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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.

The EDO paradigm


While the equal temperament paradigm is firmly established, with its benefits clearly understood, little has been done to establish a competitive EDO paradigm. Ivor Darreg is probably the first composer and writer to discuss EDOs without any explicit reference to their proximity to JI. Easley Blackwood (in his microtonal work) seems to have ignored or been otherwise ignorant of Just Intonation and its relationship to EDOs, instead focusing on the "recognizably diatonic" properties held by each EDO he worked with. Various other microtonal composers have, in recent years, produced compositions in various EDOs without making any conscious musical references to JI, but the question may still be asked: what is gained in this approach, that is not present in the ET paradigm?

One common feature discussions of EDOs outside the ET paradigm is the concept of "mood", first used in the writings of Ivor Darreg. An EDO's mood is typically described as a subjective and unquantifiable property that can only be ascertained through listening to and composing in the EDO. The moods of EDOs are often described rather prosaically, using terms such as "bright", "narcotic", "spicy", "mysterious", "strident", or "strong". In the absence of the objective quantitative dimension of the ET paradigm, such subjective qualitative approaches are really the only possible way to discuss EDOs. One of the benefits of this is that it avoids creating a hierarchy of EDOs, and thus avoids creating a cognitive bias against the use of certain EDOs.

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.