28edo

Theory

Approximation of odd harmonics in 28 EDO

Odd harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31
Error	absolute (¢)	-16.2	-0.6	+16.9	+10.4	+5.8	+16.6	-16.8	-19.2	+2.5	+0.6	+14.6	-1.2	-5.9	-1.0	+12.1
	relative (%)	-38	-1	+39	+24	+14	+39	-39	-45	+6	+2	+34	-3	-14	-2	+28
Steps (reduced)		44 (16)	65 (9)	79 (23)	89 (5)	97 (13)	104 (20)	109 (25)	114 (2)	119 (7)	123 (11)	127 (15)	130 (18)	133 (21)	136 (24)	139 (27)

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 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
	Cents	Interval	Cents
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	double-up 1sn, double-down 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	double-up 2nd, double-down 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	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	498.04	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	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	701.96	-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, 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	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	1200.00	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	Oneirotonic
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	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

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.

• Oneirotonic [5] 65656
• Oneirotonic [8] 55155151
• Oneirotonic [13] 4141141411411
• Oneirotonic [18] 311311131131113111
• Pathological Oneirotonic [23] 21112111121112111121111
• machine5
• machine6
• machine11
• machine17

Music

• 28 tone Prelude by Kosmorksy
• Purple Skyes by Userminusone
• Fantasy for Piano by Eliora