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Line 1: Line 1:
{{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 intervals]].
{| class="wikitable center-all right-2 right-4 right-5 left-6"
 
{| class="wikitable center-all"
|-
|-
| rowspan="2" | Step #
! rowspan="2" | Step #
| style="text-align:center;" | ET
! style="text-align:center;" | ET (e)
| colspan="2" | Just  
! colspan="2" | Just (j)
| Difference <br> (ET minus Just)
! rowspan="2" | Delta <br> (e-j)
| colspan="3" | [[Ups and Downs Notation]]
! rowspan="2" colspan="3" | [[Ups and downs notation]]
|-
|-
| Cents
! Cents
| Interval
! Interval
| Cents
! Cents
|
|
|
|
|-
|-
| 0
| 0
| 0¢
| 0.00
|  
| 1/1
|  
| 0.00
|  
| 0.00
| unison
| unison
| 1
| 1
Line 38: Line 39:
| 1
| 1
| 42.86
| 42.86
| 41:40
| [[41/40]]
| 42.74
| 42.74
| 0.12
| 0.12
Line 47: Line 48:
| 2
| 2
| 85.71
| 85.71
| 21:20
| [[21/20]]
| 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 56: Line 57:
| 3
| 3
| 128.57
| 128.57
| 14:13
| [[14/13]]
| 128.3
| 128.30
| 0.27
| 0.27
| down 2nd
| down 2nd
Line 65: Line 66:
| 4
| 4
| 171.43
| 171.43
| 11:10
| [[11/10]]
| 165
| 165.00
| 6.43
| 6.43
| 2nd
| 2nd
Line 74: Line 75:
| 5
| 5
| 214.29
| 214.29
| 17:15
| [[17/15]]
| 216.69
| 216.69
| -2.40
| -2.40
Line 83: Line 84:
| 6
| 6
| 257.14
| 257.14
| 7:6
| [[7/6]]
| 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
|-
|-
| 7
| 7
| 300
| 300.00
| 6:5
| [[6/5]]
| 315.64
| 315.64
| -15.64
| -15.64
Line 101: Line 102:
| 8
| 8
| 342.86
| 342.86
| 11:9
| [[11/9]]
| 347.41
| 347.41
| -4.55
| -4.55
Line 110: Line 111:
| 9
| 9
| 385.71
| 385.71
| 5:4
| [[5/4]]
| 386.31
| 386.31
| -0.60
| -0.60
Line 119: Line 120:
| 10
| 10
| 428.57
| 428.57
| 9:7
| [[9/7]]
| 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 128: Line 129:
| 11
| 11
| 471.43
| 471.43
| 21:16
| [[21/16]]
| 470.78
| 470.78
| 0.65
| 0.65
Line 137: Line 138:
| 12
| 12
| 514.29
| 514.29
| 4:3
| [[4/3]]
| 498.045
| 498.04
| 16.245
| 16.25
| 4th
| 4th
| 4
| 4
Line 146: Line 147:
| 13
| 13
| 557.14
| 557.14
| 11:8
| [[11/8]]
| 551.32
| 551.32
| 5.82
| 5.82
Line 154: Line 155:
|-
|-
| 14
| 14
| 600
| 600.00
| 7:5
| [[7/5]]
| 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 164: Line 165:
| 15
| 15
| 642.86
| 642.86
| 16:11
| [[16/11]]
| 648.68
| 648.68
| -5.82
| -5.82
Line 173: Line 174:
| 16
| 16
| 685.71
| 685.71
| 3:2
| [[3/2]]
| 701.955
| 701.96
| -16.245
| -16.25
| 5th
| 5th
| 5
| 5
Line 182: Line 183:
| 17
| 17
| 728.57
| 728.57
| 32:21
| [[32/21]]
| 729.22
| 729.22
| -0.65
| -0.65
Line 191: Line 192:
| 18
| 18
| 771.43
| 771.43
| 14:9
| [[14/9]]
| 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 200: Line 201:
| 19
| 19
| 814.29
| 814.29
| 8:5
| [[8/5]]
| 813.68
| 813.68
| 0.61
| 0.61
Line 209: Line 210:
| 20
| 20
| 857.14
| 857.14
| 18:11
| [[18/11]]
| 852.59
| 852.59
| 4.55
| 4.55
Line 217: Line 218:
|-
|-
| 21
| 21
| 900
| 900.00
| 5:3
| [[5/3]]
| 884.36
| 884.36
| 15.64
| 15.64
Line 227: Line 228:
| 22
| 22
| 942.86
| 942.86
| 12:7
| [[12/7]]
| 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 236: Line 237:
| 23
| 23
| 985.71
| 985.71
| 30:17
| [[30/17]]
| 983.31
| 983.31
| 2.40
| 2.40
Line 245: Line 246:
| 24
| 24
| 1028.57
| 1028.57
| 20:11
| [[20/11]]
| 1035
| 1035.00
| -6.43
| -6.43
| 7th
| 7th
Line 254: Line 255:
| 25
| 25
| 1071.42
| 1071.42
| 13:7
| [[13/7]]
| 1071.70
| 1071.70
| -0.27
| -0.27
Line 263: Line 264:
| 26
| 26
| 1114.29
| 1114.29
| 40:21
| [[40/21]]
| 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 272: Line 273:
| 27
