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]n[/math]-ED2 and [math]n[/math]-ET refer to a set of equal-spaced pitches, where that repeating step size is somewhere close to 1200/[math]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]\frac21[/math], [math]\frac31[/math], or [math]\frac32[/math] into [math]n[/math] equal steps. We do this using the names [math]n[/math]-ED2, [math]n[/math]-ED3, or [math]n[/math]-ED3/2, respectively. These are alternatively written as [math]n[/math]-EDO, [math]n[/math]-EDT, or [math]n[/math]-EDF, for "Octave", "Tritave", and "Fifth".
• On the other hand, [math]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]n[/math] of those steps. As for which particular interval that is, it is implicitly assumed to be the octave, [math]\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]n[/math], there's only one [math]n[/math]-ED2, but there are many [math]n[/math]-ETs. An [math]n[/math]-ED2 is already a fully-specified pitch set, but [math]n[/math]-ET is not quite there yet. For starters, there are different [math]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]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)

Thanks for your reply, Fredg999. You make very good points. In particular, yes, I'd definitely meant to make some historical notes along the lines of what you wrote up here, but I completely forgot! And in forgetting this, my statements about ETs completely assumed the modern regular temperament interpretation of that term, ignoring the fact that historically "equal temperament" was shorthand for "12-tone equal temperament", and that it didn't need to make any distinction between rounding and mapping in the way I described.

Your suggestion that temperament maps and tuning maps would be excellent keys to unlock this understanding for readers is interesting; however, I feel your concern that tying tuning maps to EDOs may be a stretch. I think that could cause more confusion than insight. Maybe there's another better way.

I support your progression from general overview to specific details, and think this page could be revised to fit that better.

You bring up another good point, which is avoiding repeating what is already written on other pages. Ideally, all these pages should be considered simultaneously, and each piece of information should end up in the ideal form in each ideal place. Though that's a bit of an endeavor, to be sure. Here's what I found on the Equal-step tuning page, which is what Equal temperament redirects to:

When a tuning is called n-tone equal temperament (abbreviated n-tet or n-et), this usually means "n divisions of 2/1, the octave, or some approximation thereof", but it also implies a mindset of temperament – that is, of a JI-approximation-based understanding of the scale. If you are wondering how equal divisions of the octave can become associated with temperaments, the page EDOs to ETs may help clarify. There are many reasons why one might choose to not consider JI approximations when dealing with equal tunings, and thus not treat equal tunings as temperaments. In such case, the less theory-laden term edo (occasionally written ed2), meaning equal divisions of the octave (or equal divisions of 2/1), leaves comparison to JI out of the picture, aside from the octave itself (which is assumed to be just).

And on the EDO page, in its History section, we currently find:

Tuning theorists first used the term "equal temperament" for edos designed to approximate low-complexity just intervals. The same term is still used today for all rank-1 temperaments. For example, 15edo can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

Interestingly, ET currently redirects to EDO; since ET is an abbreviation of equal temperament, certainly they should go to the same place, yeah? But this is another aspect to the extended problem. I suspect "Equal temperament" should probably not be a redirect page at all, but should be the place where we find the sort of historical information you and I have started writing about here and which is mentioned on the EDO page. Also, Rank-1 temperament should probably exist (as Rank-2 temperament and Rank-3 temperament do), and address the modern RTT interpretation of the term. Most of these pages can be quite brief, I think. --Cmloegcmluin (talk) 16:04, 21 June 2023 (UTC)

I'll clarify why I associate edos with tuning maps, and more specifically generator tuning maps. In non-RTT jargon, a specific edo is usually defined by its number of steps to the octave. If we assume the octave to be just, which is often the case (or not always critically important anyway), then that definition is equivalent to a definition by step size. In other words, 12edo can be defined either by its 12 tones per octave or by its 100¢ step size. Now, we know that an edo's step corresponds to its generator, from an RTT point of view. The generator tuning map for a rank-1 regular temperament is a 1x1 row vector containing, as its only element, the size of the generator, which we know is exactly the edo's step size. That is to say, the tuning map is a generalization of the concept of "step size" to higher dimensions. This idea might need a bit more work to be conveyed fluently, but I still believe it is a useful way to bridge edos and ETs.

I agree that several related pages would also need coherent work alongside the revision of the page here. The excerpts you quoted here give a good idea of the work to do. As for the pages for rank-n temperaments, it's good to keep in mind that wiki pages should be about concepts, not words or expressions. Rank-1 temperament can redirect to Equal temperament or to whatever "Equal temperament" redirects to. --Fredg999 (talk) 19:28, 21 June 2023 (UTC)

Ah ha! Yes, before I got a chance to read this, while I was drifting off to sleep last night, the thought suddenly occurred to me that perhaps you had been thinking specifically of generator tuning maps. And in that case, yes, I totally agree that this could be a great way to get at the difference between EDOs and ETs, for the reasons you describe.

