28ed5

Division of the 5th harmonic into 28 equal parts (28ED5) is related to 12EDO, but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 99.5112 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12EDO, as well as 18/17, 19/16, and 24/17.

degree	cents value	corresponding JI intervals	comments
0	0.0000	exact 1/1
1	99.5112	18/17
2	199.0224	55/49
3	298.5336	19/16
4	398.0448	34/27	pseudo-5/4
5	497.5560	4/3
6	597.0672	24/17
7	696.5784		meantone fifth (pseudo-3/2)
8	796.0896	19/12
9	895.6008	57/34	pseudo-5/3
10	995.1120	16/9
11	1094.6232	32/17
12	1194.1344	255/128	pseudo-octave
13	1293.6457	19/9
14	1393.1569	38/17, 85/38	meantone major second plus an octave
15	1492.6681	45/19
16	1592.1793	128/51	pseudo-5/2
17	1691.6905	85/32
18	1791.2017	45/16
19	1890.7129	170/57	pseudo-3/1
20	1990.2241	60/19
21	2089.7353		meantone major sixth plus an octave (pseudo-10/3)
22	2189.2465	85/24
23	2288.7577	15/4
24	2388.2689	135/34	pseudo-4/1
25	2487.7801	80/19
26	2587.2913	49/11
27	2686.8025	85/18
28	2786.3137	exact 5/1	just major third plus two octaves

28ed5 as a generator

28ED5 can also be thought of as a generator of the 2.3.5.17.19 subgroup temperament which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a cluster temperament with 12 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 1088/1083 ~ 256/255 ~ 289/288 ~ 324/323 ~ 361/360 all tempered together. This temperament is supported by 12EDO, 205EDO, and 217EDO among others.

Quindromeda (12&193)
5-limit
Comma: [56 -28 -5⟩
Mapping: [⟨1 2 0], ⟨0 -5 28]]
POTE generator: ~4428675/4194304 = 99.526
Vals: 12, 169, 181, 193, 205, 217, 422
Badness: 0.399849

2.3.5.17.19 subgroup
Comma list: 1216/1215, 1445/1444, 6144/6137
Gencom: [2 18/17; 1216/1215 1445/1444 6144/6137]
Gencom mapping: [⟨1 2 0 5 4], ⟨0 -5 28 -11 3]]
POTE generator: ~18/17 = 99.524
Vals: 12, 169, 181, 193, 205, 422
RMS error: 0.0813 cents

Quintakwai (12&193)
7-limit
Comma list: 5120/5103, 9765625/9680832
Mapping: [⟨1 2 0 -2], ⟨0 -5 28 58]]
POTE generator: ~625/588 = 99.483
Vals: 12, 169, 181, 193
Badness: 0.155536

11-limit
Comma list: 1375/1372, 4375/4356, 5120/5103
Mapping: [⟨1 2 0 -2 -4], ⟨0 -5 28 58 90]]
POTE generator: ~35/33 = 99.472
Vals: 12, 181, 193, 374, 567ce
Badness: 0.073158

13-limit
Comma list: 325/324, 1375/1372, 1575/1573, 4096/4095
Mapping: [⟨1 2 0 -2 -4 10], ⟨0 -5 28 58 90 -76]]
POTE generator: ~35/33 = 99.468
Vals: 12, 181, 193, 374, 567ce, 941bce
Badness: 0.062737

17-limit
Comma list: 325/324, 375/374, 595/594, 1275/1274, 4096/4095
Mapping: [⟨1 2 0 -2 -4 10 5], ⟨0 -5 28 58 90 -76 -11]]
POTE generator: ~18/17 = 99.469
Vals: 12, 181, 193, 374, 567ce, 941bceg
Badness: 0.037855

19-limit
Comma list: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274
Mapping: [⟨1 2 0 -2 -4 10 5 4], ⟨0 -5 28 58 90 -76 -11 3]]
POTE generator: ~18/17 = 99.469
Vals: 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh
Badness: 0.025861

Quintoneum (12&217)
7-limit
Comma list: 3136/3125, 33554432/33480783
Mapping: [⟨1 2 0 -3], ⟨0 -5 28 70]]
POTE generator: ~200/189 = 99.555
Vals: 12, 217, 229, 446, 675c
Badness: 0.142897

11-limit
Comma list: 441/440, 3136/3125, 7168000/7144929
Mapping: [⟨1 2 0 -3 -5], ⟨0 -5 28 70 102]]
POTE generator: ~35/33 = 99.539
Vals: 12, 205d, 217
Badness: 0.087157

Quintasandra (217&229)
11-limit
Comma list: 3136/3125, 19712/19683, 41503/41472
Mapping: [⟨1 2 0 -3 13], ⟨0 -5 28 70 -115]]
POTE generator: ~200/189 = 99.551
Vals: 12e, 217, 446
Badness: 0.109908

13-limit
Comma list: 2080/2079, 3136/3125, 4096/4095, 19712/19683
Mapping: [⟨1 2 0 -3 13 11], ⟨0 -5 28 70 -115 -88]]
POTE generator: ~55/52 = 99.548
Vals: 12e, 217, 446, 663c
Badness: 0.067730

