48ed5
| ← 47ed5 | 48ed5 | 49ed5 → |
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.
Intervals
| 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 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +19.0 | +13.6 | -20.0 | +0.0 | -25.4 | -2.0 | -1.0 | +27.3 | +19.0 | +28.2 | -6.4 |
| Relative (%) | +32.8 | +23.5 | -34.5 | +0.0 | -43.8 | -3.5 | -1.7 | +47.0 | +32.8 | +48.5 | -11.0 | |
| Steps (reduced) |
21 (21) |
33 (33) |
41 (41) |
48 (0) |
53 (5) |
58 (10) |
62 (14) |
66 (18) |
69 (21) |
72 (24) |
74 (26) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -28.9 | +17.0 | +13.6 | +18.0 | -28.9 | -11.8 | +10.7 | -20.0 | +11.6 | -10.9 | +28.3 |
| Relative (%) | -49.7 | +29.3 | +23.5 | +31.0 | -49.8 | -20.3 | +18.5 | -34.5 | +20.0 | -18.7 | +48.7 | |
| Steps (reduced) |
76 (28) |
79 (31) |
81 (33) |
83 (35) |
84 (36) |
86 (38) |
88 (40) |
89 (41) |
91 (43) |
92 (44) |
94 (46) | |
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).