13th-octave temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
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== Aluminium ==
== Aluminium ==
: ''See also: [[Ragismic microtemperaments #Aluminium]]''
: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].''


Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.
Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit, and sets [[135/128]] to 1/13 of an [[octave]].  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Comma list]]: {{monzo| 92 -39 -13 }}
[[Comma list]]: {{monzo| 92 -39 -13 }}


[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
{{Mapping|legend=1| 13 0 92 | 0 1 -3 }}
: mapping generators: ~135/128, ~3


: Mapping generators: ~135/128, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~135/128 = 92.3068{{c}}, ~3/2 = 701.9791{{c}}
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
: [[error map]]: {{val| -0.012 +0.012 +0.009 }}
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 701.9897{{c}}
: error map: {{val| 0.000 +0.032 +0.033 }}


{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}


[[Badness]] (Sintel): 2.893
[[Badness]] (Sintel): 2.89


== Tridecatonic ==
== Tridecatonic ==
: ''See also: [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[26th-octave temperaments]]''
: ''For extensions, see [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[Varunismic temperaments #Bosonic]].''


''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.
Tridecatonic tempers out the [[devil's tridecalimma]], the comma which associates the [[10/9]] minor tone with [[13edo|2\13]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 2541865828329/2500000000000
[[Comma list]]: {{monzo| -11 26 -13 }}


[[Mapping]]: [{{Val| 13 21 31 }}, {{Val| 0 -1 -2 }}]
{{Mapping|legend=1| 13 0 -11 | 0 1 2 }}
: mapping generators: ~531441/500000, ~3


: Mapping generators: ~531441/500000, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~531441/500000 = 92.3166{{c}}, ~3/2 = 701.2353{{c}}
[[Optimal tuning]] ([[CTE]]): ~531441/500000 = 1\13, ~3/2 = 701.2353
: [[error map]]: {{val| +0.116 -0.688 +0.737 }}
* [[CWE]]: ~531441/500000 = 92.3077{{c}}, ~3/2 = 701.2353{{c}}
: error map: {{val| 0.000 -0.797 +0.617 }}


{{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }}
{{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }}


[[Badness]] (Sintel): 10.263
[[Badness]] (Sintel): 10.3


== Triskaidekic ==
== Triskaidekic ==
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{{Mapping|legend=1| 13 0 30 | 0 1 0 }}
{{Mapping|legend=1| 13 0 30 | 0 1 0 }}
: Mapping generators: ~128/125, ~3
: Mapping generators: ~128/125, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~128/125 = 92.589, ~3/2 = 698.263
* [[WE]]: ~128/125 = 92.589{{c}}, ~3/2 = 698.263{{c}}
* [[CWE]]: ~128/125 = 92.308, ~3/2 = 698.068
: [[error map]]: {{val| +3.656 -0.036 -8.647 }}
* [[CWE]]: ~128/125 = 92.308{{c}}, ~3/2 = 698.068{{c}}
: error map: {{val| 0.000 -3.887 -17.083 }}


{{Optimal ET sequence|legend=1| 13, 26, 91cc }}
{{Optimal ET sequence|legend=1| 13, 26, 91cc }}


[[Badness]] (Sintel): 52.583
[[Badness]] (Sintel): 52.6


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}

Revision as of 03:46, 21 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

← 12th-octave • 13th-octave • 14th-octave →

A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although 13edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.

Aluminium

For extensions, see Ragismic microtemperaments #Aluminium.

Aluminium tempers out [92 -39 -13⟩ in the 5-limit, and sets 135/128 to 1/13 of an octave.

Subgroup: 2.3.5

Comma list: [92 -39 -13⟩

Mapping: [⟨13 0 92], ⟨0 1 -3]]

mapping generators: ~135/128, ~3

Optimal tunings:

  • WE: ~135/128 = 92.3068 ¢, ~3/2 = 701.9791 ¢
error map: ⟨-0.012 +0.012 +0.009]
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 701.9897 ¢
error map: ⟨0.000 +0.032 +0.033]

Optimal ET sequence: 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc

Badness (Sintel): 2.89

Tridecatonic

For extensions, see Octagar temperaments #Tridecatonic, Ragismic microtemperaments #Trideci, and Varunismic temperaments #Bosonic.

Tridecatonic tempers out the devil's tridecalimma, the comma which associates the 10/9 minor tone with 2\13.

Subgroup: 2.3.5

Comma list: [-11 26 -13⟩

Mapping: [⟨13 0 -11], ⟨0 1 2]]

mapping generators: ~531441/500000, ~3

Optimal tunings:

  • WE: ~531441/500000 = 92.3166 ¢, ~3/2 = 701.2353 ¢
error map: ⟨+0.116 -0.688 +0.737]
  • CWE: ~531441/500000 = 92.3077 ¢, ~3/2 = 701.2353 ¢
error map: ⟨0.000 -0.797 +0.617]

Optimal ET sequence: 26, 39, 65, 286, 351, 416, 481, 546

Badness (Sintel): 10.3

Triskaidekic

For extensions, see Orwellismic temperaments #Triskaidekic.

Subgroup: 2.3.5

Comma list: 1220703125/1073741824

Mapping: [⟨13 0 30], ⟨0 1 0]]

Mapping generators: ~128/125, ~3

Optimal tunings:

  • WE: ~128/125 = 92.589 ¢, ~3/2 = 698.263 ¢
error map: ⟨+3.656 -0.036 -8.647]
  • CWE: ~128/125 = 92.308 ¢, ~3/2 = 698.068 ¢
error map: ⟨0.000 -3.887 -17.083]

Optimal ET sequence: 13, 26, 91cc

Badness (Sintel): 52.6

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