13edo: Difference between revisions

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== Theory ==

In 13edo, ~~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~~.

13edo has a sharp fifth of 738c, which serves sort of an opposite role to 9edo's flat fifth of 667 cents (in fact, they are both separated from 3/2 by approximately the same amount in opposite directions). Notably for scale theory, this sharp fifth is extremely close to the [[Logarithmic phi|golden generator]] of 741 cents, and so 13edo has the MOS scales 2L 1s, 3L 2s, and 5L 3s and functions as an equalized 8L 5s.

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.

The simplest JI interpretation of 13edo is in the 2.5.11 subgroup, in which it approximates intervals such as 11/10, 121/80, and 64/55. However, it notably has very good approximations to 13, 17, and 19 as well.

Additionally, as a temperament of [[21-odd-limit]] [[just intonation]], 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. 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.

One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction|semiconvergent]]).

In 13edo, 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.

=== Odd harmonics ===

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 == Theory ==
-In 13edo, 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.
+edo has a sharp fifth of 738c, which serves sort of an opposite role to 9edo's flat fifth of 667 cents (in fact, they are both separated from 3/2 by approximately the same amount in opposite directions). Notably for scale theory, this sharp fifth is extremely close to the [[Logarithmic phi|golden generator]] of 741 cents, and so 13edo has the MOS scales 2L 1s, 3L 2s, and 5L 3s and functions as an equalized 8L 5s.
-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.
+The simplest JI interpretation of 13edo is in the 2.5.11 subgroup, in which it approximates intervals such as 11/10, 121/80, and 64/55. However, it notably has very good approximations to 13, 17, and 19 as well.
+Additionally, as a temperament of [[21-odd-limit]] [[just intonation]], 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. 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.
 One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction|semiconvergent]]).
+In 13edo, 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.
 === Odd harmonics ===