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, a multiple of both [[xenharmonic/7edo|7edo]] and [[xenharmonic/14edo|14edo]] (and of course [[xenharmonic/2edo|2edo]] and [[xenharmonic/4edo|4edo]]), has a step size of 42.857 [[xenharmonic/cent|cent]]s. It shares three intervals with [[xenharmonic/12edo|12edo]]: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it [[xenharmonic/tempering out|tempers out]] the [[xenharmonic/greater diesis|greater diesis]] [[xenharmonic/648_625|648:625]]. It does not however temper out the [[xenharmonic/128_125|128:125]] [[xenharmonic/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 [[xenharmonic/9_7|9/7]] and its inversion [[xenharmonic/14_9|14/9]] are also found in 14edo.
=Subgroups=
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 [[xenharmonic/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 [[xenharmonic/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 [[xenharmonic/augmented triad|augmented triad]] has a very low complexity, so many of them appear in the [[xenharmonic/MOS scales|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.
=Table of intervals=
28edo is the 2nd perfect number EDO.
The following table compares it to potentially useful nearby [[xenharmonic/just intervals|just intervals]].
|| Step # || ET Cents
== Intervals ==
DMS || Just Interval || Just Cents
The following table compares it to potentially useful nearby [[just interval]]s.
28 EDO tempers out the following [[xenharmonic/comma|comma]]s. (Note: This assumes the val < 28 44 65 79 97 104 |.)
=== Ups and downs notation ===
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
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.
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]].
[[xenharmonic/machine6|machine6]]
[[xenharmonic/machine11|machine11]]
=Compositions=
<imagemap>
[[http://www.youtube.com/watch?v=26UpCbrb3mE|28 tone Prelude]] by Kosmorksy</pre></div>
28edo, a multiple of both <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7edo">7edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/14edo">14edo</a> (and of course <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2edo">2edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/4edo">4edo</a>), has a step size of 42.857 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent">cent</a>s. It shares three intervals with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/12edo">12edo</a>: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tempering%20out">tempers out</a> the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/greater%20diesis">greater diesis</a> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/648_625">648:625</a>. It does not however temper out the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/128_125">128:125</a> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/lesser%20diesis">lesser diesis</a>, 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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9_7">9/7</a> and its inversion <a class="wiki_link" href="http://xenharmonic.wikispaces.com/14_9">14/9</a> are also found in 14edo.<br />
28edo can approximate the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit">7-limit</a> subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Semicomma%20family">orwell temperament</a> 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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/augmented%20triad">augmented triad</a> has a very low complexity, so many of them appear in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS%20scales">MOS scales</a> for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.<br />
<br />
Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Table of intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Table of intervals</h1>
The following table compares it to potentially useful nearby <a class="wiki_link" href="http://xenharmonic.wikispaces.com/just%20intervals">just intervals</a>.<br />
<br />
== 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.
28 EDO tempers out the following <a class="wiki_link" href="http://xenharmonic.wikispaces.com/comma">comma</a>s. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)<br />
! [[Harmonic Limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
'''[[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.
<table class="wiki_table">
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.
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.
<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=26UpCbrb3mE" rel="nofollow">28 tone Prelude</a> by Kosmorksy</body></html></pre></div>
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 ==
; 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 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_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 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|13edo]]) by stacking it's 11th degree, and they actually sound better in this temperament.
* [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])
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.
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 diesis648/625. It does not however temper out the 128/125lesser 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.
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.
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.
↑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 ¢.
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.
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.