| 27
| 1157.14
| 1157.14
| style="text-align:center;" |80:41
| [[80/41]]
| 1157.26
| 1157.26
| -0.12
| -0.12
Line 280: Line 281:
|-
|-
| 28
| 28
| 1200
| 1200.00
| 2:1
| [[2/1]]
| 1200
| 1200.00
| 0
| 0.00
| 8ve
| 8ve
| 8
| 8
Line 289: 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 303: 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 340:
| 1
| 1
| 9\28
| 9\28
| [[Würschmidt_family#Worschmidt|Worschmidt]]
| [[Worschmidt]]
|-
|-
| 1
| 1
| 11\28
| 11\28
|[[Oneirotonic]]
| [[A-team]]
|-
|-
| 1
| 1
| 13\28
| 13\28
| [[Starling_temperaments#Thuja|Thuja]]
| [[Thuja]]
|-
|-
| 2
| 2
Line 343: Line 356:
| 2
| 2
| 3\28
| 3\28
|
| [[Octokaidecal]]
|-
|-
| 2
| 2
| 5\28
| 5\28
| [[antikythera|Antikythera]]
| [[Antikythera]]
|-
|-
| 4
| 4
Line 355: Line 368:
| 4
| 4
| 2\28
| 2\28
| [[Diminished#Demolished|Demolished]]
| [[Demolished]]
|-
|-
| 4
| 4
Line 363: Line 376:
| 7
| 7
| 1\28
| 1\28
| [[Apotome_family|Whitewood]]
| [[Whitewood]]
|-
|-
| 14
| 14
Line 370: 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="wikitable center-all left-2"
|-
|-
! [[Ratio]]
! [[Harmonic Limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
! [[Color name]]
! [[Color name]]
! Name 1
! Name(s)
! Name 2
|-
|-
| 2187/2048
| 3
| {{Monzo| -11 7 }}
| [[2187/2048]]
| {{monzo| -11 7 }}
| 113.69
| 113.69
| Lawa
| Lawa
| Apotome
| Apotome
|
|-
|-
| 648/625
| 5
| {{Monzo| 3 4 -4 }}
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| 62.57
| Quadgu
| Quadgu
| Major Diesis
| Diminished comma, major diesis
| Diminished Comma
|-
|-
| 16875/16384
| 5
| {{Monzo| -14 3 4 }}
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| 51.12
| Laquadyo
| Laquadyo
| Negri Comma
| Negri comma, double augmentation diesis
| Double Augmentation Diesis
|-
|-
|  
| 5
| {{Monzo| 17 1 -8 }}
| [[393216/390625|(12 digits)]]
| {{monzo| 17 1 -8 }}
| 11.45
| 11.45
| Saquadbigu
| Saquadbigu
| Wuerschmidt Comma
| [[Würschmidt comma]]
|
|-
|-
| 7
| [[36/35]]
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal Quarter Tone
| Mint comma, septimal quartertone
|
|-
|-
| 7
| [[50/49]]
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic Diesis
| Jubilisma, tritonic diesis
| Jubilisma
|-
|-
| 3125/3087
| 7
| {{Monzo| 0 -2 5 -3 }}
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
| Gariboh
| Gariboh comma
|
|-
|-
| 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, septimal semicomma
| Starling Comma
|-
|-
| 65625/65536
| 7
| {{Monzo| -16 1 5 1 }}
| [[65625/65536]]
| {{monzo| -16 1 5 1 }}
| 2.35
| 2.35
| Lazoquinyo
| Lazoquinyo
| Horwell
| Horwell comma
|
|-
|-
|  
| 7
| {{Monzo| 47 -7 -7 -7 }}
| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>
| {{monzo| 47 -7 -7 -7 }}
| 0.34
| 0.34
| Trisa-seprugu
| Trisa-seprugu
| Akjaysma
| [[Akjaysma]]
| 5\7 Octave Comma
|-
|-
| 176/175
| 11
| {{Monzo| 4 0 -2 -1 1 }}
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
| Valinorsma
| Valinorsma
|
|-
|-
| 441/440
| 11
| {{Monzo| -3 2 -1 2 -1 }}
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozogu
| Werckisma
| Werckisma
|
|-
|-
| 4000/3993
| 11
| {{Monzo| 5 -1 3 0 -3 }}
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| 3.03
| Triluyo
| Triluyo
| Wizardharry
| Wizardharry comma
|
|}
|}
<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 487: 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 492: 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 502: 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'']
; [[Beheld]]
* [https://www.youtube.com/watch?v=0nvrUbw1VLQ ''Haze vibe'']


