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__FORCETOC__
__FORCETOC__
-----
=Theory=
 
=Basic properties=
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.
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=
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.
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=
=Intervals=
The following table compares it to potentially useful nearby [[just_intervals|just intervals]].
The following table compares it to potentially useful nearby [[just_intervals|just intervals]].


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(ET minus Just)
(ET minus Just)
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|Up/down]] [[Ups_and_Downs_Notation|Notation]]
| colspan="3" style="text-align:center;" | [[Ups and Downs Notation]]
|-
|-
|Cents
|Cents
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| style="text-align:center;" | up-unison
| style="text-align:center;" | up-unison
| style="text-align:center;" | ^1
| style="text-align:center;" | ^1
| style="text-align:center;" | D^
| style="text-align:center;" | ^D
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
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| style="text-align:center;" | 84.47
| style="text-align:center;" | 84.47
| style="text-align:center;" | 1.24
| style="text-align:center;" | 1.24
| style="text-align:center;" | double-up, double-down
| style="text-align:center;" | double-up 1sn, double-down 2nd
| style="text-align:center;" | ^^1, vv2
| style="text-align:center;" | ^^1, vv2
| style="text-align:center;" | D^^, Evv
| style="text-align:center;" | ^^D, vvE
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
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| style="text-align:center;" | down 2nd
| style="text-align:center;" | down 2nd
| style="text-align:center;" | v2
| style="text-align:center;" | v2
| style="text-align:center;" | Ev
| style="text-align:center;" | vE
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
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| style="text-align:center;" | up 2nd
| style="text-align:center;" | up 2nd
| style="text-align:center;" | ^2
| style="text-align:center;" | ^2
| style="text-align:center;" | E^
| style="text-align:center;" | ^E
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
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| style="text-align:center;" | double-up 2nd, double-down 3rd
| style="text-align:center;" | double-up 2nd, double-down 3rd
| style="text-align:center;" | ^^2, vv3
| style="text-align:center;" | ^^2, vv3
| style="text-align:center;" | E^^, Fvv
| style="text-align:center;" | ^^E, vvF
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
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| style="text-align:center;" | down 3rd
| style="text-align:center;" | down 3rd
| style="text-align:center;" | v3
| style="text-align:center;" | v3
| style="text-align:center;" | Fv
| style="text-align:center;" | vF
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
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| style="text-align:center;" | up 3rd
| style="text-align:center;" | up 3rd
| style="text-align:center;" | ^3
| style="text-align:center;" | ^3
| style="text-align:center;" | F^
| style="text-align:center;" | ^F
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
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| style="text-align:center;" | double-up 3rd, double-down 4th
| style="text-align:center;" | double-up 3rd, double-down 4th
| style="text-align:center;" | ^^3, vv4
| style="text-align:center;" | ^^3, vv4
| style="text-align:center;" | F^^, Gvv
| style="text-align:center;" | ^^F, vvG
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
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| style="text-align:center;" | down 4th
| style="text-align:center;" | down 4th
| style="text-align:center;" | v4
| style="text-align:center;" | v4
| style="text-align:center;" | Gv
| style="text-align:center;" | vG
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
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| style="text-align:center;" | up 4th
| style="text-align:center;" | up 4th
| style="text-align:center;" | ^4
