## developing a notational comma popularity metric

Dave Keenan
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### Re: developing a notational comma popularity metric

A thort:
"notationally-reduced" = "2,3-reduced" = "5-roughened"
"notational pitch ratio equivalence class" = "5-rough pitch ratio equivalence class"

Dave Keenan
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### Re: developing a notational comma popularity metric

cmloegcmluin wrote: Wed Aug 26, 2020 4:39 am Not that it particularly matters, but I do still think it could be possible for a 5-PREC [2,3-reduced pitch ratio equivalence class] to be notated by more than 3 symbols. It happens to be the case that Sagittal does not. And perhaps there may be some desire to avoid that level of redundancy, to free symbols up to notate other 5-PRECs which maybe haven't been exactly notated yet.
Good point.
But you could take that 1/35C and mirror it across the k|C size category bound to get its Pythagorean comma complement. But actually that won't be a k, it'll be an s (and even barely an s... almost an n!) ... but that makes sense because the size category bounds get denser as you approach unison (also, I dunno if I'd call it a Pythagorean comma complement proper, since the 1/35C is actually greater than the Pythagorean comma, at 25.3¢ to its 21.5¢, so this 1/35s is actually negative when it combines with the 1/35C to make the 1C... and hence why they are both 1/ commas, rather than one being 1/ and the other being /1 as we usually see with true complement pairs). Yeah, now that I draw this out, I'm realizing that the half Pythagorean comma, about 11.7¢, occurs a lot: it's the k|C bound, and also the delta between the C|S, S|M, M|L, and L|SS bounds. There's this weird little stretch between 21.7¢ and 23.5¢ where mirror-zones overlap (i.e. 11.7¢ greater than 11.7¢ and 11.7¢ less than 33.4¢). I'm quite sure what to do with all this; I should probably cut myself off and get back to N2D3P9-related work. But it is pretty interesting. I'm pretty sure if you found a comma in that little zone, it'd be a C, and would have a complement which was an n, and on the other end a complement that was an S, which in turn would have a complement which was an M, and so there you'd have 4 independent commas for the same 5-PREC.
Agreed. There are an infinite number of commas smaller than a half-apotome for any given 2,3-reduced ratio. But I seem to remember some result you proved about the number of such commas having an absolute 3-exponent in a certain range.

Dave Keenan
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### Re: developing a notational comma popularity metric

Here's a possible way of consistentising my use of "2,3-reduced" = "5-roughened" with your function
redb(x) = red(x, b) = b(frac(logb(x))) = b(logb(x) mod 1).

If b is a single positive real number and x is a real number it has the above definition, but if b is a set of primes and x is a rational number it is instead defined as:

redS(r) = red(r, S) = product-over-primes-p of p(rp×[p∉S])
where r = 2r2 × 3r3 × 5r5 × ...

So my "2,3-reduced" should be read as red{2,3}().
"Octave-reduced" or "2.0-reduced" should be read as red2().
But "2-reduced" should be read as red{2}().
"phitave-reduced" or "phi-reduced" can only be read as redφ().

Dave Keenan
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### Re: developing a notational comma popularity metric

A nuther thort:
Maybe "2,3-reduced pitch ratio equivalence class" can be shortened to "2,3-equivalent pitch ratio class".

Dave Keenan
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### Re: developing a notational comma popularity metric

I've done it. There's now only one occurrence of "5-rough" in the article. It's given as an alternative mathy term for "2,3-reduced", in the same way that "big omega" and "gpf" are given as alternative mathy terms for copfr and prime-limit.

I changed a few other things. Please review all my recent changes here:
https://en.xen.wiki/index.php?title=N2D ... ldid=48304

cmloegmcluin
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### Re: developing a notational comma popularity metric

Dave Keenan wrote: Wed Aug 26, 2020 12:42 pm But I don't think it would be helpful pedagogically, to start using 5/4 or 5/3 as the representative in our N2D3P9 article. I think it's clearer if it stays as 5/1.
Agreed.
I'd prefer to weasel and say that since no one knows what "2-reduced" and "3-reduced" are supposed to mean (as opposed to "octave-reduced" and "tritave-reduced"), they probably have no expectation of what "2,3-reduced" should mean, so I'm free to define it as equivalent to "5-rough" for pedagogical purposes in this context, without it having to fit into any complete consistent system.
I agree that 2,3-reduced is acceptable, with no further justification necessary, granted that realistically no one would ever grumble about any potential inconsistency here.
Dave Keenan wrote: Wed Aug 26, 2020 2:10 pm Here's a possible way of consistentising my use of "2,3-reduced" = "5-roughened" with your function
redb(x) = red(x, b) = b(frac(logb(x))) = b(logb(x) mod 1).

