# Do the EDO and TET acronyms mean the same thing?

In the context of dividing an octave into n intervals, I understand n EDO to mean n Equal Divisions per Octave and n TET to mean n Tones Equal Temperament. Do the EDO and TET acronyms mean the same thing or is there some subtle difference?

• I'm a lot more used to seeing 12TET than 12EDO, so maybe there is some difference in connotations. – Dekkadeci Sep 7 '17 at 13:11
• I just read that the EDO acronym makes it clear that the interval being subdivided is an Octave. The TET acronym doesn't stipulate that an Octave is the large interval - so it could be an interval other than an octave. But it seems that stating TET without further qualification implies Octave as the large interval. – Brian THOMAS Feb 22 '18 at 13:47

In addition to TET referring to temperament, which does not make sense for e.g. 11EDO, it also differs in that it does not specify what range these notes divide equally, whereas EDO explicitly states that it's the octave. Now, the vast majority of commonly used tuning systems do use the octave as their foundation, but not all, notable counterexamples being the (tempered) Bohlen-Pierce scale (13EDT) and Wendy Carlos' α (9EDF), β (11EDF) and γ (20EDF) scales. The latter can't really be considered temperaments at all, but the Bohlen-Pierce scale absolutely is a temperament: in much the same way 12EDO tempers the Ptolemaic system of just intonation (5-limit with octave equivalency, i.e. the set of intervals in the range [1,2[ which can be constructed as integral frequency ratios using only the numbers 2, 3 and 5), Bohlen-Pierce tempers the odd 7-limit with tritave equivalency (the set of intervals in the range [1,3[ which can be constructed using only the numbers 3, 5 and 7). So, one might well interpret “13TET” as Bohlen-Pierce, which is very different from 13EDO.

I personally think that non-octave based scales are actually more promising for making new microtonal music that's practical to play and somewhat easy to grasp from listening, in particular if you go to the 11-or even 13-limit, which gets extremely difficult (see Ben Johnston) to do with all the octave-derived pitches available (giving the factors {2,3,4,5,6,7,8,9,10,11}, but much more tractable when the ratio 2 is out and you only have {3,5,7,9,11}. Bohlen-Pierce itself is in fact arguably a bit too tidy, making it hard to build up any sort of direction-defining dissonance.

• You say 11EDO can't be a temperament. Is this because none of the scale intervals is just, i.e. Is related to the root note by a ratio of small whole numbers? Also would you explain the [1,2[ notation, which I haven't seen before? – Brian THOMAS Feb 27 '18 at 13:14
• Well, 11EDO probably could be a temperament for some obscure just-intonation system using only high number ration, but this would be really far-fetched and probably not work in an actually audible way. As Some_Guy said, if you use 11EDO you would probably not attempt to temper any JI intervals but work atonally. — [1,2[ is not musical notation at all, I use this for the half-open interval {x ∈ ℝ | x≥1 ∧ x<2}. Many people would write this as [1,2), but I don't like that notation because it gets ambiguous for intervals that are open at both sides – “is (1,2) supposed to be a tuple or interval?”. – leftaroundabout Feb 27 '18 at 13:21
• Thanks for the clarification. In the explanation of 12EDO, is the word 'the' used in its mathematical uniqueness sense? When you say the set of intervals, does that imply there exists only one set, or are there multiple possible sets of intervals and we choose 12 of them? – Brian THOMAS Feb 27 '18 at 13:42
• Well, that's not really mathematically rigorous. The integral ratios using only the numbers 2, 3 and 5 (or only 2 and 3) are actually dense on the real line if you allow any combination of these factors. In practice, you'll only use particularly simple combinations. There are some ways to quantify this rigorously without arbitrary “cutoff” choices, but they're a bit to mathematically involved to bring them in here. – leftaroundabout Feb 27 '18 at 13:50
• @leftaroundabout "We are sorry, but the site you are looking for no longer exists." – phoog Feb 21 '19 at 21:31

In practice, they are the same thing, but they show a slight theoretical distinction in their "purpose", which is outlined quite well here http://xenharmonic.wikispaces.com/EDO+vs+ET

EDO indicates that an octave is being subdivded, and nothing more.

A temperament is essentially an attempt to "square the circle", and get intervals that transpose well but also provide a reasonably effective approximation of "pure" (or "just") intervals (whole number frequency ratios). It is an exercise in compromise. Some (most?) temperaments rely on other methods than subdividing the octave equally: so you can have a temperament that is not equal (like meantone temperaments for example), and you could also have an equal division of the octave which doesn't really approximate any pure intervals. In fact, more of them don't.

In that sense, you could argue that, say, while 12TET/12EDO, 19TET/19EDO, 31TET/31EDO etc. are exactly identical, 11TET doesn't really exist, as 11EDO isn't really a "temperament" at all; it doesn't make any attempt to be an approximation to natural whole number frequency ratios.

That being said, if you say 11TET, it's never going to cause confusion, but 11EDO is more "technically correct".

• And don't forget 10TET, as in that Mellertion! – Tim Sep 7 '17 at 16:15
• @Tim Mad... I just assumed it was a typo and left that question alone! Thanks for drawing it to my attention :D – Some_Guy Sep 8 '17 at 11:48