ELD
An ELD, or equal length division, is a kind of arithmetic and harmonotonic tuning.
Its full specification is n-ELDp: n equal length divisions of the irrational interval p. The only difference between an n-ELDp and an n-UDp (or utonal division) is that the p for a utonal division is rational.
The analogous otonal equivalent of an ELD is an EFD (equal frequency division).
An ELD will be equivalent to some ALS (arithmetic length sequence); specifically n-ELD((p-1)/n) = n-ALSp.
It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by length. You will have 12-ELDO. 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 ELD are typically reserved for irrational tunings, such as 12-ELDφ. So it would be more appropriate to name this tuning 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f) | (1) | 1.11 | 1.24 | 1.40 | φ |
| pitch (log₂f) | (0) | 0.14 | 0.31 | 0.49 | 0.69 |
| length (1/f) | (1) | 0.90 | 0.81 | 0.71 | 1/φ |
vs. EDL
An ELD is not to be confused with EDL, equal division of length. The latter term does not take an interval parameter because it is assumed to be the length of an entire string, and then only an octave subset of that is taken.