developing a notational comma popularity metric

[cmloegcmluin]

Our metric implies a ranking that is independent of the Scala archive, which is a ranking of all possible 5-rough ratios, whether they occurred in the archive or not. The above formula is an approximation of that ranking, not the ranking of the ratios that just happen to be included in the Scala archive.

Okay, I grant that it's perfectly reasonable to entertain the idea of ranking all 5-rough ratios, not just the ones in our data set, despite clearly not finding that to be natural initially.

if we used the above as our metric, we would be justified in calling it "notational popularity rank", since fractional ranks are not unheard of. Or at least calling it "a" notational popularity rank, since I'm pretty sure I've referred to the old sopfr (SOPF>3) as "notational popularity rank" in the past.

And I do like the idea of being able to include "rank" in the name of the metric, as that short and common word does a lot of lifting in terms of its purpose. The name I gave to this forum topic is "notational comma popularity metric". If you swap out metric for rank then you get "notational comma popularity rank", which says it all. `text{ncpr}` might work then.

So should we go with the divided-by-9 version, to get more convincingly rank-like values?

[Dave Keenan]

if we used the above as our metric, we would be justified in calling it "notational popularity rank", since fractional ranks are not unheard of.

`text{ncpr}` might work then.

That's not bad. Except there's still the 3 exponent to deal with, in determining the ranking of notational commas. This is really just a ranking of the pitch ratios to be notated by those commas, with up to 3 commas competing to notate the same 5-rough ratio.

BTW, you were mistaken when you suggested that the reason you couldn't find some of those high-prime ratios in the archive was because they were only present at ratios between pitches. The database we are using does not count any interval ratios, only pitch ratios. The likely reason you couldn't find them is that they were only present as pitch ratios containing one or more factors of prime 3.

So should we go with the divided-by-9 version, to get more convincingly rank-like values?

That is what I said. The expression referred to by my "the above" was the divided-by-9 version:

n × d × gpf(n×d)
2^copfr(n) × 3^copfr(d)+2

Perhaps you missed the "+2" at the end of the last superscript.

[Dave Keenan]

Here's the other transformation I mentioned I was working on.

We start by writing it as
$$\frac{n}{2^{\text{copfr}(n)}}×\frac{d}{3^{\text{copfr}(d)}}×\frac{\text{prime-limit}(nd)}{9}\text{, where }n≥d$$
Then note that where

`n = 5^{n_5}×7^{n_7}×...`

we have

`2^{\text{copfr}(n)}=2^{n_5}×2^{n_7}×...`.

and so

`\frac{n}{2^{\text{copfr}(n)}}`

can be written as

`(\frac{5}{2})^{n_5}×(\frac{7}{2})^{n_7}×...`

and

`\frac{d}{3^{\text{copfr}(d)}}`

can be written as

`(\frac{5}{3})^{d_5}×(\frac{7}{3})^{d_7}×...`

These can be described as "product of half prime factors of the numerator (with repeats)" or pohpfr(n), and "product of one-third prime factors of the denominator (with repeats)" or potpfr(d). It may be simpler to describe it verbally as:

"Take the prime factorisation of the numerator and divide all the primes by 2, then multiply it out again. Do the same with the denominator but divide by 3 instead of 2. Multiply these two results together then multiply by the prime limit of the ratio and divide by 9",

but the quickest way to compute it is still by using the first formula in this post.

[cmloegcmluin]

if we used the above as our metric, we would be justified in calling it "notational popularity rank", since fractional ranks are not unheard of.

`text{ncpr}` might work then.

That's not bad. Except there's still the 3 exponent to deal with, in determining the ranking of notational commas. This is really just a ranking of the pitch ratios to be notated by those commas, with up to 3 commas competing to notate the same 5-rough ratio.

Okay, I understand most of that. For example, we've got a tina candidate 5832/5831. That's a comma. When we're ranking it against other candidates, we care about WBL-1/ncpr/whatever, but we also care about its abs3exp and apotome slope which involve the 3 exponent. What WBL-1 is doing by appraising the 5-roughened version of the comma 5832/5831 is appraising it indirectly by considering the usefulness/popularity/simplicity/importance of the pitch ratios that it is able to notate.

Back on page 1 of this topic you said this metric "is specifically for ranking commas according to the popularity (or relative frequency of occurrence) of the pitch ratios that they allow you to notate, as alterations from Pythagorean pitches" so that's helpful to get my head around it too.

