28edo: Difference between revisions

{{Harmonics in equal|28}}

{{Harmonics in equal|28|start=12|collapsed=1|intervals=odd}}

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 log₂5 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.

Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.41.

28edo is the 2nd perfect number EDO.

== Intervals ==

The following table compares it to potentially useful nearby [[just interval]]s.

== Notation ==

This notation uses the same sagittal sequence as EDOs [[23edo#Sagittal notation|23]] and [[33edo#Sagittal notation|33]], and is a superset of the notations for EDOs [[14edo#Sagittal notation|14]] and [[7edo#Sagittal notation|7]].

File:28-EDO_Sagittal.svg

rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 447 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]

default [[File:28-EDO_Sagittal.svg]]

</imagemap>

* 0-8-16 = C E G = C = C or C perfect

* 0-7-16 = C vE G = Cv = C down

* 0-9-16 = C ^E G = C^ = C up

* 0-8-15 = C E vG = C(v5) = C down-five

* 0-9-17 = C ^E ^G = C^(^5) = C up up-five

* 0-8-16-24 = C E G B = C7 = C seven

* 0-8-16-23 = C E G vB = C,v7 = C add down-seven

* 0-7-16-24 = C vE G B = Cv,7 = C down add seven

* 0-7-16-23 = C vE G vB = Cv7 = C down-seven

 

! Periods<br>per 8ve

! Temperaments

|-

|-

|-

|-

 

=== Commas ===

 

{| class="commatable wikitable center-all left-3 right-4 left-6"

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

! Name(s)

|-

| {{monzo| 17 1 -8 }}

|-

|-

| [[65625/65536]]

| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>

 

== Scales ==

28edo is particularly well suited to Whitewood in the same way that [[15edo|15edo]] is for Blackwood, as it has one third that is heavily tempered, but in a familiar way shared with 12edo, while the other one is significantly closer to just. (Contrast with 20 & 21edo, where one third remains the same as in 12, and the other is pushed further away, making the overall sound considerably more xenharmonic) This makes the 5ths more out of tune, but in a useful way, as you can stack major and minor thirds indefinitely until they repeat 4 octaves and 14 notes up, producing one of the largest nonrepeating harmonious chords possible in an edo this low. This produces mode of symmetry scales with two different modes and 4 different keys, making it equally easy to establish any chord in the scale as the root and modulate between them.

 

* (Whitewood neutral is also theoretically possible, stacking neutral or subminor & supermajor thirds, but in practice that works out as 22222222222222, or 14edo, so it doesn't count as a 28edo scale)

 

If you're looking for something a little more xenharmonic sounding, but still relatively low in badness, Negri [9] & [10] function very similarly to Porcupine [7] & [8] in 15edo, further cementing the idea that 28 is to 15 what 29 is to 12 - a similar structure, but with tempering errors in the opposite direction and slightly greater complexity. Both have a decent selection of familiar major and minor chords clustered towards one end of the scale, while the way they modulate between one-another and the melodies they form is quite alien to people used to the diatonic scale. Most of the compositional notes on the Porcupine page can also be applied here.

 

 

 

 

Interestingly, as it has a near perfect 21/16, 28edo can also generate Oneirotonic scales (see [[13edo|13edo]]) by stacking it's 11th degree, and they actually sound better in this temperament.

 

 

 

28edo can also be played on a [[14edo]] [[guitar]] with very little effort. See [[User:MisterShafXen/Skip fretting system 28 2 3]].

 

== Music ==

* [https://www.youtube.com/watch?v=d_55LCULX9g ''28 Mansions of the Moon'']

 

; [[Beheld]]

 

; [[Bryan Deister]]

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

 

; [[duckapus]]

 

; [[User:Eliora|Eliora]]

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

 

; [[Kosmorksy]]

 

; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=30XUKJsaINU ''Happy Birthday Canon'', 5-in-1 Canon in 28edo]

* [https://www.youtube.com/watch?v=wYdRAzp8Qi0 ''Canon on Twinkle Twinkle Little Star'', for Baroque Oboe & Viola] (2023) – ([https://www.youtube.com/watch?v=B1KZoLR8UTI for Organ])

 

; [[NullPointerException Music]]

* [https://www.youtube.com/watch?v=vw6l5b3oGk0 ''Edolian - Machinery''] (2020)

 

; [[User:Userminusone|Userminusone]]

* [https://youtu.be/NbR3i45qQVQ ''Purple Skyes'']