If b is a single positive real number and x is a real number it has the above definition, but if b is a set of primes and x is a rational number it is instead defined as:

redS(r) = red(r, S) = product-over-primes-p of p(rp×[p∉S])
where r = 2r2 × 3r3 × 5r5 × ...

So my "2,3-reduced" should be read as red{2,3}().
"Octave-reduced" or "2.0-reduced" should be read as red2().
But "2-reduced" should be read as red{2}().
"phitave-reduced" or "phi-reduced" can only be read as redφ().
Ha! When I first read b(logb(x) mod 1) form I thought "oh, that's clever... he must have suggested that back on that early Facebook thread" so I went to my records to check... and found that that's the way I myself described the operation in the original post! Who's clever now?!

Okay, seriously though. Hm. I dunno about this. They're such different operations sometimes, aren't they? Consider red2(3/5) = 6/5. It actually introduces a factor of 2 into a number where before there was none.

This gives me a thort, though: how about "2,3-removed"? So we could write rem2,3(81/80) = 1/5. They're such different operations sometimes, sure... but often they're quite similar, so I like that their three-letter math operation abbreviation shares the first two of three letters to give them a bit of kinship.
Dave Keenan wrote: Wed Aug 26, 2020 1:08 pm A thort:
"notationally-reduced" = "2,3-reduced" = "5-roughened"
"notational pitch ratio equivalence class" = "5-rough pitch ratio equivalence class"
I like "notationally reduced". Really nice.

Let's not use the "-ened" ending for roughened or smoothened, though, if we just mean rough or smooth. See, I knew as soon as I wrote it that it was too close to be any good!

So where before we had a 5-rough notational popularity rank vs. a comma notational popularity rank, we'd now have a ... well, simply a notational (pitch ratio equivalence class) popularity rank (which is N2D3P9) and a ... well ...

So I know the final layer is a "badness" metric, which will incorporate (tina) error and be used to choose commas for new symbols. And I know the intermediate layer is what we've been calling a comma-no-pop-rank, which you've said we could use for revisiting existing comma assignments. Is it really even a "popularity rank"? It will heavily leverage a popularity rank, but by including apotome slope (and possibly abs3exp) doesn't it sufficiently depart from that definition? It's not a badness rank, but perhaps it's a notational "usefulness rank" or "utility rank", since the apotome slope is there to help tell us how useful the comma is for notating EDOs as the size of the fifth varies?

Back here you said "It doesn't make sense (to me) to talk of a notational pitch ratio or notational 5-rough ratio. They aren't used for notation, they need to be notated. They are notated ratios, not notational ratios." But I suppose that pitch ratio equivalence classes are different than pitch ratios, critically in the sense that it does feel natural to speak of them as "notational".

So we might end up with notational on the left side of the word in both cases: a notational pitch-ratio-class popularity rank, and a notational comma utility rank.
Dave Keenan wrote: Wed Aug 26, 2020 2:52 pm A nuther thort:
Maybe "2,3-reduced pitch ratio equivalence class" can be shortened to "2,3-equivalent pitch ratio class".
I like the idea of rendering the "equivalence" in "equivalence class" unnecessary by the choice of word hyphenated with the 2,3. And I agree that "equivalent" itself works. But now that I've thort of "2,3-removed", I have a slight preference for it. I like that "removed" lends itself well to a math operation name; it's simple as a verb "remove" and as far as I can tell doesn't have other conflicting meanings. On the other hand, you'd have to say a "2,3-equivocated ratio", which I get what it means but I think it could be better.

So combining our two thorts we could get "2,3-removed pitch ratio class".

With that, we could relegate "5-rough" to the canon of established mathematical terms which appear in the N2D3P9 Xen Wiki article but which get secondary treatment. In other words, we mention 5-rough once, but use 2,3-removed as the go-to term throughout the article. Similarly, we should only ever need to use the word "equivalence" once (I'm not going to enumerate every re-phrasing in every context throughout the article but I expect they'll come fairly naturally).
Dave Keenan wrote: Wed Aug 26, 2020 1:24 pm Agreed. There are an infinite number of commas smaller than a half-apotome for any given 2,3-reduced ratio. But I seem to remember some result you proved about the number of such commas having an absolute 3-exponent in a certain range.
Bedtime. I know what you're talking about. I'll have to find it tomorrow.
Dave Keenan wrote: Wed Aug 26, 2020 4:27 pm I've done it. There's now only one occurrence of "5-rough" in the article. It's given as an alternative mathy term for "2,3-reduced", in the same way that "big omega" and "gpf" are given as alternative mathy terms for copfr and prime-limit.

I changed a few other things. Please review all my recent changes here:
https://en.xen.wiki/index.php?title=N2D ... ldid=48304
Seems you picked up on some of the feelings I was having above before I got a chance to finish putting them out there! I'll get my review to you soon.

