45edo: Difference between revisions

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It tempers out [[81/80]], [[525/512]], [[875/864]], and [[3125/3087]] 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 for the 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]].

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 2.27.25.63.33.65.17 subgroup to great precision; this is the part of the [[17-limit]] shared with [[270edo]].

Otherwise, it can be treated as a 2.5/3.7/3 subgroup system (borrowing 5/3 from [[15edo]] and 7/3 from [[9edo]]) and is a good tuning for [[gariberttet]], defined by tempering out [[3125/3087]] in this subgroup, approximating 2/5-comma gariberttet.

Otherwise, it can be treated as a 2.5/3.7/3-subgroup system (borrowing 5/3 from [[15edo]] and 7/3 from [[9edo]]) and is a good tuning for [[gariberttet]], defined by tempering out [[3125/3087]] in this subgroup, approximating 2/5-comma gariberttet.

== Intervals ==

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 It tempers out [[81/80]], [[525/512]], [[875/864]], and [[3125/3087]] 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 for the 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]].
+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 2.27.25.63.33.65.17 subgroup to great precision; this is the part of the [[17-limit]] shared with [[270edo]].
-Otherwise, it can be treated as a 2.5/3.7/3 subgroup system (borrowing 5/3 from [[15edo]] and 7/3 from [[9edo]]) and is a good tuning for [[gariberttet]], defined by tempering out [[3125/3087]] in this subgroup, approximating 2/5-comma gariberttet.
+Otherwise, it can be treated as a 2.5/3.7/3-subgroup system (borrowing 5/3 from [[15edo]] and 7/3 from [[9edo]]) and is a good tuning for [[gariberttet]], defined by tempering out [[3125/3087]] in this subgroup, approximating 2/5-comma gariberttet.
 == Intervals ==