45edo: Difference between revisions

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== Theory ==

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 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 525/512 [[planar]] 7-limit [[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 the [[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]].

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 {{EDO intro|45}}
 == Theory ==
-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 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 525/512 [[planar]] 7-limit [[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 the [[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]].