28edo

← 27edo 28edo 29edo →
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 42.9 ¢ 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 (%)	-37.9	-1.4	+39.4	+24.2	+13.6	+38.8	-39.3	-44.9	+5.8	+1.5	+34.0
Steps (reduced)		44 (16)	65 (9)	79 (23)	89 (5)	97 (13)	104 (20)	109 (25)	114 (2)	119 (7)	123 (11)	127 (15)

Approximation of odd harmonics in 28edo
Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	-1.2	-5.9	-1.0	+12.1	-10.4	+16.3	+5.8	+0.4	-0.5	+2.8	+9.8
Error	Relative (%)	-2.8	-13.7	-2.3	+28.3	-24.3	+38.0	+13.5	+0.9	-1.1	+6.5	+22.8
Steps (reduced)		130 (18)	133 (21)	136 (24)	139 (27)	141 (1)	144 (4)	146 (6)	148 (8)	150 (10)	152 (12)	154 (14)

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

Notation

Ups and downs notation

As 28edo tempers out the Pythagorean apotome, the traditional sharps and flats have no affect on the pitch. Therefore, with ups and downs notation, arrows are required.

Step offset	−3	−2	−1	0	+1	+2	+3
Symbol

Sagittal notation

This notation uses the same sagittal sequence as EDOs 23 and 33, and is a superset of the notations for EDOs 14 and 7.

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.

Regular temperament properties

Rank-2 temperaments

Periods per 8ve	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

28et tempers out the following commas. 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	Diminished comma, major diesis
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	Mint comma, septimal quartertone
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Jubilisma, tritonic diesis
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh comma
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Starling comma, septimal semicomma
7	65625/65536	[-16 1 5 1⟩	2.35	Lazoquinyo	Horwell comma
7	(30 digits)	[47 -7 -7 -7⟩	0.34	Trisa-seprugu	Akjaysma
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 comma

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

Octave stretch or compression

89ed9, a compressed-octaves tuning of 28edo, makes 28edo potentially useable as a dual-fifth tuning. It shares the error equally between 28edo's two perfect fifths.

So, where pure-octaves 28edo has a flat fifth with 16.2 ¢ error and a sharp fifth with 28.6 ¢ error, 89ed9 instead has a flat fifth with 21.4 ¢ error and a sharp fifth with 21.4 ¢ error.

21.4 ¢ error is slightly better than 23edo's best fifth but slightly worse than 5edo's.

Some might consider this fifth too bad to use, in which case 89ed9 would be a downgrade from pure-octaves 28edo.

Others might consider it just useable enough, in which case 89ed9 would be an upgrade because it has one more useable interval and one less wolf interval.

89ed9 approximates the no-3s 7-, 11-, 13-, 17-, 19- and 23-limits significantly better than pure-octaves 28edo.

28edo

Step size: 42.8571 ¢, octave size: 1200.00 ¢

Approximates all no-3s harmonics up to 7 within 16.9 ¢.

Approximates all no-3s harmonics up to 11 & up to 13 within 16.9 ¢.

Approximates all no-3s harmonics up to 17 & up to 19 within 19.2 ¢.

Approximation of harmonics in 28edo
Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.0	-16.2	+0.0	-0.6	-16.2	+16.9	+0.0	+10.4	-0.6	+5.8	-16.2
Error	Relative (%)	+0.0	-37.9	+0.0	-1.4	-37.9	+39.4	+0.0	+24.2	-1.4	+13.6	-37.9
Steps (reduced)		28 (0)	44 (16)	56 (0)	65 (9)	72 (16)	79 (23)	84 (0)	89 (5)	93 (9)	97 (13)	100 (16)

Approximation of harmonics in 28edo (continued)
Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+16.6	+16.9	-16.8	+0.0	-19.2	+10.4	+2.5	-0.6	+0.6	+5.8	+14.6	-16.2
Error	Relative (%)	+38.8	+39.4	-39.3	+0.0	-44.9	+24.2	+5.8	-1.4	+1.5	+13.6	+34.0	-37.9
Steps (reduced)		104 (20)	107 (23)	109 (25)	112 (0)	114 (2)	117 (5)	119 (7)	121 (9)	123 (11)	125 (13)	127 (15)	128 (16)

89ed9

Step size: 42.7406 ¢, octave size: 1196.74 ¢

Approximates all no-3s harmonics up to 7 within 8.2 ¢.

Approximates all no-3s harmonics up to 11 & up to 13 within 11.4 ¢.

Approximates all no-3s harmonics up to 17 & up to 19 within 11.4 ¢.

Approximation of harmonics in 89ed9
Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-3.3	+21.4	-6.5	-8.2	+18.1	+7.7	-9.8	+0.0	-11.4	-5.5	+14.8
Error	Relative (%)	-7.6	+50.0	-15.3	-19.1	+42.4	+18.0	-22.9	+0.0	-26.8	-12.8	+34.7
Steps (reduced)		28 (28)	45 (45)	56 (56)	65 (65)	73 (73)	79 (79)	84 (84)	89 (0)	93 (4)	97 (8)	101 (12)

Approximation of harmonics in 89ed9 (continued)
Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+4.5	+4.4	+13.2	-13.1	+10.2	-3.3	-11.4	-14.7	-13.7	-8.7	-0.2	+11.6
Error	Relative (%)	+10.5	+10.3	+30.9	-30.5	+23.9	-7.6	-26.6	-34.4	-32.0	-20.5	-0.5	+27.1
Steps (reduced)		104 (15)	107 (18)	110 (21)	112 (23)	115 (26)	117 (28)	119 (30)	121 (32)	123 (34)	125 (36)	127 (38)	129 (40)

Scales

Whitewood 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)

Negri scales

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

Diatonic scales

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

Other Scales

10-tone 4&7edo scale 4 3 1 4 2 2 4 1 3 4

Instruments

28edo can be played on the Lumatone. See Lumatone mapping for 28edo.

28edo can also be played on a 14edo guitar with very little effort. See User:MisterShafXen/Skip fretting system 28 2 3.

Music

[1] Ratios longer than 10 digits are presented by placeholders with informative hints

[1]