EFD
An EFD (equal frequency division) or AD (arithmetic division) is a periodic and arithmetic tuning system in which each period is divided to a number of steps of equal frequency difference.
Specification
Its full specification is n-efd-p or n-ad-p: n equal frequency divisions of p, or n arithmetic divisions of p .
Formula
To find the steps for an n-efd-p, 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 within each period, the ratio c of the k-th step of an n-efd-p is:
[math]\displaystyle{ c = 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].
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 |