| style="text-align:center;" | ^4
| style="text-align:center;" | G^
| style="text-align:center;" | ^G
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
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| style="text-align:center;" | double-up 4th, double-down 5th
| style="text-align:center;" | double-up 4th, double-down 5th
| style="text-align:center;" | ^^4, vv5
| style="text-align:center;" | ^^4, vv5
| style="text-align:center;" | G^^, vvA
| style="text-align:center;" | ^^G, vvA
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 15
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| style="text-align:center;" | down 5th
| style="text-align:center;" | down 5th
| style="text-align:center;" | v5
| style="text-align:center;" | v5
| style="text-align:center;" | Av
| style="text-align:center;" | vA
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
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| style="text-align:center;" | up 5th
| style="text-align:center;" | up 5th
| style="text-align:center;" | ^5
| style="text-align:center;" | ^5
| style="text-align:center;" | A^
| style="text-align:center;" | ^A
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 18
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| style="text-align:center;" | double-up 5th, double-down 6th
| style="text-align:center;" | double-up 5th, double-down 6th
| style="text-align:center;" | ^^5, vv6
| style="text-align:center;" | ^^5, vv6
| style="text-align:center;" | A^^, Bvv
| style="text-align:center;" | ^^A, vvB
|-
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 19
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| style="text-align:center;" | down 6th
| style="text-align:center;" | down 6th
| style="text-align:center;" | v6
| style="text-align:center;" | v6
| style="text-align:center;" | Bv
| style="text-align:center;" | vB
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 20
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| style="text-align:center;" | up 6th
| style="text-align:center;" | up 6th
| style="text-align:center;" | ^6
| style="text-align:center;" | ^6
| style="text-align:center;" | B^
| style="text-align:center;" | ^B
|-
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 22
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| style="text-align:center;" | double-up 6th, double-down 7th
| style="text-align:center;" | double-up 6th, double-down 7th
| style="text-align:center;" | ^^6, vv7
| style="text-align:center;" | ^^6, vv7
| style="text-align:center;" | B^^, Cvv
| style="text-align:center;" | ^^B, vvC
|-
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 23
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| style="text-align:center;" | down 7th
| style="text-align:center;" | down 7th
| style="text-align:center;" | v7
| style="text-align:center;" | v7
| style="text-align:center;" | Cv
| style="text-align:center;" | vC
|-
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 24
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| style="text-align:center;" | up 7th
| style="text-align:center;" | up 7th
| style="text-align:center;" | ^7
| style="text-align:center;" | ^7
| style="text-align:center;" | C^
| style="text-align:center;" | ^C
|-
|-
| style="text-align:center;" | 26
| style="text-align:center;" | 26
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| style="text-align:center;" | double-up 7th, double-down 8ve
| style="text-align:center;" | double-up 7th, double-down 8ve
| style="text-align:center;" | ^^7, vv8
| style="text-align:center;" | ^^7, vv8
| style="text-align:center;" | C^^, Dvv
| style="text-align:center;" | ^^C, vvD
|-
|-
| style="text-align:center;" | 27
| style="text-align:center;" | 27
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| style="text-align:center;" | down 8ve
| style="text-align:center;" | down 8ve
| style="text-align:center;" | v8
| style="text-align:center;" | v8
| style="text-align:center;" | Dv
| style="text-align:center;" | vD
|-
|-
| style="text-align:center;" | 28
| style="text-align:center;" | 28
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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 vE G = Cv = C down


