Talk:EDO vs ET

Opening statements

This page starts out by stating that ED2s inherently require a pure octave, but I don't think that this is true. "ED2" only describes that the octave is equally divided. It seems perfectly possible for this octave to be stretched or compressed (for reasons that have nothing to do with tempering or temperaments).

I also don't think it's a good idea to describe ETs as taking ED2s and adding a conceptual layer, as is done immediately afterward. One could just as easily say that an ED2 is a layer we add on top of an ET, as a tuning of it (in this context, with ED2 implying a pure octave). I think these two objects start from different places and neither one is best described as a layer on the other.

So how would I revise this opening sentence? Well, I started trying to figure this out, and I ended up with a sizeable chunk of writing. And I realized that it was generally nicer to speak about the difference between EDs and ETs, that is, leaving the equally-divided interval unspecified, for a more direct naming comparison. Anyway, so I'm not suggesting replacing the whole article, but perhaps some of the following might be incorporated into it:

Both [math]\displaystyle{ n }[/math]-ED2 and [math]\displaystyle{ n }[/math]-ET refer to a set of equal-spaced pitches, where that repeating step size is somewhere close to 1200/[math]\displaystyle{ n }[/math] cents; for example, 12-ED2 and 12-ET are both pitch systems that use a repeating step size of 1200/12 = 100 ¢. For many readers, this is all they would ever care to know, and further distinctions will seem pedantic; these readers may now feel free to excuse themselves. Other readers, however, will want to understand the different conceptual implications of these names. These readers should stay with us.

The fundamental difference between the two types of pitch system can be made clear just from the names alone: EDs name an interval that is divided, while ETs do not.

• EDs are the straightforward way to name divisions of intervals like [math]\displaystyle{ \frac21 }[/math], [math]\displaystyle{ \frac31 }[/math], or [math]\displaystyle{ \frac32 }[/math] into [math]\displaystyle{ n }[/math] equal steps. We do this using the names [math]\displaystyle{ n }[/math]-ED2, [math]\displaystyle{ n }[/math]-ED3, or [math]\displaystyle{ n }[/math]-ED3/2, respectively. These are alternatively written as [math]\displaystyle{ n }[/math]-EDO, [math]\displaystyle{ n }[/math]-EDT, or [math]\displaystyle{ n }[/math]-EDF, for "Octave", "Tritave", and "Fifth".
• On the other hand, [math]\displaystyle{ n }[/math]-ETs do not explicitly name an interval like this. Their names instead convey that they are a temperament — specifically, they a regular temperament — which has one size of step (i.e. all steps are equal) and that they map some key interval to [math]\displaystyle{ n }[/math] of those steps. As for which particular interval that is, it is implicitly assumed to be the octave, [math]\displaystyle{ \frac21 }[/math], and in cases where we may want to break from this octave-based assumption, we have no clear built-in naming convention to follow here.

Here are three other important differences between EDs and ETs:

1. Re: JI. EDs have nothing to do with it. ETs are for approximating it. An ED tells us to divide an interval into equal steps, and that's all. Certainly, you could play an approximation of JI music in an ED, but there's nothing about EDs that is inherently related to JI. On the other hand, ETs are a concept from regular temperament theory (RTT) and are thus specifically designed to approximate JI. (Technically, temperaments can approximate things other than JI, too, but JI is the overwhelmingly popular choice.)
2. Re: finding the approximation of JI intervals: EDs round, and ETs map. As we just mentioned, EDs have nothing in particular to do with JI; that said, whenever we do want to know what an ED's closest approximation of a JI interval is, we simply round each interval's pitch (as measured e.g. in cents) to the nearest step count. With ETs, however, we map each interval's frequency ratio — using its prime composition — to some step count, which is not necessarily nearest by pitch rounding.
3. For each integer [math]\displaystyle{ n }[/math], there's only one [math]\displaystyle{ n }[/math]-ED2, but there are many [math]\displaystyle{ n }[/math]-ETs. An [math]\displaystyle{ n }[/math]-ED2 is already a fully-specified pitch set, but [math]\displaystyle{ n }[/math]-ET is not quite there yet. For starters, there are different [math]\displaystyle{ n }[/math]-ETs for each prime limit, while EDs have nothing in particular to do with primes or prime limits. And more interestingly, we have cases such as [math]\displaystyle{ n }[/math] = 17 at the 5-limit where we find more than one reasonable equal temperament: we have 17p-ET with map ⟨17 27 39], but we also have 17c-ET with map ⟨17 27 40].

