28edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-----
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-26 04:51:25 UTC</tt>.<br>
: The original revision id was <tt>602812742</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
----


=Basic properties=  
=Basic properties=
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.
28edo, a multiple of both [[7edo|7edo]] and [[14edo|14edo]] (and of course [[2edo|2edo]] and [[4edo|4edo]]), has a step size of 42.857 [[cent|cent]]s. It shares three intervals with [[12edo|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|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|9/7]] and its inversion [[14/9|14/9]] are also found in 14edo.


=Subgroups=  
=Subgroups=
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.
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|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.


=Table of intervals=  
=Table of intervals=
The following table compares it to potentially useful nearby [[xenharmonic/just intervals|just intervals]].
The following table compares it to potentially useful nearby [[just_intervals|just intervals]].


||= Step # ||= ET Cents ||= Just Interval ||= Just Cents ||= Difference
{| class="wikitable"
(ET minus Just) ||||||= [[xenharmonic/Ups and Downs Notation|Up/down ]][[xenharmonic/Ups and Downs Notation|Notation]] ||
|-
||= 0 ||= 0¢ ||=  ||=  ||=  ||= unison ||= 1 ||= D ||
| style="text-align:center;" | Step #
||= 1 ||= 42.86 ||=  ||=  ||=  ||= up-unison ||= ^1 ||= D^ ||
| style="text-align:center;" | ET Cents
||= 2 ||= 85.71 ||= 21:20 ||= 84.47 ||= 1.24 ||= double-up, double-down ||= ^^1, vv2 ||= D^^, Evv ||
| style="text-align:center;" | Just Interval
||= 3 ||= 128.57 ||= 14:13 ||= 128.30 ||= 0.27 ||= down 2nd ||= v2 ||= Ev ||
| style="text-align:center;" | Just Cents
||= 4 ||= 171.43 ||= 11:10 ||= 165.00 ||= 6.43 ||= 2nd ||= 2 ||= E ||
| style="text-align:center;" | Difference
||= 5 ||= 214.29 ||= 17:15 ||= 216.69 ||= -2.40 ||= up 2nd ||= ^2 ||= E^ ||
||= 6 ||= 257.14 ||= 7:6 ||= 266.87 ||= -9.73 ||= double-up 2nd, double-down 3rd ||= ^^2, vv3 ||= E^^, Fvv ||
||= 7 ||= 300 ||= 6:5 ||= 315.64 ||= -15.64 ||= down 3rd ||= v3 ||= Fv ||
||= 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 ||= double-up 3rd, double-down 4th ||= ^^3, vv4 ||= F^^, Gvv ||
||= 11 ||= 471.43 ||= 21:16 ||= 470.78 ||= 0.65 ||= down 4th ||= v4 ||= Gv ||
||= 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 ||= 7:5 ||= 582.51 ||= 17.49 ||= double-up 4th, double-down 5th ||= ^^4, vv5 ||= G^^, vvA ||
||= 15 ||= 642.86 ||= 16:11 ||= 648.68 ||= -5.82 ||= down 5th ||= v5 ||= Av ||
||= 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 ||= double-up 5th, double-down 6th ||= ^^5, vv6 ||= A^^, Bvv ||
||= 19 ||= 814.29 ||= 5:8 ||= 813.68 ||= 0.61 ||= down 6th ||= v6 ||= Bv ||
||= 20 ||= 857.14 ||= 18:11 ||= 852.59 ||= 4.55 ||= 6th ||= 6 ||= B ||
||= 21 ||= 900 ||= 5:3 ||= 884.36 ||= 15.64 ||= up 6th ||= ^6 ||= B^ ||
||= 22 ||= 942.86 ||= 12:7 ||= 933.13 ||= 9.73 ||= double-up 6th, double-down 7th ||= ^^6, vv7 ||= B^^, Cvv ||
||= 23 ||= 985.71 ||= 30:17 ||= 983.31 ||= 2.40 ||= down 7th ||= v7 ||= Cv ||
||= 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 ||= double-up 7th, double-down 8ve ||= ^^7, vv8 ||= C^^, Dvv ||
||= 27 ||= 1157.14 ||=  ||=  ||=  ||= down 8ve ||= v8 ||= Dv ||
||= 28 ||= 1200 ||= 2:1 ||= 1200 ||= 0 ||= 8ve ||= 8 ||= D ||