* [http://www.youtube.com/watch?v=26UpCbrb3mE 28 tone Prelude] by Kosmorksy
; [[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]]

Latest revision as of 01:07, 20 August 2025

← 27edo 28edo 29edo →
Prime factorization 22 × 7
Step size 42.8571 ¢ 
Fifth 16\28 (685.714 ¢) (→ 4\7)
Semitones (A1:m2) 0:4 (0 ¢ : 171.4 ¢)
Dual sharp fifth 17\28 (728.571 ¢)
Dual flat fifth 16\28 (685.714 ¢) (→ 4\7)
Dual major 2nd 5\28 (214.286 ¢)
Consistency limit 5
Distinct consistency limit 5

28 equal divisions of the octave (abbreviated 28edo or 28ed2), also called 28-tone equal temperament (28tet) or 28 equal temperament (28et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 28 equal parts of about 42.9 ¢ each. Each step represents a frequency ratio of 21/28, or the 28th root of 2.

Theory

Approximation of odd harmonics in 28edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -16.2 -0.6 +16.9 +10.4 +5.8 +16.6 -16.8 -19.2 +2.5 +0.6 +14.6
Relative (%) -37.9 -1.4 +39.4 +24.2 +13.6 +38.8 -39.3 -44.9 +5.8 +1.5 +34.0
Steps
(reduced) 		44
(16) 		65
(9) 		79
(23) 		89
(5) 		97
(13) 		104
(20) 		109
(25) 		114
(2) 		119
(7) 		123
(11) 		127
(15)
Approximation of odd harmonics in 28edo
Harmonic 25 27 29 31 33 35 37 39 41 43 45
Error Absolute (¢) -1.2 -5.9 -1.0 +12.1 -10.4 +16.3 +5.8 +0.4 -0.5 +2.8 +9.8
Relative (%) -2.8 -13.7 -2.3 +28.3 -24.3 +38.0 +13.5 +0.9 -1.1 +6.5 +22.8
Steps
(reduced) 		130
(18) 		133
(21) 		136
(24) 		139
(27) 		141
(1) 		144
(4) 		146
(6) 		148
(8) 		150
(10) 		152
(12) 		154
(14)

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 tempers out the greater diesis 648:625. It does not however temper out the 128:125 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 subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to 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 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 intervals.

