OD

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

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

The only difference between n-ODp and n-EFDp is that the p for an EFD is irrational.

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

Your sequence will be equivalent to some OS (otonal sequence). E.g. 8-OD7 = 8-OS3/4, because to get from 1 to 7 you cover 6 overtones, and 6 divided by 8 is 3/4.

The equivalent utonal version of an OD is a UD.

If you want to describe overtones 1-9 as an OD you would need to use 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. Alternatively, you could describe this as an OS, or overtone sequence, by simply saying 8-OS.

To find the steps for an n-ODp, begin by recognizing that while the multiplicative interval relating your root position to the end position is [math]\displaystyle{ p }[/math] (or [math]\displaystyle{ \frac p1 }[/math]), if you are going to move arithmetically (by repeated addition) from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ p }[/math], then the difference in frequency space that you are dividing up is not actually [math]\displaystyle{ p }[/math], but [math]\displaystyle{ p - 1 }[/math]. And because you are dividing it into [math]\displaystyle{ n }[/math] parts, each step will have a size of [math]\displaystyle{ \frac{p-1}{n} }[/math]. So, the formula for the frequency of step [math]\displaystyle{ k }[/math] of an n-ODp is:

[math]\displaystyle{ f(k) = 1 + (\frac kn)(p-1) }[/math]

This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ f(k) }[/math] is simply [math]\displaystyle{ 1 }[/math]. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ f(k) }[/math] is simply [math]\displaystyle{ 1 + (p-1) = p }[/math].

example: 4-ODO

quantity	(0)	1	2	3	4
frequency (f)	(4/4)	5/4	6/4	7/4	8/4
pitch (log₂f)	(0)	0.32	0.58	0.81	1
length (1/f)	(4/4)	4/5	4/6	4/7	4/8