developing a notational comma popularity metric

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Re: developing a notational comma popularity metric

Post by cmloegcmluin »

Dave Keenan wrote: Mon Aug 17, 2020 10:51 pm Good work. Thanks. Here's my edit so far. I haven't got to the development section yet, but it looks pretty good.
Overall impression of your edit: I love it.
You will see that I changed my mind and decided, for pedagogical purposes, to describe the n≥d requirement as outside the N2D3P9 function.
Or perhaps you noticed that I had made no attempt myself to represent the 5-roughening of `n/d` in the formula for N2D3P9... :oops:

I can't see any good way to do without majorly clouding the more important concepts. I expect you tried and failed as well.

I think it's fine to describe it as outside N2D3P9's formula. That's far from a strict loss. The write-up will now agree with the code.
N2D3P9 or Entoo-Deethree-Peenine, is a fictional character in the Star Wars franchise. In an alternative timeline, the young Anakin Skywalker assembles the droid N2D3P9 from the parts of three other droids: R2D2, C3P0 and NR-N99. Only joking.
I know it's a joke, but I nonetheless salute your hiliting work here.
N2D3P9 is a mathematical function which was developed to help in designing the Sagittal microtonal notation. Given a pitch ratio n/d, N2D3P9 estimates its rank in popularity among all rational pitches in musical use. A low value of N2D3P9 indicates that the ratio is used often, and so should have a simple accidental symbol, while a high value indicates that the ratio is used rarely and so can have a more complex symbol if necessary. The real reason for its name will become obvious when we describe how to calculate it.
I agree with focusing on it in terms of Sagittal, rather than how I had put it, "for a JI microtonal notation system" (my italics). I wasn't too confident about this bit in the first place, since it felt unlikely that anyone else would ever actually use N2D3P9 to help them design their own JI notation. And why would they, if Sagittal already has them covered? :)

Putting the concepts of popularity and notational simplicity together right away in the third sentence, rather than one after another in two separate sentences as I did, is a marked improvement in clarity from my version.

My only concern here is that the "the real reason for its name will become obvious when we describe how to calculate it" sentence feels a tad informal compared to the rest. And I don't think the word "real" is right, since no reason for its name has been addressed at all yet. I might instead write "Its name is taken from key components of its formula, which will be described in detail later."
Because factors of 2 and 3 in pitch ratios are already notated by moving along the chain of fifths (... B♭♭ F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ Fx ...) and by changing octaves, N2D3P9 only operates on ratios that have had their factors of 2 and 3 removed. For example, there are various numbers of factors of 2 and 3 in the following ratios: 16/15, 10/9, 6/5, 5/4, 27/20, 45/32, 64/45, 40/27, 8/5, 5/3, 9/5, 15/8, but when their factors of 2 and 3 are removed, they all reduce to 1/5 or 5/1, and so they can all be notated using the same microtonal accidental, pointing either up or down, combined with different letters and sharps or flats. We say that 1/5 or 5/1 is the "2,3-reduced" or "5-rough" form of these pitch ratios.

Because 1/5 and 5/1 use the same accidental pointing either up or down, and because N2D3P9 only operates on ratios whose numerator is larger than their denominator, 5/1 can represent this entire equivalence class for the purpose of notation design.
This mini-lesson in microtonal notation system design is excellent. I just can't say enough good things about it. You're the man now, dog.
Formula

Before describing how to calculate N2D3P9, we describe two simpler functions that are used in calculating it.

We call the first one "copfr", which stands for "Count Of Prime Factors with Repeats". It applies to any positive integer. For example 175 has the prime factorisation 5 x 5 x 7, which has 3 factors, so copfr(175) = 3. copfr(1) = 0. It is also called the "big omega" function, `Ω`.

The second is "prime-limit", which is also known as "gpf", which stands for greatest prime factor. Prime-limit(175) = 7. Normally, Prime-limit(1) = 1, but because we are dealing with 2,3-reduced or 5-rough numbers, we take prime-limit(1) to be 3, although this does not alter the ranking of ratios by N2D3P9.
I think putting explanations of the submetrics first is a great idea.

I also agree that re-ordering each submetric's list of multiple names such that the official mathematical name comes second, while the more pedagogically accessible name comes first, is better than how I had it. Thanks.

