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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>2</sup> × 7
{{ED intro}}
| Step size = 42.857¢
| Fifth = 16\28 = 685.714¢ (&rarr;[[7edo|4\7]])
| Major 2nd = 4\28 = 171.428¢
| Semitones = 0:4 (0¢:171.428¢)
}}
'''28 equal divisions of the octave''' ('''28edo''') is the [[tuning system]] derived by dividing the [[octave]] into 28 equal steps of approximately 42.857{{cent}} each.
 
== Theory ==
== Theory ==
{{Harmonics in equal|28}}
{{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 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|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]].
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! colspan="2" | Just (j)
! colspan="2" | Just (j)
! rowspan="2" | Delta <br> (e-j)
! rowspan="2" | Delta <br> (e-j)
! rowspan="2" colspan="3" | [[Ups and Downs Notation]]
! rowspan="2" colspan="3" | [[Ups and downs notation]]
|-
|-
! Cents
! Cents
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| 84.47
| 84.47
| 1.24
| 1.24
| double-up 1sn, double-down 2nd
| dup 1sn, dud 2nd
| ^^1, vv2
| ^^1, vv2
| ^^D, vvE
| ^^D, vvE
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| 266.87
| 266.87
| -9.73
| -9.73
| double-up 2nd, double-down 3rd
| dup 2nd, dud 3rd
| ^^2, vv3
| ^^2, vv3
| ^^E, vvF
| ^^E, vvF
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| 435.08
| 435.08
| -6.51
| -6.51
| double-up 3rd, double-down 4th
| dup 3rd, dud 4th
| ^^3, vv4
| ^^3, vv4
| ^^F, vvG
| ^^F, vvG
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| 582.51
| 582.51
| 17.49
| 17.49
| double-up 4th, double-down 5th
| dup 4th, dud 5th
| ^^4, vv5
| ^^4, vv5
| ^^G, vvA
| ^^G, vvA
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| 764.92
| 764.92
| 6.51
| 6.51
| double-up 5th, double-down 6th
| dup 5th, dud 6th
| ^^5, vv6
| ^^5, vv6
| ^^A, vvB
| ^^A, vvB
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| 933.13
| 933.13
| 9.73
| 9.73
| double-up 6th, double-down 7th
| dup 6th, dud 7th
| ^^6, vv7
| ^^6, vv7
| ^^B, vvC
| ^^B, vvC
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| 1028.57
| 1028.57
| [[20/11]]
| [[20/11]]
| 1035
| 1035.00
| -6.43
| -6.43
| 7th
| 7th
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| 1115.53
| 1115.53
| -1.24
| -1.24
| double-up 7th, double-down 8ve
| dup 7th, dud 8ve
| ^^7, vv8
| ^^7, vv8
| ^^C, vvD
| ^^C, vvD
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|}
|}


== Chord Names ==
== Notation ==
=== Sagittal 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]].
 
<imagemap>
File:28-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
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>
 
== Chord names ==
Ups and downs can be used to name 28edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.
Ups and downs can be used to name 28edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.


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


For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].


== Rank two temperaments ==
== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Periods <br> per octave
! Periods<br>per 8ve
! Generator
! Generator
! Temperaments
! Temperaments
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| 1
| 1
| 11\28
| 11\28
| [[Oneirotonic]]
| [[A-team]]
|-
|-
| 1
| 1
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| 2
| 2
| 3\28
| 3\28
| [[Trienstonic_clan#Octokaidecal|Octokaidecal]]
| [[Octokaidecal]]
|-
|-
| 2
| 2
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|}
|}


== Commas ==
=== Commas ===
28 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 28 44 65 79 97 104 }}.)
28et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 28 44 65 79 97 104 }}.


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic Limit|Prime<br>Limit]]
! [[Harmonic Limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
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| 62.57
| 62.57
| Quadgu
| Quadgu
| Major diesis, diminished comma
| Diminished comma, major diesis
|-
|-
| 5
| 5
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| 48.77
| 48.77
| Rugu
| Rugu
| Septimal quartertone
| Mint comma, septimal quartertone
|-
|-
| 7
| 7
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| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, Jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
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| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
| Gariboh
| Gariboh comma
|-
|-
| 7
| 7
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| 13.79
| 13.79
| Zotrigu
| Zotrigu
| Septimal semicomma, Starling comma
| Starling comma, septimal semicomma
|-
|-
| 7
| 7
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| 2.35
| 2.35
| Lazoquinyo
| Lazoquinyo
| Horwell
| Horwell comma
|-
|-
| 7
| 7
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| 0.34
| 0.34
| Trisa-seprugu
| Trisa-seprugu
| [[Akjaysma]], 5\7-octave comma
| [[Akjaysma]]
|-
|-
| 11
| 11
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| 3.03
| 3.03
| Triluyo
| Triluyo
| Wizardharry
| Wizardharry comma
|}
|}
<references/>
<references/>
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* Diatonic Major [7] 5434552
* Diatonic Major [7] 5434552
* Diatonic Minor [7] 5254345
* Diatonic Minor [7] 5254345
* Diatonic Naive Major [7] 4534543
* Diatonic [[Naive_scale|Naive]] Major [7] 4534543
* Diatonic Naive Minor [7] 4354345
* Diatonic Naive Minor [7] 4354345
* Diatonic Major [10] 3243432322
* Diatonic Major [10] 3243432322
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* [[machine11]]
* [[machine11]]
* [[machine17]]
* [[machine17]]
== Instruments ==
28edo can be played on the [[Lumatone]]. See [[Lumatone mapping for 28edo]].
28edo can also be played on a [[14edo]] [[guitar]] with very little effort. See [[User:MisterShafXen/Skip fretting system 28 2 3]].


== Music ==
== Music ==
* [http://www.youtube.com/watch?v=26UpCbrb3mE 28 tone Prelude] by Kosmorksy
; [[Ambient Esoterica]]
* [https://youtu.be/NbR3i45qQVQ Purple Skyes] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=d_55LCULX9g ''28 Mansions of the Moon'']
* [https://www.youtube.com/watch?v=ghVCGlm7yOk Fantasy for Piano] by [[User:Eliora|Eliora]]
 
; [[Beheld]]
* [https://www.youtube.com/watch?v=0nvrUbw1VLQ ''Haze vibe'']
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/--BIQKJ9uvI ''minuet in 28edo''] (2025)
 
; [[duckapus]]
* [https://www.youtube.com/watch?v=F74B9qUpYi8 ''G.27 Variations in 28edo''] (2023)
 
; [[User:Eliora|Eliora]]
* [https://www.youtube.com/watch?v=ghVCGlm7yOk ''Fantasy for Piano'']
 
; [[Kosmorksy]]
* [https://www.youtube.com/watch?v=26UpCbrb3mE ''28 tone Prelude'']
 
; [[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'']


[[Category:28edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Twentuning]]
[[Category:Twentuning]]
 
[[Category:Listen]]
[[Category:Todo:unify precision]]
Retrieved from "https://en.xen.wiki/w/28edo"