Also agreed re: pages being for concepts, not terms. So the RTT interpretation of equal temperament should be a section on this new non-redirect Equal temperament page, and Rank-1 temperament should redirect to that section.

If I don't get any pushback here or further suggestions, I can take a crack at making these improvements next week.

--Cmloegcmluin (talk) 17:07, 22 June 2023 (UTC)

Hello. I hope it is written in the style of RTT how to describe the EDO as a non-JI abstract temperament (even if it doesn't make much sense to do so). e. g. 12edo is "2^1/12 domain basis" (and 2^1/12.5 is for "no-threes compton temperament.") e. g. Whereas 5-limit 12-ET maps 5/4 to a stack of 4 generators, 12-EDO maps 2^1/3 to a stack of 4 generators.--Dummy index (talk) 06:54, 24 June 2023 (UTC)

Hello, Dummy index. Sorry, but I don't understand most of what you're saying. I do think you're correct to say that EDOs do not make much sense to describe as temperaments. But then the rest of your message seems to go against that, so I'm confused.

When you say "2^1/12 domain basis", that conveys that 2^1/12 is a system you're tempering, but that's not the case, at least as far as I understand things. (On the other hand, it does make sense to say "2.3.5 domain basis" or "5-limit domain basis", because 5-limit JI is a system that we do often temper, e.g. by 12-ET.)

I think that using the verb "map" for an EDO has more potential to create confusion than it does to increase clarity. It would be better to reserve that word for temperaments — which are based on linear mappings — and then use "round" for EDOs, as I've described above. What does "map" bring to the table that "round" doesn't, in your point of view? Why couldn't you just say that 12-EDO rounds 2^1/3 to a stack of 4 generators? Though I'm not sure what insights this statement is supposed to provide in the first place.

--Cmloegcmluin (talk) 20:19, 24 June 2023 (UTC)

I think Dummy index is trying to convey the idea that an EDO is a generated tuning, and therefore it can be visualized on a lattice with a domain basis of 2^1/n. In comparison, 5-limit JI can be visualized on a lattice with a domain basis of 2.3.5, or possibly 2.3/2.5/4. Generated tunings can easily be confused with regular temperaments, especially because there are common elements shared by the two concepts, so I think it's worth distinguishing them here and referring to the appropriate pages as necessary. --Fredg999 (talk) 05:22, 25 June 2023 (UTC)

Yes, it's a trivial temperament where no tempering is happening: no commas are tempered out. Well... EDO is simple and appear in many places in theory. We must say this first. Next, I think about it according to the title EDO vs ET, I dare to position them on the same layer (as a regular temperament) for a clear comparison. Clear comparison means simply and mathematically pointing out the difference, not long descriptions of different layers.

Sorry I can't judge if it will be an easy-to-understand explanation for beginners ...　--Dummy index (talk) 14:12, 25 June 2023 (UTC)

Ah, ok. Thanks for explaining, Fredg999. It was the "domain" part that threw me off.

Dummy index, what you wrote would have made sense to me if you had simply written "basis" instead of "domain basis". That's because "domain" is a mathematical term in that context, and what it refers to is a space before some mapping occurs, or the space that something gets mapped from. The opposite word of domain in this sense is "range"; that's the space after mapping, or the space that we map into. So when we apply this mathematical concept to our xenharmonic topic, the domain is our JI lattice, the mapping is tempering, and the range is our tempered lattice. Does that make sense? So you can see that in an EDO, no mapping happens; there's no domain or range, or said another way, it is a system that doesn't require any before-and-after aspect to understand. Yes, we can certainly look at an EDO on a 1D lattice, and say that 12-EDO has a basis of 2^1/12, but this basis wouldn't be called a domain basis.

As for your most recent message:

* I do not think it is reasonable to say that an EDO is "a trivial temperament where no tempering is happening: no commas are tempered out"; that's a description I've used often for JI. I would say that JI does not need to be looked at as a temperament, but it's possible to look at it as a trivial temperament if you want. EDOs, on the other hand, cannot be looked at as temperaments in any meaningful way, trivial or otherwise. They just don't have anything to do with tempering at all.

* I also think it would be a bad idea to describe both EDOs and ETs as regular temperaments. I agree we should clearly/simply/mathematically point out their differences, but this way would be inaccurate and confusing.

