Talk:Optimal ET sequence

Is "Val list" the best name for this thing?

I note that this isn't really a list of maps ("vals"), like ⟨12 19 28], but actually a list of ETs given in wart notation. So I could see this being called an "ET list", with the other distinguishing facts left opaque (those being that they are [a] only ETs whose maps are uniform maps ("GPVs") and [b] where each subsequent map improves upon TE error). Unless some people don't consider e.g. 17p to be a different ET from 17c, but only different maps for the same ET, but then that's getting pretty philosophical I think. Or the title could attempt to convey both those facts, such as "Error-decreasing GPV sequence" (though of course I would prefer my own term "uniform map", I understand I should defer to the convention here.) --Cmloegcmluin (talk) 17:47, 18 November 2021 (UTC)

You're right. Val list isn't the best name for this thing. Imo list should be avoided and sequence is a good substitute. ET sequence is a name I reckon very proper. I'm afraid further constraints like error-decreasing or GPV are less essential, but are somewhat heuristic choices that help to shape a neat sequence. The error-decreasing constraint is good at limiting the length of the sequence. It's possible to make another sequence that's quite similar but different – the MOS numbers for the optimal tuning. The GPV constraint is handy in that it makes the sequence terminate. FloraC (talk) 19:14, 18 November 2021 (UTC)

Oh, interesting — I didn't know the GPV constraint made the sequence terminate. Reviewing your code, it looks more like one of a couple arbitrary thresholds/ranges are ending the search; even `find_next_gpv` doesn't seem to have an exit condition related to having found the final GPV. But perhaps I'm misinterpreting one or many things here :) When you have a minute, would you be able to explain how/why this is or what you meant? And also, does the article need a correction with regards to that fact, then? I wrote, "No standard beginning or ending cutoff to the list has been specified", but if the GPV constraint forces termination, then I'd think that'd constitute such a "standard ending" to the list.

After I understand this particular detail, I'll make a brief post on the Facebook Xenwiki Work Group and the Discord wiki channel to see if anyone has any dispute with renaming this "ET sequence", which I agree should suit this feature well (and better than "val list"). --Cmloegcmluin (talk) 19:36, 23 November 2021 (UTC)

The sequence "terminates" as no more GPVs turn up after a sufficiently large index. Any temperament has a fixed error, whereas the sequence of all GPV's errors converges to zero, so all GPVs has less error than the temperament past a certain point. Obviously my code doesn't know it, and you must set the range yourself. FloraC (talk) 21:28, 23 November 2021 (UTC)

I expect that most readers who see "optimal GPV sequence" in temperament catalogs, if they can figure it out at all, will assume that "GPV" must mean "supporting ET" (i.e. "ET that supports this temperament"), which of course it does not. So I think these should be called "optimal supporting-ET sequence" or if that's considered too long then "optimal ET sequence" since why would it be listed against the temperament if it didn't support it. Is there such a thing as an ET that doesn't have a uniform map/GPV? If so, the uniform map/GPV requirement can simply be made part of the optimality requirement. Dave Keenan (talk) 02:13, 4 May 2023 (UTC)

I recall reading that there are infinitely many ET for each EDO (or equal tuning) out there, alluding to the fact that an ET is defined by a temperament map (val), and you could technically use any kind of map, even wonky ones, with any tuning. The use of wart notation in this sequence confirms that we're really enumerating equal temperaments, not equal tunings (both of which can abbreviate to ET, that's annoying, but I'm using ET for equal temperament here). After all, putting something like 17c in a sequence is exactly the same as spelling out the corresponding map in full, it's just that wart notation makes the sequence look like a list of equal tunings when it actually isn't. Now, uniform maps (GPVs) are a special kind of ET that excludes the "wonky ones", and in a given prime limit (or domain), there are finitely many uniform maps that map a given prime to the same number of steps; File:Generalized Patent Vals.png offers a nice visualization of this. I believe it's reasonable to only include uniform maps in these lists, because most people interested in temperament data want maps that actually try to approximate JI logically, and it would only be less clear what exactly is being listed if we switched from GPV to ET. If people understand it as "supporting ET", while it's not the full picture, it's not completely off track either, unless I am myself off track of course. My current concern would be "GPV" vs. "uniform map" (UM?). "Optimal uniform map sequence" is a mouthful, and "Optimal UM sequence" looks weird because it's new although I bet I could get used to it quickly. On the other hand, since it's an "optimal sequence", one might expect that the ETs listed would also be uniform maps. From that perspective, I agree that "optimal ET sequence" is the most straightforward choice. However, I wonder if non-uniform maps can make it in the sequences even with the restriction of decreasing error, so maybe someone could help me clear that up? --Fredg999 (talk) 03:10, 4 May 2023 (UTC)

I've never heard of ET standing for "equal tuning", only "equal temperament". I do not say that ETs are maps (vals), but rather that ETs have maps (vals), because an ET can just as well be defined by its comma basis (monzo list). So, to me, those sequences are not sequences of any kind of map or val. That would look like: ⟨12 19 28], ⟨17 27 40], ... They are sequences of ETs. I agree it would be interesting to know if an ET with a non-uniform map could ever make it into such list, given only the requirement of decreasing error with increasing ET number. I'm sorry I can't answer that. I expect that Flora could easily perform some experiments in that regard. But we don't need to know that to resolve this issue, since we agree we can (and would want to) simply add the uniformity (GPV) requirement as part of what it means to be "optimal" here. Dave Keenan (talk) 04:06, 4 May 2023 (UTC)

