# 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 [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-EFDp 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].

## 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

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 |