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.

Furthermore, one step of 13edo is very close to 135/128, a notable 5-limit interval, in fact it is a semiconvergent to the said representation. Hence, considered below is aluminium.

Temperaments discussed elsewhere are tridecatonic, trideci, and triskaidekic.

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