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).

example: 4-UDO = 4th undertone mode (arranged so that the pitches are in ascending order and still begin on 1/1)

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

example: 4-UDO = 4th undertone mode (descending pitches)

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