I don't know exactly where you get this "up to 3 commas competing" figure from. I think it probably has something to do with the maximum count of pitch ratios separated by Pythagorean commas (23.46¢) which can fit inside the 68.57¢ stretch Sagittal concerns itself with.

What do you suggest instead, then? "Notatable pitch ratio popularity rank", or `text{nprpr}`? "Rank of notatable pitch ratios"... I'll keep thinking on it.

BTW, you were mistaken when you suggested that the reason you couldn't find some of those high-prime ratios in the archive was because they were only present at ratios between pitches. The database we are using does not count any interval ratios, only pitch ratios. The likely reason you couldn't find them is that they were only present as pitch ratios containing one or more factors of prime 3.

That makes sense. When scraping the Scala stats, you didn't gather all internal intervals, because composers do not notate intervals, they notate pitches.

I did, in fact, try searching for powers of 3 times those two numbers. I think I gave up around 3⁷ without finding anything. *shrug*

So should we go with the divided-by-9 version, to get more convincingly rank-like values?

That is what I said. The expression referred to by my "the above" was the divided-by-9 version:

n × d × gpf(n×d)
2^copfr(n) × 3^copfr(d)+2

Perhaps you missed the "+2" at the end of the last superscript.

I'm confused. Clearly you're interpreting my words in some way I didn't mean them and I can't figure out what it is.

No, I did not miss the +2, no. I understand that, being in the exponent on a 3 in the denominator, is the divided-by-9 effect, and I understand that it is that which makes the resultant values more convincingly rank-like.

You say you used the words "the above" to refer to this divided-by-9-version, and I did understand that. A few words earlier, though, you used the word "if", and I interpreted that to mean "maybe we could go with it", not "we should go with it". So, I had not understood you to have decided between the divided-by-9-version and the not-divided-by-9-version.

Maybe we don't have to drag out what who meant and understood what when... why don't you just say which of the two metrics you prefer. I think you prefer the divided-by-9-version. I am leaning that way myself.

Edit: you beat me to it. I see you indeed prefer the divided-by-9-version.

"Take the prime factorisation of the numerator and divide all the primes by 2, then multiply it out again. Do the same with the denominator but divide by 3 instead of 2. Multiply these two results together then multiply by the prime limit of the ratio and divide by 9",

I dig it!

[Dave Keenan]

... All that's cool and all, but it doesn't address the main issue which is the failure of WBL-1 to differentiate between 77/5, 55/7, and 35/11. Though I note that this differentiation was not among those enumerated by myself a couple days ago. It's similar to the fourth task, balance, except it's not "prime balance", but just plain old "balance", and thus it would be the only one of the five tasks which is not with respect to primes (i.e. it would be "prime repetitions", "prime content", "prime limit", "prime balance", "balance") but simplify magnitude of the numbers in the numerator and denominator. That makes me a tiny bit suspicious of it. But let's see if it holds for any other permutations of 3 primes:

[35/11: 55 votes
55/7: 61 votes
77/5: 92 votes]

35/13: 25 votes
65/7: 11 votes
91/5: 11 votes

55/13: 4 votes
65/11: 6 votes
143/5: 13 votes

77/13: 8 votes
95/11: 1 votes
143/7: 9 votes

I'm not seeing enough of a pattern to think this is something our metric must capture. What do you think?

I think you've just done a brilliant job of convincing me that WBL-1 is a fine choice of metric, as the data also fails to distinguish these cases in any consistent way. Thanks.

Oh — so have you for certain decided that even though wybl beat wbl on SoS(-1), wbl beat wybl on SoS(1) by enough that it's better? I'm just a bit surprised by how much you seem to care about SoS(1) now, when everything I'd spent days running scripts for was to desperately minimize SoS(-1) (why wasn't I running that script for 3 days on SoS(1) and then in the end checking on SoS(-1) to see how that looked? Why was it this way and not the other?)

You are simply mistaken when you assume I care about SoS(1) now. Once we had SoS(-1) for all ratios, I looked only at that, and complexity/fitability. I thought that the small improvement in SoS(-1) did not justify the added complexity/fitability of wybl over wbl, or of wbl over wbl-1, given that they're all roughly 2.5 times better than sopfr.

Ah ok. I'd never heard the phrase product complexity before (and web searches don't readily turn up evidence of it being an accepted or popular mathematic term).

Not mathematics generally — only music theory. Maybe it was just me and George Secor, but I'd be surprised if it didn't occur in the tuning list archive you resurrected.