Perhaps, though, all you mean is, again, not that EDOs and ETs are both regular temperaments, but that they are both generated tunings, as Fredg999 described, or in other words, that they can be visualized on a lattice. Even more specifically, they can both be visualized on 1D lattices. And one other thing they have in common is that both lattices' single dimension's generators represent an interval 2^1/n, whether that's for n-ET or n-EDO, and whether or not this interval is tuned exactly. That much I can certainly agree with.

--Cmloegcmluin (talk) 15:55, 25 June 2023 (UTC)

> the domain is our JI lattice

I first said it was a non-JI. However, "completely non-JI" "temperament" may have been crazy idea. At that time, I was thinking about the validity of naming the pitches 2^i/n in near-octave n-EDO.

And thanks for the discussion. I feel like I'm starting to understand something, but let me confirm.

• Is just intonation a RT?
  1. Yes. It maps ideal 2/1 to actual 2:1.
  2. No. It doesn't have any tempered out comma.
  3. No. It is not a temperament because it does not call itself a temperament.
  4. ...But yes, it is useful for introductory.
• Is 12-EDO a RT?
  1. ...Unknown. How do you think of "Octave" as?
  2. No. It doesn't have any tempered out comma.
  3. No. It is not a temperament because it does not call itself a temperament.
• Is 2-limit 12-ET (= contorted (order 12) 2-limit) a RT?
  1. Yes. It maps ideal 2/1 to actual 2:1.
  2. No. It doesn't have any tempered out comma.
  3. Yes. It is a temperament because it calls itself a temperament and well-defined.

--Dummy index (talk) 14:27, 26 June 2023 (UTC)

Just intonation and 12-EDO are tuning systems (sets of concrete pitches or intervals), not regular temperaments. However, for both of these tuning systems, you could choose a domain and calculate the simple map associated with that domain, and now you would have a regular temperament. For instance, p-limit just intonation with the identity matrix as the temperament map is a regular temperament (usually considered trivial for obvious reasons). 12-ET is indeed a regular temperament. --Fredg999 (talk) 15:55, 26 June 2023 (UTC)

Thanks for that, Fredg999. Though I'd tweak that slightly. The simple map for an n-EDO is one simple way to get a regular temperament from an EDO, specifically an n-ET, but there are other possible maps you might want (optimal with respect to the tunings of a consonance set, for one of the very many definitions of optimal popularly used), and I think this is more straightforwardly understood as finding an n-ET from an n-EDO so much as it is finding an n-ET from n itself. Also, I don't think there's any sense in which one could calculate a simple map for JI, but maybe you didn't intend to suggest that. (Oh, except that the 1×1 identity matrix that interprets 2-limit JI as a trivial regular temperament would qualify as the simple map for an equal temperament, but that's decidedly an edge case!)

Yes, Dummy index, thank you for the discussion too. This is helping me refine my understanding of the similarities and differences between all of these structures, too. Fredg999 has already given direct answers to your questions, but I'll supplement what they wrote by answering using your own choices:

Is just intonation a RT? No. It is not a temperament because it does not call itself a temperament. (But I'll add: we can interpret JI as a trivial regular temperament, i.e. one that doesn't make any commas vanish.)

Is 12-EDO a RT? No. It is not a temperament because it does not call itself a temperament. (But I'll add: more importantly, it's not a regular temperament because it doesn't map a domain such as JI.)

Is 2-limit 12-ET (= contorted (order 12) 2-limit) a RT? Yes. It is a temperament because it calls itself a temperament and well-defined. (But I'll add: I've suggested we call this sort of thing a "temperoid"; I prefer to use the term "enfactored" rather than "contorted", and the point is that ⟨12] doesn't bring anything more to the table as a regular temperament than ⟨1] does, so if we wanted to make a list of all the unique regular temperaments in existence, we would include ⟨1] but we might not want to include ⟨12].)

Let me know if you have further questions, and I'll do my best to answer. --Cmloegcmluin (talk) 20:49, 26 June 2023 (UTC)

So I tried to get a one-to-one correspondence with the output by "enfactoring" the input by 12, and the result was the 2^1/12 domain basis. But I have just come to the conclusion that this has no practical significance. Mapping an exponential function to the equal-step tuning is no different from the generator chain, and there is nothing to do in 1D domain. Barely the 2-limit case is only useful for explaining as a edge case. --Dummy index (talk) 15:00, 28 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)

Agreed, except that (as I mentioned in the other topic of this discussion page) I don't think the other type of map — the tuning map — is a good idea to bring up with respect to EDOs. And that's okay because I don't think we even need to mention it here. We can simply explain that ETs have a temperament map and EDOs don't. If you like, I can take a first attempt at improving this section now. Let me know. --Cmloegcmluin (talk) 16:04, 21 June 2023 (UTC)