# OD

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

## Specification

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

## Formula

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

$f(k) = 1 + (\frac kn)(p-1)$

This way, when $k$ is $0$, $f(k)$ is simply $1$. And when $k$ is $n$, $f(k)$ is simply $1 + (p-1) = p$.

## Tips

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.

### Vs. EFD

The only difference between n-ODp and n-EFDp is that the p for an EFD (equal frequency division) is irrational, and therefore its pitches and intervals are all irrational too.

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).

## Examples

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