The prime-limit(1) = 3 bit I addressed in my previous reply. We'll see how that plays out and then update here accordingly.

I would just insert something like "So the formula for N2D3P9(`n`/`d`) is:" right before dropping the formula.
$$\text{N2D3P9}(n/d)=\frac{n}{2^{\text{copfr}(n)}}×\frac{d}{3^{\text{copfr}(d)}}×\frac{\text{prime-limit}(nd)}{9} $$
$$\text{where }n\text{ and }d\text{ are 5-rough positive integers and }n≥d$$
Note that where

\(n = 5^{n_5}×7^{n_7}×11^{n_{11}}×...\)

we have

\(2^{\text{copfr}(n)}=2^{n_5}×2^{n_7}×2^{n_{11}}×...\)

and so

\(\frac{n}{2^{\text{copfr}(n)}}=(\frac{5}{2})^{n_5}×(\frac{7}{2})^{n_7}×(\frac{11}{2})^{n_{11}}×...\)

and similarly

\(\frac{d}{3^{\text{copfr}(d)}}=(\frac{5}{3})^{d_5}×(\frac{7}{3})^{d_7}×(\frac{11}{3})^{d_{11}}×...\)
I really like putting this version of the formula first, and working up to the condensed one. Great call.

I also notice and approve of the change to use the pedagogically accessible names (copfr and prime-limit) in the first part and the official mathematical names (Ω and gpf) in the second part.

Using `n`/`d` rather than ever bringing `r` into it is an improvement too.

And one-lining your explanatory breakdowns is a big aid in following it. Those were already amazingly helpful but you found a way to improve them.
The division by 9 does not affect the ranking, but it has the effect that there are approximately `N` 5-rough ratios with `\text{N2D3P9}≤N`.
Would you mind if we returned the example I gave of this feature to the write-up, here? So just append the sentence "For example, N2D3P9(80/77) ≈ 39, suggesting that approximately 38 other undirected 5-rough ratios exist which are more popularly used than 77/5."

Woo! This is so exciting to see coming together like this.
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Re: developing a notational comma popularity metric

Post by Dave Keenan »

cmloegcmluin wrote: Tue Aug 18, 2020 1:11 am
Dave Keenan wrote: Mon Aug 17, 2020 5:30 pm I've now updated the 0.11 to 0.33 in your earlier list. However I don't really see it as a "special value". I think that 3 simply is the prime limit of a 5-rough monzo with all zeros.
You have a high standard of "special", my friend! :lol: I think 1/1 having a prime limit of 3 is pretty special.
OK. Fair enough. :lol:
Well, how about we do give N2D3P9(1/1) a special value explicitly, i.e. we say it is defined as 1?
At first I resisted this because
(a) I thought it made it too close to N2D3P9(5/1), whereas I thought the 1/1 equivalence class was far more popular than the 5/1 class, and
(b) I could at least vaguely justify prime-limit(1) = 3 in the context of 5 rough ratios, whereas this N2D3P9(1/1) = 1 seemed to be pulled out of thin air, and seemed to be implying that prime-limit(1) = 9.
Okay, maybe we should talk about this a bit more. The reason I think you want to make the prime limit of 1/1 as 3 in a 5-roughened environment is this: "take the next smallest prime number after the p you have roughened to, and that should be the prime limit of 1 in a p-rough world".
That is correct. Except I'd add: And when there is no lower prime, you're left with the multiplicative identity (or "empty product"), 1.
As for why the greatest prime factor (or largest prime divisor, or prime limit) gives 1 for 1 i.e. gpf(1) = 1, I had to look that one up too. It turns out I was wrong; I built that case into my code, for whatever reason, however it looks like it is simply not defined for numbers below 2.
That's clearly a matter of taste, or of practical utility, as your OEIS quote indicates. Good find.
My suggestion, then, is that N2D3P9 is simply undefined for 1/1, inheriting this lack of definition from gpf.