Dave Keenan
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### Re: developing a notational comma popularity metric

See https://www.mathworks.com/help/matlab/ref/rem.html
I suggest "rmv".

But I just realised there's an aspect of "5-rough" that is lost in going to "2,3-reduced" or "2,3-removed" and that is that "5-rough" is agnostic about whether the ratio started out with 2's and 3's and had them removed, or was simply generated without them from the start.

I figure the equivalent term is "2,3-free", which I have used previously, on occasion.

Thanks for reminding me of the ambiguity of "notational pitch ratio equivalence class". Forget that. And I've gone right off "notationally-reduced" (and "notationally-removed"). It's better to have the 2,3 right there in the term and not have to remember that's what "notational" means here.

I don't know why you think the use of "2,3-equivalent" requires we use "2,3-equivocated" instead of "2,3-removed" or "2,3-reduced". Of course there would no longer be such a thing as "2,3-reduced", but there would still be 2-reduced, as distinct from 2-removed. And I pointed out earlier (using other terminology) that it doesn't matter whether you 2-reduce or 2-remove your ratios, you end up with exactly the same equivalence classes. So either way it makes sense to call them "2-equivalent pitch ratio classes", or "2-equivalence-classes" for short, in an appropriate context.

I have no idea what it could mean to "2-equivocate" a ratio.

So I have replaced all occurrences of 2,3-reduced in the article with either 2,3-removed or 2,3-free as appropriate. Occurrences of "2,3-equivalent" or "2,3-equivalence-class" remain as they were.

If you haven't yet begun the review I requested in my previous post, you should now use this link:
https://en.xen.wiki/index.php?title=N2D ... ldid=48304
otherwise use this link for the most recent changes only:
https://en.xen.wiki/index.php?title=N2D ... ldid=48306

I'm really hoping we can announce this soon, so we can move on to other things -- you to your friend's VR project and me to reviewing your Metallic MOS article.

cmloegmcluin
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### Re: developing a notational comma popularity metric

Okay, I think I've found a link to the discussion about a proof of you were looking for: viewtopic.php?f=12&t=484&p=1723&hilit=proof#p1723

Perhaps we can use this -21 to 20 3-exp range built into the comma names Sagittal uses to guide us somehow in balancing apotome slope with N2D3P9.
Dave Keenan wrote: Wed Aug 26, 2020 5:22 pm I like your "2,3-removed" better than "2,3-reduced" for what we're talking about in this article, and I like your "red" vs "rem" distinction. However "rem" already means "remainder".
See https://www.mathworks.com/help/matlab/ref/rem.html
I suggest "rmv".
Glad you like it Dang, I looked to make sure "removed" wasn't used but I neglected to search for "rem". Thanks for checking that. Yes, I agree "rmv" is the right choice for 3-letter math function abbreviation.

I'll add a summary of our discussion here about red vs rmv to the original topic about red.
But I just realised there's an aspect of "5-rough" that is lost in going to "2,3-reduced" or "2,3-removed" and that is that "5-rough" is agnostic about whether the ratio started out with 2's and 3's and had them removed, or was simply generated without them from the start.

I figure the equivalent term is "2,3-free", which I have used previously, on occasion.
Good point. Subtle but I agree it renders 2,3-removed (and 2,3-reduced) subpar. Indeed you have used 2,3-free before... I think you're right, it's the winner.

We still have use for 2,3-removed when indeed 2's and 3's have been removed. And we still have use for 2,3-equivalent in the context of the pitch ratio classes, which also includes the equivalence of 5/1 with 1/5.
So I have replaced all occurrences of 2,3-reduced in the article with either 2,3-removed or 2,3-free as appropriate. Occurrences of "2,3-equivalent" or "2,3-equivalence-class" remain as they were.

If you haven't yet begun the review I requested in my previous post, you should now use this link:
https://en.xen.wiki/index.php?title=N2D ... ldid=48304
otherwise use this link for the most recent changes only:
https://en.xen.wiki/index.php?title=N2D ... ldid=48306
I approve of all your changes. Thanks for all that. That was an especially excellent insight to group the explanation of 2,3-free in with the explanation of copfr and prime-limit.
I'm really hoping we can announce this soon, so we can move on to other things -- you to your friend's VR project and me to reviewing your Metallic MOS article.
I suggest share it out at around 5pm my time on the 27th / 10am your time on the 28th, so we both have plenty of time to respond the first day if people have questions.

Dave Keenan
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### Re: developing a notational comma popularity metric

Why not 5pm your time on the 26th? I'll be unavailable, working with Warrick on the EV, tomorrow.

cmloegmcluin