28edo
Basic properties[edit]
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[edit]
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[edit]
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[edit]
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[edit]
| 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[edit]
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[edit]
Compositions[edit]
28 tone Prelude by Kosmorksy