Who needs N2D3P9(1/1), anyway? Aren't you using it wrong if you're wondering whether or not it's important to be able to notate 1/1 in a simple way in the Sagittal JI notation system? : D
You could look at it that way, but surely the simplest symbol is no symbol, and I already explained, with a graph, why it is useful to keep 1/1 in the set of ratios being ranked.
As a bonus, we could then simplify the ≥ to >.
But we'd then have to complexify "where n and d are positive integers" to "where n and d are integers, with n > 1 and d > 0" or equivalent.

Instead I accept your first suggestion, to define N2D3P9(1/1) as 1, because
(a) it actually makes the ratio between the N2D3P9's of 1/1 and 5/1 approximately equal to the inverse (Zipf's law) of the ratio of their occurrence counts in the Scala archive (about 1.4),
(b) I can justify N2D3P9(1/1) = 1 by saying that, for 5-rough ratios, 1/1 is equivalent to 3/1 and N2D3P9(3/1) = 3/1 × 1/1 × 3/9 = 1, which does not imply that prime-limit(1) = 9, but instead has 3 appearing twice, once as numerator and once as prime-limit.
Omigosh what is wrong with me. I really ran with that thinko...
Welcome to the human race. ;)
No there's no convention in American English to accompany disagreement with an affirmation, as in "I disagree, yes," if that's what you're asking ;)
I couldn't fail to disagree with you less. ;)

Ha! I actually wondered if there might be some weird thing, in say Java Script documentation, where "round to X digits" (as opposed to "round to X significant digits") meant the same as "round to X decimal places". IMHO that would be a really bad convention, just inviting confusion. Strangely, I actually found such a convention (kind of) in documentation for the language "R", used for statistical computing. https://www.rdocumentation.org/packages ... pics/Round
Alright, shall we stick with two decimal places by convention, then?
Lets just say that's the default. There may well be contexts where it makes sense to round them to one or zero decimal places.
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Re: developing a notational comma popularity metric

Post by cmloegcmluin »

Dave Keenan wrote: Tue Aug 18, 2020 8:57 am ...I already explained, with a graph, why it is useful to keep 1/1 in the set of ratios being ranked.
Ah, so you did. I think at that time I didn't understand rank 0 to mean exclusion from the set, and so perhaps I didn't quite get your point with that last sentence... which I would reproduce here, except it's almost verbatim what you just said above.
Instead I accept your first suggestion, to define N2D3P9(1/1) as 1, because
(a) it actually makes the ratio between the N2D3P9's of 1/1 and 5/1 approximately equal to the inverse (Zipf's law) of the ratio of their occurrence counts in the Scala archive (about 1.4),
7624/5371 ≈ 1.419, yes, I see. Interesting. Cool! (Also kind of close to that 1.37 exponent we were looking at for a while).

Plus, 1.39 in the ballpark of the qoppas of the first dozen or so ranks (1.39, 1.96, 1.28, 1.31, 1.48, 1.01, 1.27, 1.08, 1.01, 1.18, 1.01, 1.38, 1.00...). Whereas if we kept N2D3P9(1/1) as 0.33, then that qoppa would be 4.21 instead, which is outside that ballpark.
(b) I can justify N2D3P9(1/1) = 1 by saying that, for 5-rough ratios, 1/1 is equivalent to 3/1 and N2D3P9(3/1) = 3/1 × 1/1 × 3/9 = 1, which does not imply that prime-limit(1) = 9, but instead has 3 appearing twice, once as numerator and once as prime-limit.
That works too. I think your previous argument that prime-limit(1) = 3 in 5-rough land is slightly more compelling, but I'll take this.
I actually wondered if there might be some weird thing, in say Java Script documentation, where "round to X digits" (as opposed to "round to X significant digits") meant the same as "round to X decimal places". IMHO that would be a really bad convention, just inviting confusion. Strangely, I actually found such a convention (kind of) in documentation for the language "R", used for statistical computing. https://www.rdocumentation.org/packages ... pics/Round
Looks like JavaScript uses "digits" too: https://developer.mozilla.org/en-US/doc ... er/toFixed

I expect this is something like the whole exponent vs. power deal. It's just gonna be a mess.
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Re: developing a notational comma popularity metric

Post by Dave Keenan »

Thanks for the review. Your ability to appreciate pedagogical nuances is what made me want to encourage your work, when you first appeared on the facebook tuning groups.

