EFD

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

Its full specification is n-EFDp: n equal frequency divisions of irrational interval p. The only difference between n-ODp and n-EFDp is that the p for an EFD is irrational.

Instead of equally dividing the octave into 12 equal parts by pitch, as is done for 12-EDO, standard tuning, you could divide it into 12 equal parts by frequency. This would give you 12-EFDO. 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 analogous utonal equivalent of an EFD is an ELD (equal length division).

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).

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]\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-EFDp 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-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