developing a notational comma popularity metric

[Dave Keenan]

Hm, should we add columns for sopfr and estimated rank by sopfr?

No. I think that would just muddy the water.

Would you like to be the one share the page on the Facebook group? I think your name has a bit more sway there. Of course please feel free to tag me in it.

Can do. But it's nice to have someone other than me or George post things that are Sagittal-related.

I noticed that 65/1 is the first 5–rough ratio with an estimated rank which is quite different than its real rank (22 vs 50). This struck me as interesting because 65/1 is really close to 2/1, so perhaps in practice it’s not a particularly popular ratio despite being relatively simple by the numeric calculations within N2D3P9’s formula. In other words, N2D3P9 does not account for harmonic concepts like this. We only briefly discussed dealing with such things. Just wondering if your feelings have changed. I assume not. But perhaps we should mention this somewhere on the Xen Wiki article in the justification section.

Yes. It would be good to point out that our approach was not able to consider all possible psychoacoustic reasons for a ratio's popularity. For example it was not able to consider whether some member of a 5-rough equivalence class might be very close in pitch to some member of another equivalence class, such as 65/64 being very close to 1/1.

But I note that 31 is close to 32 (1/1) and 41 is close to 40 (5/1) etc and yet the difference between N2D3P9 and the Scala archive goes in the other direction for them, compared to what it does in the case of 65.

[Dave Keenan]

Perhaps "checked Sagittal against it, to see how well they'd been served by sopfr." would be better as:

"checked the ratios for the existing Sagittal symbols against it, to see how well they'd been served by the Scala archive stats and the earlier sopfr metric."

[Dave Keenan]

How's this? The "N2D3P9 predicted occurrences" are 9021/N2D3P9^1.37. That sums to the same number of total occurrences as the Scala archive occurrences, namely 29 403.

[Dave Keenan]

And this?



I've added both graphs to the article.
https://en.xen.wiki/w/N2D3P9#Justification

[cmloegcmluin]

I'd prefer the heading "Scala archive rank" rather than "actual rank". Who's to say what's "actual" or "real" here? And I'd prefer "N2D3P9 rank" rather than 'estimated rank", since there are other ways to estimate the Scala arcive rank (like sopfr), and who knows if estimating the Scala archive rank is even the right thing to do, in the case of less popular ratios.

Yup, totally agree. Done.

While we're at it, how do you feel about "5-Rough Ratio Equivalence Class" for the column title (and section heading)? Or just "5-rough equivalence class"? Anecdotally, I think that if we had mixed the term "equivalence class" in to our discussions more earlier, I might have been less confused and/or not suggested taking the N2D3P9 of commas like I did back here (and even as recently as my first draft of the Wiki post here). As you said (here and here), each comma only notates one 5-rough ratio equivalence class, but a given 5-rough ratio equivalence class ("FREC"?) may be notated by several commas. It turns out to be at least up to four commas; I found that 35/1 is the single ratio in the Scala archives for which more three Sagittal symbol primary commas notate it (1/35C, 35S, 1/35M, 35L). There could be others. The latter pair are apotome complements, or in other words mirror images across the half apotome mirror (which is also the M|L size category bound); similarly, the former pair are Pythagorean large diesis complements, or in other words mirror images across the half Pythagorean large diesis mirror (which is also the C|S size category bound). Well actually now that I look into it, the middle two are complements too: limma complements, mirror images about the half limma mirror. Hm. There's probably a cool visualization or even interactive widget in this. Pretty much you drop one comma down and you get a hall of mirrors effect, or maybe a kaleidoscope is a better metaphor, but in any case a bunch of other commas follow, as ricocheting reflections.

Can you tell it's my first Monday without a job? Having a bit of a honeymoon period with this I'm not actually going to build this widget. Well, at least certainly not until muuuch later.

So I'm pretty sure now that this is not the same effect as how the existing spreadsheet calculator returns 2-3 results. I had to go back to the calculator and fool around with it a bit. Almost immediately I was reminded that the calculator doesn't always return exact results. You give it 13/1 and it'll give you a for the 35L. So that 2-3 result count really has little to do with the exact value of the ratio you're notating and more to do with its cents value and where it falls within the octave and where within that octave the double-apotome ranges emanating from each Pythagorean nominal anchor and how many of them are overlapping in that particular swath.

