# EDO vs ET

(Redirected from Edovset)

# EDOs vs. Equal Temperaments

Equal divisions of the octave and equal temperaments are not the same thing, at least not in concept. An equal division of the octave is just that--a division of the pure 2/1 octave of 1200 cents into some number of equal parts. An equal temperament, on the other hand, is what you get when you take an EDO and declare its intervals to be approximations to Just Intonation, thus adding a new conceptual layer on top of the bare equal division.

## Why bother making this distinction?

There are many EDOs which, by virtue of some freak miracle of mathematics, happen to sound a lot like Just Intonation. Some say this is the case with 12-EDO, and that one of the reasons for its popularity is because it sounds enough like 5-limit (or perhaps even 7-limit) JI to please the masses, while being incredibly practical and convenient. Some also say that EDOs like 19, 22, 31, 34, 41, 46 or 53 sound even more like JI than 12-EDO, and this indeed seems to be the case. Because of this acoustic similarity between equal tunings and Just tunings, lots of people like to treat equal tunings as approximations to JI--in other words, temperaments. With EDOs such as these, it is possible to gain some level of insight into their harmonic structures by thinking in terms of approximate JI, and thus this approach has been very useful to many people.

However, it is not always true that those using EDOs are interested in approximating JI, nor is it true that describing all EDOs in terms of approximate JI is universally helpful or illuminating. As an example of the former, consider the atonalists, a loose school of 20th-century composers who sought to embrace the "equality" of equal temperament by treating every note as having equal musical importance, and thereby escape connotations of tonality that had previously defined Western classical music.

Consider also 7-EDO: there is not a single triadic sonority within the EDO that is concordant enough to plausibly be conflated with Just Intonation, and attempts to describe its harmonic structures in terms of Just ratios is often more confusing than it is illuminating. It is not impossible to treat 7-EDO as an equal temperament, but the question arises of what is being gained in the process. One possible answer to that is that 7-EDO treated as a temperament, even though it is not actually used as one, is basic to Western musical theory. One step is a tone, two steps a third, three steps a fourth, four steps a fifth, five steps a sixth, six steps a seventh, and seven steps an octave. These can be major, minor, diminished or augmented, which are all the same to 7-EDO. It is understood that the perfect octave is a 2, and the perfect fifth must approximate 3/2; if the perfect major third approximates 5/4 then we have the <7 11 16| val of 7-EDO as a temperament lying behind the terminology of Western music. While true, this is also not really relevant to the use of 7-EDO itself as a musical scale.

Consider also 9-EDO; this has nearly pure intervals of 7/6 and 12/7, so close (one fifth of a cent) that they really cannot be heard as other than JI. However that is not enough to give 9-EDO the overall character of approximate JI. Nor does the fact that it possesses the same 400 cent major thirds as 12-EDO really do it, and the attempt to hear 667 cents as a fifth is at best marginally successful. What we find is a peculiar hybrid, a chimera neither fish nor fowl.