=[[#Chord Names]]Chord Names=  
(ET minus Just)
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|Up/down ]][[Ups_and_Downs_Notation|Notation]]
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0¢
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | unison
| style="text-align:center;" | 1
| style="text-align:center;" | D
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 42.86
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | up-unison
| style="text-align:center;" | ^1
| style="text-align:center;" | D^
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 85.71
| style="text-align:center;" | 21:20
| style="text-align:center;" | 84.47
| style="text-align:center;" | 1.24
| style="text-align:center;" | double-up, double-down
| style="text-align:center;" | ^^1, vv2
| style="text-align:center;" | D^^, Evv
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 128.57
| style="text-align:center;" | 14:13
| style="text-align:center;" | 128.30
| style="text-align:center;" | 0.27
| style="text-align:center;" | down 2nd
| style="text-align:center;" | v2
| style="text-align:center;" | Ev
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 171.43
| style="text-align:center;" | 11:10
| style="text-align:center;" | 165.00
| style="text-align:center;" | 6.43
| style="text-align:center;" | 2nd
| style="text-align:center;" | 2
| style="text-align:center;" | E
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 214.29
| style="text-align:center;" | 17:15
| style="text-align:center;" | 216.69
| style="text-align:center;" | -2.40
| style="text-align:center;" | up 2nd
| style="text-align:center;" | ^2
| style="text-align:center;" | E^
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 257.14
| style="text-align:center;" | 7:6
| style="text-align:center;" | 266.87
| style="text-align:center;" | -9.73
| style="text-align:center;" | double-up 2nd, double-down 3rd
| style="text-align:center;" | ^^2, vv3
| style="text-align:center;" | E^^, Fvv
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 300
| style="text-align:center;" | 6:5
| style="text-align:center;" | 315.64
| style="text-align:center;" | -15.64
| style="text-align:center;" | down 3rd
| style="text-align:center;" | v3
| style="text-align:center;" | Fv
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 342.86
| style="text-align:center;" | 11:9
| style="text-align:center;" | 347.41
| style="text-align:center;" | -4.55
| style="text-align:center;" | 3rd
| style="text-align:center;" | 3
| style="text-align:center;" | F
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 385.71
| style="text-align:center;" | 5:4
| style="text-align:center;" | 386.31
| style="text-align:center;" | -0.60
| style="text-align:center;" | up 3rd
| style="text-align:center;" | ^3
| style="text-align:center;" | F^
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 428.57
| style="text-align:center;" | 9:7
| style="text-align:center;" | 435.08
| style="text-align:center;" | -6.51
| style="text-align:center;" | double-up 3rd, double-down 4th
| style="text-align:center;" | ^^3, vv4
| style="text-align:center;" | F^^, Gvv
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 471.43
| style="text-align:center;" | 21:16
| style="text-align:center;" | 470.78
| style="text-align:center;" | 0.65
| style="text-align:center;" | down 4th
| style="text-align:center;" | v4
| style="text-align:center;" | Gv
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 514.29
| style="text-align:center;" | 4:3
| style="text-align:center;" | 498.04
| style="text-align:center;" | 16.25
| style="text-align:center;" | 4th
| style="text-align:center;" | 4
| style="text-align:center;" | G
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 557.14
| style="text-align:center;" | 11:8
| style="text-align:center;" | 551.32
| style="text-align:center;" | 5.82
| style="text-align:center;" | up 4th
| style="text-align:center;" | ^4
| style="text-align:center;" | G^
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 600
| style="text-align:center;" | 7:5
| style="text-align:center;" | 582.51
| style="text-align:center;" | 17.49