From an anecdotal standpoint, Benedetti height seems to come up all the time when I find myself in discussions of xenharmonics (and not always brought up by myself). Those things both said, if you do not think the appeal to xenharmonicists of seeing BH in our metric will be worth the cost of some people potentially not knowing or caring about it, then I am certainly fine sticking with n·d.

OK. I accept that the term "Benedetti height" should get a mention, along with gpf and Ω.

You're talking to the guy who, when folks on the tuning list wanted to name a temperament "Keenan", begged them to instead give it a name with some mnemonic value. I think I made some suggestions. And did they? No. They only honoured the "don't name it after me" part of my request. Completely missing the point of that, they called it "Keemun" instead. Sigh. So now, I can't remember anything about it, and have to look it up whenever it gets mentioned.

Yeah, speaking of exposition, who's writing our findings up for Xenharmonikôn?!

I thought of asking that earlier, but I wasn't sure you'd think there'd be wider interest in what is essentially an inner mechanic to Sagittal.

I think you're right. I think it's only of interest to any who want to know how we decided what ratios should get symbols in Sagittal. As such, the writeup probably belongs only on this forum, or perhaps on the static pages of the Sagittal website. At most a Xen Wiki article.

[Dave Keenan]

I don't know exactly where you get this "up to 3 commas competing" figure from. I think it probably has something to do with the maximum count of pitch ratios separated by Pythagorean commas (23.46¢) which can fit inside the 68.57¢ stretch Sagittal concerns itself with.

That may be right. It's something like that. You may remember the max of 3 alternate spellings in the JI notation calculator spreadsheet. It might go up to a choice of 4 commas per 5-rough ratio if we allow a wider range of 3 exponents, as I think we have been doing for tina definition candidates. Sorry I'm a bit vague on this ATM. We'll sort it out when we get to it.

What do you suggest instead, then? "Notatable pitch ratio popularity rank", or `text{nprpr}`? "Rank of notatable pitch ratios"... I'll keep thinking on it.

Do that. Thanks.

I'm confused. Clearly you're interpreting my words in some way I didn't mean them and I can't figure out what it is.

Sorry about that. Good that we're sorted now.

[cmloegcmluin]

I think you've just done a brilliant job of convincing me that WBL-1 is a fine choice of metric, as the data also fails to distinguish these cases in any consistent way. Thanks.

Oh — so have you for certain decided that even though wybl beat wbl on SoS(-1), wbl beat wybl on SoS(1) by enough that it's better? I'm just a bit surprised by how much you seem to care about SoS(1) now, when everything I'd spent days running scripts for was to desperately minimize SoS(-1) (why wasn't I running that script for 3 days on SoS(1) and then in the end checking on SoS(-1) to see how that looked? Why was it this way and not the other?)

You are simply mistaken when you assume I care about SoS(1) now. Once we had SoS(-1) for all ratios, I looked only at that, and complexity/fitability. I thought that the small improvement in SoS(-1) did not justify the added complexity/fitability of wybl over wbl, or of wbl over wbl-1, given that they're all roughly 2.5 times better than sopfr.

Okay, I guess I got a bit confused. Back in this post you said "We care about both SoSs. This indicates that wybl is overfitting to the most popular ratios, and not as good a model overall. So wbl is to be preferred over wybl, so far," which I took to mean that SoS(1) was thing indicating wybl was overfitting. You then said, "But we eventually need to compare how well the shortlist of metrics do on the whole database of ratios. Or at least a larger set than the 80 we're using now," and then a few posts later you said, "SoS(1) with the set of 80 was a quick'n'dirty approximation of doing SoS(-1) for a larger set." I'll go ahead and trust you on this one. Just wanted a little clarification is all. Which I've got now. Thank you.

Ah ok. I'd never heard the phrase product complexity before (and web searches don't readily turn up evidence of it being an accepted or popular mathematic term).

Not mathematics generally — only music theory. Maybe it was just me and George Secor, but I'd be surprised if it didn't occur in the tuning list archive you resurrected.

From an anecdotal standpoint, Benedetti height seems to come up all the time when I find myself in discussions of xenharmonics (and not always brought up by myself). Those things both said, if you do not think the appeal to xenharmonicists of seeing BH in our metric will be worth the cost of some people potentially not knowing or caring about it, then I am certainly fine sticking with n·d.

OK. I accept that the term "Benedetti height" should get a mention, along with gpf and Ω.

