UD
A UD, or utonal division, is a kind of arithmetic and harmonotonic tuning.
Its full specification is n-UDp: n utonal divisions of rational interval p. An n-UDO is equivalent to the nth undertone mode, or under-n scale.
The only difference between n-UDp and n-ELDp (equal length division) is that the p for UD is rational, while the p for ELD is irrational.
Your sequence will be equivalent to some US (utonal sequence). E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.
An n-EDL is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f) | (1/1) | 8/7 | 4/3 | 8/5 | 2/1 |
| pitch (log₂f) | (0) | 0.19 | 0.42 | 0.68 | 1.00 |
| length (1/f) | (8/8) | 7/8 | 6/8 | 5/8 | 4/8 |
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f) | (1/1) | 4/5 | 2/3 | 4/7 | 1/2 |
| pitch (log₂f) | (0) | -0.32 | -0.58 | -0.81 | -1.00 |
| length (1/f) | (4/4) | 5/4 | 6/4 | 7/4 | 8/4 |