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 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~~. It is also the unique equal temperament tuning that [[tempers out]] both the [[syntonic comma]] and the [[ennealimma]]~~.

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.

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]].

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 that 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]].

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.

=== Odd harmonics ===

=== Octave stretch ===

~~45edo’s~~ approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[Gallery of arithmetic pitch sequences#APS of farabs|APS3.21farab]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1.

45edo's approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[Gallery of arithmetic pitch sequences #APS of farabs|APS3.21farab]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1.

The tuning [[126ed7]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1.

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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 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. It is also the unique equal temperament tuning that [[tempers out]] both the [[syntonic comma]] and the [[ennealimma]].
+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.
-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]].
+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 that 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]].
 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.
 === Odd harmonics ===
-{{harmonics in equal|45}}
+{{Harmonics in equal|45}}
 === Octave stretch ===
-edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[Gallery of arithmetic pitch sequences#APS of farabs|APS3.21farab]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1.
+edo's approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[Gallery of arithmetic pitch sequences #APS of farabs|APS3.21farab]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1.
 The tuning [[126ed7]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1.