You're talking to the guy who, when folks on the tuning list wanted to name a temperament "Keenan", begged them to instead give it a name with some mnemonic value. I think I made some suggestions. And did they? No. They only honoured the "don't name it after me" part of my request. Completely missing the point of that, they called it "Keemun" instead. Sigh. So now, I can't remember anything about it, and have to look it up whenever it gets mentioned.

Ah, gotcha. You are correct: I just searched the archive, and "product complexity" turns up 18 times.

However, they are indeed all from you or George. And "Benedetti height" beats it with 40 hits (these from Mike, Gene, Carl, Graham). It is said somewhere that Paul has a strong preference for the term "Benedetti height". So yes, Benedetti height should definitely at least get mentioned.

(also 17 hits for BH on the Xen wiki while 0 for product complexity)

To be clear, now that "product complexity" has been brought to my attention, I strongly prefer it, for the same reason you do. Though I prefer n·d the most.

Ugh, I am sorry to hear about Keemun. Worst.

Yeah, speaking of exposition, who's writing our findings up for Xenharmonikôn?!

I thought of asking that earlier, but I wasn't sure you'd think there'd be wider interest in what is essentially an inner mechanic to Sagittal.

I think you're right. I think it's only of interest to any who want to know how we decided what ratios should get symbols in Sagittal. As such, the writeup probably belongs only on this forum, or perhaps on the static pages of the Sagittal website. At most a Xen Wiki article.

Agreed. Let's plan on editing the first post of this topic to link to a later post which will summarize our process, findings, and decisions. I'm pretty flexible on how we approach it, i.e. whether I draft and you edit, vice versa, or some other strategy altogether. It's not a project I want to spend a week on. But I do think I would regret it if I didn't — while all of this is so fresh in my head — distill something lucid down for posterity.

[cmloegcmluin]

How about `text{ranc}`, for “rank with respect to Ratios Able to be Notated by this Comma”?

[Dave Keenan]

Thanks for that terminology research.

Agreed. Let's plan on editing the first post of this topic to link to a later post which will summarize our process, findings, and decisions. I'm pretty flexible on how we approach it, i.e. whether I draft and you edit, vice versa, or some other strategy altogether. It's not a project I want to spend a week on. But I do think I would regret it if I didn't — while all of this is so fresh in my head — distill something lucid down for posterity.

Great idea. I'm happy for you to draft and me to edit, ASAP. Let's get that done before we go back to the tina thread.

I'm still working on the naming problem. I think we need two kinds of name. One for what it is attempting to measure, or its purpose, and another for the specific mathematical function we've chosen for the job.

For the former, how about "5-rough notational popularity rank (5-no-pop-rank)" as opposed to "3-rough notational popularity rank (3-no-pop-rank)"? Or "ratio notational popularity rank (rat-no-pop-rank)" as opposed to "comma notational popularity rank (com-no-pop-rank)"?

For the latter, how about something like "N2D3L9" or "N2D3P9" or just "N2D3"? Pronunciation: "EN-too-DEETH-ree"? Kinda like "R2D2" and "C3P0"?

[cmloegcmluin]

I'm happy for you to draft and me to edit, ASAP. Let's get that done before we go back to the tina thread.

Okay, I’ll get started.

I'm still working on the naming problem. I think we need two kinds of name. One for what it is attempting to measure, or its purpose, and another for the specific mathematical function we've chosen for the job.

Agreed.

For the former, how about "5-rough notational popularity rank (5-no-pop-rank)" as opposed to "3-rough notational popularity rank (3-no-pop-rank)"? Or "ratio notational popularity rank (rat-no-pop-rank)" as opposed to "comma notational popularity rank (com-no-pop-rank)"?

I like those. Are you suggesting that we will eventually also prepare all-encompassing 3-no-pop-rank/com-no-pop-rank metrics too, folding in abs3exp, apotome slope, etc (what we were referring to as a "consolidated badness metric")?

For the latter, how about something like "N2D3L9" or "N2D3P9" or just "N2D3"? Pronunciation: "EN-too-DEETH-ree"? Kinda like "R2D2" and "C3P0"?

I like N2D3P9. Captures the critical insight of the metric, and weirdly memorable and rolls off the tongue — probably in no small part to its similarity to the Star Wars droids

What did you think about “ranc” though? I was kinda excited to wake up and see what you thought about that one. I’m totally fine if you don’t like it — it’s just that my Dave model suggested a nontrivial chance that you’d love it.