Talk:Meet and join

Additional Updates

Since you can either join/meet two temperaments' vals or their kernels, and both are useful - and the larger picture of how both relate is what is the really mathematically interesting thing - I've just added both to the page. This way things are permissive and people simply can be clear in their own writings if they are joining/meeting vals or kernels.

For now I left the symbols as they are but I am not married to them and happy to make them more ASCII-friendly; I would just like to pick something simple that hasn't been also used in some incompatible way. I still thought && and || was a good suggestion. Mike Battaglia (talk) 21:34, 20 December 2021 (UTC)

Interesting stuff. I'll need to think on it a while. I'm wondering what Dave thinks; he's usually got an opinion on this sort of thing, but he hasn't added his voice to the conversation yet.

I don't think anyone has suggested && and || are bad suggestions here. Like you, I think they're good suggestions, and have supported them below. Again, I encourage you to get more input about this stuff on Facebook and Discord. --Cmloegcmluin (talk) 21:44, 20 December 2021 (UTC)

I've spoken to people on Facebook and added && and || as the symbols for kernel-meet and kernel-join. Again, I don't really participate on the Discord, but you are free to ask them what they think. Mike Battaglia (talk) 21:19, 21 December 2021 (UTC)

Lots of Updates

I've added lots of additional material on meet and join of subgroup temperaments, some poset-theoretic stuff, embellished some of the original material, corrected some strange inconsistencies and errors, etc. Not a total overhaul but much of the below was addressed in this. One noteworthy thing is that I flipped the convention for meet and join, after posting on FB, as the original one was chosen at random and doesn't generalize to subgroup temperaments - the "join" of two subgroup temperaments would have been the "meet" of the kernels and subgroups under the old convention. As the person who went with the initial convention said it didn't matter either way in the original post, I went with this instead. For notation I went with ⊓ and ⊔ for meet and join as they are easier to differentiate from the wedge product than the curvy one used. Anyway, always more to do, but I'm mostly happy with it for now. Mike Battaglia (talk) 10:49, 19 December 2021 (UTC)

Mike Battaglia (talk) 10:49, 19 December 2021 (UTC)

First of all, thanks for all this Mike! Your new changes on the whole are great.

I don't want to leave you with the misconception that "much of the below was addressed in this", however. If you re-read the concerns I expressed below, you should find that your latest changes actually address none of them.

That said, and as you know, I've recently begun work on a new page presenting a general audience version of this topic. So I'm not going to get too hung up on detailed criticism of this mathematician-targeted page anymore. So if anyone wants to address those some day, great. Otherwise, no big deal.

There is one thing, though, that I would still like to discuss. So I was particularly disappointed to find that you had agreed with me that it was a good idea to replace "curly logical and" ⋏ and "curly logical or" ⋎, however, you decided not to replace them with my suggestions, the & and | symbols, instead deciding to replace them with "square cap" ⊓ and "square cup" ⊔. Which to me neither makes it better or worse. So I'll repeat my request to go with & and | just in case you missed it when making your changes yesterday. And I'll repeat my arguments:

1) they are well-known symbols for and/intersection and or/union. Perhaps particularly by software engineers, but probably more widely than that, and in any case many people working on RTT have familiarity with computer code conventions like this.

2) they are already used by the xen community in web tools like Graham Breed's x31eq, my own extensive RTT writings, and notations like Inthar's for naming scales, which I can't seem to find at the moment.

3) they are easy to type on keyboards without web searching and copying-and-pasting or using special tools like Wincompose.

4) they don't have the historical hang ups that up/down based symbols have as described below.

So, if you intentionally chose not to use the symbols for & and | for meet and join on this page as I had suggested, then I won't press you on the matter further. Though I would at least like to know what your reasons were. And also, if that is the case, would you mind at least — for outwards compatibility — adding a note that & and | can be used as well? --Cmloegcmluin (talk) 17:24, 19 December 2021 (UTC)

The way I have been using & for years is as a generic "join" operator, either to join vals, such as 12p & 19p, or also to join monzos, such as 2.3.5.7 81/80 & 64/63. If you are using & to join vals then it is a "meet" of the temperament kernels, and if you are using & to join kernels then you get a "meet" of vals. It seems like your proposal wouldn't be compatible with this existing usage so I used another symbol. ​Mike Battaglia (talk) 22:10, 19 December 2021 (UTC)

Oh! Fascinating. Thanks for explaining your reasons here. I totally see your concern now. Personally, I haven't seen anyone use & to union commas/vectors like that, as in "81/80 & 64/63", but I agree my proposal would be clearly incompatible with that usage. The possibility hadn't occurred to me that Graham chose & for 5&7 = meantone not because it represented a meet of two temperaments, but rather because it represented a join of two maps, but I will ask him to clarify his thinking on that. Do you think this usage of & as a generic join/union operator is popular enough that you would dissuade others from using & for meeting temperaments and | for joining temperaments elsewhere? Or, to the best of your knowledge, is that just your personal practice? --Cmloegcmluin (talk) 22:54, 19 December 2021 (UTC)

Join has long been used in the sense where 7edo JOIN 12edo = meantone. So we should be consistent with this.