| style="text-align:center;" | double-up 4th, double-down 5th
| style="text-align:center;" | ^^4, vv5
| style="text-align:center;" | G^^, vvA
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 642.86
| style="text-align:center;" | 16:11
| style="text-align:center;" | 648.68
| style="text-align:center;" | -5.82
| style="text-align:center;" | down 5th
| style="text-align:center;" | v5
| style="text-align:center;" | Av
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 685.71
| style="text-align:center;" | 3:2
| style="text-align:center;" | 701.96
| style="text-align:center;" | -16.25
| style="text-align:center;" | 5th
| style="text-align:center;" | 5
| style="text-align:center;" | A
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 728.57
| style="text-align:center;" | 32:21
| style="text-align:center;" | 729.22
| style="text-align:center;" | -0.65
| style="text-align:center;" | up 5th
| style="text-align:center;" | ^5
| style="text-align:center;" | A^
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 771.43
| style="text-align:center;" | 14:9
| style="text-align:center;" | 764.92
| style="text-align:center;" | 6.51
| style="text-align:center;" | double-up 5th, double-down 6th
| style="text-align:center;" | ^^5, vv6
| style="text-align:center;" | A^^, Bvv
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 814.29
| style="text-align:center;" | 5:8
| style="text-align:center;" | 813.68
| style="text-align:center;" | 0.61
| style="text-align:center;" | down 6th
| style="text-align:center;" | v6
| style="text-align:center;" | Bv
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 857.14
| style="text-align:center;" | 18:11
| style="text-align:center;" | 852.59
| style="text-align:center;" | 4.55
| style="text-align:center;" | 6th
| style="text-align:center;" | 6
| style="text-align:center;" | B
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 900
| style="text-align:center;" | 5:3
| style="text-align:center;" | 884.36
| style="text-align:center;" | 15.64
| style="text-align:center;" | up 6th
| style="text-align:center;" | ^6
| style="text-align:center;" | B^
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 942.86
| style="text-align:center;" | 12:7
| style="text-align:center;" | 933.13
| style="text-align:center;" | 9.73
| style="text-align:center;" | double-up 6th, double-down 7th
| style="text-align:center;" | ^^6, vv7
| style="text-align:center;" | B^^, Cvv
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 985.71
| style="text-align:center;" | 30:17
| style="text-align:center;" | 983.31
| style="text-align:center;" | 2.40
| style="text-align:center;" | down 7th
| style="text-align:center;" | v7
| style="text-align:center;" | Cv
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 1028.57
| style="text-align:center;" | 20:11
| style="text-align:center;" | 1035.00
| style="text-align:center;" | -6.43
| style="text-align:center;" | 7th
| style="text-align:center;" | 7
| style="text-align:center;" | C
|-
| style="text-align:center;" | 25
| style="text-align:center;" | 1071.42
| style="text-align:center;" | 13:7
| style="text-align:center;" | 1071.70
| style="text-align:center;" | -0.27
| style="text-align:center;" | up 7th
| style="text-align:center;" | ^7
| style="text-align:center;" | C^
|-
| style="text-align:center;" | 26
| style="text-align:center;" | 1114.29
| style="text-align:center;" | 40:21
| style="text-align:center;" | 1115.53
| style="text-align:center;" | -1.24
| style="text-align:center;" | double-up 7th, double-down 8ve
| style="text-align:center;" | ^^7, vv8
| style="text-align:center;" | C^^, Dvv
|-
| style="text-align:center;" | 27
| style="text-align:center;" | 1157.14
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | down 8ve
| style="text-align:center;" | v8
| style="text-align:center;" | Dv
|-
| style="text-align:center;" | 28
| style="text-align:center;" | 1200
| style="text-align:center;" | 2:1
| style="text-align:center;" | 1200
| style="text-align:center;" | 0
| style="text-align:center;" | 8ve
| style="text-align:center;" | 8
| style="text-align:center;" | D
|}
 