--Cmloegcmluin (talk) 18:55, 20 June 2023 (UTC)

I'd first like to point out that a historical note would also help clarify this page. For several centuries, "equal temperaments" were not conceived as regular temperaments like we do today, and it is likely that still many people use ET to designate a concrete tuning. Similarly, I realize that the article suddenly jumps in with RTT notation in the middle of what looks like a layman's explanation. I wouldn't expect everyone reading this page to be familiar with RTT, let alone temperament maps. This is the sort of issues that need to be fix if we don't want to scare away people who are less familiar with relatively advanced math.

I agree that the different "layers" of EDOs and ETs should be laid out more carefully in this article. The RTT point of view certainly helps, although it's good to keep in mind that RTT didn't invent everything. For instance, in the lead section, I would mention that the modern usage of n-ET implies a regular temperament, with a link to the relevant article, leaving the reader to check it out if they don't know about it already, and giving a hint about the way the article will be developed.

RTT distinguishes "temperament map" and "tuning map". An abstract ET only has information on the temperament map (defaulting to the simple map), while a concrete ET also assumes a tuning map. On the other hand, an EDO only has information on the tuning map (defaulting to a generator of 1200 ¢/n), even though calling it a tuning map is a bit of a stretch. From there, I believe these are the two key concepts that should be used to explain in more detail the points you raised: stretched/compressed octaves, relation to JI (or other pitch set to be approximated), rounding (or direct mapping) vs. regular mapping, the existence of multiple ETs for each EDO, the ambiguity contained in statements like "n-EDO supports", and so on. I don't think the lead section should mention all of these topics, but it should at least summarize why the concepts are similar and why they are distinct (which is already what the page does, but I think it could be improved).

To summarize, this page should do its best to answer the question "what is the difference between an EDO and an ET" without needlessly repeating what's already on the other pages (by referencing these pages instead) while bringing a sense of progression that goes from general to specific. --Fredg999 (talk) 20:43, 20 June 2023 (UTC)

"Supports"

Toward the bottom of this page it brings up EDOs "supporting" temperaments. This could be a bit confusing because it's using "support" in the informal, generic sense (see: https://en.xen.wiki/w/Support#Other_informal_usage) while "support" also has a technical, specific sense for ETs (https://en.xen.wiki/w/Support).

In other words, answering the question, "Does 11-ED2 support hanson?" is indeed ambiguous. However, answering the question "Does 11-ET support hanson?" is not. This is just cold hard numbers. Hanson is a 5-limit temperament, and there are three different reasonable 5-limit 11-ETs (according to http://x31eq.com/cgi-bin/rt.cgi?ets=11&limit=5), so this is different than the case for 12 where there's only one reasonable 5-limit 12-ET. Right, so we have 11c with map ⟨11 17 25], 11b with map ⟨11 18 26], and 11p with map ⟨11 17 26]. To know whether these support hanson, we just need to ask whether or not they make the hanson comma vanish. The hanson comma is also known as the kleisma, and is 15625/15552, which as a vector is [-6 -5 6⟩. 11c maps this interval to -1 steps, 11b maps it to 0 steps, and 11p maps it to 5 steps. So 11b-ET supports hanson, but 11c and 11p do not. There are infinitely more 5-limit 11-ETs (which are increasingly unreasonable), and some of these support hanson, and some of them don't.

I suggest that some explanation along these lines should be added to this part of the page, in place of where it says "has been debated without a consensus having been reached", which is not substantiated with links to discussion, and I think the answer is clear, as I've just described.

--Cmloegcmluin (talk) 18:54, 20 June 2023 (UTC)

Following by reply to the topic above, I think this can be explained by the two kinds of map; an ET implies a particular temperament map, while an EDO doesn't, so it's ambiguous for EDO (unless everyone agrees to assume patent val, but I'd rather say it's ambiguous). I agree that the so-called "debate" doesn't really make sense if you look at the definitions properly. --Fredg999 (talk) 20:43, 20 June 2023 (UTC)