Now some people like joining commas, such that 81/80 JOIN 128/125 = 12edo. This might be confusing because people use "augmented" for both 128/125 and the temperament defined by it.

If we define join to only work on temperaments, there is no ambiguity:

7edo JOIN 12edo = meantone

meantone MEET augmented = 12edo

I have pursposefully avoided symbols above because they are irrelevant to that point.

But I will suggest this: "join" has always been denoted with &, so it would be best to just use this symbol.

So:

7edo & 12edo = meantone

meantone | augmented = 12edo

81/80 | 128/125 = 12edo

In my opinion this is the least ambiguous.

Now for JI subspaces, I propose to define it such that:

2.3.7 & 2.3.5 = 2.3

2.3.7 | 2.3.5 = 2.3.5.7

This neatly follows the meaning of the symbols when used as logical operators: with A | B takes all the elements in A or B. A & B takes only the elements in A and B. It also means that joining subspace temperaments joins both the subspace and the temperament, which seems nice.

I am not in favor of introducing new symbols, especially since & has been in wide use specifically to denote the operation "join" on temperaments.

- Sintel (talk) 23:27, 19 December 2021 (UTC)

People have been talking about joining and meeting both vals and kernels for quite some time. The way you are proposing to use & for kernels would conflict with the more general usage of &, where it denotes the join of either vals or monzos. The symbols I chose weren't really an attempt to introduce something new, just a response to Douglas's suggestion to replace the curvy cup/wedge symbols with something easier to differentiate in text. I'd be willing to go with something like && and || though, if you'd like. Mike Battaglia (talk) 00:20, 20 December 2021 (UTC)

For Mike:

I'll repeat my question, "Do you think this usage of & as a generic join/union operator is popular enough that you would dissuade others from using & for meeting temperaments and | for joining temperaments elsewhere? Or, to the best of your knowledge, is that just your personal practice?" In your reply to Sintel, it seems like you may be implicitly saying, "yes," but I'm not sure.

I've asked Graham how he thinks of the & operator. He made a strong argument in favor of it meaning the join of maps as opposed to the meet of temperaments (i.e. the intersection of comma bases): 5&19, in his x31eq tool, does not give meantone; it gives 2-enfactored meantone. In other words, it's used simply to concatenate the maps, not the more complex operation that is described on this page, which helps you find a basis for only the commas tempered out by both temperaments, which would involve eliminating any enfactoring. So Graham agrees with you on this one, and I'm back to the drawing board.

Your suggestion to use && and || is intriguing, though. I like it. It's how AND and OR are done in very many popular programming languages. And the doubling of the symbols is suggestive of the higher order of the operation, i.e. it applies to abstract temperaments rather than literal vectors or matrices. Personally I would prefer this over the caps and cups you picked in your recent edit.

But perhaps we should give some time for other people to weigh in. This swap of the words "meet" and "join" was done without going through the standard protocols of posting in the Xenwiki Work Group on Facebook and the wiki channel of the Discord, so I've been doing some damage control and trying to bring people up to speed there. You did post in the Mathematical Theory group on Facebook, but I think the Xenwiki Work Group is always important when editing the wiki, even if you're its super-admin or whatever your role is exactly. So perhaps it would be good to make a follow-up post in these places, and at that time solicit input from the community re: the symbol as well. --Cmloegcmluin (talk) 17:24, 20 December 2021 (UTC)

The answer to your first question is that I would recommend using & as a generic join of either monzos or vals. Thus, I wouldn't recommend using & in the way that you're suggesting or else we'd have 81/80 & 64/63 = 1/1. Beyond that you are twisting things pretty severely: I am not a "super-admin," and I did post the "meet"/"join" swap on Facebook; people seemed agreeable to it, so I went with it. We could swap it back, although that would, as previously mentioned, complicate the definition for subgroup temperaments.

I expect you to post your proposal on Facebook that all changes to the Mathematical Theory pages be validated by the Discord group first. Mike Battaglia (talk) 18:30, 20 December 2021 (UTC)

Thank you for answering my question. That's good to know and I agree with your conclusion.

