119edo: Difference between revisions

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== Theory ==

119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

~~== Theory ==~~

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo) and 7-limit [[mothra]], with the 119be val supporting 11-limit mothra with a flat tendency.

It is ~~notable~~ for ~~offering~~ as ~~many~~ as ~~four different options for fifths:~~

~~68 steps: 7edo superflat fifth: 17 17 17 17 17 17 17~~

=== Odd harmonics ===

{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

~~69 steps: Approximation~~ of ~~2/7 comma meantone: 19 19 19 12 19 19 19 19 12~~

=== Subsets and supersets ===

Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

~~70 steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1~~

== Intervals ==

~~71 steps~~: Approximation of half comma ~~eventone~~: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

== Scales ==

* Approximation of 2/7 comma [[meantone]]: 19 19 12 19 19 19 12

* Approximation of half comma [[archy]]: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

~~=== Harmonics ===~~

~~{{Harmonics in equal|119}}~~

[[Category:Equal divisions of the octave|###]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
+{{ED intro}}
+== Theory ==
+edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.
-== Theory ==
+Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo) and 7-limit [[mothra]], with the 119be val supporting 11-limit mothra with a flat tendency.
-It is notable for offering as many as four different options for fifths:
-steps: 7edo superflat fifth: 17 17 17 17 17 17 17
+=== Odd harmonics ===
+{{Harmonics in equal|119|columns=12}}
+{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}
-steps: Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
+=== Subsets and supersets ===
+Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.
-steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1
+== Intervals ==
+{{Interval table}}
-steps: Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2
+== Scales ==
+* Approximation of 2/7 comma [[meantone]]: 19 19 12 19 19 19 12
+* Approximation of half comma [[archy]]: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2
-=== Harmonics ===
-{{Harmonics in equal|119}}
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->