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{{Harmonics in equal|28|start=12|collapsed=1|intervals=odd}}
{{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]]. 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.
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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! 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 291: Line 291:


== Notation ==
== Notation ==
=== Ups and downs notation ===
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}}
=== Sagittal 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]].
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]].
Line 316: Line 320:
* 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]].


== Regular temperament properties ==
== Regular temperament properties ==
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|}
|}
<references/>
<references/>
== Octave stretch or compression ==
'''[[equal tuning |89ed9]]''', a [[octave shrinking|compressed-octaves]] tuning of 28edo, makes 28edo potentially useable as a [[dual-fifth]] tuning. It shares the error equally between 28edo's two perfect fifths.
So, where pure-octaves 28edo has a flat fifth with 16.2{{c}} error and a sharp fifth with 28.6{{c}} error, 89ed9 instead has a flat fifth with 21.4{{c}} error and a sharp fifth with 21.4{{c}} error.
21.4{{c}} error is slightly better than [[23edo]]'s best fifth but slightly worse than [[5edo]]'s.
Some might consider this fifth too bad to use, in which case 89ed9 would be a downgrade from pure-octaves 28edo.
Others might consider it just useable enough, in which case 89ed9 would be an upgrade because it has one more useable interval and one less [[wolf interval]].
89ed9 approximates the no-3s 7-, 11-, 13-, 17-, 19- and 23-limits significantly better than pure-octaves 28edo.
; 28edo
* Step size: 42.8571{{c}}, octave size: 1200.00{{c}}
Approximates all no-3s harmonics up to 7 within ''16.9''{{c}}.
Approximates all no-3s harmonics up to 11 & up to 13 within ''16.9''{{c}}.
Approximates all no-3s harmonics up to 17 & up to 19 within ''19.2''{{c}}.
{{Harmonics in equal|28|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 28edo}}
{{Harmonics in equal|28|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 28edo (continued)}}
; [[equal tuning|89ed9]]
* Step size: 42.7406{{c}}, octave size: 1196.74{{c}}
Approximates all no-3s harmonics up to 7 within ''8.2''{{c}}.
Approximates all no-3s harmonics up to 11 & up to 13 within ''11.4''{{c}}.
Approximates all no-3s harmonics up to 17 & up to 19 within ''11.4''{{c}}.
{{Harmonics in equal|89|9|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 89ed9}}
{{Harmonics in equal|89|9|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 89ed9 (continued)}}


== 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.
; Whitewood 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 Major [14] 13131313131313
* Whitewood Major [14] 13131313131313
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* (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)


; Negri scales
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.
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.


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* Negri [19] 2121212121212121211
* Negri [19] 2121212121212121211


; Diatonic scales
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 539: Line 579:
* Melodic Major [16] 2122221222122212
* Melodic Major [16] 2122221222122212


; Oneirotonic scales
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 550: Line 591:
* [[machine11]]
* [[machine11]]
* [[machine17]]
* [[machine17]]
; Other Scales
* 10-tone 4&7edo scale 4 3 1 4 2 2 4 1 3 4


== Instruments ==
== Instruments ==
28edo can be played on the [[Lumatone]]. See: [[Lumatone mapping for 28edo]]
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 ==
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; [[Beheld]]
; [[Beheld]]
* [https://www.youtube.com/watch?v=0nvrUbw1VLQ ''Haze vibe'']
* [https://www.youtube.com/watch?v=0nvrUbw1VLQ ''Haze vibe'']
; [[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]]
; [[duckapus]]
Line 565: Line 617:


; [[User:Eliora|Eliora]]
; [[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]]
* [https://www.youtube.com/watch?v=8dubV7STjJA ''According to a Tomato''] (2025)


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


; [[Claudi Meneghin]]
; [[User:Phanomium|Phanomium]]
* ''[https://www.youtube.com/watch?v=zZJ5b4SkT7Q Euxenite]'' (2025)
 
;[[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=30XUKJsaINU ''Happy Birthday Canon'', 5-in-1 Canon in 28edo]
* [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])
* [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])

Latest revision as of 16:57, 29 May 2026

← 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

Ups and downs notation

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

Step offset −3 −2 −1 0 +1 +2 +3
Symbol
   
  
 
  
 
  
   

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

Octave stretch or compression

89ed9, a compressed-octaves tuning of 28edo, makes 28edo potentially useable as a dual-fifth tuning. It shares the error equally between 28edo's two perfect fifths.

