28edo: Difference between revisions

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28edo is a multiple of both [[7edo]] and [[14edo]] (and of course [[2edo]] and [[4edo]]). It shares three intervals with [[12edo]]: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it [[tempering_out|tempers out]] the [[greater diesis]] [[648/~~625|648:~~625]]. It does not however temper out the [[128/~~125|128:~~125]] [[lesser_diesis|lesser diesis]], as 28 is not divisible by 3. It has the same perfect fourth and fifth as 7edo. It also has decent approximations of several septimal intervals, of which [[9/7]] and its inversion [[14/9]] are also found in 14edo. Its approximation to [[5/4]] is unusually good for an edo of this size, being the next convergent to log<sub>2</sub>5 after [[3edo]].

28edo is a multiple of both [[7edo]] and [[14edo]] (and of course [[2edo]] and [[4edo]]). It shares three intervals with [[12edo]]: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it [[tempering_out|tempers out]] the [[greater diesis]] [[648/625]]. It does not however temper out the [[128/125]] [[lesser_diesis|lesser diesis]], as 28 is not divisible by 3. It has the same perfect fourth and fifth as 7edo. It also has decent approximations of several septimal intervals, of which [[9/7]] and its inversion [[14/9]] are also found in 14edo. Its approximation to [[5/4]] is unusually good for an edo of this size, being the next convergent to log<sub>2</sub>5 after [[3edo]].

28edo can approximate the [[7-limit|7-limit]] subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to [[Semicomma_family|orwell temperament]] now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the [[Marvel_chords|augmented triad]] has a very low complexity, so many of them appear in the [[MOS scales]] for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.

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== Notation ==

=== Ups and downs notation ===

As 28edo tempers out the [[2187/2048|Pythagorean apotome]], the traditional sharps and flats have no ~~affect~~ on the pitch. Therefore, with [[ups and downs notation]], arrows are required.

As 28edo tempers out the [[2187/2048|Pythagorean apotome]], the traditional sharps and flats have no effect on the pitch. Therefore, with [[ups and downs notation]], arrows are required.

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; [[Bryan Deister]]

* [https://www.youtube.com/shorts/1nWL2qEcI-Q ''28edo improvisation''] (2022)

* [https://www.youtube.com/shorts/sCE0MjUyRUk ''28edo blues''] (2023)

* [https://www.youtube.com/shorts/--BIQKJ9uvI ''minuet in 28edo''] (2025)

* [https://www.youtube.com/shorts/vEcihDrmddg ''Fantasy in 28edo''] (2026)

; [[duckapus]]

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; [[User:Eliora|Eliora]]

* [https://www.youtube.com/watch?v=ghVCGlm7yOk ''Fantasy for Piano'']

* [https://www.youtube.com/watch?v=ghVCGlm7yOk ''Fantasy for Piano (and strings)''] (2021)

; [[Francium]]

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 {{Harmonics in equal|28|start=12|collapsed=1|intervals=odd}}
-edo is a multiple of both [[7edo]] and [[14edo]] (and of course [[2edo]] and [[4edo]]). It shares three intervals with [[12edo]]: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it [[tempering_out|tempers out]] the [[greater diesis]] [[648/625|648:625]]. It does not however temper out the [[128/125|128:125]] [[lesser_diesis|lesser diesis]], as 28 is not divisible by 3. It has the same perfect fourth and fifth as 7edo. It also has decent approximations of several septimal intervals, of which [[9/7]] and its inversion [[14/9]] are also found in 14edo. Its approximation to [[5/4]] is unusually good for an edo of this size, being the next convergent to log<sub>2</sub>5 after [[3edo]].
+edo is a multiple of both [[7edo]] and [[14edo]] (and of course [[2edo]] and [[4edo]]). It shares three intervals with [[12edo]]: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it [[tempering_out|tempers out]] the [[greater diesis]] [[648/625]]. It does not however temper out the [[128/125]] [[lesser_diesis|lesser diesis]], as 28 is not divisible by 3. It has the same perfect fourth and fifth as 7edo. It also has decent approximations of several septimal intervals, of which [[9/7]] and its inversion [[14/9]] are also found in 14edo. Its approximation to [[5/4]] is unusually good for an edo of this size, being the next convergent to log<sub>2</sub>5 after [[3edo]].
 edo can approximate the [[7-limit|7-limit]] subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to [[Semicomma_family|orwell temperament]] now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the [[Marvel_chords|augmented triad]] has a very low complexity, so many of them appear in the [[MOS scales]] for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.
@@ Line 292: / Line 292: @@
 == Notation ==
 === Ups and downs notation ===
-As 28edo tempers out the [[2187/2048|Pythagorean apotome]], the traditional sharps and flats have no affect on the pitch. Therefore, with [[ups and downs notation]], arrows are required.
+As 28edo tempers out the [[2187/2048|Pythagorean apotome]], the traditional sharps and flats have no effect on the pitch. Therefore, with [[ups and downs notation]], arrows are required.
 {{ups and downs sharpness}}
@@ Line 609: / Line 609: @@
 ; [[Bryan Deister]]
 * [https://www.youtube.com/shorts/1nWL2qEcI-Q ''28edo improvisation''] (2022)
+* [https://www.youtube.com/shorts/sCE0MjUyRUk ''28edo blues''] (2023)
 * [https://www.youtube.com/shorts/--BIQKJ9uvI ''minuet in 28edo''] (2025)
+* [https://www.youtube.com/shorts/vEcihDrmddg ''Fantasy in 28edo''] (2026)
 ; [[duckapus]]
@@ Line 615: / Line 617: @@
 ; [[User:Eliora|Eliora]]
-* [https://www.youtube.com/watch?v=ghVCGlm7yOk ''Fantasy for Piano'']
+* [https://www.youtube.com/watch?v=ghVCGlm7yOk ''Fantasy for Piano (and strings)''] (2021)
 ; [[Francium]]