=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.


0-8-16 = C E G = C = C or C perfect
0-8-16 = C E G = C = C or C perfect
0-7-16 = C Ev G = C(v3) = C down-three
0-7-16 = C Ev G = C(v3) = C down-three
0-9-16 = C E^ G = C(^3) = C up-three
0-9-16 = C E^ G = C(^3) = C up-three
0-8-15 = C E Gv = C(v5) = C down-five
0-8-15 = C E Gv = C(v5) = C down-five
0-9-17 = C E^ G^ = C(^3,^5) = C up-three up-five
0-9-17 = C E^ G^ = C(^3,^5) = C up-three up-five


0-8-16-24 = C E G B = C7 = C seven
0-8-16-24 = C E G B = C7 = C seven
0-8-16-23 = C E G Bv = C(v7) = C down-seven
0-8-16-23 = C E G Bv = C(v7) = C down-seven
0-7-16-24 = C Ev G B = C7(v3) = C seven down-three
0-7-16-24 = C Ev G B = C7(v3) = C seven down-three
0-7-16-23 = C Ev G Bv = C.v7 = C dot down seven
0-7-16-23 = C Ev G Bv = C.v7 = C dot down seven


For a more complete list, see [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
 
 
=&lt;span style="background-color: #ffffff;"&gt;Rank two temperaments&lt;/span&gt;=
 
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\28 ||  ||
|| 1 || 3\28 || [[xenharmonic/Negri|Negri]] ||
|| 1 || 5\28 || [[xenharmonic/Machine|Machine]] ||
|| 1 || 9\28 || [[xenharmonic/Würschmidt family#Worschmidt|Worschmidt]] ||
|| 1 || 11\28 ||  ||
|| 1 || 13\28 || &lt;span style="background-color: #ffffff;"&gt;[[xenharmonic/Thuja|Thuja]]&lt;/span&gt; ||
|| 2 || 1\28 ||  ||
|| 2 || 3\28 ||  ||
|| 2 || 5\28 || [[antikythera|Antikythera]] ||
|| 4 || 1\28 ||  ||
|| 4 || 2\28 || [[xenharmonic/Diminished#Demolished|Demolished]] ||
|| 4 || 3\28 ||  ||
|| 7 || 1\28 || [[xenharmonic/Apotome family|Whitewood]] ||
|| 14 || 1\28 ||  ||
 
=Commas=
28 EDO tempers out the following [[xenharmonic/comma|comma]]s. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)
 
||~ Comma ||~ Monzo ||~ Cents ||~ Name 1 ||~ Name 2 ||
||= 2187/2048 ||&lt; | -11 7 &gt; ||= 113.69 ||= Apotome ||=  ||
||= 648/625 ||&lt; | 3 4 -4 &gt; ||= 62.57 ||= Major Diesis ||= Diminished Comma ||
||= 16875/16384 ||&lt; | -14 3 4 &gt; ||= 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||
||=  ||&lt; | 17 1 -8 &gt; ||= 11.45 ||= Wuerschmidt Comma ||=  ||
||= 36/35 ||&lt; | 2 2 -1 -1 &gt; ||= 48.77 ||= Septimal Quarter Tone ||=  ||
||= 50/49 ||&lt; | 1 0 2 -2 &gt; ||= 34.98 ||= Tritonic Diesis ||= Jubilisma ||
||= 3125/3087 ||&lt; | 0 -2 5 -3 &gt; ||= 21.18 ||= Gariboh ||=  ||
||= 126/125 ||&lt; | 1 2 -3 1 &gt; ||= 13.79 ||= Septimal Semicomma ||= Starling Comma ||
||= 65625/65536 ||&lt; | -16 1 5 1 &gt; ||= 2.35 ||= Horwell ||=  ||
||=  ||&lt; | 47 -7 -7 -7 &gt; ||= 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||
||= 176/175 ||&lt; | 4 0 -2 -1 1 &gt; ||= 9.86 ||= Valinorsma ||=  ||
||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||= 3.93 ||= Werckisma ||=  ||
||= 4000/3993 ||&lt; | 5 -1 3 0 -3 &gt; ||= 3.03 ||= Wizardharry ||=  ||
 
=Some scales=
[[xenharmonic/machine5|machine5]]
[[xenharmonic/machine6|machine6]]
[[xenharmonic/machine11|machine11]]
 
=Compositions=
[[http://www.youtube.com/watch?v=26UpCbrb3mE|28 tone Prelude]] by Kosmorksy</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;28edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Basic properties"&gt;Basic properties&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Subgroups"&gt;Subgroups&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Table of intervals"&gt;Table of intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Chord Names"&gt;Chord Names&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Some scales"&gt;Some scales&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;hr /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basic properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Basic properties&lt;/h1&gt;
28edo, a multiple of both &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7edo"&gt;7edo&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/14edo"&gt;14edo&lt;/a&gt; (and of course &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/2edo"&gt;2edo&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/4edo"&gt;4edo&lt;/a&gt;), has a step size of 42.857 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent"&gt;cent&lt;/a&gt;s. It shares three intervals with &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/12edo"&gt;12edo&lt;/a&gt;: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/tempering%20out"&gt;tempers out&lt;/a&gt; the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/greater%20diesis"&gt;greater diesis&lt;/a&gt; &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/648_625"&gt;648:625&lt;/a&gt;. It does not however temper out the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/128_125"&gt;128:125&lt;/a&gt; &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/lesser%20diesis"&gt;lesser diesis&lt;/a&gt;, 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/9_7"&gt;9/7&lt;/a&gt; and its inversion &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/14_9"&gt;14/9&lt;/a&gt; are also found in 14edo.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Subgroups&lt;/h1&gt;
28edo can approximate the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit"&gt;7-limit&lt;/a&gt; subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Semicomma%20family"&gt;orwell temperament&lt;/a&gt; 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/augmented%20triad"&gt;augmented triad&lt;/a&gt; has a very low complexity, so many of them appear in the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS%20scales"&gt;MOS scales&lt;/a&gt; for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.&lt;br /&gt;
&lt;br /&gt;
Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Table of intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Table of intervals&lt;/h1&gt;
The following table compares it to potentially useful nearby &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/just%20intervals"&gt;just intervals&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;