Now that I see the table, with its 604.5 and 329 Scala archive ranks, I think it would be good to include another column to the right which is "Scala archive occurrences". So people can see they shouldn't take too much notice of such ranks, given that they correspond to 1 or 2 occurrences.

Great idea. Done.

I'm having thoughts about a graph where I plot Scala archive occurrences against N2D3P9 notional occurrences or impled occurrences or something, but as well as not being sure what to call it, I don't know whether it should be proportional to 1/N2D3P9^1.37 or 1/N2D3P9_rank^1.37. Any thortz?

Whoa, interesting idea. I can't figure out the motivation. Maybe I would get it if I saw it. At least, I don't get the purpose well enough to help suggest a graph name.

I do, though, feel confident enough in suggesting proportioning to 1/N2D3P9^1.37, not 1/N2D3P9_rank^1.37. Since we used Spearman's rank correlation as our selection strategy, the ranks are our real results, and the actual N2D3P9 values are somewhat secondary, no? If you were to proportion to the actual values, I feel it may ever so slightly undercut the message we've got in the text of the article already about correlating by ranks being the proper degree of correlation given the problem.

Of course I wrote the above before you dropped a couple new exciting posts. I see that you did indeed go with 1/N2D3P9^1.37. Good stuff. Both of your charts are just dandy. Really gives the article the pizzazz it deserved!

I suppose we could also say "introducing subset" versus "introducing precision level". Thanks for consistentising that terminology. Sorry I'm still learning it.

No worries. I think "introducing" is a term I started using in my own materials for my own sense of clarity. Let me know if you like it or have a better suggestion. In many situations it can just be dropped and assumed I think.

I think you mean  , not  .

Ha! Actually, I meant and I know why I made the mistake: because in my codebase I always have to escape backslashes so I'm used to typing it with two of them.

Also added. Last paragraph of the development / discovery section if you want to review it.

The paragraph in question is:
"After deciding upon N2D3P9, the Sagittal forum members checked Sagittal against it, to see how well they'd been served by sopfr. Each symbol in Sagittal's JI notations has a default value, or primary comma, which allows it to exactly notate ratios in a 5-rough ratio equivalence class, and based on N2D3P9, it was found that only a couple of these commas should be changed (these were among the rarest-used symbols in Sagittal). This was as expected; N2D3P9 was developed primarily in order to add new symbols to Sagittal, to enable it to exactly notate even rarer JI pitches than it already does."

The phrase "the Sagittal forum members" strikes me as very odd here, and I'd prefer it was replaced with "we" or "the authors" or "Blumeyer and Keenan".. This also makes me realise we have a mixture of active and passive voice in this article. I think we should choose one and be consistent about it. I am conflicted between the fact that it is a wiki post so in theory anyone can edit, in which case "we" or "the authors" may not be the people who did the research described, and the fact that it is intended to educate, in which case the use of "we" results in a much more engaging story.

Oof, you're right.

It looks like — generally speaking — the stuff I wrote is passive while you opted for active voice.

I expect you may find similar inconsistency in the voice used on my metallic MOS work which I've asked you to scrutinize for me; it's shared on the wiki, but definitely leans more toward a journal article than an encyclopedia entry.

Unlike Wikipedia, the Xen Wiki has a pretty inconsistent and often extremely informal tone; it doesn't struggle as much with misinformation as it does with just really messy and impenetrable information. It seems so unlikely to me that anyone would contribute to this page besides us that I think we should just embrace the storytelling "we".

Though I will point out that many instances of "we" in the article could be interpreted as the type of we acceptable in an encyclopedic setting: the we meaning "me the author, you the reader, and everyone else", often found when working through mathematical or scientific processes.

Yes. I think that while the word "we" occurs throughout the article, it's really only in the Justification section where it's clearly being used in the sense of "we the authors". It looks like it might be a lot of effort to rework the Development/Discovery section. So what I did for now was actually keep the "the Sagittal forum members" phrase, and doubled down on the passive voice in that section by removing the first sentence "How did we come up with that particular 5-rough notational-popularity ranking function?" which, if one's taking the drier and more academic angle rather than the fun & juicy storytelling angle, is basically redundant with the section heading. And then I changed the title of the Justification section to "Authors' Justification" to distinguish it and sort of give it a mandate to use the different voice.

