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.
| 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 |