Talk:Normal forms

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Smith normal form

I noticed "Smith normal form" was added to this page as a name for the form of the normal val list. I've done a couple examples and I'm pretty sure that Smith normal form is not equivalent to this process; e.g. meantone's normal val list would be [⟨1 0 -4] ⟨0 1 4]] while I suppose you could say meantone's val list in Smith normal form (taking the first k rows only, as is demonstrated in the penultimate paragraph of the page on saturation) would be [⟨1 0 4] ⟨0 1 -4]]. If this is an attempt to name this other form for Gene Ward Smith, I think it's not a great idea, because of the preexistence of the linked Smith normal form which was named for Henry John Stephen Smith. --Cmloegcmluin (talk) 18:24, 23 June 2021 (UTC)

I saw it somewhere and thought this was what it referred to. I was really sorry about that. FloraC (talk) 07:31, 24 June 2021 (UTC)

Oh, no big deal at all. Thanks for updating the page. --Cmloegcmluin (talk) 15:17, 24 June 2021 (UTC)

how best to handle the new canonical form for RTT matrices w/r/t normal form

I recently published a page on the xen wiki proposing a canonical form for RTT mappings (or comma-bases). This new page of mine solves a problem that this page seems to have set out to solve but not finished the job: establishing a form for RTT matrices which uniquely identifies them, for a definition of uniqueness that is appropriate to the RTT domain. And so my page refers to this page in several places, mostly critiquing it.

I note that this page doesn't even present a unified front. It presents both HNF and IRREF as potential normal forms. But it is not clear whether a given matrix on a temperament page marked as "normal" is its HNF or IRREF (sometimes they are the same, other times different).

But in particular I note that neither the HNF nor IRREF methods discussed here reliably defactor matrices (or in other words, saturate them, or remove contorsion, though those are confusing terms for the issue that my new page sets out to eliminate). This is what I'm referring to when I speak of a definition of uniqueness that is appropriate to the RTT domain. The HNF of a matrix is unique, yes, as is the IRREF. But these definitions of uniqueness treat e.g. ⟨12 19 28] and ⟨24 38 56] as distinct, when from a strict RTT perspective the latter is not distinct insofar as how it tempers JI from the former. At least when enfactoring is found in mappings, it has musical reality, but in comma-bases it's meaningless and confusing (I can still play music in 24-ET that sounds different than 12-ET, but I can't pump the comma [-8 8 -2⟩ any differently than the comma [-4 4 -1⟩)). So: enfactored matrices are pathological. (If you're interested in this issue, my new page discusses it in detail.)

My concern is that the normal forms for RTT matrices which are discussed here have proliferated widely, but I believe that now that a canonical form has been developed (by Dave Keenan, in collaboration with myself, Douglas Blumeyer, inspired in no small part by many insights from Gene Ward Smith) should be the primary form of RTT matrices used throughout the wiki. Of course I don't plan to do this myself immediately, for numerous reasons. For starters, that'd be a Herculean task. But mostly I wouldn't do something that impactful without soliciting input from the community first.

And as for this page itself: I am not saying that it is harmful or that there is nothing of value on it. Far from it! There's some good thinking here. But I do wonder what people here think about what the best approach should be for recognizing its relationship with the new canonical form. --Cmloegcmluin (talk) 18:10, 27 September 2021 (UTC)

First of all we must make clear what purposes these pages serve. Imo this page should always be kept up to date and used as the canonical reference. I'm not sure about your view of your canonical form article. To me it acts more like a development note thereof.

Currently the form in all temp pages is the form defined in the normal val list section, and they are in canonical form already since they are not enfactored to start with. (More precisely, there is not a start but only the end and the end is correct.) So they are in canonical form and we don't need to do anything about them.

The important part is the amendment of the definition in the normal val list section so that we'll convert the form to the canonical form, to ensure the correct end even if we start with an enfactored map. It can be a one-liner amendment, as simple as "defactor it", or rewritten to focus on the new reduction method.

Finally, the irref form should be removed as it's almost never used. We cover it in another page i.e. generator size manipulation, just like other forms. FloraC (talk) 15:07, 28 September 2021 (UTC)

My view of my new canonical form article is that it is mostly about defactoring, or in other words, the main element that is missing from the existing normal form and its wiki page. It is only partially about a canonical form for matrices which uses defactoring. I certainly agree with you that the extreme level of detail I included on my page, and in particular the documentation of failed experiments and tangential information, gives it a "development notes" character. Totally fair. :) You may have noticed that I also initiated similar discussion on the page for saturation/contorsion here: https://en.xen.wiki/w/Talk:Saturation, i.e. discussion re: how best to integrate this material about defactoring and canonical form in with what already exists on the wiki.

I think that at the end of the day, only two pages are necessary: one page about defactoring, and one page about the matrix form which uses defactoring along with Hermite normalization in order to achieve a unique identifier for a temperament. Due to recent freaky experiences proposing changes to the wiki on Facebook, my confidence about making major contributions directly to existing pages has been shaken. That's why I added all my new material to one new page: perhaps only as a staging ground. If I were to have felt more confident, I would have directly:

1. added most of that new material to the page for saturation/contortion, added redirect pages to it for "enfactored", "defactoring", etc., and then in the Talk page drawn people's attention to the parts of the new material explaining the major flaws in the existing terminology for the concept and recommending that the primary name of the page be updated to "defactoring".
2. added a small amount of material to this existing normal lists page, added a redirect page to it for "canonical form", and then in the Talk page drawn people's attention to the parts of the new material explaining the preference for "canonical" over "normal" as the term for this form and recommend that the primary name of the page be updated to "canonical form".

