10ed5
In general, 10ed5 is simply a smashing tuning. The relatively large steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.
| ← 9ed5 | 10ed5 | 11ed5 → |
(semiconvergent)
It is especially important as a structural framework for the 5.7.11.13 subgroup.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -85 | +48 | +108 | +0 | -37 | -25 | +22 | +97 | -85 | +28 | -122 |
| Relative (%) | -30.7 | +17.4 | +38.6 | +0.0 | -13.3 | -9.1 | +8.0 | +34.8 | -30.7 | +10.1 | -44.0 | |
| Steps (reduced) |
4 (4) |
7 (7) |
9 (9) |
10 (0) |
11 (1) |
12 (2) |
13 (3) |
14 (4) |
14 (4) |
15 (5) |
15 (5) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +18 | -111 | +48 | -63 | +110 | +11 | -82 | +108 | +23 | -57 | -134 |
| Relative (%) | +6.3 | -39.7 | +17.4 | -22.7 | +39.6 | +4.1 | -29.5 | +38.6 | +8.3 | -20.6 | -48.2 | |
| Steps (reduced) |
16 (6) |
16 (6) |
17 (7) |
17 (7) |
18 (8) |
18 (8) |
18 (8) |
19 (9) |
19 (9) |
19 (9) |
19 (9) | |
Intervals
| Degree | Cents | 5.7.11.13 intervals |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 278.631 | 13/11, 55/49 |
| 2 | 557.263 | 7/5 |
| 3 | 835.894 | 11/7 |
| 4 | 1114.525 | 13/7, 25/13 |
| 5 | 1393.157 | 11/5, 25/11 |
| 6 | 1671.788 | 13/5, 35/13 |
| 7 | 1950.420 | 35/11 |
| 8 | 2229.051 | 49/13 |
| 9 | 2507.682 | 49/11 |
| 10 | 2786.314 | 5/1 |
Subsets and supersets
Half of 20ed5.
As 5ed5 is the simplest hyperpyth tuning (analogous to 5edo and 4edt in their own spheres) this, its double, can be compared structurally to 10edo. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.
Adding octaves relates this tuning to 13edo, which divides the step in three, although the octaves are 7 cents sharp. If octaves are instead made just, everything else (especially prime 7) becomes flatter. Alternatively, the step can be divided in 10 to get 43edo.
Music
Weird Blues -- Kosmorsky