13th-octave temperaments

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.

Fractional-octave temperaments

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← 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

See also: Ragismic microtemperaments #Aluminium

Aluminium is named after the 13th element, and tempers out the [92 -39 -13⟩ comma which sets 135/128 interval to be equal to 1/13th of the octave.

Subgroup: 2.3.5

Comma list: [92 -39 -13⟩

Mapping: [⟨13 0 92], ⟨0 1 -3]]

Mapping generators: ~135/128, ~3

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 701.9897

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

Badness (Sintel): 2.893

Tridecatonic

See also: Octagar temperaments #Tridecatonic, Ragismic microtemperaments #Trideci, and 26th-octave temperaments

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

Subgroup: 2.3.5

Comma list: 2541865828329/2500000000000

Mapping: [⟨13 21 31], ⟨0 -1 -2]]

Mapping generators: ~531441/500000, ~3

Optimal tuning (CTE): ~531441/500000 = 1\13, ~3/2 = 701.2353

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

Badness (Sintel): 10.263

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
• CWE: ~128/125 = 92.308, ~3/2 = 698.068

Optimal ET sequence: 13, 26, 91cc

Badness (Sintel): 52.583