So, where pure-octaves 28edo has a flat fifth with 16.2 ¢ error and a sharp fifth with 28.6 ¢ error, 89ed9 instead has a flat fifth with 21.4 ¢ error and a sharp fifth with 21.4 ¢ error.

21.4 ¢ error is slightly better than 23edo's best fifth but slightly worse than 5edo's.

Some might consider this fifth too bad to use, in which case 89ed9 would be a downgrade from pure-octaves 28edo.

Others might consider it just useable enough, in which case 89ed9 would be an upgrade because it has one more useable interval and one less wolf interval.

89ed9 approximates the no-3s 7-, 11-, 13-, 17-, 19- and 23-limits significantly better than pure-octaves 28edo.

28edo
  • Step size: 42.8571 ¢, octave size: 1200.00 ¢

Approximates all no-3s harmonics up to 7 within 16.9 ¢.

Approximates all no-3s harmonics up to 11 & up to 13 within 16.9 ¢.

Approximates all no-3s harmonics up to 17 & up to 19 within 19.2 ¢.

Approximation of harmonics in 28edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -16.2 +0.0 -0.6 -16.2 +16.9 +0.0 +10.4 -0.6 +5.8 -16.2
Relative (%) +0.0 -37.9 +0.0 -1.4 -37.9 +39.4 +0.0 +24.2 -1.4 +13.6 -37.9
Steps
(reduced) 		28
(0) 		44
(16) 		56
(0) 		65
(9) 		72
(16) 		79
(23) 		84
(0) 		89
(5) 		93
(9) 		97
(13) 		100
(16)
Approximation of harmonics in 28edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.6 +16.9 -16.8 +0.0 -19.2 +10.4 +2.5 -0.6 +0.6 +5.8 +14.6 -16.2
Relative (%) +38.8 +39.4 -39.3 +0.0 -44.9 +24.2 +5.8 -1.4 +1.5 +13.6 +34.0 -37.9
Steps
(reduced) 		104
(20) 		107
(23) 		109
(25) 		112
(0) 		114
(2) 		117
(5) 		119
(7) 		121
(9) 		123
(11) 		125
(13) 		127
(15) 		128
(16)
89ed9
  • Step size: 42.7406 ¢, octave size: 1196.74 ¢

Approximates all no-3s harmonics up to 7 within 8.2 ¢.

Approximates all no-3s harmonics up to 11 & up to 13 within 11.4 ¢.

Approximates all no-3s harmonics up to 17 & up to 19 within 11.4 ¢.

Approximation of harmonics in 89ed9
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.3 +21.4 -6.5 -8.2 +18.1 +7.7 -9.8 +0.0 -11.4 -5.5 +14.8
Relative (%) -7.6 +50.0 -15.3 -19.1 +42.4 +18.0 -22.9 +0.0 -26.8 -12.8 +34.7
Steps
(reduced) 		28
(28) 		45
(45) 		56
(56) 		65
(65) 		73
(73) 		79
(79) 		84
(84) 		89
(0) 		93
(4) 		97
(8) 		101
(12)
Approximation of harmonics in 89ed9 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.5 +4.4 +13.2 -13.1 +10.2 -3.3 -11.4 -14.7 -13.7 -8.7 -0.2 +11.6
Relative (%) +10.5 +10.3 +30.9 -30.5 +23.9 -7.6 -26.6 -34.4 -32.0 -20.5 -0.5 +27.1
Steps
(reduced) 		104
(15) 		107
(18) 		110
(21) 		112
(23) 		115
(26) 		117
(28) 		119
(30) 		121
(32) 		123
(34) 		125
(36) 		127
(38) 		129
(40)

Scales

Whitewood 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)
Negri scales

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
Diatonic scales

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
Oneirotonic scales

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
Other Scales
  • 10-tone 4&7edo scale 4 3 1 4 2 2 4 1 3 4

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
Francium
Kosmorksy
Phanomium
Claudi Meneghin
NullPointerException Music
Userminusone
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