45edo: Difference between revisions

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45edo effectively has two approximate 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 a flat-tending system in the [[7-limit]], with 3, 5, and 7 all flat, but the 11 is sharp.

It provides the [[optimal patent val]] for [[flattone]] temperament, 7-limit rank-3 [[avicennmic]] temperament [[tempering out]] [[525/512]], the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank-4 temperament tempering out ~~45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and~~ 45/44. It is also the unique equal temperament tuning whose patent val tempers out both the syntonic comma and the [[ennealimma]].

It tempers out 81/80, 3125/3087, 525/512, and 875/864 in the 7-limit, and 45/44 in the [[11-limit]]. It provides the [[optimal patent val]] for 7- and 11-limit [[flattone]] temperament, and the 45f val is an excellent tuning for 13-limit flattone. It also provides the optimal patent val 7-limit rank-3 [[avicennmic]] temperament [[tempering out]] [[525/512]], the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank-4 temperament tempering out 45/44. It is also the unique equal temperament tuning whose patent val tempers out both the syntonic comma and the [[ennealimma]].

45edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.7/3.33 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. A bit more broadly, it maps the 2.17.25.27.33.63.65 subgroup to great precision; this is the part of the [[17-limit]] shared with [[270edo]].

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 edo effectively has two approximate 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 a flat-tending system in the [[7-limit]], with 3, 5, and 7 all flat, but the 11 is sharp.
-It provides the [[optimal patent val]] for [[flattone]] temperament, 7-limit rank-3 [[avicennmic]] temperament [[tempering out]] [[525/512]], the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank-4 temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is also the unique equal temperament tuning whose patent val tempers out both the syntonic comma and the [[ennealimma]].
+It tempers out 81/80, 3125/3087, 525/512, and 875/864 in the 7-limit, and 45/44 in the [[11-limit]]. It provides the [[optimal patent val]] for 7- and 11-limit [[flattone]] temperament, and the 45f val is an excellent tuning for 13-limit flattone. It also provides the optimal patent val 7-limit rank-3 [[avicennmic]] temperament [[tempering out]] [[525/512]], the 11-limit [[calliope]] temperament tempering out [[45/44]] and [[81/80]], and the rank-4 temperament tempering out 45/44. It is also the unique equal temperament tuning whose patent val tempers out both the syntonic comma and the [[ennealimma]].
 edo tempers out the [[quartisma]] and provides an excellent tuning for the 2.7/3.33 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. A bit more broadly, it maps the 2.17.25.27.33.63.65 subgroup to great precision; this is the part of the [[17-limit]] shared with [[270edo]].