I note that I said in response to your original question, that the 5-roughening should be outside the N2D3P9 function, for the writeup (and could also be so for the code). It was only whether the n≥d requirement should be outside the function, that I changed my mind on (to yes), when I came to edit your writeup.

And I note that I really intend the Star Wars joke to be the start of the writeup. It serves several pedagogical purposes.

I accept all of your excellent suggestions and include them below, along with some minor tweaks of my own.


N2D3P9 or Entoo-Deethree-Peenine, is a fictional character in the Star Wars franchise. In an alternative timeline, the young Anakin Skywalker assembles the droid N2D3P9 from the parts of three other droids: R2D2, C3P0 and NR-N99. We're only joking, but we hope this helps with remembering and pronouncing the name.

N2D3P9 is a mathematical function which was developed to help in designing the Sagittal microtonal notation. Given a pitch ratio n/d, N2D3P9 estimates its rank in popularity among all rational pitches in musical use. A low value of N2D3P9 indicates that the ratio is used often, and so should have a simple accidental symbol, while a high value indicates that the ratio is used rarely and so can have a more complex symbol if necessary. Its name is an abbreviation of key components of its formula, which will be described in detail later.

Because factors of 2 and 3 in pitch ratios are already notated by changing octaves or moving along the chain of fifths (... B♭♭ F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ Fx ...), N2D3P9 only operates on ratios that have had their factors of 2 and 3 removed. For example, there are various numbers of factors of 2 and 3 in the following ratios: 16/15, 10/9, 6/5, 5/4, 27/20, 45/32, 64/45, 40/27, 8/5, 5/3, 9/5, 15/8, but when their factors of 2 and 3 are removed, they all reduce to 1/5 or 5/1, and so they can all be notated using the same microtonal accidental, pointing either up or down, combined with different letters and sharps or flats. We say that 1/5 or 5/1 is the "2,3-reduced" or "5-rough" form of these pitch ratios.

Because 1/5 and 5/1 use the same accidental pointing either up or down, and because N2D3P9 only operates on ratios whose numerator is larger than their denominator (superunison ratios), 5/1 can represent this entire equivalence class for the purpose of notation design.

Formula

Before describing how to calculate N2D3P9, we describe two simpler functions that are used in calculating it.

We call the first one copfr, which stands for "Count Of Prime Factors with Repeats". It applies to any positive integer. For example 175 has the prime factorisation 5 x 5 x 7, which has 3 factors including the repeat of 5, so copfr(175) = 3. copfr(1) = 0. Copfr is also called the "big omega" function, `Ω`.

The second is prime-limit, which is also known as "gpf", which stands for greatest prime factor. Prime-limit(175) = 7. Some authors leave prime-limit(1) undefined. We avoid the question because we define N2D3P9(1/1) ≡ N2D3P9(3/1) = 1. This is because the ratios in the equivalence class represented by the 5-rough 1/1 actually have a prime limit of 3.

So the formula for N2D3P9(n/d) is:
$$\text{N2D3P9}(n/d)=\frac{n}{2^{\text{copfr}(n)}}×\frac{d}{3^{\text{copfr}(d)}}×\frac{\text{prime-limit}(nd)}{9}$$
$$\text{where }n\text{ and }d\text{ are 5-rough positive integers and }n>d.$$
$$\text{N2D3P9}(1/1)=1$$
Note that where

\(n = 5^{n_5}×7^{n_7}×11^{n_{11}}×...\)

we have

\(2^{\text{copfr}(n)}=2^{n_5}×2^{n_7}×2^{n_{11}}×...\)

and so

\(\frac{n}{2^{\text{copfr}(n)}}=(\frac{5}{2})^{n_5}×(\frac{7}{2})^{n_7}×(\frac{11}{2})^{n_{11}}×...\)

and similarly

\(\frac{d}{3^{\text{copfr}(d)}}=(\frac{5}{3})^{d_5}×(\frac{7}{3})^{d_7}×(\frac{11}{3})^{d_{11}}×...\)

These can be described respectively as "product of half prime factors of the numerator (with repeats)" and "product of one-third prime factors of the denominator (with repeats)". So we can describe the procedure for calculating N2D3P9(n/d) as:

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

N2D3P9 can also be written as:
$$\text{N2D3P9}(n/d)=\frac{nd⋅\text{gpf}(nd)}{2^{Ω(n)}3^{Ω(d) + 2}} $$
where `nd` is established in music theory as a ratio's "product complexity" or Benedetti height.

