13edo: Difference between revisions

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13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. It is the sixth [[prime_numbers|prime]] edo, following [[11edo|11edo]] and coming before [[17edo|17edo]]. The steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.

As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, 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 temperament on the '''2.5.9.11.13.17.19.21''' subgroup~~, and~~ has a substantial repertoire of complex consonances for its small size.

As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, 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.

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 edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. It is the sixth [[prime_numbers|prime]] edo, following [[11edo|11edo]] and coming before [[17edo|17edo]]. The steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.
-As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, 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 temperament on the '''2.5.9.11.13.17.19.21''' subgroup, and has a substantial repertoire of complex consonances for its small size.
+As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, 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.
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