17-limit
Comma list: 936/935, 1156/1155, 1377/1375, 3136/3125, 4096/4095
Mapping: [⟨1 2 0 -3 13 11 5], ⟨0 -5 28 70 -115 -88 -11]]
POTE generator: ~18/17 = 99.548
Vals: 12e, 217, 446, 663c
Badness: 0.038153

19-limit
Comma list: 476/475, 936/935, 1156/1155, 1216/1215, 1377/1375, 1729/1728
Mapping: [⟨1 2 0 -3 13 11 5 4], ⟨0 -5 28 70 -115 -88 -11 3]]
POTE generator: ~18/17 = 99.547
Vals: 12e, 217, 446, 663ch
Badness: 0.026654

Quintasandroid (229&241)
11-limit
Comma list: 3136/3125, 8019/8000, 15488/15435
Mapping: [⟨1 2 0 -3 -6], ⟨0 -5 28 70 114]]
POTE generator: ~200/189 = 99.570
Vals: 12, 217e, 229, 470cd, 699cd
Badness: 0.093971

13-limit
Comma list: 351/350, 2080/2079, 3136/3125, 10648/10647
Mapping: [⟨1 2 0 -3 -6 -8], ⟨0 -5 28 70 114 141]]
POTE generator: ~55/52 = 99.578
Vals: 12f, 217ef, 229, 241, 470cd, 711ccd
Badness: 0.065701

17-limit
Comma list: 351/350, 442/441, 561/560, 3136/3125, 7744/7735
Mapping: [⟨1 2 0 -3 -6 -8 5], ⟨0 -5 28 70 114 141 -11]]
POTE generator: ~18/17 = 99.574
Vals: 12f, 217ef, 229, 241, 470cd
Badness: 0.046624

19-limit
Comma list: 351/350, 442/441, 476/475, 561/560, 627/625, 6144/6137
Mapping: [⟨1 2 0 -3 -6 -8 5 4], ⟨0 -5 28 70 114 141 -11 3]]
POTE generator: ~18/17 = 99.575
Vals: 12f, 217ef, 229, 241, 470cd
Badness: 0.033145

Quintasand (12&229)
13-limit
Comma list: 1573/1568, 3136/3125, 4096/4095, 4459/4455
Mapping: [⟨1 2 0 -3 -6 11], ⟨0 -5 28 70 114 -88]]
POTE generator: ~200/189 = 99.556
Vals: 12, 217e, 229, 446e, 675ceef
Badness: 0.100195

17-limit
Comma list: 561/560, 715/714, 1701/1700, 3136/3125, 4096/4095
Mapping: [⟨1 2 0 -3 -6 11 5], ⟨0 -5 28 70 114 -88 -11]]
POTE generator: ~18/17 = 99.556
Vals: 12, 217e, 229, 446e, 675ceef
Badness: 0.057851

19-limit
Comma list: 286/285, 476/475, 561/560, 627/625, 1216/1215, 1729/1728
Mapping: [⟨1 2 0 -3 -6 11 5 4], ⟨0 -5 28 70 114 -88 -11 3]]
POTE generator: ~18/17 = 99.557
Vals: 12, 217e, 229, 446e, 675ceefh
Badness: 0.040410

Semiquindromeda (12&422)
7-limit
Comma list: 102760448/102515625, 1220703125/1219784832
Mapping: [⟨2 4 0 -5], ⟨0 -5 28 64]]
POTE generator: ~1323/1250 = 99.521
Vals: 12, 398, 410, 422, 832, 1254d, 2086bd
Badness: 0.233140

11-limit
Comma list: 5632/5625, 9801/9800, 85937500/85766121
Mapping: [⟨2 4 0 -5 -10], ⟨0 -5 28 64 102]]
POTE generator: ~1323/1250 = 99.525
Vals: 12, 410, 422
Badness: 0.093926

13-limit
Comma list: 1716/1715, 2080/2079, 5632/5625, 831875/830466
Mapping: [⟨2 4 0 -5 -10 -13], ⟨0 -5 28 64 102 123]]
POTE generator: ~1323/1250 = 99.523
Vals: 12f, 410, 422, 1254df, 1676bdff, 2098bcddff
Badness: 0.053361

17-limit
Comma list: 1716/1715, 2080/2079, 2500/2499, 5632/5625, 15895/15876
Mapping: [⟨2 4 0 -5 -10 -13 10], ⟨0 -5 28 64 102 123 -11]]
POTE generator: ~18/17 = 99.522
Vals: 12f, 410, 422, 832, 1254df, 1676bdff
Badness: 0.034659

19-limit
Comma list: 1216/1215, 1445/1444, 1716/1715, 2080/2079, 2376/2375, 2500/2499
Mapping: [⟨2 4 0 -5 -10 -13 10 8], ⟨0 -5 28 64 102 123 -11 3]]
POTE generator: ~18/17 = 99.523
Vals: 12f, 410, 422, 1254dfhh, 1676bdffhh
Badness: 0.025439

See also

• 12edo: relative EDO
• 19ed3: relative ED3
• 31ed6: relative ED6
• 34ed7: relative ED7
• 40ed10: relative ED10
• 42ed11: relative ED11