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Catalog of rank-4 temperaments: Difference between revisions

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Added azazoth the temperament
 
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[[Badness]] (Sintel): 0.456
[[Badness]] (Sintel): 0.456


== Azazoth ( [−20 −16 12 5 1⟩ ) ==
== Azazoth ({{monzo| -20 -16 12 5 1 }}) ==
Named by [[Eufalesio]] in 2026, azazoth has a very natural extension reached by tempering out the [[pontigailimma]], which alternatively also tempers out the [[chalmersia]] and the [[insanisma]]; three of the smallest commas of the 13-limit which results in one of the least logflat badness, most accurate 13-limit temperaments.
 
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [−20 −16 12 5 1⟩
[[Comma list]]: {{monzo| -20 -16 12 5 1 }}


{{Mapping|legend=1|1 0 0 0 20|0 1 0 0 16|0 0 1 0 −12|0 0 0 1 −5}}
{{Mapping|legend=1| 1 0 0 0 20 | 0 1 0 0 16 | 0 0 1 0 -12 | 0 0 0 1 -5 }}
: mapping generators: ~2, ~3, ~5, ~7
: mapping generators: ~2, ~3, ~5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.999330{{c}}, ~3/2 = 701.954324{{c}}, ~5/4 = 386.317222{{c}}, ~7/4 = 968.828567{{c}}
* [[WE]]: ~2 = 1199.999330{{c}}, ~3/2 = 701.954324{{c}}, ~5/4 = 386.317222{{c}}, ~7/4 = 968.828567{{c}}
: [[error map]]: {{val|−0.000670 −0.001347 +0.002168 +0.001320 +0.000401}}
: [[error map]]: {{val| -0.000670 -0.001347 +0.002168 +0.001320 +0.000401 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.954096{{c}}, ~5/4 = 386.316982{{c}}, ~7/4 = 968.828415{{c}}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.954096{{c}}, ~5/4 = 386.316982{{c}}, ~7/4 = 968.828415{{c}}
: error map: {{val|0.000000 −0.000905 +0.003268 +0.002509 +0.001731}}
: error map: {{val| 0.000000 -0.000905 +0.003268 +0.002509 +0.001731 }}


{{Optimal ET sequence|legend=1|7, 39, 42e, 46, 53, 95, 99, 148d, 168d, 171, 217, 224, 270, 494, 764, 1205, 1426, 1475, 1650, 1696, 1920, 2190, 2684, 3395, 4874, 5315, 5585, 6079, 8269, 14348, 14618…}}
{{Optimal ET sequence|legend=1| 46, 53, 95, 99, 148d, 168d, 171, 217, 224, 270, 494, 764, 1205, 1426, 1475, 1650, 1696, 1920, 2190, 2684, 3395, 4874, 5315, 5585, 6079, 8269, 14348, 14618, 22887, 34821, 43090, 57708, 150237, 193327c, … }}


Badness (Sintel): 2.92
Badness (Sintel): 2.92


=== 13-limit ===
=== Tridecimal azazoth ===
A very natural extension of azazoth reached by tempering out the [[pontigailimma]], which alternatively also tempers out the [[chalmersia]] and the [[insanisma]]; three of the smallest commas of the 13-limit which results in one of the least logflat badness, most accurate 13-limit temperaments.
 
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 123201/123200
[[Comma list]]: 123201/123200, 1990656/1990625


{{Mapping|legend=1|1 0 0 0 20 13|0 1 0 0 16 5|0 0 1 0 −12 -5|0 0 0 1 −5 -2}}
{{Mapping|legend=1| 1 0 0 0 20 13 | 0 1 0 0 16 5 | 0 0 1 0 -12 -5 | 0 0 0 1 -5 -2 }}
: mapping generators: ~2, ~3, ~5, ~7
: mapping generators: ~2, ~3, ~5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.999330{{c}}, ~3/2 = 701.953924{{c}}, ~5/4 = 386.316850{{c}}, ~7/4 = 968.828214{{c}}
* [[WE]]: ~2 = 1199.999330{{c}}, ~3/2 = 701.953924{{c}}, ~5/4 = 386.316850{{c}}, ~7/4 = 968.828214{{c}}
: [[error map]]: {{val|−0.000471 −0.001548 +0.002193 +0.001365 +0.000627 −0.000606}}
: [[error map]]: {{val| -0.000471 -0.001548 +0.002193 +0.001365 +0.000627 -0.000606 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.954028{{c}}, ~5/4 = 386.316921{{c}}, ~7/4 = 968.828353{{c}}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.954028{{c}}, ~5/4 = 386.316921{{c}}, ~7/4 = 968.828353{{c}}
: error map: {{val|0.000000 −0.000972 +0.003207 +0.002446 +0.001697 +0.001170}}
: error map: {{val| 0.000000 -0.000972 +0.003207 +0.002446 +0.001697 +0.001170 }}


{{Optimal ET sequence|legend=1|7, 39, 42e, 46, 53, 95, 99, 148d, 168d, 171, 217, 224, 270, 494, 764, 1205, 1426, 1475, 1650, 1696, 1920, 2190, 2684, 3395, 4874, 5315, 5585, 6079, 8269, 14348, 14618, 20203…}}
{{Optimal ET sequence|legend=1| 46, 53, 99, 171, 217, 224, 270, 494, 764, 935, 1205, 1696, 1920, 2190, 2684, 3395, 5144, 5585, 6079, 8269, 8539, 11664, 14124, 14348, 14618, 20203, 22887, 34821, 37505, 43090, 100798, 143888, 166775, … }}


Badness (Sintel): 0.107
Badness (Sintel): 0.107
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