mapping: [{{val|35 0 0 0 0 0 36}}, {{val|0 1 0 0 0 0 -1}}, {{val|0 0 1 0 0 0 2}}, {{val|0 0 0 1 0 0 0}}, {{val|0 0 0 0 1 0 0}}, {{val|0 0 0 0 0 1 0}}] | |||
: mapping generators: ~51/50 = 1\35, ~3, ~5, ~7, ~11, ~13, ~17 | |||
[[Support]]ing [[ET]]s: {{EDOs|140, 525, 665, 2730, 3395, 3920, 7980, 11375, 15960, 16625, 24605, 28000, 32585}}, ... | |||
=== Insanobromic === | |||
Subgroup: 2.3.5.17 | |||
Comma list: {{monzo|36 -35 70 -35}} | |||
{{Mapping|legend=2|35 0 0 36|0 1 0 -1|0 0 1 2}} | |||
: mapping generators: ~51/50 = 1\35, ~3, ~5, | |||
[[Support]]ing [[ET]]s: {{EDOs|140, 525, 665, 2730, 3395, 3920, 7980, 11375, 15960, 16625, 24605, 28000, 32585}}, ... | |||
== Bromine == | == Bromine == | ||
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. | |||
If a strong 11th harmonic is needed, [[6125edo]] is the tuning for that. | |||
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}}] | |||
Mapping generators: ~1203125/1179648 = 1\35, ~3 | |||
Optimal tuning (CTE): ~3/2 = 701.975 | |||
{{Optimal ET sequence|legend=1|665, 2730, 3395, 6125, 9520}} | |||
=== 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, ... | |||
=== 17-limit === | |||
The period is mapped to ~51/50. | |||
Subgroup: 2.3.5.7.11.13.17 | |||
{{Optimal | 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 = 701.97... | |||
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) == | ||
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Optimal tuning (CTE): ~5/4 = 386.174 | Optimal tuning (CTE): ~5/4 = 386.174 | ||
{{Navbox fractional-octave}} | |||
[[Category:35edo]] | [[Category:35edo]] | ||