It's nice to have someone other than me or George post things that are Sagittal-related.

Ah, that's a good point. Okay, how about something like this:

• Over the past couple months, @Dave Keenan and I — along with other members of the Sagittal forum — have developed a function called "N2D3P9" which can estimate the popularity of a pitch ratio. More specifically, it gives an estimated rank among 5-rough ratio equivalence classes such as 5/1, 7/1, or 55/13.
  
  For example, N2D3P9(5/1) = 1.39, because 5/1 is the simplest possible 5-rough ratio equivalence class (besides N2D3P9(1/1) which is defined as 1). For another example, N2D3P9(77/5) = 39.21, suggesting there are approximately 38 other 5-rough pitch ratios more popular than it.
  
  We fit our function to data from the Scala scale archives. The goal was to help choose the best commas to assign to symbols being introduced in a new and higher precision level of Sagittal's JI notation (coming soon!). You can read more on the Xenharmonic Wiki here: https://en.xen.wiki/w/N2D3P9

It would be good to point out that our approach was not able to consider all possible psychoacoustic reasons for a ratio's popularity. For example it was not able to consider whether some member of a 5-rough equivalence class might be very close in pitch to some member of another equivalence class, such as 65/64 being very close to 1/1.

Added (close to verbatim).

But I note that 31 is close to 32 (1/1) and 41 is close to 40 (5/1) etc and yet the difference between N2D3P9 and the Scala archive goes in the other direction for them, compared to what it does in the case of 65.

Oh, good point. I meant to check that but I forgot to before I had to hop in the car.

Perhaps 31 is popular because La Monte Young liked it so much. Whether I'm remembering that specific fact correctly or not, it's certainly the case that historical forces like this may have had observable effects on the scale archives. So even if we added another layer of complexity to the metric to account for such psychoacoustic phenomena as harmonic entropy or whatnot, we still couldn't account for the whole picture of ratio popularity. But perhaps this is starting to get into the territory of "probably no one would expect a notation system designer to fret about such subtleties, so it doesn't even bear mentioning."

Perhaps "checked Sagittal against it, to see how well they'd been served by sopfr." would be better as:

"checked the ratios for the existing Sagittal symbols against it, to see how well they'd been served by the Scala archive stats and the earlier sopfr metric."

I like it. Done.

[Dave Keenan]

While we're at it, how do you feel about "5-Rough Ratio Equivalence Class" for the column title (and section heading)?

Fine. And feel free to throw the word "Pitch" in there too, otherwise people who don't read the article may think it applies to intervals. i.e. "5-Rough Pitch Ratio Equivalence Class". And if that's too many words, you could drop "Equivalence".

Or just "5-rough equivalence class"? Anecdotally, I think that if we had mixed the term "equivalence class" in to our discussions more earlier, I might have been less confused and/or not suggested taking the N2D3P9 of commas like I did back here (and even as recently as my first draft of the Wiki post here).

Yes. Sorry I didn't think of it sooner. I guess I had to be focussing on the pedagogy before it occurred to me.

As you said (here and here), each comma only notates one 5-rough ratio equivalence class, but a given 5-rough ratio equivalence class ("FREC"?) may be notated by several commas. It turns out to be at least up to four commas; I found that 35/1 is the single ratio in the Scala archives for which more three Sagittal symbol primary commas notate it (1/35C, 35S, 1/35M, 35L).

OK. But there are only 3 independent comma definitions or comma choices there. i.e. those less than the half-apotome. We always have apotome complements.

There could be others. The latter pair are apotome complements, or in other words mirror images across the half apotome mirror (which is also the M|L size category bound); similarly, the former pair are Pythagorean large diesis complements, or in other words mirror images across the half Pythagorean large diesis mirror (which is also the C|S size category bound). Well actually now that I look into it, the middle two are complements too: limma complements, mirror images about the half limma mirror. Hm. There's probably a cool visualization or even interactive widget in this. Pretty much you drop one comma down and you get a hall of mirrors effect, or maybe a kaleidoscope is a better metaphor, but in any case a bunch of other commas follow, as ricocheting reflections.