Step # ET (e) Just (j) Delta
(e-j) 		Ups and downs notation
Cents Interval Cents
0 0.00 1/1 0.00 0.00 unison 1 D
1 42.86 41/40 42.74 0.12 up-unison ^1 ^D
2 85.71 21/20 84.47 1.24 dup 1sn, dud 2nd ^^1, vv2 ^^D, vvE
3 128.57 14/13 128.30 0.27 down 2nd v2 vE
4 171.43 11/10 165.00 6.43 2nd 2 E
5 214.29 17/15 216.69 -2.40 up 2nd ^2 ^E
6 257.14 7/6 266.87 -9.73 dup 2nd, dud 3rd ^^2, vv3 ^^E, vvF
7 300.00 6/5 315.64 -15.64 down 3rd v3 vF
8 342.86 11/9 347.41 -4.55 3rd 3 F
9 385.71 5/4 386.31 -0.60 up 3rd ^3 ^F
10 428.57 9/7 435.08 -6.51 dup 3rd, dud 4th ^^3, vv4 ^^F, vvG
11 471.43 21/16 470.78 0.65 down 4th v4 vG
12 514.29 4/3 498.04 16.25 4th 4 G
13 557.14 11/8 551.32 5.82 up 4th ^4 ^G
14 600.00 7/5 582.51 17.49 dup 4th, dud 5th ^^4, vv5 ^^G, vvA
15 642.86 16/11 648.68 -5.82 down 5th v5 vA
16 685.71 3/2 701.96 -16.25 5th 5 A
17 728.57 32/21 729.22 -0.65 up 5th ^5 ^A
18 771.43 14/9 764.92 6.51 dup 5th, dud 6th ^^5, vv6 ^^A, vvB
19 814.29 8/5 813.68 0.61 down 6th v6 vB
20 857.14 18/11 852.59 4.55 6th 6 B
21 900.00 5/3 884.36 15.64 up 6th ^6 ^B
22 942.86 12/7 933.13 9.73 dup 6th, dud 7th ^^6, vv7 ^^B, vvC
23 985.71 30/17 983.31 2.40 down 7th v7 vC
24 1028.57 20/11 1035.00 -6.43 7th 7 C
25 1071.42 13/7 1071.70 -0.27 up 7th ^7 ^C
26 1114.29 40/21 1115.53 -1.24 dup 7th, dud 8ve ^^7, vv8 ^^C, vvD
27 1157.14 80/41 1157.26 -0.12 down 8ve v8 vD
28 1200.00 2/1 1200.00 0.00 8ve 8 D

Notation

Sagittal notation

This notation uses the same sagittal sequence as EDOs 23 and 33, and is a superset of the notations for EDOs 14 and 7.

Sagittal notationPeriodic table of EDOs with sagittal notationlimma-fraction notation

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.

  • 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

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

Regular temperament properties

Rank-2 temperaments

Periods
per 8ve 		Generator Temperaments
1 1\28
1 3\28 Negri
1 5\28 Machine
1 9\28 Worschmidt
1 11\28 A-team
1 13\28 Thuja
2 1\28
2 3\28 Octokaidecal
2 5\28 Antikythera
4 1\28
4 2\28 Demolished
4 3\28
7 1\28 Whitewood
14 1\28

Commas

28et tempers out the following commas. This assumes the val ⟨28 44 65 79 97 104].

Prime
limit 		Ratio[1] Monzo Cents Color name Name(s)
3 2187/2048 [-11 7⟩ 113.69 Lawa Apotome
5 648/625 [3 4 -4⟩ 62.57 Quadgu Diminished comma, major diesis
5 16875/16384 [-14 3 4⟩ 51.12 Laquadyo Negri comma, double augmentation diesis
5 (12 digits) [17 1 -8⟩ 11.45 Saquadbigu Würschmidt comma
7 36/35 [2 2 -1 -1⟩ 48.77 Rugu Mint comma, septimal quartertone
7 50/49 [1 0 2 -2⟩ 34.98 Biruyo Jubilisma, tritonic diesis
7 3125/3087 [0 -2 5 -3⟩ 21.18 Triru-aquinyo Gariboh comma
7 126/125 [1 2 -3 1⟩ 13.79 Zotrigu Starling comma, septimal semicomma
7 65625/65536 [-16 1 5 1⟩ 2.35 Lazoquinyo Horwell comma
7 (30 digits) [47 -7 -7 -7⟩ 0.34 Trisa-seprugu Akjaysma
11 176/175 [4 0 -2 -1 1⟩ 9.86 Lorugugu Valinorsma
11 441/440 [-3 2 -1 2 -1⟩ 3.93 Luzozogu Werckisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.03 Triluyo Wizardharry comma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Scales

28edo is particularly well suited to Whitewood in the same way that 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 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)

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.

  • Negri [9] 333343333
  • 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.

  • Diatonic Major [7] 5434552
  • Diatonic Minor [7] 5254345
  • Diatonic 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 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 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) by stacking it's 11th degree, and they actually sound better in this temperament.

  • Oneirotonic [5] 65656
  • Oneirotonic [8] 55155151
  • Oneirotonic [13] 4141141411411
  • Oneirotonic [18] 311311131131113111
  • Pathological Oneirotonic [23] 21112111121112111121111
  • machine5
  • machine6
  • 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

Ambient Esoterica
Beheld
Bryan Deister
duckapus
Eliora
Kosmorksy
Claudi Meneghin
NullPointerException Music
Userminusone
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