# EFD

An EFD, or equal frequency division, is a kind of arithmetic and harmonotonic tuning.

## Specification

Its full specification is n-EFDp: n equal frequency divisions of irrational interval p.

## Formula

To find the steps for an n-EFDp, 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-EFDp 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 $1 + (p-1) = p$.

## Relationship to other tunings

### vs. EPD

Instead of equally dividing the octave into 12 equal parts by pitch, as is done for 12-EPDO, or 12-EDO (because pitch can be assumed), standard tuning, you could divide it into 12 equal parts by frequency. This would give you 12-EFDO.

### vs. ODO

However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD is typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name this tuning 12-ODO, for otonal divisions of the octave.

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

### vs. ELD

The analogous utonal equivalent of an EFD is an ELD (equal length division).

### vs. AFS

An EFD will be equivalent to some AFS, or arithmetic frequency sequence, which has had its count of pitches specified by prefixing "n-"; specifically, n-EFDp = n-AFS((p-1)/n).

## Examples

example: 4-EFDφ
quantity (0) 1 2 3 4
frequency (f) (1+(0/4)(φ-1)) = (0φ + 4)/4 = 1 1+(1/4)(φ-1) = (1φ + 3)/4 1+(2/4)(φ-1) = (2φ + 2)/4 1+(3/4)(φ-1) = (3φ + 1)/4 1+(4/4)(φ-1) = (4φ + 0)/4 = φ
pitch (log₂f) (0) 0.21 0.39 0.55 0.69
length (1/f) (1) 0.87 0.76 0.68 1/φ