28edo

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	41:40	42.74	0.12	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	80:41	1157.26	-0.12	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