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 | |||
{{Optimal ET sequence|legend=1|35f, 140, 280, 385, 525, 665, 805, 1190, 1330, 2205, 2590, 2730, 3395, 7315, 10045, 10710, 10850, 14420, 16625, 17955, 18480, 20020, 20895, 21875, 24605, 25270, 28000, 31920, ...}} | |||
\* optimal patent val: [[589400edo|589400]] | |||
=== 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, 16625, 28000}}, ... | |||
== Bromine == | == Bromine == | ||
Bromine is named after the 35th chemical element. | Bromine is named after the 35th chemical element, and is made by combining two notable high-accuracy edos divisible by 35, [[665edo]] and [[3395edo]] - described as the 665 & 3395 temperament. | ||
In the 7-limit, [[1400edo]] is also a tuning. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: {{monzo|47 -7 -7 -7}}, {{monzo|6 -37 13 8}} | |||
[[Mapping]]: [{{val|35 0 -418 653}}, {{val|0 1 9 -10}}] | |||
: mapping generators: ~38263752/37515625 = 1\35, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~38263752/37515625 = 34.286{{c}}, ~3/2 = 701.969{{c}} | |||
: [[error map]]: {{val| -0.006, 0.009, -0.001, 0.002 }} | |||
* [[CWE]]: ~38263752/37515625 = 34.286{{c}}, ~3/2 = 701.973{{c}} | |||
: error map: {{val| -0.000, 0.018, 0.012, 0.018 }} | |||
Optimal | {{Optimal ET sequence|legend=1|70d, 665, 2065, 2730, 3395, 10850, 14245, 17640}} | ||
{{Optimal ET sequence|legend=1|665, | Badness: 4.681 | ||
=== 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]]s: | |||
* [[WE]]: ~1203125/1179648 = 34.286{{c}}, ~3/2 = 701.970{{c}} | |||
: [[error map]]: {{val| -0.007, 0.008, 0.012, -0.020, 0.014 }} | |||
* [[CWE]]: ~1203125/1179648 = 34.286{{c}}, ~3/2 = 701.975{{c}} | |||
: error map: {{val| 0.000, 0.020, 0.029, -0.000, 0.035 }} | |||
{{Optimal ET sequence|legend=1|665, 2065e, 2730, 3395, 6125, 9520, 12915}} | |||
[[Badness]]: 4.891 | |||
=== 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]]s: | |||
* [[WE]]: ~1485/1456 = 34.286{{c}}, ~3/2 = 701.970{{c}} | |||
: [[error map]]: {{val| -0.006, 0.010, 0.013, -0.014, 0.027, -0.035 }} | |||
* [[CWE]]: ~1485/1456 = 34.286{{c}}, ~3/2 = 701.975{{c}} | |||
: error map: {{val| 0.000, 0.019, 0.027, 0.002, 0.043, -0.016 }} | |||
{{Optimal ET sequence|legend=1|665, 2065e, 2730, 3395, 9520, 12915}} | |||
[[Badness]]: 2.328 | |||
=== 17-limit === | |||
The period is finally mapped to true ~51/50. | |||
Subgroup: 2.3.5.7.11.13.17 | |||
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) == | ||
| Line 42: | Line 154: | ||
Optimal tuning (CTE): ~5/4 = 386.174 | Optimal tuning (CTE): ~5/4 = 386.174 | ||
{{Navbox fractional-octave}} | |||
[[Category:35edo]] | [[Category:35edo]] | ||