The division by 9 does not affect the ranking, but it has the convenient effect that N2D3P9 values are almost the same as the ranks they produce when applied to all 5-rough superunison ratios. Putting it another way, there are approximately `N` 5-rough pitch ratios with `\text{N2D3P9}≤N`. For example, N2D3P9(77/5) = 7/2 × 11/2 × 5/3 × 11/9 ≈ 39, suggesting there are approximately 38 other 5-rough pitch ratios more popular than 77/5. There are actually about 4% fewer than that on average. In this case there are 36.


Justification
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Re: developing a notational comma popularity metric

Post by cmloegcmluin »

Dave Keenan wrote: Tue Aug 18, 2020 10:53 am Thanks for the review. Your ability to appreciate pedagogical nuances is what made me want to encourage your work, when you first appeared on the facebook tuning groups.
:) Thanks.

Anyway, I can sit here and appreciate it all day — you consistently produce it!

Seriously though, I do think I did an alright job with the first draft. The core if it's still there. I think we've got a pretty good thing going on together here and I look forward to assembling the educational materials for Sagittal with you, potentially using a similar back-and-forth process.
I note that I said in response to your original question, that the 5-roughening should be outside the N2D3P9 function, for the writeup (and could also be so for the code). It was only whether the n≥d requirement should be outside the function, that I changed my mind on (to yes), when I came to edit your writeup.
Gah! Yes yes yes. You're totally right. Arugments to N2D3P9 have two very different requirements.
  1. they must be 5-rough, and
  2. they must be super.
We could build the n≥d requirement into the formula by replacing `n` and `d` with `\text{max}(n,d)` and `\text{min}(n,d)` respectively. This would only be required for the `n` and `d` in the exponents in the denominator . But then if you didn't include them in the corresponding numerators in the broken up version which is presented first, the parallelism between the fractional parts which clarifies "half prime factors" and "one third prime factors" would be less obvious. But then if you did include them, the parallelism between the first version of the formula and the second version of the formula would be less obvious! So either way you'd be forced to cloud the more important concepts. Which'd be such a shame, because I think the way it is now is excellent! Excellent, as in: I plan to share this wiki page with friends and family, because I think that even if one has little to no background with this material, it's been articulated well enough that it may be a compelling read regardless.

Also, as long as there's already one requirement to the ratios you put in, it's much less of an ask for users to be mindful also of the second requirement. So, I think we should feel less pressed to squeeze it into the formula.
And I note that I really intend the Star Wars joke to be the start of the writeup. It serves several pedagogical purposes.
Ha! Wow. Yes, I clearly assumed the "Only joking" part was you joking with me, not with the Wiki readers :) Otherwise I wouldn't have criticized your sentence "the real reason for the name" for feeling "a tad informal", since joking about Star Wars is tremendously more informal than that. Also, wow, okay, I'm realizing now that when you said "the real reason for its name" and I complained that "no reason for its name has been addressed at all yet" that this might have been a good signal for me that the non-real name reason you were referencing was the Star Wars one... :man_facepalming:

