OD

An OD, or otonal division, is a kind of arithmetic and monotonic tuning.

Its full specification is n-ODp: n otonal divisions of interval p.

The nth overtone mode, or over-n scale is equivalent to n-ODO. So is n-ADO.

An OD is a specific (rational) type of EFD, or equal frequency division.

note there's a kinda tricky aspect which is that if you just want overtones 1-9 you need 8-OD9 because there are only 8 steps from 1 to 9. You could think of it like 9 is the 8th overtone, so you're really dividing 8 by 8. You're dividing the number of overtones.

example: 4-ODO

quantity	1	2	3	4	(5)
frequency (f)	4/4	5/4	6/4	7/4	(8/4)
pitch (log₂f)	0.00	0.32	0.58	0.81	(1.00)
length (1/f)	4/4	4/5	4/6	4/7	(4/8)