=<span style="background-color: #ffffff;">Rank two temperaments</span>=


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td style="text-align: center;"&gt;Step #&lt;br /&gt;
! | Periods
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;ET Cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Just Interval&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Just Cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Difference&lt;br /&gt;
(ET minus Just)&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="3" style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;Up/down &lt;/a&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;Notation&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;42.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up-unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;85.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;21:20&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;84.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1.24&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up, double-down&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^1, vv2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D^^, Evv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;128.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;14:13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;128.30&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.27&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;down 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ev&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;171.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11:10&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;165.00&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;214.29&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;17:15&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;216.69&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-2.40&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;257.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7:6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;266.87&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-9.73&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up 2nd, double-down 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^2, vv3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E^^, Fvv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;300&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6:5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;315.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-15.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;down 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;342.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11:9&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;347.41&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-4.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;385.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5:4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;386.31&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-0.60&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;428.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9:7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;435.08&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-6.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up 3rd, double-down 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^3, vv4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F^^, Gvv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;471.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;21:16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;470.78&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.65&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;down 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;514.29&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4:3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;498.04&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;16.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;557.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11:8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;551.32&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5.82&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;600&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7:5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;582.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;17.49&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up 4th, double-down 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^4, vv5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G^^, vvA&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;642.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;16:11&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;648.68&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-5.82&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;down 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Av&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;685.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3:2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;701.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-16.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;728.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;32:21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;729.22&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-0.65&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;771.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;14:9&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;764.92&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up 5th, double-down 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^5, vv6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A^^, Bvv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;814.29&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5:8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;813.68&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.61&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;down 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;857.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;18:11&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;852.59&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;900&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5:3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;884.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;15.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;942.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;12:7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;933.13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9.73&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up 6th, double-down 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^6, vv7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B^^, Cvv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;23&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;985.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;30:17&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;983.31&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.40&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;down 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Cv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1028.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;20:11&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1035.00&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-6.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;25&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1071.42&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;13:7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1071.70&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-0.27&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;26&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1114.29&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;40:21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1115.53&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;-1.24&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up 7th, double-down 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^7, vv8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C^^, Dvv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;27&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1157.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;down 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Dv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;28&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2:1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
per octave
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Chord Names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:26:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Chord Names&amp;quot; title=&amp;quot;Anchor: Chord Names&amp;quot;/&amp;gt; --&gt;&lt;a name="Chord Names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:26 --&gt;Chord Names&lt;/h1&gt;
! | Generator
&lt;br /&gt;
! | Temperaments
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.&lt;br /&gt;
|-
&lt;br /&gt;
| | 1
0-8-16 = C E G = C = C or C perfect&lt;br /&gt;
| | 1\28
0-7-16 = C Ev G = C(v3) = C down-three&lt;br /&gt;
| |
0-9-16 = C E^ G = C(^3) = C up-three&lt;br /&gt;
|-
0-8-15 = C E Gv = C(v5) = C down-five&lt;br /&gt;
| | 1
0-9-17 = C E^ G^ = C(^3,^5) = C up-three up-five&lt;br /&gt;
| | 3\28
&lt;br /&gt;
| | [[Negri|Negri]]
0-8-16-24 = C E G B = C7 = C seven&lt;br /&gt;
|-
0-8-16-23 = C E G Bv = C(v7) = C down-seven&lt;br /&gt;
| | 1
0-7-16-24 = C Ev G B = C7(v3) = C seven down-three&lt;br /&gt;
| | 5\28
0-7-16-23 = C Ev G Bv = C.v7 = C dot down seven&lt;br /&gt;
| | [[Machine|Machine]]
&lt;br /&gt;
|-
For a more complete list, see &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs"&gt;Ups and Downs Notation - Chord names in other EDOs&lt;/a&gt;.&lt;br /&gt;
| | 1
&lt;br /&gt;
| | 9\28
&lt;br /&gt;
| | [[Würschmidt_family#Worschmidt|Worschmidt]]
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;span style="background-color: #ffffff;"&gt;Rank two temperaments&lt;/span&gt;&lt;/h1&gt;
|-
&lt;br /&gt;
| | 1
| | 11\28
| |
|-
| | 1
| | 13\28
| | <span style="background-color: #ffffff;">[[Thuja|Thuja]]</span>
|-
| | 2
| | 1\28
| |
|-
| | 2
| | 3\28
| |
|-
| | 2
| | 5\28
| | [[antikythera|Antikythera]]
|-
| | 4
| | 1\28
| |
|-
| | 4
| | 2\28
| | [[Diminished#Demolished|Demolished]]
|-
| | 4
| | 3\28
| |
|-
| | 7
| | 1\28
| | [[Apotome_family|Whitewood]]
|-
| | 14
| | 1\28
| |
|}


