What is a "compact lattice"? I don't find a definition for this concept on the wiki. Does this relate to a specific mathematical concept of compactness, e.g. https://en.wikipedia.org/wiki/Compact_space? I'm embarrassed to note that the phrase has been on the page since the day I created it, so I'm responsible for this. Often I add stuff to the wiki at the behest of old-school RTT friends who don't participate in the wiki, and this has the air of something someone might have emailed, messaged, or texted me on a recommendation, but I can't find a record of it. It's possible this is just an informal usage, but I feel like I do see people sometimes use this "compact" term in a technical-seeming way, so if it is an objectively-defined property of importance to xen, I'd like to see it explicitly defined at least here and possibly in another centralized place on the wiki. --Cmloegcmluin (talk) 22:51, 24 January 2022 (UTC)
Incorrect usage of "transversal"?
I set out to edit this page for clarity just now, and as I did so, I ended up finding what appeared to be a mistake, and so I removed it. But then I noticed this mistake in another place, which got me thinking that perhaps this was an intentional use of the word in a way I was unfamiliar with. (For now, it has still been removed from that page, but not declared to have been a mistake.)
Here's what it is. Inthar modified the page to read "detempering is the process of taking a tempered pitch system and replacing each of its pitches with one or more JI pitches that the pitch represents (called a transversal)}}" (my highlighting). I thought this was a mistake because I figured "called a transversal" was referring to the entire phrase, a "[replacement] of each of [a tempered pitch system]'s pitches with one ore more JI pitches that the pitch represents". And a transversal is defined to select only exactly one pitch that the tempered pitch represents, never more than one.
But then I noticed the following sentence: "The first tval maps the first generator (for which 2/1 is a transversal) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval." It is found on this page, toward the bottom: https://en.xen.wiki/w/Temperament_mapping_matrices So here "transversal" is being used apparently to mean "member of its preimage". (This edit was from Mike Battaglia back in 2012, but it looks like he may have just imported it in an automated move along with a ton of other stuff wholesale from the previous version of the wiki, so I'm not attributing any personal intentionality to him here: https://en.xen.wiki/index.php?title=Temperament_mapping_matrices&diff=27089&oldid=27090)
And when you come back to the Detempering page and review Inthar's edit, it now is clear that he may also have intended "called a transversal" not to refer to the entire resulting detempering, but actually just to the phrase "JI [pitch] that the pitch represents," or in other words "member of its preimage."
So to me, this usage is quite confusing. It no longer has anything to do with the actual meaning of "transversal". A transversal transverses a list of sets. Such as a transversal of preimages of generators for a temperament, or of pitches in a tempered scale. It seems that "transversal", via its typical association in RTT with lists of preimages, has come to be used simply for a member of a preimage, out of any context of transversing. I don't think we should use "transversal" in this way, but I'd like to hear what other people think. Maybe I'm all wet on this, or maybe there's something else I don't know yet.