28edo: Difference between revisions

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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 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 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.

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! colspan="2" | Just (j)

! rowspan="2" | Delta (e-j)

! rowspan="2" colspan="3" | [[Ups and ~~Downs Notation~~]]

! rowspan="2" colspan="3" | [[Ups and downs notation]]

|-

! Cents

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== 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.

=== 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]].

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* 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 ==

=== Rank-2 temperaments ===

~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

{| class="wikitable center-1 center-2"

|-

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|}

== 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 ==

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

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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)

; 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.

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* 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.

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* 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.

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* [[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 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 ==

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; [[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]]

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; [[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]]

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

@@ Line 5: / Line 5: @@
 {{Harmonics in equal|28|start=12|collapsed=1|intervals=odd}}
-edo 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]].
+edo 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]].
 edo 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.
@@ Line 22: / Line 22: @@
 ! colspan="2" | Just (j)
 ! rowspan="2" | Delta <br> (e-j)
-! rowspan="2" colspan="3" | [[Ups and Downs Notation]]
+! rowspan="2" colspan="3" | [[Ups and downs notation]]
 |-
 ! Cents
@@ Line 291: / Line 291: @@
 == 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 ===
 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
-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 ==
 === Rank-2 temperaments ===
-''See [[regular temperament]] for more about what all this means and how to use it.''
 {| class="wikitable center-1 center-2"
 |-
@@ Line 489: / Line 491: @@
 |}
 <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.
+.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]].
+ed9 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 ==
-edo 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
+edo 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
@@ Line 500: / Line 536: @@
 * (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.
@@ Line 506: / Line 543: @@
 * 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.
@@ Line 541: / Line 579: @@
 * 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.
@@ Line 552: / Line 591: @@
 * [[machine11]]
 * [[machine17]]
+; Other Scales
+* 10-tone 4&7edo scale 4 3 1 4 2 2 4 1 3 4
 == Instruments ==
-edo can be played on the [[Lumatone]]. See: [[Lumatone mapping for 28edo]]
+edo can be played on the [[Lumatone]]. See [[Lumatone mapping for 28edo]].
+edo can also be played on a [[14edo]] [[guitar]] with very little effort. See [[User:MisterShafXen/Skip fretting system 28 2 3]].
 == Music ==
@@ Line 564: / Line 608: @@
 ; [[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]]
@@ Line 570: / Line 617: @@
 ; [[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]]
 * [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=wYdRAzp8Qi0 ''Canon on Twinkle Twinkle Little Star'', for Baroque Oboe & Viola] (2023) – ([https://www.youtube.com/watch?v=B1KZoLR8UTI for Organ])