ELD

An ELD, or equal length division, is a kind of arithmetic and harmonotonic tuning.

Specification

Its full specification is n-ELDp: n equal length divisions of the irrational interval p.

Formula

To find the steps for an n-ELDp, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch 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 string length that you need to cover is not actually [math]\displaystyle{ p }[/math], but only [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 length of step [math]\displaystyle{ k }[/math] of an n-ELDp is:

[math]\displaystyle{ L(k) = 1 + (\frac kn)(p-1) }[/math]

This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ L(k) }[/math] is simply [math]\displaystyle{ 1 }[/math]. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ L(k) }[/math] is simply [math]\displaystyle{ 1 + (p-1) = p }[/math].

Tip about tunings based on length

Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

Relationship to other tunings

vs. EPD

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 utonal divisions of the octave.

vs. UD

The only difference between an n-ELDp and an n-UDp (or utonal division) is that the p for a utonal division is rational.

vs. EFD

The analogous otonal equivalent of an ELD is an EFD (equal frequency division).

vs. ALS

An ELD will be equivalent to some ALS (arithmetic length sequence); specifically n-ELD((p-1)/n) = n-ALSp.

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.

Examples

example: 4-ELDφ (arranged so that the pitches are in ascending order and still begin on 1/1)

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

example: 4-ELDφ (descending pitches)

quantity	(0)	1	2	3	4
frequency (f)	(1)	0.87	0.76	0.68	1/φ
pitch (log₂f)	(0)	-0.21	-0.39	-0.55	-0.69
length (1/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