A cool visualisation indeed.

Can you tell it's my first Monday without a job? Having a bit of a honeymoon period with this I'm not actually going to build this widget. Well, at least certainly not until muuuch later.

Congrats. No need to build it. The mental image was enough.

So I'm pretty sure now that this is not the same effect as how the existing spreadsheet calculator returns 2-3 results. I had to go back to the calculator and fool around with it a bit. Almost immediately I was reminded that the calculator doesn't always return exact results. You give it 13/1 and it'll give you a for the 35L. So that 2-3 result count really has little to do with the exact value of the ratio you're notating and more to do with its cents value and where it falls within the octave and where within that octave the double-apotome ranges emanating from each Pythagorean nominal anchor and how many of them are overlapping in that particular swath.

Good point. Thanks for explaining.

Oof, you're right [about there being inconsistent use of active and passive voice].

It looks like — generally speaking — the stuff I wrote is passive while you opted for active voice.

Right. Although you're also sometimes writing 3rd-person active, "<named person or group> did X", where I'm writing 1st-person active, "we (the authors) did X". But other times you're writing passive, "X was done". I think the 3rd-person active is a good compromise. I see it in other articles in the Wiki, like the Kite guitar article.

Unlike Wikipedia, the Xen Wiki has a pretty inconsistent and often extremely informal tone; it doesn't struggle as much with misinformation as it does with just really messy and impenetrable information. It seems so unlikely to me that anyone would contribute to this page besides us that I think we should just embrace the storytelling "we".

Though I will point out that many instances of "we" in the article could be interpreted as the type of we acceptable in an encyclopedic setting: the we meaning "me the author, you the reader, and everyone else", often found when working through mathematical or scientific processes.

I'd like to rewrite my parts to be 3rd person active (and maybe sometimes passive), and to only use "we" when it means "you and I, dear reader" and sometimes "everyone". Is that OK with you.

Yes. I think that while the word "we" occurs throughout the article, it's really only in the Justification section where it's clearly being used in the sense of "we the authors". It looks like it might be a lot of effort to rework the Development/Discovery section. So what I did for now was actually keep the "the Sagittal forum members" phrase, and doubled down on the passive voice in that section by removing the first sentence "How did we come up with that particular 5-rough notational-popularity ranking function?" which, if one's taking the drier and more academic angle rather than the fun & juicy storytelling angle, is basically redundant with the section heading. And then I changed the title of the Justification section to "Authors' Justification" to distinguish it and sort of give it a mandate to use the different voice.

I'd prefer to reword it, rather than have that strange title. Is that OK with you?

Ah, that's a good point. Okay, how about something like this:

I give a suggested edit below:

• Over the past couple months, @Dave Keenan and I — along with other members of the Sagittal forum — have developed a function we call "N2D3P9" which can estimate the popularity of a pitch ratio. More specifically, it gives an estimated rank among 5-rough ratio equivalence classes such as 5/1, 7/1, or 55/13.
  
  [deleted sentence] For example, N2D3P9(77/5) = 39.21, suggesting there are approximately 38 other 5-rough pitch ratios more popular than it.
  
  We fit our function to data from the Scala scale archives. The goal was to help choose the best commas to assign to symbols being introduced in a new and higher precision level of Sagittal's JI notation [deleted parenthetic comment]. You can read more on the Xenharmonic Wiki here: https://en.xen.wiki/w/N2D3P9

I had suggested "We fitted our function to data ...", but then I saw that "fit" is the standard American past tense of "fit" — completely indistinguishable from its present tense.

Perhaps 31 is popular because La Monte Young liked it so much. Whether I'm remembering that specific fact correctly or not, it's certainly the case that historical forces like this may have had observable effects on the scale archives. So even if we added another layer of complexity to the metric to account for such psychoacoustic phenomena as harmonic entropy or whatnot, we still couldn't account for the whole picture of ratio popularity. But perhaps this is starting to get into the territory of "probably no one would expect a notation system designer to fret about such subtleties, so it doesn't even bear mentioning."

