6ed5
6 equal divisions of the 5th harmonic (abbreviated 6ed5) is a nonoctave tuning system that divides the interval of 5/1 into 6 equal parts of about 464 ¢ each. Each step represents a frequency ratio of 51/6, or the 6th root of 5.
Harmonics
6ed5 does not approximate any sensible subgroup of integer harmonics. If one does try to interpret it using an integer subgroup, then it looks something like a 5.19.33 subgroup: which has severely limited use cases.
6ed5 does however have some chords and intervals that sound good for its size, despite its poor approximations of pure harmonics. These shine when it is interpreted using a fractional subgroup.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +193 | -44 | -78 | +0 | +149 | -118 | +115 | -89 | +193 | +28 | -122 |
| Relative (%) | +41.6 | -9.6 | -16.8 | +0.0 | +32.0 | -25.4 | +24.8 | -19.1 | +41.6 | +6.1 | -26.4 | |
| Steps (reduced) |
3 (3) |
4 (4) |
5 (5) |
6 (0) |
7 (1) |
7 (1) |
8 (2) |
8 (2) |
9 (3) |
9 (3) |
9 (3) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +203 | +75 | -44 | -156 | +203 | +104 | +11 | -78 | -163 | +221 | +144 | +71 | +0 | -68 |
| Relative (%) | +43.8 | +16.2 | -9.6 | -33.6 | +43.8 | +22.5 | +2.3 | -16.8 | -35.0 | +47.7 | +31.1 | +15.2 | +0.0 | -14.6 | |
| Steps (reduced) |
10 (4) |
10 (4) |
10 (4) |
10 (4) |
11 (5) |
11 (5) |
11 (5) |
11 (5) |
11 (5) |
12 (0) |
12 (0) |
12 (0) |
12 (0) |
12 (0) | |
Intervals
| Step | Interval (¢) | JI approximated (subgroup A) |
JI approximated (subgroup B) |
JI approximated (subgroup C) |
|---|---|---|---|---|
| 1 | 424.39 | 9/7 | 14/11 | 23/18 |
| 2 | 928.77 | 12/7 | 12/7 | 53/31 |
| 3 | 1393.16 | 9/4 | 29/13 | 38/17 |
| 4 | 1857.54 | 29/10 | 35/12 | 38/13 |
| 5 | 2321.93 | 19/5 | 23/6 | 65/17 |
| 6 | 2786.31 | 5/1 | 5/1 | 5/1 |
- Subgroup A = low complexity subgroup: 5/1.9/4.9/7.12/7.19/5.29/10
- Subgroup B = compromise subgroup: 5/1.12/7.14/11.23/6.29/13.35/12
- Subgroup C = low error subgroup: 5/1.23/18.38/13.38/17.53/31.65/17
Other interpretations are possible.