ELD
An ELD, or equal length division, is a kind of arithmetic and harmonotonic tuning.
A UD is a specific (rational) type of ELD.
So far we've looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.
The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal pitch divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by frequency, or length. In the former case, you will have 12-EFDO, and in the latter case, 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 EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 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.