24ed5
| ← 23ed5 | 24ed5 | 25ed5 → |
Division of the 5th harmonic into 24 equal parts (24ed5) is related to the miracle temperament. The step size about 116.0964 cents. It is similar to every third step of 31edo, 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.
Theory
From a no-twos-or-threes point of view, 24ed5 offers a particularly good tuning of the very low-badness 5.7.11 subgroup temperament named as juggernaut, tempering out 125/121. This has a CTE generator of exactly 7/5 (in 24ed5 approximated as 5 steps) and a period of 1\2ed5 or the square root of five (which is equated to 11/5).
Interval table
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 116.0964 | 16/15, 15/14 | |
| 2 | 232.1928 | 8/7 | |
| 3 | 348.2892 | 11/9 | |
| 4 | 464.3856 | 17/13 | |
| 5 | 580.4820 | 7/5 | |
| 6 | 696.5784 | meantone fifth (pseudo-3/2) | |
| 7 | 812.6748 | 8/5 | |
| 8 | 928.7712 | 65/38 | |
| 9 | 1044.8676 | 11/6 | |
| 10 | 1160.9640 | 45/23 | |
| 11 | 1277.0605 | 23/11 | |
| 12 | 1393.1569 | 38/17, 85/38 | meantone major second plus an octave |
| 13 | 1509.2533 | 55/23 | |
| 14 | 1625.3497 | 23/9 | |
| 15 | 1741.4461 | 30/11 | |
| 16 | 1857.5425 | 38/13 | |
| 17 | 1973.6389 | 25/8 | |
| 18 | 2089.7353 | meantone major sixth plus an octave (pseudo-10/3) | |
| 19 | 2205.8317 | 25/7 | |
| 20 | 2321.9281 | 65/17 | |
| 21 | 2438.0245 | 45/11 | |
| 22 | 2554.1209 | 35/8 | |
| 23 | 2670.2173 | 14/3 | |
| 24 | 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 (¢) | -39.0 | -44.4 | +38.0 | +0.0 | +32.6 | -2.0 | -1.0 | +27.3 | -39.0 | +28.2 | -6.4 |
| Relative (%) | -33.6 | -38.3 | +32.8 | +0.0 | +28.1 | -1.7 | -0.9 | +23.5 | -33.6 | +24.2 | -5.5 | |
| Steps (reduced) |
10 (10) |
16 (16) |
21 (21) |
24 (0) |
27 (3) |
29 (5) |
31 (7) |
33 (9) |
34 (10) |
36 (12) |
37 (13) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -28.9 | -41.1 | -44.4 | -40.0 | -28.9 | -11.8 | +10.7 | +38.0 | -46.4 | -10.9 | +28.3 |
| Relative (%) | -24.9 | -35.4 | -38.3 | -34.5 | -24.9 | -10.1 | +9.2 | +32.8 | -40.0 | -9.4 | +24.3 | |
| Steps (reduced) |
38 (14) |
39 (15) |
40 (16) |
41 (17) |
42 (18) |
43 (19) |
44 (20) |
45 (21) |
45 (21) |
46 (22) |
47 (23) | |
24ed5 as a generator
24ed5 can also be thought of as a generator of the 2.3.5.7.11.23 subgroup temperament which tempers out 225/224, 243/242, 385/384, and 529/528, which is a cluster temperament with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44 ~ 46/45 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together. This temperament is supported by 31edo, 82edo, 113edo, and 144edo.