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Edo

An equal division of the octave (edo or EDO) is a tuning obtained by dividing the octave in a certain number of equal steps. This means that each step corresponds to the same interval.

A tuning with [math]\displaystyle{ n }[/math] equal divisions of the octave is usually called "[math]\displaystyle{ n }[/math]edo" ("[math]\displaystyle{ n }[/math]-EDO"). For instance, the predominant tuning system in the world today is 12edo (12-EDO).

An edo is a specific case of equal pitch division, which is a kind of equal-step tuning. Therefore, it is also a kind of arithmetic and harmonotonic tuning.

History

For a long time, tuning theorists used the term "equal temperament" for edos designed to approximate low-complexity just intervals. The same term is still used today to designate more generally all rank-1 temperaments. For example, 15edo can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

The acronym "EDO" (EE-dee-oh) was coined by Daniel Anthony Stearns^{[year needed]}. More recently, the anacronym "edo" (EE-doh), spelled in lowercase, has become increasingly widespread.

With the development of equal divisions of non-octave intervals (edonoi), some musicians started using "ed2" ("ED2"), especially when naming a specific tuning. Furthermore, in order to distinguish equal pitch division from equal frequency division and equal length division, "epd" ("EPD") is sometimes used in place of "ed" ("ED").

Several alternate notations have been devised by some musicians more recently, including "edd" ("EDD"; equal divisions of the ditave), "DIV," and "EQ."

Formula

To find the step size for an [math]\displaystyle{ n }[/math]edo, take the [math]\displaystyle{ n }[/math]th root of 2. For example, the step of 12edo is [math]\displaystyle{ 2^{\frac{1}{12}} }[/math] ([math]\displaystyle{ \approx 1.059 }[/math]). So the formula for the [math]\displaystyle{ k }[/math]th step of an [math]\displaystyle{ n }[/math]edo is:

[math]\displaystyle{ c(k) = 2^{\frac{k}{n}} }[/math]

This way, when [math]\displaystyle{ k }[/math] is 0, [math]\displaystyle{ k }[/math] is simply 1, because any number to the 0th power is 1. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ c(k) }[/math] is simply 2, because any number to the 1st power is itself.