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


0-8-15 = C E Gv = C(v5) = C down-five
0-8-15 = C E vG = 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^(^5) = C up 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 vB = C,v7 = C add down-seven


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


0-7-16-23 = C Ev G Bv = C.v7 = C dot down seven
0-7-16-23 = C vE G vB = Cv7 = C down-seven


For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|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]].
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{| class="wikitable"
{| class="wikitable"
|-
|-
! | Comma
! | [[Ratio]]
! | Monzo
! | [[Monzo]]
! | Cents
! | [[Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 1
! | Name 2
! | Name 2
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| |<nowiki> | -11 7 </nowiki>&gt;
| |<nowiki> | -11 7 </nowiki>&gt;
| style="text-align:center;" | 113.69
| style="text-align:center;" | 113.69
| style="text-align:center;" |Lawa
| style="text-align:center;" | Apotome
| style="text-align:center;" | Apotome
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 3 4 -4 </nowiki>&gt;
| |<nowiki> | 3 4 -4 </nowiki>&gt;
| style="text-align:center;" | 62.57
| style="text-align:center;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | Diminished Comma
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| |<nowiki> | -14 3 4 </nowiki>&gt;
| |<nowiki> | -14 3 4 </nowiki>&gt;
| style="text-align:center;" | 51.12
| style="text-align:center;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| style="text-align:center;" | Double Augmentation Diesis
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| |<nowiki> | 17 1 -8 </nowiki>&gt;
| |<nowiki> | 17 1 -8 </nowiki>&gt;
| style="text-align:center;" | 11.45
| style="text-align:center;" | 11.45
| style="text-align:center;" |Saquadbigu
| style="text-align:center;" | Wuerschmidt Comma
| style="text-align:center;" | Wuerschmidt Comma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
| style="text-align:center;" | 48.77
| style="text-align:center;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 1 0 2 -2 </nowiki>&gt;
| |<nowiki> | 1 0 2 -2 </nowiki>&gt;
| style="text-align:center;" | 34.98
| style="text-align:center;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | Jubilisma
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| |<nowiki> | 0 -2 5 -3 </nowiki>&gt;
| |<nowiki> | 0 -2 5 -3 </nowiki>&gt;
| style="text-align:center;" | 21.18
| style="text-align:center;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | Gariboh
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| style="text-align:center;" | 13.79
| style="text-align:center;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | Starling Comma
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| |<nowiki> | -16 1 5 1 </nowiki>&gt;
| |<nowiki> | -16 1 5 1 </nowiki>&gt;
| style="text-align:center;" | 2.35
| style="text-align:center;" | 2.35
| style="text-align:center;" |Lazoquinyo
| style="text-align:center;" | Horwell
| style="text-align:center;" | Horwell
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 47 -7 -7 -7 </nowiki>&gt;
| |<nowiki> | 47 -7 -7 -7 </nowiki>&gt;
| style="text-align:center;" | 0.34
| style="text-align:center;" | 0.34
| style="text-align:center;" |Trisa-seprugu
| style="text-align:center;" | Akjaysma
| style="text-align:center;" | Akjaysma
| style="text-align:center;" | 5\7 Octave Comma
| style="text-align:center;" | 5\7 Octave Comma
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| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| style="text-align:center;" | 9.86
| style="text-align:center;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| style="text-align:center;" | 3.93
| style="text-align:center;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | Werckisma
| style="text-align:center;" |  
| style="text-align:center;" |  
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| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
| style="text-align:center;" | 3.03
| style="text-align:center;" | 3.03
| style="text-align:center;" |Triluyo
| style="text-align:center;" | Wizardharry
| style="text-align:center;" | Wizardharry
| style="text-align:center;" |  
| style="text-align:center;" |  
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[[machine11|machine11]]
[[machine11|machine11]]


=Compositions=
=Music=
[http://www.youtube.com/watch?v=26UpCbrb3mE 28 tone Prelude] by Kosmorksy
[http://www.youtube.com/watch?v=26UpCbrb3mE 28 tone Prelude] by Kosmorksy
[[Category:28edo]]
[[Category:28edo]]

Revision as of 09:14, 14 December 2019

Theory

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.

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.

Intervals

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

Step # ET Just Difference

(ET minus Just)

Ups and Downs Notation
Cents Interval Cents
0 0¢ unison 1 D
1 42.86 up-unison ^1 ^D
2 85.71 21:20 84.47 1.24 double-up 1sn, double-down 2nd ^^1, vv2 ^^D, vvE
3 128.57 14:13 128.3 0.27 down 2nd v2 vE
4 171.43 11:10 165 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, vvF
7 300 6:5 315.64 -15.64 down 3rd v3 vF
8 342.86 11:9 347.41 -4.55 3rd 3 F
9 385.71 5:4 386.31 -0.60 up 3rd ^3 ^F
10 428.57 9:7 435.08 -6.51 double-up 3rd, double-down 4th ^^3, vv4 ^^F, vvG
11 471.43 21:16 470.78 0.65 down 4th v4 vG
12 514.29 4:3 498.045 16.245 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 vA
16 685.71 3:2 701.955 -16.245 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, vvB
19 814.29 8:5 813.68 0.61 down 6th v6 vB
20 857.14 18:11 852.59 4.55 6th 6 B
21 900 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, vvC
23 985.71 30:17 983.31 2.40 down 7th v7 vC
24 1028.57 20:11 1035 -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, vvD
27 1157.14 down 8ve v8 vD
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 vE G = Cv = C down

0-9-16 = C ^E G = C^ = C up

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

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

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

0-8-16-23 = C E G vB = C,v7 = C add down-seven

0-7-16-24 = C vE G B = Cv,7 = C down add seven

0-7-16-23 = C vE G vB = Cv7 = C down-seven

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

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 |.)

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

Some scales

machine5

machine6

machine11

Music

28 tone Prelude by Kosmorksy

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