Okay. So I don't actually have anything against the Star Wars joke. Despite numerous moral and aesthetic exceptions I take with the franchise, I count myself a pretty big fan. There may actually be nothing cooler ever invented than lightsabers. So I like that the Star Wars joke helps with pronunciation (I see your pronunciation spelling corresponds to my favored stress pattern, so, thank you for that nod; or perhaps it is just the case that my favored stress pattern corresponds to the key components of its formula). And it does also add some much appreciated mnemonic spice to the whole affair. Let's keep it.
This is because the ratios in the equivalence class represented by the 5-rough 1/1, actually have a prime limit of 3.
Orthographic quibble: I believe the comma in this sentence is unnecessary and possibly incorrect. Unless you think otherwise I'll strike it when I post to the wiki.
The division by 9 does not affect the ranking, but it has the convenient effect that N2D3P9 values are almost the same as the ranks they produce when applied to all 5-rough pitch ratios. Putting it another way, there are approximately `N` 5-rough pitch ratios with `\text{N2D3P9}≤N`. There are actually about 4% fewer than that on average. For example, N2D3P9(77/5) = 7/2 × 11/2 × 5/3 × 11/9 ≈ 39, suggesting there are approximately 38 other 5-rough pitch ratios more popular than 77/5.
I'm glad you've included the factlet about there being actually 4% fewer fewer 5-rough pitch ratios with `\text{N2D3P9}≤N`. I'm concerned, however, that inserting it between the main statement it disclaims and that main statement's illustrative example might confuse the issue. What I mean is: if the "For example" sentence comes after the "4% fewer" sentence then to me, purely based on proximity, it suggests an example is being given of the "4% fewer" bit rather than the main statement about `\text{N2D3P9}≤N`. And unfortunately this effect is exacerbated by the fact that the example sentence contains both the number 39 and the number 38, which are weirdly close to 4% off from each other. Of course the point is that 38 is one less than 39 — because the ratio with rank 39 is in the 39th position of a one-indexed list — but that may not be obvious to first time readers. Probably there's some way to refactor the example sentence to evade this snag, but I think the best solution is even simpler: just switch the last two sentences. I think this reads better:

  • The division by 9 does not affect the ranking, but it has the convenient effect that N2D3P9 values are almost the same as the ranks they produce when applied to all 5-rough pitch ratios. Putting it another way, there are approximately `N` 5-rough pitch ratios with `\text{N2D3P9}≤N`. For example, N2D3P9(77/5) = 7/2 × 11/2 × 5/3 × 11/9 ≈ 39, suggesting there are approximately 38 other 5-rough pitch ratios more popular than 77/5. There are actually about 4% fewer than that on average.

Let me know if you agree.

Am I looking forward to edits to the Development section, or when you said "it looks pretty good" did you mean "should be basically fine as is"?
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Re: developing a notational comma popularity metric

Post by Dave Keenan »

cmloegcmluin wrote: Tue Aug 18, 2020 11:57 am Anyway, I can sit here and appreciate it all day — you consistently produce it!
Ha! Thanks.
Seriously though, I do think I did an alright job with the first draft. The core if it's still there.
Absolutely. I mostly just rearranged things and padded out a few explanations. And its soo much easier editing existing stuff than creating it ex nihilo. So thanks for doing the hard part of looking at the blank page and answering the hard questions of where do I start? How do I organise this? What do we need to say?
I think we've got a pretty good thing going on together here and I look forward to assembling the educational materials for Sagittal with you, potentially using a similar back-and-forth process.
Cool.
Gah! Yes yes yes. You're totally right. Arugments to N2D3P9 have two very different requirements.
  1. they must be 5-rough, and
  2. they must be super.
I wanted that word "super" for this article, where "undirected" seemed wrong, but I couldn't think of it. I eventually convinced myself that by saying "pitch" ratio I was implying > 1. But that's not true. However this use of "super" isn't exactly well known and I don't want to have to explain it here. Yes, I could link to an explanation, but it would still break the flow.

How about "superunison" as an adjective? Even if it doesn't work for numbers in general, it should work for pitch ratios. I tried "superunitary" and "superunital" but they already have (high falutin') math definitions which are not the one we want.
We could build the n≥d requirement into the formula by replacing `n` and `d` with `\text{max}(n,d)` and `\text{min}(n,d)` respectively. This would only be required for the `n` and `d` in the exponents in the denominator . But then if you didn't include them in the corresponding numerators in the broken up version which is presented first, the parallelism between the fractional parts which clarifies "half prime factors" and "one third prime factors" would be less obvious. But then if you did include them, the parallelism between the first version of the formula and the second version of the formula would be less obvious! So either way you'd be forced to cloud the more important concepts. Which'd be such a shame, because I think the way it is now is excellent!
I totally agree, Putting max and min in there would be hideous.
Excellent, as in: I plan to share this wiki page with friends and family, because I think that even if one has little to no background with this material, it's been articulated well enough that it may be a compelling read regardless.
Good luck with that. :)
Also, as long as there's already one requirement to the ratios you put in, it's much less of an ask for users to be mindful also of the second requirement. So, I think we should feel less pressed to squeeze it into the formula.
Agreed.
Okay. So I don't actually have anything against the Star Wars joke. Despite numerous moral and aesthetic exceptions I take with the franchise, I count myself a pretty big fan. There may actually be nothing cooler ever invented than lightsabers. So I like that the Star Wars joke helps with pronunciation (I see your pronunciation spelling corresponds to my favored stress pattern, so, thank you for that nod; or perhaps it is just the case that my favored stress pattern corresponds to the key components of its formula). And it does also add some much appreciated mnemonic spice to the whole affair. Let's keep it.
It was both. I realised your stress pattern was best because it matches the grouping of components in the formula.