Definitely in that territory.

[cmloegcmluin]

While we're at it, how do you feel about "5-Rough Ratio Equivalence Class" for the column title (and section heading)?

Fine. And feel free to throw the word "Pitch" in there too, otherwise people whop don't read the article may think it applies to intervals. i.e. "5-Rough Pitch Ratio Equivalence Class". And if that's too many words, you could drop "Equivalence".

Good point.

I think I got timid about this vicinity, because you'd asked me earlier to drop the "ratio" in the context of "5-rough (ratio) notational popularity rank" versus "comma notational popularity rank". But this is a different context.

I agree. We say "equivalence class" enough in the text of the article that I think by the time you make it to the table, "class" suffices.

I've changed it in both tables (on the forum, and on the wiki).

I guess that'd make it a FRPRC then, instead of a FREC. Well, maybe we could keep the E for the acronym... it's quite helpful in that situation. Not that anyone asked for an acronym for this. Maybe 5-PREC works okay, where the word "rough" doesn't get a letter, because the "pitch ratio equivalence class" bit sort of carries that weight for it.

The term "pitch class" is well known in music theory, where it typically means to contain pitches with the same name, e.g. A  , i.e. octave equivalent pitches. So these would be 3-PRECs. Which is kind of weird; when people think about octave equivalence they are thinking about the 2's, not the first prime after 2 which makes differences.

So now I'm doubting whether roughness is the right direction to come at this from, or whether we should have stuck with 2,3-reduced as I think you may have preferred from the beginning (and we included it in the xen wiki article as an alternative).

Although I do not find "k-reduced" in the math literature.

And it is not the same meaning of "reduced" as we proposed here. By that definition, 5/1 would not be 3-reduced or 2-reduced. red₃(5/1) = 5/3 and red₂ = 5/4. A 2,3-reduced number I suppose would depend on which reduction was applied first.

If there were to be a term for it, I'd like it to not require listing all the numbers (i.e. it would imply all the number on the other side of the one number) and I'd like it to play well in a set with the concept of k-smooth numbers too. I'll give a really basic proposition which I do not love: 3-roughened ≡ 5-rough, and 3-smoothened ≡ 2-smooth. I really don't like this though, because it seems sooo easy to confuse "roughened" with "rough".

It's almost like we'd want to change the number part of the term rather than the word part of the term, like ">3-rough" or something.

There are synonyms for rough and smooth: jagged and friable, respectively, but I don't think that helps here.

And so I guess at this point I propose we just remove the 2,3-reduced bit. But this problem only just occurred to me and perhaps you have some other thoughts. I do think it's helpful there to reinforce, one way or another, the idea that all the 2's and 3's have been divided out. And in the preceding sentence where we say "when their factors of 2 and 3 are removed, they all reduce to 1/5 or 5/1" the word 'reduced' seems to work absolutely fine. Let me know what you think.

...I think that if we had mixed the term "equivalence class" in to our discussions more earlier, I might have been less confused...

Yes. Sorry I didn't think of it sooner. I guess I had to be focussing on the pedagogy before it occurred to me.

No big deal. Only meant the anecdote to underline the effectiveness of the term now that you've found it for us.

As you said (here and here), each comma only notates one 5-rough ratio equivalence class, but a given 5-rough ratio equivalence class ("FREC"?) may be notated by several commas. It turns out to be at least up to four commas; I found that 35/1 is the single ratio in the Scala archives for which more three Sagittal symbol primary commas notate it (1/35C, 35S, 1/35M, 35L).

OK. But there are only 3 independent comma definitions or comma choices there. i.e. those less than the half-apotome. We always have apotome complements.

Mm, indeed. So in the sense of comma assignment to Sagittal symbols, the half-apotome is the highest position necessary to consider, because every comma assignment between it and the maximum position AKA the L|SS size category bound — or in other words, every L — is dependent on the commas already assigned between the half-apotome and the S|M size category bound — or in other words, every M.

Not that it particularly matters, but I do still think it could be possible for a 5-PREC 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. 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.

