Talk:Meet and join

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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'm including this introductory section in both of my replies here to make sure that you see it, and because it applies to both places.) Sheesh, Mike. Please take it a little easier on me. I hope I will eventually be able to convince you that I'm your friend. Tone can be hard to read on the internet. You frequently react to the things I write in a combative way that I think is uncalled for. I think I'm a pretty chill and nice guy who thinks highly of you and hopes you will come to enjoy my presence and contributions here one day. Perhaps we should have another video chat session soon to clear some things up.
I believe I made myself clear in the above that I appreciate the efforts that you have already made to solicit feedback on this change. But I will express it again: I recognize those efforts and applaud you for them. Thank you for checking in with folks on Facebook about these changes.
Beyond this, I have suggested some addition actions I thought were appropriate for you to take, namely, posting in the Facebook group specifically designed for that purpose, and finding some way to get feedback from Discord as well. Those suggestions still stand.
I will also take this opportunity to again express my gratitude to you for the role you've played in the creation and maintenance of this wiki. Thank you, a thousand times over. Without your efforts, countless people in our community, including myself, would be significantly less enlightened about this fascinating theory that we all share a love for.
I now feel like you're twisting my actions crazily far from reality:
1) I never indicated that I expected anyone to use the Discord as a prerequisite to making changes on the wiki. When, per your request, I posted on the Xenwiki Work Group on Facebook to specifically voice my suggestion to the wider group, that people should consider getting feedback from Discord because it is such an active community, I specifically said that I did not expect anyone to join Discord, that it was not in any way meant as a guideline but merely a suggestion if you wanted to selfishly de-risk your own time investment in this sort of thing, it might be a good idea. Furthermore, I personally volunteered there to be the middle man and help anyone who wanted an agent on Discord to get feedback for them. What else could I possibly do on that front?
2) The fact that you perceive this as a debate that one of us will win is telling. I do not see it that way at all. I can't understand how you could get that impression. All I'm asking is that you give people who care about this sufficient opportunity to add their voices to the discussion, so that in the end we can come up with the best solution, which may or may not be one that you or I suggest ourselves. Getting my way is not an end in itself. I just want what we end up with to be high quality, and fair. I hope that makes sense now. --Cmloegcmluin (talk) 04:00, 24 December 2021 (UTC)
Oops, I said I wasn't going to respond to this, but I guess one fragment of this line somehow got through anyway (but dedented). So since it was already half-posted, what I said was in response to this: "I never indicated that I expected anyone to use the Discord as a prerequisite to making changes on the wiki," "I specifically said that I did not expect anyone to join Discord, that it was not in any way meant as a guideline" etc. If I read your earlier post, quoting from it directly and interpreting at face value, your stated position is that it is "standard protocol" to "go through" the Discord, that "the president shouldn't be above the law" with respect to this "standard protocol" and so on - these are all the exact words you used, it's pretty strong language that can't be interpreted in any other way I can see, and what I took offense to. Beyond that, I've spent I don't know how many hours fielding feedback from you and others weighing on it and given you all the changes you wanted. Mike Battaglia (talk) 07:34, 24 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)
(I'm including this introductory section in both of my replies here to make sure that you see it, and because it applies to both places.) Sheesh, Mike. Please take it a little easier on me. I hope I will eventually be able to convince you that I'm your friend. Tone can be hard to read on the internet. You frequently react to the things I write in a combative way that I think is uncalled for. I think I'm a pretty chill and nice guy who thinks highly of you and hopes you will come to enjoy my presence and contributions here one day. Perhaps we should have another video chat session soon to clear some things up.
I can see that I wrote things in a rather matter-of-fact, not sugar-coated way, but I didn't mean to come across "demanding" evidence, and I see nothing in the words I chose that suggest I was writing in an aggressive, "demanding" type way. I hope it's clear that in order to make the best decision we can make here, it would help the community here to have some further information, specifically about the extent of this usage you've described. I was merely seeking to gauge the extent of the situation: how common is this. It seemed quite likely to me that you could produce an extensive body of work with this usage, in which case I would seriously reconsider my position.
Also, I didn't mean to come across like I didn't believe you. I very much believe you, and I don't see any words I wrote that indicate otherwise.
It seems like the usage you describe is somewhat extensive (at least your subgroup temperament page, https://en.xen.wiki/w/Subgroup_Temperament_Families,_Relationships,_and_Genes, as you mentioned on Facebook), but apparently not extensive enough to completely problematize our proposal. So it does make me happy that you have added & and | to the page as the predominant convention. Thank you for doing that. All my best. --Cmloegcmluin (talk) 04:00, 24 December 2021 (UTC)
The feeling about combativeness is certainly mutual. Thanks for your assessment that I didn't use & enough to problematize your proposal. I'm not going to respond to the other post. Mike Battaglia (talk) 07:08, 24 December 2021 (UTC)
I just stated that I'm not intending to be combative. You've here chosen to implicitly reject believing me about that, and furthermore, recognize that you are behaving combatively. And you say these things in a snide way. So, I'll say this explicitly: I wish you would not act combatively with me in the future, and also behave friendlily.
This wasn't my proposal, if you recall; It was Sintel's, and he persuaded me and Dave with his strong, well-presented ideas. You're clearly thanking me sarcastically here, but I can't understand why you would be upset that we successfully identified that the scope of a potential obstacle to this proposal was limited enough so as not to problematize it, and therefore we had a good solution to this pedagogical issue. I know that of the four of us discussing here, this solution poses the greatest inconvenience to you, and and I can see that I haven't sufficiently expressed my sympathy and gratitude that you're taking this one for the team. So: thank you for your sacrifice here. I appreciate it.
I'll just go ahead and reply to your other comment here so we don't have multiple threads going on at once.
The fact that you're spent your time quibbling about the strength of the language I used at one point or another to recommend that you get input from the community in the ways it has specifically positioned itself to invite such input, rather than just actually getting this feedback, indicates to me that really don't really respect me or the community or care if we end up with the best results for everyone here. You just have your own inflexible ideas about exactly which and how much feedback is appropriate.
You also ignored all the stuff I said about collaboration and tone and friendship, etc. so this suggests to me that have no intention to behave more friendlily toward me. I hope that's not the case.
I think it would be best for me to excuse myself from this conversation at this point. Feel free to say whatever you want to say to clarify things on your end, but I don't think I have anything more to say at this time. Thanks again for your contributions to this topic. --Cmloegcmluin (talk) 14:13, 24 December 2021 (UTC)
Of course I certainly never intended to be combative and apologize if anything came off this way. What I am saying is that the the feeling on most of these is mutual: many of the things you have accused me of doing are things I also feel you have done to me, such as misinterpreting my tone, quibbling over language, and generally being very harsh and unfriendly in the things you write sometimes. I am sure none of this was intentional for either of us and we both do our best; this is simply my feeling as it is yours.
If you want me to respond point by point I will, as some of the things you have just written are I think quite hurtful, but I don't know if it would do any good. I will instead just say that I respect you and the community a great deal, which is why I have spent so much time listening to your feedback and changing things the way you want. The only "inflexibility" here is that there are only 24 hours in a day and I don't have any of those hours available to put into participating in the Discord. Anyway, I hope you understand and would also like to move past this. Mike Battaglia (talk) 19:16, 24 December 2021 (UTC)
I said I wasn't going to respond, but this comment of yours deserves one. Thanks for being the bigger person here, being kind, and extending an olive branch. Your assertions are very fair. I have misinterpreted your tone on many occasions. I have quibbled over language on many occasions. I haven't tried to be harsh or unfriendly, but if that's how I've come across, then I've got some stuff to work on. I'm certainly no master of self-awareness. I know I get angry and upset and I think sometimes even when I put on my politeness cap, it still bleeds out. You're a smart guy and you can pick up on how I really feel even if I'm trying to control it. I don't think it would do us good to analyze to death everything wrong we've said to each other here, point by point. I agree to let this one go and will try to start the next discussion off on a better foot. Thanks again for all your help on this and countless other things this year. Happy holidays. --Cmloegcmluin (talk) 19:38, 24 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)

