28edo: Difference between revisions

__FORCETOC__

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 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.

The following table compares it to potentially useful nearby [[just_intervals|just intervals]].

(ET minus Just)

|-

|Cents

| style="text-align:center;" | up-unison

| style="text-align:center;" | ^1

|-

| style="text-align:center;" | 2

| style="text-align:center;" | 84.47

| style="text-align:center;" | 1.24

| style="text-align:center;" | ^^1, vv2

|-

| style="text-align:center;" | 3

| style="text-align:center;" | down 2nd

| style="text-align:center;" | v2

|-

| style="text-align:center;" | 4

| style="text-align:center;" | up 2nd

| style="text-align:center;" | ^2

|-

| style="text-align:center;" | 6

| style="text-align:center;" | double-up 2nd, double-down 3rd

| style="text-align:center;" | ^^2, vv3

|-

| style="text-align:center;" | 7

| style="text-align:center;" | down 3rd

| style="text-align:center;" | v3

|-

| style="text-align:center;" | 8

| style="text-align:center;" | up 3rd

| style="text-align:center;" | ^3

|-

| style="text-align:center;" | 10

| style="text-align:center;" | double-up 3rd, double-down 4th

| style="text-align:center;" | ^^3, vv4

|-

| style="text-align:center;" | 11

| style="text-align:center;" | down 4th

| style="text-align:center;" | v4

|-

| style="text-align:center;" | 12

| style="text-align:center;" | up 4th

| style="text-align:center;" | ^4

|-

| style="text-align:center;" | 14

| style="text-align:center;" | double-up 4th, double-down 5th

| style="text-align:center;" | ^^4, vv5

|-

| style="text-align:center;" | 15

| style="text-align:center;" | down 5th

| style="text-align:center;" | v5

|-

| style="text-align:center;" | 16

| style="text-align:center;" | up 5th

| style="text-align:center;" | ^5

|-

| style="text-align:center;" | 18

| style="text-align:center;" | double-up 5th, double-down 6th

| style="text-align:center;" | ^^5, vv6

|-

| style="text-align:center;" | 19

| style="text-align:center;" | down 6th

| style="text-align:center;" | v6

|-

| style="text-align:center;" | 20

| style="text-align:center;" | up 6th

| style="text-align:center;" | ^6

|-

| style="text-align:center;" | 22

| style="text-align:center;" | double-up 6th, double-down 7th

| style="text-align:center;" | ^^6, vv7

|-

| style="text-align:center;" | 23

| style="text-align:center;" | down 7th

| style="text-align:center;" | v7

|-

| style="text-align:center;" | 24

| style="text-align:center;" | up 7th

| style="text-align:center;" | ^7

|-

| style="text-align:center;" | 26

| style="text-align:center;" | double-up 7th, double-down 8ve

| style="text-align:center;" | ^^7, vv8

|-

| style="text-align:center;" | 27

| style="text-align:center;" | down 8ve

| style="text-align:center;" | v8

|-

| style="text-align:center;" | 28

0-8-16 = C E G = C = C or C perfect

0-8-16-24 = C E G B = C7 = C 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]].

{| class="wikitable"

|-

! | Name 1

! | Name 2

| |<nowiki> | -11 7 </nowiki>&gt;

| style="text-align:center;" | 113.69

| style="text-align:center;" | Apotome

| style="text-align:center;" |  

| |<nowiki> | 3 4 -4 </nowiki>&gt;

| style="text-align:center;" | 62.57

| style="text-align:center;" | Major Diesis

| style="text-align:center;" | Diminished Comma

| |<nowiki> | -14 3 4 </nowiki>&gt;

| style="text-align:center;" | 51.12

| style="text-align:center;" | Negri Comma

| style="text-align:center;" | Double Augmentation Diesis

| |<nowiki> | 17 1 -8 </nowiki>&gt;

| style="text-align:center;" | 11.45

| style="text-align:center;" | Wuerschmidt Comma

| style="text-align:center;" |  

| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;

| style="text-align:center;" | 48.77

| style="text-align:center;" | Septimal Quarter Tone

| style="text-align:center;" |  

| |<nowiki> | 1 0 2 -2 </nowiki>&gt;

| style="text-align:center;" | 34.98

| style="text-align:center;" | Tritonic Diesis

| style="text-align:center;" | Jubilisma

| |<nowiki> | 0 -2 5 -3 </nowiki>&gt;

| style="text-align:center;" | 21.18

| style="text-align:center;" | Gariboh

| style="text-align:center;" |  

| |<nowiki> | 1 2 -3 1 </nowiki>&gt;

| style="text-align:center;" | 13.79

| style="text-align:center;" | Septimal Semicomma

| style="text-align:center;" | Starling Comma

| |<nowiki> | -16 1 5 1 </nowiki>&gt;

| style="text-align:center;" | 2.35

| style="text-align:center;" | Horwell

| style="text-align:center;" |  

| |<nowiki> | 47 -7 -7 -7 </nowiki>&gt;

| style="text-align:center;" | 0.34

| style="text-align:center;" | Akjaysma

| style="text-align:center;" | 5\7 Octave Comma

| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;

| style="text-align:center;" | 9.86

| style="text-align:center;" | Valinorsma

| style="text-align:center;" |  

| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;

| style="text-align:center;" | 3.93

| style="text-align:center;" | Werckisma

| style="text-align:center;" |  

| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;

| style="text-align:center;" | 3.03

| style="text-align:center;" | Wizardharry

| style="text-align:center;" |  

[[machine11|machine11]]

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

[[Category:28edo]]