28ed5

Division of the 5th harmonic into 28 equal parts (28ed5) is related to 12 edo, 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.

5-limit 12&193 (quinsa-quingu)

Comma: |56 -28 -5>

POTE generator: ~4428675/4194304 = 99.526

Map: [<1 2 0|, <0 -5 28|]

EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603, 627, 639, 784, 808, 832, 856, 989, 1001, 1013, 1037, 1049, 1061, 1242

Badness: 0.399849

7-limit 12&193

Commas: 5120/5103, 9765625/9680832

POTE generator: ~625/588 = 99.483

Map: [<1 2 0 -2|, <0 -5 28 58|]

EDOs: 12, 169, 181, 193, 205, 374

Badness: 0.155536

11-limit 12&193

Commas: 1375/1372, 4375/4356, 5120/5103

POTE generator: ~35/33 = 99.472

Map: [<1 2 0 -2 -4|, <0 -5 28 58 90|]

EDOs: 12, 169e, 181, 193, 205e, 374

Badness: 0.073158

13-limit 12&193

Commas: 325/324, 1375/1372, 1575/1573, 4096/4095

POTE generator: ~35/33 = 99.468

Map: [<1 2 0 -2 -4 10|, <0 -5 28 58 90 -76|]

EDOs: 12, 181, 193, 374

Badness: 0.062737

17-limit 12&193

Commas: 325/324, 375/374, 595/594, 1275/1274, 4096/4095

POTE generator: ~18/17 = 99.469

Map: [<1 2 0 -2 -4 10 5|, <0 -5 28 58 90 -76 -11|]

EDOs: 12, 181, 193, 374

Badness: 0.037855

19-limit 12&193

Commas: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274

POTE generator: ~18/17 = 99.469

Map: [<1 2 0 -2 -4 10 5 4|, <0 -5 28 58 90 -76 -11 3|]

EDOs: 12, 181, 193, 374

Badness: 0.025861

7-limit 12&229

Commas: 3136/3125, 33554432/33480783

POTE generator: ~200/189 = 99.555

Map: [<1 2 0 -3|, <0 -5 28 70|]

EDOs: 12, 217, 229, 241, 446

Badness: 0.142897

11-limit 12&229

Commas: 3136/3125, 8019/8000, 15488/15435

POTE generator: ~200/189 = 99.570

Map: [<1 2 0 -3 -6|, <0 -5 28 70 114|]

EDOs: 12, 217e, 229, 241, 446e

Badness: 0.093971

13-limit 12&229

Commas: 1573/1568, 3136/3125, 4096/4095, 4459/4455

POTE generator: ~200/189 = 99.556

Map: [<1 2 0 -3 -6 11|, <0 -5 28 70 114 -88|]

EDOs: 12, 217e, 229, 241f, 446e

Badness: 0.100195

17-limit 12&229

Commas: 561/560, 715/714, 1701/1700, 3136/3125, 4096/4095

POTE generator: ~18/17 = 99.556

Map: [<1 2 0 -3 -6 11 5|, <0 -5 28 70 114 -88 -11|]

EDOs: 12, 217e, 229, 241f, 446e

Badness: 0.057851

19-limit 12&229

Commas: 286/285, 476/475, 561/560, 627/625, 1216/1215, 1729/1728

POTE generator: ~18/17 = 99.557

Map: [<1 2 0 -3 -6 11 5 4|, <0 -5 28 70 114 -88 -11 3|]

EDOs: 12, 217e, 229, 241f, 446e

Badness: 0.040410

7-limit 12&422

Commas: 102760448/102515625, 1220703125/1219784832

POTE generator: ~1323/1250 = 99.521

Map: [<2 4 0 -5|, <0 -5 28 64|]

EDOs: 12, 398, 410, 422, 808, 832, 1242

Badness: 0.233140

11-limit 12&422

Commas: 5632/5625, 9801/9800, 85937500/85766121

POTE generator: ~1323/1250 = 99.525

Map: [<2 4 0 -5 -10|, <0 -5 28 64 102|]

EDOs: 12, 410, 422, 832

Badness: 0.093926

13-limit 12f&422

Commas: 1716/1715, 2080/2079, 5632/5625, 831875/830466

POTE generator: ~1323/1250 = 99.523

Map: [<2 4 0 -5 -10 -13|, <0 -5 28 64 102 123|]

EDOs: 12f, 410, 422, 832

Badness: 0.053361

17-limit 12f&422

Commas: 1716/1715, 2080/2079, 2500/2499, 5632/5625, 15895/15876

POTE generator: ~18/17 = 99.522

Map: [<2 4 0 -5 -10 -13 10|, <0 -5 28 64 102 123 -11|]

EDOs: 12f, 410, 422, 832

Badness: 0.034659

19-limit 12f&422

Commas: 1216/1215, 1445/1444, 1716/1715, 2080/2079, 2376/2375, 2500/2499

POTE generator: ~18/17 = 99.523

Map: [<2 4 0 -5 -10 -13 10 8|, <0 -5 28 64 102 123 -11 3|]

EDOs: 12f, 410, 422, 832h

Badness: 0.025439

2.3.5.17.19 subgroup 12&193

Commas: 1216/1215, 1445/1444, 6144/6137

POTE generator: ~18/17 = 99.524

Map: [<1 2 0 5 4|, <0 -5 28 -11 3|]

EDOs: 12, 169, 181, 193, 205, 217, 229, 241, 374, 398, 422, 446, 591, 603

See also

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