...you're also sometimes writing 3rd-person active, "<named person or group> did X", where I'm writing 1st-person active, "we (the authors) did X". But other times you're writing passive, "X was done". I think the 3rd-person active is a good compromise.

I admit that I recognized the difference between voice (active vs passive) and person (1st vs 3rd) when I replied yesterday, but I was simply too lazy to disentangle the two issues. Sorry for imposing that on you.

I agree with your assessment of the current state of the article, and also that 3-person active should be our goal to standardize around.

I'd like to rewrite my parts to be 3rd person active (and maybe sometimes passive), and to only use "we" when it means "you and I, dear reader" and sometimes "everyone". Is that OK with you.

... I changed the title of the Justification section to "Authors' Justification" to distinguish it and sort of give it a mandate to use the different voice.

I'd prefer to reword it, rather than have that strange title. Is that OK with you?

Totes magotes!

I give a suggested edit below:

I like all your edits to the introductory message I'll post on Facebook when we're ready. No need to explain them; I get where they're coming from. One more thing though — I'll change 5-rough ratio to 5-rough pitch ratio.

[Dave Keenan]

I think I got timid about this vicinity, because you'd asked me earlier to drop the "ratio" in the context of "5-rough (ratio) notational popularity rank" versus "comma notational popularity rank". But this is a different context.

Yes. That was specifically about abbrevations, for our convenience in discussions, namely: "comma-no-pop-rank" vs "5-rough-no-pop-rank".

I agree with your assessment of the current state of the article, and also that 3-person active should be our goal to standardize around.

OK. That's done. Partly 3rd person active, partly passive. The main goal is to remove all 1st person writing except where it means "the authors of this article", who are not necessarily the developers/discoverers of N2D3P9. For example "We're only joking, but we hope this helps with remembering and pronouncing the name." and "we describe two simpler functions that are used in calculating it.".

Another fine detail in the article. I notice you changed the heading "Development/Discovery" to "Development and Discovery".

My intention with the slash was kind of "You might say N2D3D9 was Developed or you might say it was Discovered". As described in this article:
https://www.huffpost.com/entry/is-mathe ... _b_3895622

[cmloegcmluin]

I agree with your assessment of the current state of the article, and also that 3-person active should be our goal to standardize around.

OK. That's done. Partly 3rd person active, partly passive. The main goal is to remove all 1st person writing except where it means "the authors of this article", who are not necessarily the developers/discoverers of N2D3P9. For example "We're only joking, but we hope this helps with remembering and pronouncing the name." and "we describe two simpler functions that are used in calculating it.".

Excellent point. The kind of "we" which is troublesome is not necessarily the kind that excludes the reader/everyone — as I'd characterized it — but is rather the kind which excludes the authors of the article (namely, the kind which refers to the authors of N2D3P9).

I've read the changes to the Justification section and approve.

Another fine detail in the article. I notice you changed the heading "Development/Discovery" to "Development and Discovery".

My intention with the slash was kind of "You might say N2D3D9 was Developed or you might say it was Discovered". As described in this article:
https://www.huffpost.com/entry/is-mathe ... _b_3895622

Blast! I thought I could sneak that one by you!

Ha, just kidding. I didn't do it deviously, or even mischievously. Perhaps obliviously, though.

I appreciate the development/discovery ambiguity. I've always loved this Michaelangelo quote:

Every block of stone has a statue inside it and it is the task of the sculptor to discover it.

I think I lean more non-Platonist myself, but for a different reason than the author of your linked article gives for his own non-Platonism. My attitude is that we devalue the utility of the word "discovery" the more things we qualify to be discoverable. Something like quaternions? Sure, Hamilton totally discovered them that day walking around town when he carved them into the bridge or however that story goes. But N2D3P9? I feel like yeah, it's pretty damn good at what it does, but I think it is sufficiently removed from being some previously obscure but now plain-to-see fundamental truth that I hesitate to consider it a discovery. There are just too many other things it could have been, unlike, for another example, the golden section, which was unknown to all humans for some time, then eventually understood and widely established, so in that sense it was totally discovered, and there's just no substitute for it at all. What makes us wonder if "development" might be the better word involves questioning some pretty deep foundations of our knowledge. So I certainly think this development/discovery ambiguity is strong in some math, but not all of it. But I'm definitely dwelling less on mathematical issues here and more on semantic ones.

