28edo

Prime factorization

2² × 7

Step size

42.8571¢

Fifth

16\28 (685.714¢) (→4\7)

Semitones (A1:m2)

0:4 (0¢ : 171.4¢)

Dual sharp fifth

17\28 (728.571¢)

Dual flat fifth

16\28 (685.714¢) (→4\7)

Dual major 2nd

5\28 (214.286¢)

Consistency limit

5

Distinct consistency limit

5

28 equal divisions of the octave (abbreviated 28edo or 28ed2), also called 28-tone equal temperament (28tet) or 28 equal temperament (28et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 28 equal parts of about 42.857 ¢ each. Each step represents a frequency ratio of 2^1/28, or the 28th root of 2.

Theory

Approximation of odd harmonics in 28edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-16.2	-0.6	+16.9	+10.4	+5.8	+16.6	-16.8	-19.2	+2.5	+0.6	+14.6
Error	relative (%)	-38	-1	+39	+24	+14	+39	-39	-45	+6	+2	+34
Steps (reduced)		44 (16)	65 (9)	79 (23)	89 (5)	97 (13)	104 (20)	109 (25)	114 (2)	119 (7)	123 (11)	127 (15)

28edo is a multiple of both 7edo and 14edo (and of course 2edo and 4edo). 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 28 is not divisible by 3. 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. Its approximation to 5/4 is unusually good for an edo of this size, being the next convergent to log₂5 after 3edo.

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 2.5.11.19.21.27.29.39.41.

28edo is the 2nd perfect number EDO.

Intervals

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

Step #	ET (e)	Just (j)		Delta (e-j)	Ups and Downs Notation
Step #	Cents	Interval	Cents	Delta (e-j)	Ups and Downs Notation
0	0.00	1/1	0.00	0.00	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	dup 1sn, dud 2nd	^^1, vv2	^^D, vvE
3	128.57	14/13	128.30	0.27	down 2nd	v2	vE
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	dup 2nd, dud 3rd	^^2, vv3	^^E, vvF
7	300.00	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	dup 3rd, dud 4th	^^3, vv4	^^F, vvG
11	471.43	21/16	470.78	0.65	down 4th	v4	vG
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	600.00	7/5	582.51	17.49	dup 4th, dud 5th	^^4, vv5	^^G, vvA
15	642.86	16/11	648.68	-5.82	down 5th	v5	vA
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	dup 5th, dud 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	900.00	5/3	884.36	15.64	up 6th	^6	^B
22	942.86	12/7	933.13	9.73	dup 6th, dud 7th	^^6, vv7	^^B, vvC
23	985.71	30/17	983.31	2.40	down 7th	v7	vC
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	dup 7th, dud 8ve	^^7, vv8	^^C, vvD
27	1157.14	80/41	1157.26	-0.12	down 8ve	v8	vD
28	1200.00	2/1	1200.00	0.00	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	A-team
1	13\28	Thuja
2	1\28
2	3\28	Octokaidecal
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].)

Prime Limit	Ratio^[1]	Monzo	Cents	Color name	Name(s)
3	2187/2048	[-11 7⟩	113.69	Lawa	Apotome
5	648/625	[3 4 -4⟩	62.57	Quadgu	Major diesis, diminished comma
5	16875/16384	[-14 3 4⟩	51.12	Laquadyo	Negri comma, double augmentation diesis
5	(12 digits)	[17 1 -8⟩	11.45	Saquadbigu	Würschmidt comma
7	36/35	[2 2 -1 -1⟩	48.77	Rugu	Septimal quartertone
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Tritonic diesis, Jubilisma
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Septimal semicomma, Starling comma
7	65625/65536	[-16 1 5 1⟩	2.35	Lazoquinyo	Horwell
7	(30 digits)	[47 -7 -7 -7⟩	0.34	Trisa-seprugu	Akjaysma, 5\7-octave comma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Triluyo	Wizardharry

↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Scales

28edo is particularly well suited to Whitewood in the same way that 15edo is for Blackwood, as it has one third that is heavily tempered, but in a familiar way shared with 12edo, while the other one is significantly closer to just. (Contrast with 20 & 21edo, where one third remains the same as in 12, and the other is pushed further away, making the overall sound considerably more xenharmonic) This makes the 5ths more out of tune, but in a useful way, as you can stack major and minor thirds indefinitely until they repeat 4 octaves and 14 notes up, producing one of the largest nonrepeating harmonious chords possible in an edo this low. This produces mode of symmetry scales with two different modes and 4 different keys, making it equally easy to establish any chord in the scale as the root and modulate between them.

Whitewood Major [14] 13131313131313
Whitewood Minor [14] 31313131313131
Whitewood Major [21] 121121121121121121121
Whitewood Minor [21] 211211211211211211211
Whitewood Diminished [21] 112112112112112112112
(Whitewood neutral is also theoretically possible, stacking neutral or subminor & supermajor thirds, but in practice that works out as 22222222222222, or 14edo, so it doesn't count as a 28edo scale)

If you're looking for something a little more xenharmonic sounding, but still relatively low in badness, Negri [9] & [10] function very similarly to Porcupine [7] & [8] in 15edo, further cementing the idea that 28 is to 15 what 29 is to 12 - a similar structure, but with tempering errors in the opposite direction and slightly greater complexity. Both have a decent selection of familiar major and minor chords clustered towards one end of the scale, while the way they modulate between one-another and the melodies they form is quite alien to people used to the diatonic scale. Most of the compositional notes on the Porcupine page can also be applied here.

Negri [9] 333343333
Negri [10] 3333333331
Negri [19] 2121212121212121211

However, unlike 15, 28 is complex enough to do recognisable approximations of various diatonic scales and their modes, although they will sound noticeably out of tune and it's obviously not the best method of using the temperament.

Diatonic Major [7] 5434552
Diatonic Minor [7] 5254345
Diatonic Naive Major [7] 4534543
Diatonic Naive Minor [7] 4354345
Diatonic Major [10] 3243432322
Diatonic Minor [10] 3223243432
Diatonic Major [12] 322232232322
Diatonic Minor [12] 322322232232
Diatonic Major [16] 2122221222122122
Diatonic Minor [16] 2122212222122212
Harmonic Minor [7] 5254372
Harmonic Major [7] 5434372
Harmonic Minor [8] 52543522, 52543432
Harmonic Major [8] 54343522, 54343432
Harmonic Minor [10] 3223243432
Harmonic Minor [11] 32232433222
Harmonic Major [9] 324343432
Harmonic Major [10] 3243433222
Harmonic Minor [12] 322322232232, 322322233222
Harmonic Major [12] 322232232232, 322232233222
Harmonic Minor [16] 2122212222122212, 212221222212121222
Harmonic Major [16] 2122221222122212, 212221222212121222
Melodic Minor [7] 5254552
Melodic Major [7] 5434345
Melodic Minor [11] 32232432322
Melodic Major [9] 324343432
Diasem (Right-handed) 414434143
Diasem (Left-handed) 441434143
Melodic Minor [12] 322322232322
Melodic Major [12] 322232232232
Melodic Minor [16] 2122212222122122
Melodic Major [16] 2122221222122212

Interestingly, as it has a near perfect 21/16, 28edo can also generate Oneirotonic scales (see 13edo) by stacking it's 11th degree, and they actually sound better in this temperament.