45edo: Difference between revisions

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== Theory ==

45edo has two major thirds, each almost equally far from ~~Just~~, but as the flat one is slightly closer, it qualifies as a meantone temperament, forming a good approximation to 2/5 comma meantone. It is the [[optimal patent val]] for [[flattone]] temperament, the [[7-limit]] 525/512 planar [[Avicennmic_temperaments|avicennmic]] temperament, the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp. ~~Also~~ [[~~support|supports~~]] ~~a very inaccurate version of~~ [[~~ennealimmal~~]], ~~if you want to view~~ it that ~~way~~.

45edo effectively has two major thirds, each almost equally far from just, but as the flat one is slightly closer, it qualifies as a meantone temperament, forming a good approximation to 2/5 comma meantone. It is the [[optimal patent val]] for [[flattone]] temperament, the [[7-limit]] 525/512 planar [[Avicennmic_temperaments|avicennmic]] temperament, the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp.

45edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.33/32.7/6 subgroup [[The Quartercache#Direct quartismic|direct quartismic]] temperament, in which it approximates 33/32 quartertone with 2 steps and 7/6 with 10 steps. It is also the unique equal temperament tuning that tempers out both the syntonic comma and the [[ennealimma]].

45edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.33/32.7/6 subgroup [[The Quartercache#Direct quartismic|direct quartismic]] temperament, in which it approximates 33/32 quartertone with 2 steps and 7/6 with 10 steps.

=== Odd harmonics ===

== Intervals ==

{| class="wikitable mw-collapsible mw-collapsed right-all center-3 left-6 center-7"

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 {{EDO intro|45}}
 == Theory ==
-edo has two major thirds, each almost equally far from Just, but as the flat one is slightly closer, it qualifies as a meantone temperament, forming a good approximation to 2/5 comma meantone. It is the [[optimal patent val]] for [[flattone]] temperament, the [[7-limit]] 525/512 planar [[Avicennmic_temperaments|avicennmic]] temperament, the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp. Also [[support|supports]] a very inaccurate version of [[ennealimmal]], if you want to view it that way.
+edo effectively has two major thirds, each almost equally far from just, but as the flat one is slightly closer, it qualifies as a meantone temperament, forming a good approximation to 2/5 comma meantone. It is the [[optimal patent val]] for [[flattone]] temperament, the [[7-limit]] 525/512 planar [[Avicennmic_temperaments|avicennmic]] temperament, the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp.
+edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.33/32.7/6 subgroup [[The Quartercache#Direct quartismic|direct quartismic]] temperament, in which it approximates 33/32 quartertone with 2 steps and 7/6 with 10 steps. It is also the unique equal temperament tuning that tempers out both the syntonic comma and the [[ennealimma]].
-edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.33/32.7/6 subgroup [[The Quartercache#Direct quartismic|direct quartismic]] temperament, in which it approximates 33/32 quartertone with 2 steps and 7/6 with 10 steps.
 === Odd harmonics ===
 {{harmonics in equal|45}}
 == Intervals ==
 {| class="wikitable mw-collapsible mw-collapsed right-all center-3 left-6 center-7"