35th-octave temperaments: Difference between revisions

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→Bromine: supporting ETs -> optimal ET sequence now that I have Canou's script working in my Jupyter
→Insanobromismic: remove ~17 as a mapping generator
 
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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: [{{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
: mapping generators: ~51/50 = 1\35, ~3, ~5, ~7, ~11, ~13


[[Support]]ing [[ET]]s: {{EDOs|140, 525, 665, 2730, 3395, 3920, 7980, 11375, 15960, 16625, 24605, 28000, 32585}}, ...
{{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 ===
=== Insanobromic ===
Line 27: Line 29:
: mapping generators: ~51/50 = 1\35, ~3, ~5,
: 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}}, ...
[[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
[[Subgroup]]: 2.3.5.7
Line 38: Line 42:
[[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]]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 ET sequence|legend=1|70d, 665, 2065, 2730, 3395, 10850, 14245, 17640}}
{{Optimal ET sequence|legend=1|70d, 665, 2065, 2730, 3395, 10850, 14245, 17640}}
Badness: 4.681


=== 11-limit ===
=== 11-limit ===
Line 56: Line 66:
Mapping: [{{val|35 0 -418 653 2451}}, {{val|0 1 9 -10 -42}}]
Mapping: [{{val|35 0 -418 653 2451}}, {{val|0 1 9 -10 -42}}]


Mapping generators: ~1203125/1179648 = 1\35, ~3
: mapping generators: ~1203125/1179648 = 1\35, ~3


Optimal tuning (CTE): ~3/2 = 701.975
[[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, 2730, 3395, 6125, 9520}}
{{Optimal ET sequence|legend=1|665, 2065e, 2730, 3395, 6125, 9520, 12915}}
 
[[Badness]]: 4.891


=== 13-limit ===
=== 13-limit ===
Line 72: Line 88:
Mapping generators: ~1485/1456 = 1\35, ~3
Mapping generators: ~1485/1456 = 1\35, ~3


Optimal tuning (CTE): ~3/2 = 701.975
[[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}}


Supporting ETs: 665, 3395, ...
[[Badness]]: 2.328


=== 17-limit ===
=== 17-limit ===


The period is mapped to ~51/50.
The period is finally mapped to true ~51/50.


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17

Latest revision as of 21:20, 3 October 2026

← 34th-octave • 35th-octave • 36th-octave →

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 article: Insanobromisma

This is the comma that identifies 51/50 with 1 step of 35edo.

Subgroup: 2.3.5.7.11.13.17

Comma list: [36 -35 70 0 0 0 -35⟩

Mapping: [⟨35 0 0 0 0 0 36], ⟨0 1 0 0 0 0 -1], ⟨0 0 1 0 0 0 2], ⟨0 0 0 1 0 0 0], ⟨0 0 0 0 1 0 0], ⟨0 0 0 0 0 1 0]]

mapping generators: ~51/50 = 1\35, ~3, ~5, ~7, ~11, ~13

Optimal ET sequence: 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: 589400

Insanobromic

Subgroup: 2.3.5.17

Comma list: [36 -35 70 -35⟩

Subgroup-val mapping: [⟨35 0 0 36], ⟨0 1 0 -1], ⟨0 0 1 2]]

mapping generators: ~51/50 = 1\35, ~3, ~5,

Supporting ETs: 140, 525, 665, 2730, 3395, 3920, 7980, 16625, 28000, ...

Bromine

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: [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 tunings:

  • WE: ~38263752/37515625 = 34.286 ¢, ~3/2 = 701.969 ¢
error map: ⟨-0.006, 0.009, -0.001, 0.002]
  • CWE: ~38263752/37515625 = 34.286 ¢, ~3/2 = 701.973 ¢
error map: ⟨-0.000, 0.018, 0.012, 0.018]

Optimal ET sequence: 70d, 665, 2065, 2730, 3395, 10850, 14245, 17640

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: [⟨35 0 -418 653 2451], ⟨0 1 9 -10 -42]]

mapping generators: ~1203125/1179648 = 1\35, ~3

Optimal tunings:

  • WE: ~1203125/1179648 = 34.286 ¢, ~3/2 = 701.970 ¢
error map: ⟨-0.007, 0.008, 0.012, -0.020, 0.014]
  • CWE: ~1203125/1179648 = 34.286 ¢, ~3/2 = 701.975 ¢
error map: ⟨0.000, 0.020, 0.029, -0.000, 0.035]

Optimal ET sequence: 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: [⟨35 0 -418 653 2451 1239], ⟨0 1 9 -10 -42 -20]]

Mapping generators: ~1485/1456 = 1\35, ~3

Optimal tunings:

  • WE: ~1485/1456 = 34.286 ¢, ~3/2 = 701.970 ¢
error map: ⟨-0.006, 0.010, 0.013, -0.014, 0.027, -0.035]
  • CWE: ~1485/1456 = 34.286 ¢, ~3/2 = 701.975 ¢
error map: ⟨0.000, 0.019, 0.027, 0.002, 0.043, -0.016]

Optimal ET sequence: 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: [⟨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

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