I don't think I'm twisting things at all. Please re-read what I wrote carefully. I specifically acknowledged that you did post on Facebook. My concern was that you did not post to a group that was important to post to: the Xenwiki Work Group. Perhaps I should have taken a moment to acknowledge that — as you point out here — in the group that you did post to on Facebook, there indeed was no disagreement with your proposal, which was a good sign.

Re: "super-admin", what I meant was that you have more user roles than anyone else on this wiki, and no one has any role that you do not have. I know "super-admin" is not a technical term. That's why I accompanied it with "or whatever your role is exactly". I'm not an expert in the exact roles' powers. I suppose I wasn't doing this in the most polite way, so I apologize about that. I was only trying to convey the whole "the president shouldn't be above the law" sort of sentiment, that's all. I'm sure you can understand that as a major contributor to the wiki who tries to be methodical and careful about getting wide approval for major changes I try to make, I'd like to see that other major contributors follow the same process.

I don't want you to swap meet and join back. I hope I've been clear that I support the change, and I'm very glad that you made this proposal, and that you did the work to make it happen. But I'll praise and express my gratitude to you for it again. I'm honestly really, really excited that we've got it the way it is now. You're the man. Thanks again.

Also, thanks for the recommendation: yes, that is a good idea to make a dedicated post on the Xenwiki Work Group to recommend that people find a way to run big changes by the very active XA Discord server in addition to posting there. I believe this practice has been recommended frequently in recent discussions in that group. I don't think of it as my own proposal so much as something that people have reminded me was good practice, if one wants to avoid thrashing in the edits on the wiki.

And I still think it would be a good idea for you to post to the Xenwiki Work Group about this meet/join swap, and its symbiology. --Cmloegcmluin (talk) 19:01, 20 December 2021 (UTC)

Douglas: I didn't see the above post. I think you are clearly trying to twist things into a picture whereby I am abusing some kind of authority to make unilateral changes to the page without going fielding community feedback. As I've noted several times, I did post it on Facebook to get feedback. I got some initial feedback, so I made some changes. Then you gave some feedback, so I made more changes. Then people on here gave feedback, so I made even more changes. The feedback was good and I incorporated it into a new version of the page. Now there is even more feedback and I'm sure there will be even more changes. For someone who is unilaterally abusing whatever authority you imagine me to have, I seem to be spending an awful lot of time fielding feedback and making changes on what was originally intended to be a minor edit.

Lastly, the issue of my "user roles": is that they are set the way they are because the Wiki used to be hosted on Wikispaces. When that went down, Tyler reached out to me to see if we could migrate everything elsewhere before it collapsed. So we took it on ourselves to do that. While and my largest role was in the initial migration, the ongoing maintenance of the sysadmin stuff is really Tyler's thing, but I still contribute to occasional minor technical maintenance from time to time when I can. This is not something I really enjoyed doing, but I spent many hours doing all of it for free, or sometimes at personal cost, just to make sure the information was preserved. But none of this has anything to do with anything, because my ability to help install some new MediaWiki extension has no bearing on the contents of this article, and certainly it does not matter either way in whether or not I participate in a Discord server, and I think it's in poor taste to try to somehow use it against me to win a notation debate about the use of & vs &&. Mike Battaglia (talk) 21:07, 23 December 2021 (UTC)

I totally agree with Sintel. That's how I've always thought of it. I've never seen "&" used for the join of monzos. Can anyone point to some places on the wiki where it's used that way? In any case, what would be the harm in swapping those to "|" as in Sintel's and my scheme. Has "|" been used previously in a way that would conflict. So I'm supporting "&" for val-join, "|" for comma-join and subgroup-join. Then "&" is temperament join and "|" is temperament meet. Then "12 & 19" can be reads as vals or as temperaments, "&" works either way. Dave Keenan (talk) 09:06, 23 December 2021 (UTC) Dave Keenan

I agree with Dave and Sintel's proposal, at least as far as making it so that & is a concatenation (and reduction) of maps (meaning an intersection of commas and subspaces) and | is a concatenation (and reduction) of commas and subspaces (meaning an intersection of maps). I'm not concerned about which one is which, being called "meet" or "join" anymore. If I had clearly realized it was as simple as this earlier, i.e. that the only thing standing in the way of this simple of a solution was dismissing the use of & for concatenating commas as Mike claims has been done historically, I would have pushed for it. So I echo Dave's request for specific evidence of this usage, but more importantly his request for clarification on what the problem would be (if any) in changing them to use |. --Cmloegcmluin (talk) 15:26, 23 December 2021 (UTC)

