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.

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.

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.

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.

=== Odd harmonics ===

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

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

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.

== Intervals ==

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! #

! Cents

! colspan="3" |[[Ups and ~~Downs Notation~~|Up/down notation]] using the wide 5th of 8\13

! colspan="3" |[[Ups and downs notation|Up/down notation]] using the wide 5th of 8\13

|-

| 0

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! #

! Cents

! colspan="3" |[[Ups and ~~Downs Notation~~|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor

! colspan="3" |[[Ups and downs notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor

! colspan="3" | Up/down notation using the narrow 5th of 7\13, <br> with major narrower than minor

|-

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Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.

~~=== Pathological Modes ===~~

~~2 1 1 1 1 2 1 1 1 1 1 [[2L 9s]] MOS~~

~~3 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS~~

~~2 1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS~~

== Harmony in 13edo ==

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== Regular temperament properties ==

=== Uniform maps ===

=== Commas ===

@@ Line 9: / Line 9: @@
 == 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.
+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.
+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.
-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.
 === Odd harmonics ===
 {{Harmonics in equal|13}}
@@ Line 18: / Line 26: @@
 edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].
-One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction|semiconvergent]]). 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.
+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.
 == Intervals ==
@@ Line 216: / Line 224: @@
 ! #
 ! Cents
-! colspan="3" |[[Ups and Downs Notation|Up/down notation]] using the wide 5th of 8\13
+! colspan="3" |[[Ups and downs notation|Up/down notation]] using the wide 5th of 8\13
 |-
 | 0
@@ Line 319: / Line 327: @@
 ! #
 ! Cents
-! colspan="3" |[[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor
+! colspan="3" |[[Ups and downs notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor
 ! colspan="3" | Up/down notation using the narrow 5th of 7\13, <br> with major narrower than minor
 |-
@@ Line 1,165: / Line 1,173: @@
 Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.
-=== Pathological Modes ===
-1 1 1 1 2 1 1 1 1 1 [[2L 9s]] MOS
-1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
-1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS
 == Harmony in 13edo ==
@@ Line 1,428: / Line 1,429: @@
 == Regular temperament properties ==
 === Uniform maps ===
-{{Uniform map|11|12.9|13.1}}
+{{Uniform map|edo=13}}
 === Commas ===