Talk:Defactoring terminology proposal

Some thoughts

Torsion indeed does not have much relevance to music, as you note. While it is possible to produce a temperament that tempers out (81/80)^2 and not 81/80 by going to a subgroup that doesnt include 81/80 (e.g.: 4.9.25), this is extremely dubious, in my opinion. (The wiki talk about this here, see my note in the talk page.) However, torsion is simply a fact of life: torsion modules capture exactly how Z-modules are different from vector spaces, and why we need to be very careful when porting over ideas that apply to vector spaces. In most cases things will simply work out, but modules are weird. This is why temperaments should be defined as maps between free Z-modules. So torsion is something that should definitely be discussed on the wiki, since it will come up when trying to build a temperament calculator or some other app that tries to deal with temperaments automatically. It is my view that if you are building such a program, the gritty maths should be hidden from the user, and dealt with accordingly. However, if you want to do that, you (the developer) need to know about all this in the first place!

Contorsion is slightly less problematic, because by increasing the rank of the subgroup, it usually gets fixed (or you can just defactor them). So I like the idea of calling them temperoids!

As for terminology: torsion is an established term in module theory, so I think it's fine. And contorsion might be a bit of a stretch, but it does fit the idea of using co- as the dual thing. (Enfactored seems good too, but contorsion might be hard to get rid of at this point. I don't really care how you want to call it.) I do think it's a bad idea to use enfactored to cover both torsion and contorsion, if that's what's being proposed here.

Finally, I strongly disagree with this:

While it can be argued that it is theoretically possible to interpret RTT using mathematical structures like quotient subgroups, lattices, and free abelian groups, [...]

In my mind, those things are *exactly* the kind of objects RTT deals with, and there's no way around it. You are very pragmatic and I respect that, but I don't see how else you can precisely define all these notions, let alone make provable statements about them.

- Sintel (talk) 22:07, 20 December 2021 (UTC)

Thank you Sintel for being the first person to engage with this proposal :)

I recognize that a great deal of the writings on RTT found here on the wiki use these more advanced mathematical concepts. My work on RTT deals with making the simplest and most musically practical stuff in RTT easier to understand, so I stick to basic linear algebra (and therefore vector spaces, not modules or groups). There may be RTT ideas that cannot be explained with LA, but I don't think they should cause us to sacrifice pedagogical clarity re: the basics.

I'm glad you like the term "temperoids".

"Enfactored" is proposed to cover both "contortion" and "torsion". Why do you think that's a bad idea? I think it's better to illuminate their similarity, and if it's important to distinguish them, you can simply say "enfactored mapping" or "enfactored comma basis".

I have slightly edited the statement you disagree with to make it less strongly opposed to your perspective, but I otherwise stand by it in this context. --Cmloegcmluin (talk) 19:39, 3 January 2022 (UTC)

Torsion refers specifically to the structure of a module. That is, the module has torsion when some of its elements have finite order (they are cyclic subgroups). By the way you're wording things, it seems like defactoring refers to properties of the matrices involved. That way it's fine though: "enfactored mapping" and "enfactored comma basis" are clear enough.

Surely you're not claiming the work you've done on exterior algebra is basic! I guess I'm not always sure what level of depth/complexity an article should fall on. I do think it's possible to explain 'advanced' concepts clearly, as you have certainly demonstrated.

- Sintel (talk) 22:58, 3 January 2022 (UTC)

Thanks for the encouragement, Sintel. I'm glad to hear you've found my explanations clear. I'll return the compliment — your explanations have consistently been quite helpful for me too. I can't wait to find the time to look into your recent thoughts on "transversal generators".

Good point: no, ​I don't count the exterior algebra article as "basic" (and don't call me "Shirley", haha.) Recently I've been writing some other definitely-not-basic theory on the wiki, too. But I still feel like I'm only scratching the surface. In any case, with regards to this torsion issue, I think that for the vast majority of xen musicians interested in working with regular temperaments, it's possible to boil it down to very basic terms. And so I think considering "enfactor" and "defactor" to be properties of matrices is fine; no need to know or care if these matrices are modules, bases, etc. Though it may give additional insight to eventually learn about those concepts, of course... I'm not trying to altogether suppress the links to these higher-level concepts, just get them away from the beginners, so as to not scare them off. For many people, matrices are scary enough already. --Cmloegcmluin (talk) 23:25, 3 January 2022 (UTC)