Douglas: Sorry, but I'm not going to waste endless hours somehow trying to search Facebook for posts of the form "ratio & ratio." I don't even know if that kind of thing is possible given Facebook's limited search feature, and if the primary reason you want me to do so is to because you demand "evidence" are unwilling to take my word for it, then I think there's no point in discussing anything further. Much like you don't intend to call a "map" a "val," I also don't intend to never use the phrase "81/80 & 64/63" ever again, but if it makes you happy, I have added & and | to the page as the predominant convention. Mike Battaglia (talk) 21:07, 23 December 2021 (UTC)

Suggestion to add a helpful image

This image which is used on the fokker block page might be an excellent inclusion here as well, with a caption something like "movements downward are joins with the ET that is added; movements upwards are meets with the ET that is removed". I think this could help paint a compelling picture of how this concept is so exciting:

https://en.xen.wiki/w/File:1000px-Pajmagorpor22_temperament_support_lattice.svg.png

--Cmloegcmluin (talk) 00:33, 29 June 2021 (UTC)

Suggestion to change symbols, and simultaneously strengthen conceptual connection with the existing & operator used by Graham Breed's app

I like this meet and join concept a lot! But I have some concerns about the symbols used. They’re used nowhere else on the wiki, so I think this shouldn’t be that big of a deal. I know that Mike Battaglia is working on other materials using meet and join but other than that this would have little impact on anyone I think.

My initial concern is that these symbols ⋏ and ⋎ are easy to confuse with the symbols used for the wedge and vee products, ∧ and ∨ (the latter is currently used as the symbol for an interior product but that's another issue; see: Talk:Interior_product#Questions.2C_observations.2C_suggestions). Gene states these curvy versions were chosen over the straight versions to avoid confusion, which is well-intentioned, but I'm concerned that it's not enough. Especially since there is tremendous historical confusion between the symbols ∧ and ∨ as they are used in exterior algebra (from where Gene brought to us the wedge and vee product we commonly use now) and ∧ and ∨ as they are used in order theory (from where Gene draws the meet and join concept). It surprises many to find that wedge ∧ is akin to join ∨, in that they both increase grade (rank or nullity), and that veeing ∨ is akin to meet ∧ in that they both decreases grade. We suspect that the symbols for wedge and vee were chosen by the direction they point, while the symbols for join and meet were chosen for their resemblance to the union and intersection symbols (join ∨ is like union ∪; meet ∧ is like intersection ∩), and that’s how similar operations ended up with opposite symbols. Even Gene seemed to get this confused (see my other suggestions here). So I suggest that it would be best to leave the ∧ and ∨ type symbols in the RTT domain to the one job they already popularly do for us — wedge (and maybe vee or interior product) — and find some other symbols for meet and join.

And there is an excellent choice for join already available! The join operation may be understood as an extension of a well-established operation in RTT which was given the symbol & and is sometimes called "cross-breeding" because it is used to combine equal temperament maps, or "breeds". With & you concat rank-1 mappings into higher rank mappings. And with "join" you concat mapping matrices of any rank into a higher rank mapping matrix. So it’s the same idea, now generalized to any rank. Therefore, & should be used for join: 19&31 could be read "19 join 31" and meantone&marvel would be read "meantone join marvel".

What about the symbol for “meet” then? We suggest the pipe, |. This symbol is commonly paired with & in logical systems. And there is a visual motivation for this: because commas are represented by vectors, which are vertical columns, when they come together into matrices, the pipe resembles the seam between their meeting. So meantone|marvel would be read "meantone meet marvel".

Mostly I'm just excited to bring this connection between join and the & operator to people’s attention, because it seems probable that Gene hadn't noticed it. I understand that I'm not the first person to become aware of this connection — Paul Erlich has told me that Keenan Pepper has discussed it on Facebook groups in the past.

If anyone still likes the ⋏ and ⋎, I see no reason why they couldn't be retained as options. They don't conflict with anything.

--Cmloegcmluin (talk) 00:33, 29 June 2021 (UTC)

Are the order theory symbols as given in this article implicitly incorrect?

This sentence from the page reads:

In mathematical order theory, meet and join are denoted by ∨ and ∧.

Typically in English, parallelism is assumed, so this implies that in mathematical order theory, meet is denoted by ∨ and join is denoted by ∧. But as far as I can tell from my research, it's actually the opposite case. So I suggest this should be changed to:

In mathematical order theory, meet and join are denoted by ∧ and ∨.

Or if this is not the case, maybe the meaning should be clarified.