Come to think of it, I may have subconsciously realized that this was what you were getting at with the slashed title, and, feeling self-conscious about positing N2D3P9 as a discovery, backtracked it a bit, but keeping "discovery" there, because there's certainly no doubt that we discovered N2D3P9 within a certain scope, whether within the 2 billion functions I collided, or you discovering the beautiful metamorphosis from wbl₁ into N2D3P9 which makes it suddenly incredibly intelligible.

Okay, well that was a bunch of words, but it amounts to this: I like it as "Development/Discovery" now that you've prompted me to think it through consciously. Thanks for pushing back.

[Dave Keenan]

The term "pitch class" is well known in music theory, where it typically means to contain pitches with the same name, e.g. A  , i.e. octave equivalent pitches. So these would be 3-PRECs. Which is kind of weird; when people think about octave equivalence they are thinking about the 2's, not the first prime after 2 which makes differences.

Part of the weirdness is due to you omitting the word "rough" and its initial letter, from your just-invented acronym. So, forget the acronym.

And "pitch class" is not equivalent to "3-rough pitch ratio equivalence class" because pitches in general may be irrational, whereas "3-rough pitch ratios" are strictly rational. But yes, "rational pitch class" is equivalent to "3-rough pitch ratio equivalence class". And yes, I agree it is strange to name it after the lowest of the large number of remaining primes, rather than the single prime that is ignored to produce the equivalence class.

This is made starkly obvious in the Bohlen Peirce world, where we could talk of "tritave-equivalent pitch classes". We don't even have the option of using the P-rough terminology there, unless we invent something like 2,5-rough, which would seem bizarre when we can just call them 3-reduced.

So now I'm doubting whether roughness is the right direction to come at this from, or whether we should have stuck with 2,3-reduced as I think you may have preferred from the beginning (and we included it in the xen wiki article as an alternative).

I agree. Thanks for calling that out. I think I was seduced by how much shorter "5-rough" is than "2,3-reduced".

Although I do not find "k-reduced" in the math literature.

No. But certainly we find "octave-reduced" in the music-theory literature and it seems obvious to equate that to "2-reduced" and to generalise it, to equate "tritave-reduced" to "3-reduced" for BP.

And it is not the same meaning of "reduced" as we proposed here. By that definition, 5/1 would not be 3-reduced or 2-reduced. red₃(5/1) = 5/3 and red₂ = 5/4. A 2,3-reduced number I suppose would depend on which reduction was applied first.

That's an excellent point. I have been quite inconsistent in not treating "2,3-reduced" as equivalent to either red₂(red₃()) or red₃(red₂()). But I note that any rational octave-reduced equivalence class is identical to a 3-rough equivalence class. The only difference is in the single ratio that all its members are reduced to, and hence the single ratio that is used to represent the class. e.g. 5/4 versus 5/1.

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.

If there were to be a term for it, I'd like it to not require listing all the numbers (i.e. it would imply all the number on the other side of the one number) and I'd like it to play well in a set with the concept of k-smooth numbers too. I'll give a really basic proposition which I do not love: 3-roughened ≡ 5-rough, and 3-smoothened ≡ 2-smooth. I really don't like this though, because it seems sooo easy to confuse "roughened" with "rough".

It's almost like we'd want to change the number part of the term rather than the word part of the term, like ">3-rough" or something.

There are synonyms for rough and smooth: jagged and friable, respectively, but I don't think that helps here.

And so I guess at this point I propose we just remove the 2,3-reduced bit. But this problem only just occurred to me and perhaps you have some other thoughts. I do think it's helpful there to reinforce, one way or another, the idea that all the 2's and 3's have been divided out. And in the preceding sentence where we say "when their factors of 2 and 3 are removed, they all reduce to 1/5 or 5/1" the word 'reduced' seems to work absolutely fine. Let me know what you think.

In this latter section, I suspect you've departed from the question of how best to explain N2D3P9, and have switched over to how best to have a consistent system of names for functions that can do everything we might ever want to do.

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.