=Commas=
28 EDO tempers out the following [[Comma|comma]]s. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;th&gt;Periods&lt;br /&gt;
! | Comma
per octave&lt;br /&gt;
! | Monzo
&lt;/th&gt;
! | Cents
        &lt;th&gt;Generator&lt;br /&gt;
! | Name 1
&lt;/th&gt;
! | Name 2
        &lt;th&gt;Temperaments&lt;br /&gt;
|-
&lt;/th&gt;
| style="text-align:center;" | 2187/2048
    &lt;/tr&gt;
| | | -11 7 &gt;
    &lt;tr&gt;
| style="text-align:center;" | 113.69
        &lt;td&gt;1&lt;br /&gt;
| style="text-align:center;" | Apotome
&lt;/td&gt;
| style="text-align:center;" |
        &lt;td&gt;1\28&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 648/625
        &lt;td&gt;&lt;br /&gt;
| | | 3 4 -4 &gt;
&lt;/td&gt;
| style="text-align:center;" | 62.57
    &lt;/tr&gt;
| style="text-align:center;" | Major Diesis
    &lt;tr&gt;
| style="text-align:center;" | Diminished Comma
        &lt;td&gt;1&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 16875/16384
        &lt;td&gt;3\28&lt;br /&gt;
| | | -14 3 4 &gt;
&lt;/td&gt;
| style="text-align:center;" | 51.12
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Negri"&gt;Negri&lt;/a&gt;&lt;br /&gt;
| style="text-align:center;" | Negri Comma
&lt;/td&gt;
| style="text-align:center;" | Double Augmentation Diesis
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| style="text-align:center;" |
        &lt;td&gt;1&lt;br /&gt;
| | | 17 1 -8 &gt;
&lt;/td&gt;
| style="text-align:center;" | 11.45
        &lt;td&gt;5\28&lt;br /&gt;
| style="text-align:center;" | Wuerschmidt Comma
&lt;/td&gt;
| style="text-align:center;" |
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Machine"&gt;Machine&lt;/a&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 36/35
    &lt;/tr&gt;
| | | 2 2 -1 -1 &gt;
    &lt;tr&gt;
| style="text-align:center;" | 48.77
        &lt;td&gt;1&lt;br /&gt;
| style="text-align:center;" | Septimal Quarter Tone
&lt;/td&gt;
| style="text-align:center;" |
        &lt;td&gt;9\28&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 50/49
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/W%C3%BCrschmidt%20family#Worschmidt"&gt;Worschmidt&lt;/a&gt;&lt;br /&gt;
| | | 1 0 2 -2 &gt;
&lt;/td&gt;
| style="text-align:center;" | 34.98
    &lt;/tr&gt;
| style="text-align:center;" | Tritonic Diesis
    &lt;tr&gt;
| style="text-align:center;" | Jubilisma
        &lt;td&gt;1&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 3125/3087
        &lt;td&gt;11\28&lt;br /&gt;
| | | 0 -2 5 -3 &gt;
&lt;/td&gt;
| style="text-align:center;" | 21.18
        &lt;td&gt;&lt;br /&gt;
| style="text-align:center;" | Gariboh
&lt;/td&gt;
| style="text-align:center;" |
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| style="text-align:center;" | 126/125
        &lt;td&gt;1&lt;br /&gt;
| | | 1 2 -3 1 &gt;
&lt;/td&gt;
| style="text-align:center;" | 13.79
        &lt;td&gt;13\28&lt;br /&gt;
| style="text-align:center;" | Septimal Semicomma
&lt;/td&gt;
| style="text-align:center;" | Starling Comma
        &lt;td&gt;&lt;span style="background-color: #ffffff;"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Thuja"&gt;Thuja&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 65625/65536
    &lt;/tr&gt;
| | | -16 1 5 1 &gt;
    &lt;tr&gt;
| style="text-align:center;" | 2.35
        &lt;td&gt;2&lt;br /&gt;
| style="text-align:center;" | Horwell
&lt;/td&gt;
| style="text-align:center;" |
        &lt;td&gt;1\28&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" |
        &lt;td&gt;&lt;br /&gt;
| | | 47 -7 -7 -7 &gt;
&lt;/td&gt;
| style="text-align:center;" | 0.34
    &lt;/tr&gt;
| style="text-align:center;" | Akjaysma
    &lt;tr&gt;
| style="text-align:center;" | 5\7 Octave Comma
        &lt;td&gt;2&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 176/175
        &lt;td&gt;3\28&lt;br /&gt;
| | | 4 0 -2 -1 1 &gt;
&lt;/td&gt;
| style="text-align:center;" | 9.86
        &lt;td&gt;&lt;br /&gt;
| style="text-align:center;" | Valinorsma
&lt;/td&gt;
| style="text-align:center;" |
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| style="text-align:center;" | 441/440
        &lt;td&gt;2&lt;br /&gt;
| | | -3 2 -1 2 -1 &gt;
&lt;/td&gt;
| style="text-align:center;" | 3.93
        &lt;td&gt;5\28&lt;br /&gt;
| style="text-align:center;" | Werckisma
&lt;/td&gt;
| style="text-align:center;" |
        &lt;td&gt;&lt;a class="wiki_link" href="/antikythera"&gt;Antikythera&lt;/a&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| style="text-align:center;" | 4000/3993
    &lt;/tr&gt;
| | | 5 -1 3 0 -3 &gt;
    &lt;tr&gt;
| style="text-align:center;" | 3.03
        &lt;td&gt;4&lt;br /&gt;
| style="text-align:center;" | Wizardharry
&lt;/td&gt;
| style="text-align:center;" |
        &lt;td&gt;1\28&lt;br /&gt;
|}
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2\28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Diminished#Demolished"&gt;Demolished&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Apotome%20family"&gt;Whitewood&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
=Some scales=
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Commas&lt;/h1&gt;
[[machine5|machine5]]
28 EDO tempers out the following &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 28 44 65 79 97 104 |.)&lt;br /&gt;
&lt;br /&gt;