I don't think ET is used often for "equal tuning" by the way, I was just noticing the coincidence. Also, I would rather say that ETs can be expressed as maps, rather than "have" maps, but I might just be playing with words here. Anyway, since optimal GPV sequences enumerate ETs with warts, and since 17c is a shorthand for ⟨17 27 40], it means that including 17c is equivalent to including ⟨17 27 40] in the list. A list of comma bases would essentially serve the same purpose, but we don't choose that form for essentially the same reason we don't write out the maps in full. In any case, that doesn't change that "optimal ET sequence" still looks like the best option to me so far. --Fredg999 (talk) 06:51, 4 May 2023 (UTC)

Alright, so to recap:

1. My initial three suggestions were to change "val" to "ET", to acknowledge explicitly in the name the "unform" / "general patent" aspect, and to also acknowledge explicitly in the name of this list the fact that it is formed by adding the simplest possible ET which improves upon the error of the previous ET. To be clear, these suggestions were made when the feature on temperament pages was called "Val list".
2. Flora agreed with the first of my two suggestions and softly disagreed with the other two. She then added a new suggestion that we change "list" to "sequence".
3. Between comments on this thread, several things changed. Flora acted on her suggestion to change "list" to "sequence". And despite agreeing with my first suggestion to change "val" to "ET" and softly disagreeing with my second suggestion to add "uniform" / "general patent", she changed "val" to "general patent val" ("GPV"). Also, "optimal" was prefixed to the name, perhaps in some attempt to address my third suggestion re: the decreasing error of these ETs.
4. When Dave resurrected this thread, he questioned whether readers will understand "GPV", and pointed out another aspect of these sequences that would perhaps be best to acknowledge explicitly in their name: the fact that this is a sequence of ETs which each support the given temperament. This comment also involved coaxing us back toward my original suggestion to use "ET" rather than "val"/"map".
5. Then, Fredg999 softly disagreed with Dave's suggestion re: supports, but agreed with his repetition of my original suggestion to change "val" to "ET". He also brought up the issue of switching from Gene's 2016 term "generalized patent" (GPV) to Dave's and my 2021 term "uniform", if we are to keep this part of the concept explicit in the name.
6. Next, Dave suggested that "optimal" could and should be made to capture both the "uniform" / "general patent" aspect (my 2nd original suggestion) as well as the error-decreasing aspect (my 3rd original suggestion).
7. Finally, Fredg999 agrees with Dave.

Alright, so now it's back to me. I agree with Dave and Fredg999 that we should change "Optimal GPV sequence" to "Optimal ET sequence". And I'll go even another step further: I think that "optimal" here can and should also be made to further capture a third idea, the one Dave pointed out: that these are ETs which support the given temperament. I think there's simply too much detail to this concept to capture it all explicitly in the name, and "optimal" does a suitable job at encapsulating all three of these ideas.

--Cmloegcmluin (talk) 19:11, 4 May 2023 (UTC)

Moved. To be clear, I was thinking about rigorously defining it and back then GPV was indeed used to make the sequence, before the new proposal arrived that the GPV constraint may be assimilated into "optimal". FloraC (talk) 07:54, 5 May 2023 (UTC)

I agree that "supporting" should be explicitly mentioned in any description of the requirements of the ETs in an "optimal ET sequence". Dave Keenan (talk) 20:46, 4 May 2023 (UTC)

Yes, Flora, I should have mentioned, that even back when it was simply called "val list", GPVs / uniform maps were used to make the sequence.

Dave, I looked into it, and it seems that "supporting" was already explicitly on the page, but Flora has just now improved things by making it link to the article I wrote about the concept.

Okay. So the page has been moved. Thanks Flora. And thanks Dave and Fredg999 for helping us sort this all out. But work remains. We still have many occurrences of "Optimal GPV sequence" across the wiki. I offer to do the work myself to change every occurrence to "Optimal ET sequence" across the wiki. I will do this over the weekend unless I hear back otherwise from anyone.

--Cmloegcmluin (talk) 17:06, 5 May 2023 (UTC)

Okay, I updated all occurrences. As I went, I standardized them all to use the template, to catch many occurrences of even older names used for this information, and also so that if we ever need to make changes again, that time we can do it in one place. --Cmloegcmluin (talk) 20:16, 7 May 2023 (UTC)

Sorted by complexity?

I think that almost always (perhaps simply always) this error-decreasing constraint results in a list whose member ETs are sorted by increasing complexity. I would be interested to see if this is true, and if so, note it in the article. Does anyone have an idea how to prove this one way or the other? --Cmloegcmluin (talk) 19:12, 4 May 2023 (UTC)

The set of all GPVs can be sorted by complexity and is used here so that a "bigger ET" comes later than a "smaller ET". There's nothing to prove. FloraC (talk) 07:54, 5 May 2023 (UTC)

Can ETs with non-uniform maps ever be optimal?

On another discussion thread on this page, the question arose as to whether an ET with a non-uniform map could ever make it into an optimal ET sequence. I wanted to set this question aside on its own topic for further discussion. I agree with Fredg999 and Dave that it is an interesting question, but at this time, I don't have any particular ideas on how to answer it. --Cmloegcmluin (talk) 19:14, 4 May 2023 (UTC)

Hard to answer. Guess it could happen in the higher ranks. FloraC (talk) 07:54, 5 May 2023 (UTC)