13edo: Difference between revisions

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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, 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.

Additionally, 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 2.9.5.21.11.13.17.19 subgroup being a particularly good example. In this subgroup, all 21-odd-limit intervals have less than 25% relative error (23.1{{c}}), except for 22/19 and its [[octave complement]], which barely miss with 25.045% relative error. 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 (135/128 is a [[Wikipedia:Continued_fraction|semiconvergent]] to 2<sup>1/13</sup>).

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13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].

The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes ~~this~~ proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.) A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.

The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes the proximity of 135/128 to 1\13 through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.) A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.

== Intervals ==

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! #

! Cents

! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref>

! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.9.5.21.11.13.17.19 subgroup temperament; other approaches are possible.</ref>

![[Erv Wilson's Linear Notations|Erv Wilson]]

! Archaeotonic

@@ Line 13: / Line 13: @@
 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, 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.
+Additionally, 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 2.9.5.21.11.13.17.19 subgroup being a particularly good example. In this subgroup, all 21-odd-limit intervals have less than 25% relative error (23.1{{c}}), except for 22/19 and its [[octave complement]], which barely miss with 25.045% relative error. 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 (135/128 is a [[Wikipedia:Continued_fraction|semiconvergent]] to 2<sup>1/13</sup>).
@@ Line 26: / Line 26: @@
 edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].
-The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes this proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.)  A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.
+The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes the proximity of 135/128 to 1\13 through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.)  A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.
 == Intervals ==
@@ Line 36: / Line 36: @@
 ! #
 ! Cents
-! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref>
+! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.9.5.21.11.13.17.19 subgroup temperament; other approaches are possible.</ref>
 ![[Erv Wilson's Linear Notations|Erv Wilson]]
 ! Archaeotonic