--Cmloegcmluin (talk) 00:33, 29 June 2021 (UTC)

Suggestion to clarify usage of "abstract" with respect to temperament

There's a bit on this page that reads "meet and join are operations on abstract temperaments; ordering by increasing size of the group of commas and decreasing size of the group of vals is regarded and notated as the same." In this case, I think "abstract" may be being used for a reason other than the most common one we find it used for with respect to temperaments.

That is, most commonly "abstract" is used to make unambiguous that a temperament has not been resolved to a specific tuning; usually, in xen wiki or RTT contexts, the audience accepts that this is already the meaning of "temperament", but sometimes it is still decided by the author that specifying "abstract temperament" is important to make absolutely sure the meaning is clear.

But here I think a different meaning is being used. It could be changed to this, in order to clarify the difference: "meet and join are operations on temperaments, in the abstract, i.e. while calculating them may indeed involve manipulation of lists of vals or intervals, the two objects on either side of the meet or join operation symbol should be understood as referring to the entire temperament in general, not any individual one of the many such specific mathematical representations of it."

But I could be totally off about this. --Cmloegcmluin (talk) 00:33, 29 June 2021 (UTC)

Suggestion to explain the <S> notation earlier on the page

On this page, "<S> for a set of commas S denotes the temperament of the group G tempering out the given commas", but that explanation is only found mixed in with the examples at the bottom, and that notation gets used a couple of times in the top section before it is defined, which I think is pretty confusing. Especially since it essentially gives you the opposite of what's inside, so a sentence that contains "everything is tempered out, and we may also call it <JI>" is quite surprising at first. I suggest that the definition of this notation should be extracted from the examples section and given earlier on.

There are some other interesting statements made in the examples section which probably deserve to be addressed in outside of a list of examples, so they don't get lost down there.

--Cmloegcmluin (talk) 00:33, 29 June 2021 (UTC)

Suggestion to not use G to represent a temperament

Gene defines the variable G as "a JI group", e.g. all the interval vectors of some prime limit, and writes about "the temperaments of G".

So I think it might be a little confusing when he later describes G as "the maximal temperament" and says "in G everything is tempered out". If G is the prime limit, then how can it also be a temperament, let alone the one that tempers out everything?

If "<1>" was Gene's way of indicating JI (where the angle brackets say "my insides are tempered out", so in this case, nothing is tempered out except the unison which already was the unison), then we could use "1" as the name for this everything-tempered-out temperament currently called "G", or maybe even better, this temperament could be called "unison", because in it, the only interval is the unison.

--Cmloegcmluin (talk) 00:33, 29 June 2021 (UTC)

Suggestions to potentially improve the penultimate paragraph of the Definition section

I think in the second-to-last paragraph of this section the wrong symbol is being used, i.e. Gene uses join ⋏ where he means meet ⋎, and vice versa. To be specific, this sentence:

There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B.

Should be:

There is a partial order on the temperaments of G, given by A≤B iff A⋎B = A and A≤B iff A⋏B = B.

I think this could be further clarified if we just add parantheticals to spell out the operations used, since the symbols aren't well-known. Like this:

There is a partial order by rank on the temperaments of G, given by A≤B iff A⋎B = A (their meet is A) and A≤B iff A⋏B = B (their join is B).

I might be totally wrong about that, though. If so, please let me know.

Per the above suggestion not to use G as a temperament name, I suggest replacing these two sentences:

Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it <JI>; and in the temperament defined by G^, nothing is tempered out, and we may also call it <1>

with the more parallelized and clear:

There is a maximal temperament we may call "JI", where A⋏JI = JI for all A; JI is the temperament where nothing is tempered out. There is a minimal temperament we may call "1", where A⋎1 = 1 for all A; 1 is the temperament where everything is tempered out, leaving only the unison.

This also involves avoiding using Gene's angle bracket notation on "JI" and the unison "1". I think it's an excellent notation when used on lists of commas, but in these extreme cases where it just means the opposite of what it encloses, I think it's potentially more confusing than it is helpful.

Finally, I think this sentences:

A≤B may be expressed by "A is supported by B".

might be improved by revising it to read:

A≤B may be expressed by "A supports B" in the sense that an ET may support a rank-2 temperament.

in order to tie it better together with familiar lingo.

--Cmloegcmluin (talk) 00:33, 29 June 2021 (UTC)

Duality

Is there a proof somewhere of the duality, ie that join on kernel <=> meet on map? (And vice versa) I'm convinced this is true, but it is not obvious at all.

- Sintel (talk) 13:51, 23 December 2021 (UTC)