[[machine6|machine6]]


&lt;table class="wiki_table"&gt;
[[machine11|machine11]]
    &lt;tr&gt;
        &lt;th&gt;Comma&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Monzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 1&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 2&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2187/2048&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -11 7 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;113.69&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Apotome&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;648/625&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 3 4 -4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;62.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Major Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diminished Comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;16875/16384&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -14 3 4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;51.12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Negri Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Double Augmentation Diesis&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 17 1 -8 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.45&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Wuerschmidt Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;36/35&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 2 2 -1 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;48.77&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Quarter Tone&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;50/49&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 1 0 2 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;34.98&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Tritonic Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Jubilisma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3125/3087&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 0 -2 5 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;21.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gariboh&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;13.79&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Semicomma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Starling Comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;65625/65536&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -16 1 5 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.35&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Horwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 47 -7 -7 -7 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.34&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Akjaysma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5\7 Octave Comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;441/440&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3.93&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Werckisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4000/3993&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 5 -1 3 0 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3.03&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Wizardharry&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
=Compositions=
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Some scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Some scales&lt;/h1&gt;
[http://www.youtube.com/watch?v=26UpCbrb3mE 28 tone Prelude] by Kosmorksy
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine5"&gt;machine5&lt;/a&gt;&lt;br /&gt;
[[Category:28edo]]
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine6"&gt;machine6&lt;/a&gt;&lt;br /&gt;
[[Category:edo]]
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine11"&gt;machine11&lt;/a&gt;&lt;br /&gt;
[[Category:theory]]
&lt;br /&gt;
[[Category:todo:unify_precision]]
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Compositions&lt;/h1&gt;
[[Category:twentuning]]
&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=26UpCbrb3mE" rel="nofollow"&gt;28 tone Prelude&lt;/a&gt; by Kosmorksy&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

Basic properties

28edo, a multiple of both 7edo and 14edo (and of course 2edo and 4edo), has a step size of 42.857 cents. 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 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.