It follows from the basic properties of matrix multiplication. If you have two row vectors v and w, then the null space of the concatenated matrix [v;w] is defined as those vectors mapping to (0,0), which means any such vector needs to map to 0 for both v and w. Mike Battaglia (talk) 07:38, 24 December 2021 (UTC)

Link back to new Temperament merging page

I've added a new page Temperament merging that attempts to give a plainer language, more pragmatic take on some concepts that are discussed on this page, supplemented with diagrams and examples. This new page is directed toward a general audience — readers with less mathematical background, but who still want to understand this topic well enough to do something with it. The first line of it links here: "This page gives a general introduction to this concept; for a more mathematical take on this, see Meet and join."

I'm writing here because I think it'd be great if the Meet and join page returned the favor, linking back to the new Temperament merging page, perhaps also as its first line, something like "This page gives a more mathematical approach to this topic; for a simpler introduction, see Temperament merging."

If there is agreement, I will not be able to do it myself, because I don't have sufficient privileges to modify this page, since it has been recently restricted.

This pattern of linking via the first lines between the pairs of general and mathematical takes on the same topic is fairly common, by the way. You can find it with comma basis and dual list, mapping and temperament mapping matrices, or monzo and monzos and interval space, for examples.

Anyway, the ink is still quite wet on the new page, so constructive criticism is very welcome there. --Cmloegcmluin (talk) 20:07, 9 January 2022 (UTC)

I've just remembered that at the bottom of the page, there's a See Also section that links to the ampersand operator page which has since been turned into a redirect page to Temperament merging. So if this change is made, that link should be removed, as it'd just be duplicating things. --Cmloegcmluin (talk) 06:40, 11 January 2022 (UTC)

OM should be Om

Per discussion here: Talk:Trivial_temperament#OM it seems like "OM" should actually be capitalized as "Om", because it's a reference to a Hindu concept that is capitalized that way, and also it looks confusingly like two variables O and M being multiplied together. This is the only page besides the Trivial temperament page where it is used currently. --Cmloegcmluin (talk) 20:11, 9 January 2022 (UTC)

Also, it's unclear to me why the majority (but not all) of the occurrences of "OM" on this page have the subscript "G". It seems like they can all simply be "Om". --Cmloegcmluin (talk) 06:40, 11 January 2022 (UTC)

<1/1> would be better than <1>

The <...> notation is interesting. But I think it would be clearer if <1/1> was used rather than <1>, to make it clear that this is a frequency ratio, like how you have <225/224> etc., as opposed to logarithmic pitch space where it would be a <0> instead. --Cmloegcmluin (talk) 06:57, 11 January 2022 (UTC)

Mismatched bracket count

It looks like in one of the examples at the end, for Meantone & Magic, we've got four angle brackets on the left but only two closing brackets on the right. I think it should be four on both sides. --Cmloegcmluin (talk) 06:58, 11 January 2022 (UTC)