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 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)