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


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


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

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

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


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.

Relationship to other tunings

Vs. ED

It is possible to — instead of equally dividing the octave in 12 equal parts by pitch, or 12-EDO — divide it into 12 equal parts by length. You will have 12-UDO.


The only difference between n-ODp and n-EFDp (equal frequency division) is that the p for an OD must be rational.


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

Vs. OS

Any ODO 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.

Vs. UD

The equivalent utonal version of an OD is a UD (utonal sequence).


example: 4-ODO
quantity (0) 1 2 3 4
frequency (f, ratio) (4/4) 5/4 6/4 7/4 8/4
pitch (log₂f, octaves) (0) 0.32 0.58 0.81 1
length (1/f, ratio) (4/4) 4/5 4/6 4/7 4/8