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Basic properties | Subgroups | Table of intervals | Chord Names | Rank two temperaments | Commas | Some scales | Compositions

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




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