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{{Infobox ET}}
{{ED intro}}
== Theory ==
== Theory ==
{{Harmonics in equal|28}}
{{Harmonics in equal|28|start=12|collapsed=1|intervals=odd}}


28edo, a multiple of both [[7edo]] and [[14edo]] (and of course [[2edo]] and [[4edo]]), has a step size of 42.857 [[cent]]s. 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 its major third is less than 1 cent flat (and its inversion the minor sixth less than 1 cent sharp). 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.
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 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 [[augmented_triad|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.
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.
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 ==
== Intervals ==
The following table compares it to potentially useful nearby [[just interval]]s.
The following table compares it to potentially useful nearby [[just interval]]s.


Line 17: Line 22:
! 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
Line 46: Line 51:
| 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
Line 82: Line 87:
| 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
Line 118: Line 123:
| 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
Line 154: Line 159:
| 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
Line 190: Line 195:
| 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
Line 226: Line 231:
| 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
Line 242: Line 247:
| 1028.57
| 1028.57
| [[20/11]]
| [[20/11]]
| 1035
| 1035.00
| -6.43
| -6.43
| 7th
| 7th
Line 262: Line 267:
| 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
Line 285: Line 290:
|}
|}


== 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.


Line 299: Line 316:
* 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
Line 327: Line 344:
| 1
| 1
| 11\28
| 11\28
| [[Oneirotonic]]
| [[A-team]]
|-
|-
| 1
| 1
Line 339: Line 356:
| 2
| 2
| 3\28
| 3\28
|
| [[Octokaidecal]]
|-
|-
| 2
| 2
Line 366: Line 383:
|}
|}


== Commas ==
=== Commas ===
 
28et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 28 44 65 79 97 104 }}.
28 EDO tempers out the following [[comma]]s. (Note: 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]]
Line 381: Line 397:
| 3
| 3
| [[2187/2048]]
| [[2187/2048]]
| {{Monzo| -11 7 }}
| {{monzo| -11 7 }}
| 113.69
| 113.69
| Lawa
| Lawa
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| 5
| 5
| [[648/625]]
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| {{monzo| 3 4 -4 }}
| 62.57
| 62.57
| Quadgu
| Quadgu
| Major diesis, diminished comma
| Diminished comma, major diesis
|-
|-
| 5
| 5
| [[16875/16384]]
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| {{monzo| -14 3 4 }}
| 51.12
| 51.12
| Laquadyo
| Laquadyo
Line 402: Line 418:
| 5
| 5
| [[393216/390625|(12 digits)]]
| [[393216/390625|(12 digits)]]
| {{Monzo| 17 1 -8 }}
| {{monzo| 17 1 -8 }}
| 11.45
| 11.45
| Saquadbigu
| Saquadbigu
Line 409: Line 425:
| 7
| 7
| [[36/35]]
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal quartertone
| Mint comma, septimal quartertone
|-
|-
| 7
| 7
| [[50/49]]
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, Jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
| [[3125/3087]]
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| {{monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
| Gariboh
| Gariboh comma
|-
|-
| 7
| 7
| [[126/125]]
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotrigu
| Septimal semicomma, Starling comma
| Starling comma, septimal semicomma
|-
|-
| 7
| 7
| [[65625/65536]]
| [[65625/65536]]
| {{Monzo| -16 1 5 1 }}
| {{monzo| -16 1 5 1 }}
| 2.35
| 2.35
| Lazoquinyo
| Lazoquinyo
| Horwell
| Horwell comma
|-
|-
| 7
| 7
| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>
| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>
| {{Monzo| 47 -7 -7 -7 }}
| {{monzo| 47 -7 -7 -7 }}
| 0.34
| 0.34
| Trisa-seprugu
| Trisa-seprugu
| [[Akjaysma]], 5\7-octave comma
| [[Akjaysma]]
|-
|-
| 11
| 11
| [[176/175]]
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
Line 458: Line 474:
| 11
| 11
| [[441/440]]
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozogu
Line 465: Line 481:
| 11
| 11
| [[4000/3993]]
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3 }}
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| 3.03
| Triluyo
| Triluyo
| Wizardharry
| Wizardharry comma
|}
|}
<references/>
<references/>


