13edo: Difference between revisions

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As a temperament of [[21-odd-limit]] [[just intonation]], 13edo has excellent approximations to the 11th and 21st [[harmonic]]s, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.

~~13edo is the smallest EDO that shares the step size of its fifth with the next EDO–8\13 and 8\14 are both patent val fifths for their respective EDOs.~~

=== Odd harmonics ===

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 As a temperament of [[21-odd-limit]] [[just intonation]], 13edo has excellent approximations to the 11th and 21st [[harmonic]]s, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.
-edo is the smallest EDO that shares the step size of its fifth with the next EDO–8\13 and 8\14 are both patent val fifths for their respective EDOs.
 === Odd harmonics ===
 {{Harmonics in equal|13}}