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35th-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|35}}
{{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 ==
== Insanobromismic ==
{{Main|Insanobromisma}}
{{Main|Insanobromisma}}
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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 ===
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: 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
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[[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]]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 tuning]] ([[CTE]]): ~3/2 = 701.973
{{Optimal ET sequence|legend=1|70d, 665, 2065, 2730, 3395, 10850, 14245, 17640}}


[[Support]]ing [[ET]]s: {{EDOs|665, 2065, 2730, 3395, 4060, 7455}}
Badness: 4.681


=== 11-limit ===
=== 11-limit ===
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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 ===
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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
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