48ed5

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Division of the 5th harmonic into 48 equal parts (48ed5) is related to the semimiracle temperament. The step size is about 58.0482 cents. It is similar to every third step of 62edo, but with the 5/1 rather than the 2/1 being just. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.

degree cents value corresponding
JI intervals
comments
0 0.0000 exact 1/1
1 58.0482 31/30 ~ 30/29
2 116.0964 16/15 ~ 15/14
3 174.1446
4 232.1928 8/7
5 290.2410 13/11 ~ 45/38
6 348.2892 11/9
7 406.3374
8 464.3856 17/13
9 522.4338 23/17
10 580.4820 7/5
11 638.5302 13/9 ~ 55/38
12 696.5784 meantone fifth
(pseudo-3/2)
13 754.6266 17/11
14 812.6748 8/5
15 870.7230 38/23
16 928.7712 65/38
17 986.8194 23/13
18 1044.8676 11/6
19 1102.9158 17/9
20 1160.9640 45/23
21 1219.0122
22 1277.0605 23/11
23 1335.1087 13/6
24 1393.1569 38/17 ~ 85/38 meantone major second plus an octave
25 1451.2051 30/13
26 1509.2533 55/23
27 1567.3015
28 1625.3497 23/9
29 1683.3979 45/17
30 1741.4461 30/11
31 1799.4943 65/23
32 1857.5425 38/13
33 1915.5907 115/38
34 1973.6389 25/8
35 2031.6871 55/17
36 2089.7353 meantone major sixth plus an octave
(pseudo-10/3)
37 2147.7835 38/11 ~ 45/13
38 2205.8317 25/7
39 2263.8799 85/23
40 2321.9281 65/17
41 2379.9763
42 2438.0245 45/11
43 2496.0727 38/9 ~ 55/13
44 2554.1209 35/8
45 2612.1691
46 2670.2173 14/3
47 2728.2655 29/6
48 2786.3137 exact 5/1 just major third plus two octaves

48ed5 as a generator

48ed5 can also be thought of as a generator of the 23-limit temperament which tempers out 169/168, 225/224, 243/242, 286/285, 385/384, 598/595, and 1445/1444 which is a cluster temperament with 21 clusters of notes in an octave. The large chroma interval between adjacent notes in each cluster is very versatile, representing 77/75 ~ 39/38 ~ 165/161 ~ 253/247 ~ 169/165 ~ 128/125 ~ 306/299 ~ 45/44 ~ 136/133 ~ 46/45 ~ 608/595 ~ 1292/1265 ~ 49/48 ~ 50/49 ~ 352/345 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together. This temperament is supported by 62edo, 82edo (82gh val), and 144edo (144f val).