EDO vs ET: Difference between revisions

'''Equal divisions of the octave''' ('''[[EDO]]''') and '''equal [[temperament]]s''' ('''ET''') are not the same thing, at least not in concept. An equal division of the octave is a concrete tuning — 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.  

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.

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.

There are potentially many ways in which an EDO can be viewed without any reference to JI, or at least without the specific RTT framework of "linear map from JI subgroup".

Another benefit to the temperament-free approach to EDOs is that it can avoid confusion that sometimes comes when applying the ET paradigm to tunings that provide questionable approximations to JI. It is a common topic of debate within the microtonal community whether a given EDO supports a given temperament, or even what it means for an EDO to support a temperament. For example, the question of whether or not 11-EDO supports Hanson temperament has been debated without a consensus having been reached. Another source of confusion in many EDOs is that the chords which are closest to a Just sonority are not always the most pleasant. A triad of 0-5-9 degrees of 14-EDO can be said to approximate 7:9:11, and is the lowest-error triad in 14-EDO, yet its comparative pleasantness to, say, 0-5-8 or 0-6-9 is definitely debatable. When temperament is left out of the picture, there is nothing to debate--EDOs simply "are what they are", and can be taken or left according solely to the whims of the composer.

[[Category:Terms]]