== Scales ==
== 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.
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 Major [14] 13131313131313
* Whitewood Major [14] 13131313131313
* Whitewood Minor [14] 31313131313131
* Whitewood Minor [14] 31313131313131
* Whitewood Major [21] 121121121121121121121
* Whitewood Minor [21] 211211211211211211211
* Whitewood Diminished [21] 112112112112112112112
* (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)
* (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)


Line 484: Line 502:
* Negri [9] 333343333
* Negri [9] 333343333
* Negri [10] 3333333331
* Negri [10] 3333333331
* Negri [19] 2121212121212121211


However, unlike 15, 28 is complex enough to do recognisable approximations of various diatonic scales and their modes, although they will sound noticeably out of tune and it's obviously not the best method of using the temperament.  
However, unlike 15, 28 is complex enough to do recognisable approximations of various diatonic scales and their modes, although they will sound noticeably out of tune and it's obviously not the best method of using the temperament.  
Line 489: Line 508:
* Diatonic Major [7] 5434552
* Diatonic Major [7] 5434552
* Diatonic Minor [7] 5254345
* Diatonic Minor [7] 5254345
* Diatonic [[Naive_scale|Naive]] Major [7] 4534543
* Diatonic Naive Minor [7] 4354345
* Diatonic Major [10] 3243432322
* Diatonic Minor [10] 3223243432
* Diatonic Major [12] 322232232322
* Diatonic Minor [12] 322322232232
* Diatonic Major [16] 2122221222122122
* Diatonic Minor [16] 2122212222122212
* Harmonic Minor [7] 5254372
* Harmonic Minor [7] 5254372
* Harmonic Major [7] 5434372
* Harmonic Major [7] 5434372
* Harmonic Minor [8] 52543522, 52543432
* Harmonic Major [8] 54343522, 54343432
* Harmonic Minor [10] 3223243432
* Harmonic Minor [11] 32232433222
* Harmonic Major [9] 324343432
* Harmonic Major [10] 3243433222
* Harmonic Minor [12] 322322232232, 322322233222
* Harmonic Major [12] 322232232232, 322232233222
* Harmonic Minor [16] 2122212222122212, 212221222212121222
* Harmonic Major [16] 2122221222122212, 212221222212121222
* Melodic Minor [7] 5254552
* Melodic Minor [7] 5254552
* Melodic Major [7] 5434345
* Melodic Major [7] 5434345
* Melodic Minor [11] 32232432322
* Melodic Major [9] 324343432
* [[Diasem]] (Right-handed) 414434143
* [[Diasem]] (Left-handed) 441434143
* Melodic Minor [12] 322322232322
* Melodic Major [12] 322232232232
* Melodic Minor [16] 2122212222122122
* Melodic Major [16] 2122221222122212


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.
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.
Line 499: Line 544:
* Oneirotonic [8] 55155151
* Oneirotonic [8] 55155151
* Oneirotonic [13] 4141141411411
* Oneirotonic [13] 4141141411411
* Oneirotonic [18] 311311131131113111
* Pathological Oneirotonic [23] 21112111121112111121111
* [[machine5]]
* [[machine5]]
* [[machine6]]
* [[machine6]]
* [[machine11]]
* [[machine11]]
* [[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 ==
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=d_55LCULX9g ''28 Mansions of the Moon'']


* [http://www.youtube.com/watch?v=26UpCbrb3mE 28 tone Prelude] by Kosmorksy
; [[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:Theory]]
[[Category:28edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave]]
[[Category:Twentuning]]
[[Category:Twentuning]]
 
[[Category:Listen]]
[[Category:Todo:unify precision]]
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