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)