I believe that there are some people who will never accept proposals to rename concepts like "saturation" and "contorsion", nor would they appreciate the reworking and relegation of most of the existing material on that page to a "mathematical theory" subsection of it as I would prefer. So I think I'll never accomplish the consolidation of the defactoring material into the saturation/contorsion page. However, I do think we can migrate relevant information from my page into the existing normal lists page. In other words, extract all the information from my page about canonical form into the normal lists until I can rename my new page to be focused exclusively on "defactoring" independent of the canonical form it's used for, and then rename the normal lists page to "canonical form". The amount of migrated material to accomplish that may indeed literally be a one-liner, as you suggest. :)

You make an excellent point that while people exploring on their own are likely to encounter enfactored temperaments, probably most of those that have managed to get documented here are not enfactored. And also if you think that the IRREF form is almost never used, then there's not that risk of mismatch either. Therefore I agree that we probably could simply override the existing normal form with the new canonical form, or in other words, conflate the two while keeping the name "normal form", without causing a ton of inaccuracies across the wiki.

However, I don't think that's a good idea. While I personally agree that there may be little value in maintaining "normal form" as a term which refers to an RTT matrix which has been normalized but not also defactored, I think it is smarter and safer to allow for the possibility that there are people for whom the existing normal form w/o defactoring does hold some importance which we don't see ourselves at this time. Just for backwards compatibility's sake, I mean. There's also an argument that switching to the new term "canonical form" would be important because it helps signal to the community that the form has changed conceptually.

I agree that the IRREF form should be removed from this page. It's not mentioned in my new page for "generator size manipulation" though, so I'm not sure what you mean by that. Perhaps you mean that we should extract it to its own dedicated page? --Cmloegcmluin (talk) 17:42, 28 September 2021 (UTC)

So we've basically reached agreement that this page should be updated first. We definitely want to keep the original normal form (for both val list and monzo list), with the new canonical form added probably as a separate section. I'll try taking care of this.

Hmmm I remember at one point seeing irref in the page generator size manipulation. If not, let's move it there or somewhere else (like Mathematical theory of regular temperaments). FloraC (talk) 00:50, 29 September 2021 (UTC)

That sounds good to me so far. Re: IRREF, I know it's not anywhere on the page generator size manipulation, because I created that page myself only a week or two ago, and it doesn't really have anything to do with IRREF. Around the same time as I was creating that page, though, I was working on the canonical form page, and I do have a section about IRREF there: canonical form#IRREF I believe my section contains every bit of information re: IRREF and its relationship to HNF that is presently on the normal list page, and supplements it with visual diagrams that make comparison easy. So I think we could simply remove the IRREF stuff from this page. --Cmloegcmluin (talk) 02:21, 29 September 2021 (UTC)

Thanks for making your latest changes to the page. I'm glad to have learned about the Databox template and syntax highlighting! I've gone ahead and used it myself on other pages now.

I think the technique to link out to the canonical form page works fine. Maybe I'll keep it the way it is, i.e. not work to rename it to simply "defactoring" (I did add a redirect page for that, though).

Re: the beep example. I recently fixed that example myself. But I've since noticed it's not quite perfect. Normal form (and canonical form) require the pivots to be positive. And you find the pivots for comma-bases by anti-transposing them, i.e. flipping them across the anti-diagonal, between top-right and bottom-left, so that when the HNF tries to put all the zeros in the bottom-left corner, it gravitates them toward where we want them: the higher primes, and commas earlier in the list. Technically, then, the canonical commas for beep should be 25/27 and 35/36, even though with n < d those are negative in pitch and that's not the typical way we write commas. It looks less off-putting when the canonical form is presented as a matrix, i.e. ⟨[0 -3 2 0⟩ [2 2 -1 -1⟩], so I suggest we write them like that. Or I'm open to other suggestions.

Speaking of lists vs. matrices, I would like to rename the page from "normal lists" to "normal form". I see that this reflects my preference to think of RTT structures as matrices rather than lists of vectors or covectors. Because we are using linear algebra extensively here, I think this is the natural and appropriate way to think of them. What do you think?

Re: the new "Tenney minimal" section. I think it's an interesting idea to present a comma list in a different normal form than HNF (or defactored + HNF = canonical form), namely, some definition of the simplest possible ratios. However, I have several questions.

1. You state that this is already the case that temperament pages use this form. I have no reason to believe they're not. But I didn't know that was the case. How do you know this?
2. Do you have a definition for this normal form somewhere? If you don't yet, I would recommend excluding it from this page until it's better developed. I can easily see how the product complexity of ratios can be easily calculated individually, but minimizing the simplicity of multiple ratios may be somewhat subtle.
3. Why name it "Tenney minimal"? I do not see that this term has wide use on the wiki or Discord already. It seems like an unnecessary eponym. If it is related to the Tenney height of the ratios, wouldn't it be equivalent and simpler to just refer to their product complexity?

That's all for now. Thanks for helping with this. --Cmloegcmluin (talk) 18:54, 29 September 2021 (UTC)