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 (quindromeda temperament). 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.
Equating 225/224 with 256/255 leads quintakwai (12&193), which tempers out 400/399 in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads quintagar (12&217), which tempers out 476/475 in the 2.3.5.7.17.19 subgroup.
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
Quintagar (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
Quintoneum (12&217)
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
13-limit
Comma list: 364/363, 441/440, 3136/3125, 13720/13689
Mapping: [⟨1 2 0 -3 -5 -7], ⟨0 -5 28 70 102 129]]
POTE generator: ~35/33 = 99.541
Vals: 12f, 205df, 217
Badness: 0.052361
17-limit
Comma list: 364/363, 441/440, 595/594, 3136/3125, 3757/3750
Mapping: [⟨1 2 0 -3 -5 -7 5], ⟨0 -5 28 70 102 129 -11]]
POTE generator: ~18/17 = 99.540
Vals: 12f, 205df, 217
Badness: 0.035653
19-limit
Comma list: 364/363, 441/440, 476/475, 595/594, 1216/1215, 3757/3750
Mapping: [⟨1 2 0 -3 -5 -7 5 4], ⟨0 -5 28 70 102 129 -11 3]]
POTE generator: ~18/17 = 99.541
Vals: 12f, 205df, 217
Badness: 0.025782
Quintoneoid (12&217)
13-limit
Comma list: 441/440, 1001/1000, 3136/3125, 59150/59049
Mapping: [⟨1 2 0 -3 -5 11], ⟨0 -5 28 70 102 -88]]
POTE generator: ~35/33 = 99.537
Vals: 12, 205d, 217
Badness: 0.072826
17-limit
Comma list: 441/440, 595/594, 1001/1000, 2601/2600, 3136/3125
Mapping: [⟨1 2 0 -3 -5 11 5], ⟨0 -5 28 70 102 -88 -11]]
POTE generator: ~18/17 = 99.537
Vals: 12, 205d, 217
Badness: 0.042339
19-limit
Comma list: 441/440, 476/475, 595/594, 1001/1000, 1216/1215, 2601/2600
Mapping: [⟨1 2 0 -3 -5 11 5 4], ⟨0 -5 28 70 102 -88 -11 3]]
POTE generator: ~18/17 = 99.537
Vals: 12, 205d, 217
Badness: 0.028983
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