Subgroups

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.

Table of intervals

The following table compares it to potentially useful nearby just intervals.

Step # ET Cents Just Interval Just Cents Difference

(ET minus Just)

Up/down Notation
0 0¢ unison 1 D
1 42.86 up-unison ^1 D^
2 85.71 21:20 84.47 1.24 double-up, double-down ^^1, vv2 D^^, Evv
3 128.57 14:13 128.30 0.27 down 2nd v2 Ev
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 double-up 2nd, double-down 3rd ^^2, vv3 E^^, Fvv
7 300 6:5 315.64 -15.64 down 3rd v3 Fv
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 double-up 3rd, double-down 4th ^^3, vv4 F^^, Gvv
11 471.43 21:16 470.78 0.65 down 4th v4 Gv
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 7:5 582.51 17.49 double-up 4th, double-down 5th ^^4, vv5 G^^, vvA
15 642.86 16:11 648.68 -5.82 down 5th v5 Av
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 double-up 5th, double-down 6th ^^5, vv6 A^^, Bvv
19 814.29 5:8 813.68 0.61 down 6th v6 Bv
20 857.14 18:11 852.59 4.55 6th 6 B
21 900 5:3 884.36 15.64 up 6th ^6 B^
22 942.86 12:7 933.13 9.73 double-up 6th, double-down 7th ^^6, vv7 B^^, Cvv
23 985.71 30:17 983.31 2.40 down 7th v7 Cv
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 double-up 7th, double-down 8ve ^^7, vv8 C^^, Dvv
27 1157.14 down 8ve v8 Dv
28 1200 2:1 1200 0 8ve 8 D

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 Ev G = C(v3) = C down-three

0-9-16 = C E^ G = C(^3) = C up-three

0-8-15 = C E Gv = C(v5) = C down-five

0-9-17 = C E^ G^ = C(^3,^5) = C up-three up-five

0-8-16-24 = C E G B = C7 = C seven

0-8-16-23 = C E G Bv = C(v7) = C down-seven

0-7-16-24 = C Ev G B = C7(v3) = C seven down-three

0-7-16-23 = C Ev G Bv = C.v7 = C dot down seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Rank two temperaments

Periods

per octave

Generator Temperaments
1 1\28
1 3\28 Negri
1 5\28 Machine
1 9\28 Worschmidt
1 11\28
1 13\28 Thuja
2 1\28
2 3\28
2 5\28 Antikythera
4 1\28
4 2\28 Demolished
4 3\28
7 1\28 Whitewood
14 1\28

Commas

28 EDO tempers out the following commas. (Note: This assumes the val < 28 44 65 79 97 104 |.)

Comma Monzo Cents Name 1 Name 2
2187/2048 | -11 7 > 113.69 Apotome
648/625 | 3 4 -4 > 62.57 Major Diesis Diminished Comma
16875/16384 | -14 3 4 > 51.12 Negri Comma Double Augmentation Diesis
| 17 1 -8 > 11.45 Wuerschmidt Comma
36/35 | 2 2 -1 -1 > 48.77 Septimal Quarter Tone
50/49 | 1 0 2 -2 > 34.98 Tritonic Diesis Jubilisma
3125/3087 | 0 -2 5 -3 > 21.18 Gariboh
126/125 | 1 2 -3 1 > 13.79 Septimal Semicomma Starling Comma
65625/65536 | -16 1 5 1 > 2.35 Horwell
| 47 -7 -7 -7 > 0.34 Akjaysma 5\7 Octave Comma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
441/440 | -3 2 -1 2 -1 > 3.93 Werckisma
4000/3993 | 5 -1 3 0 -3 > 3.03 Wizardharry

Some scales

machine5

machine6

machine11

Compositions

28 tone Prelude by Kosmorksy

Retrieved from "https://en.xen.wiki/index.php?title=28edo&oldid=573"