You may not have seen where I smoothed the transition from informal to formal by changing the "Only joking" to "We're only joking, but we hope this helps with remembering and pronouncing the name."
This is because the ratios in the equivalence class represented by the 5-rough 1/1, actually have a prime limit of 3.
Orthographic quibble: I believe the comma in this sentence is unnecessary and possibly incorrect. Unless you think otherwise I'll strike it when I post to the wiki.
Agreed.
I'm glad you've included the factlet about there being actually 4% fewer fewer 5-rough pitch ratios with `\text{N2D3P9}≤N`. I'm concerned, however, that inserting it between the main statement it disclaims and that main statement's illustrative example might confuse the issue. What I mean is: if the "For example" sentence comes after the "4% fewer" sentence then to me, purely based on proximity, it suggests an example is being given of the "4% fewer" bit rather than the main statement about `\text{N2D3P9}≤N`. And unfortunately this effect is exacerbated by the fact that the example sentence contains both the number 39 and the number 38, which are weirdly close to 4% off from each other. Of course the point is that 38 is one less than 39 — because the ratio with rank 39 is in the 39th position of a one-indexed list — but that may not be obvious to first time readers. Probably there's some way to refactor the example sentence to evade this snag, but I think the best solution is even simpler: just switch the last two sentences. I think this reads better:

  • The division by 9 does not affect the ranking, but it has the convenient effect that N2D3P9 values are almost the same as the ranks they produce when applied to all 5-rough pitch ratios. Putting it another way, there are approximately `N` 5-rough pitch ratios with `\text{N2D3P9}≤N`. For example, N2D3P9(77/5) = 7/2 × 11/2 × 5/3 × 11/9 ≈ 39, suggesting there are approximately 38 other 5-rough pitch ratios more popular than 77/5. There are actually about 4% fewer than that on average.
Agreed. And even better when you include my later-added final sentence, slightly modified: "In this case there are 36.".

I will go back and edit the previous version.
Am I looking forward to edits to the Development section, or when you said "it looks pretty good" did you mean "should be basically fine as is"?
Sorry. I do want to edit the Development section. Starting now.
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Re: developing a notational comma popularity metric

Post by cmloegcmluin »

Dave Keenan wrote: Tue Aug 18, 2020 2:01 pm How about "superunison" as an adjective? Even if it doesn't work for numbers in general, it should work for pitch ratios. I tried "superunitary" and "superunital" but they already have (high falutin') math definitions which are not the one we want.
Whoa, I really like "superunison", yes! We should add that, and "subunison", to our topic in the just-for-fun subforum about multiplicative equivalents of additive concepts. Actually, I think I like these even more than the more succinct "super" and "sub", which may be effective in circles where they've become accepted, sure, but until then, may be a bit too succinct... I expect they would beg some "huh?"s from folks. "Superunison", five syllables though it may be, just rolls right off the tongue, and is fairly memorable. Unless you meant you thought "superunison" was only suitable in musical contexts, and not general numeric ones, because "unison" is not a standard mathematical term for 1/1.

I wasn't actually suggesting we put "super" in the N2D3P9 write-up, by the way. Now that you've upgraded it to "superunison", though, if we did decide to put it in, I think we'd still want to include n≥d in a parenthetical.

Oh crap, though... now that begs the question: how should we distinguish between n≥d and n>d? I suppose it only matters if n≥d can't be called "superunison". Perhaps it can, though, while the latter gets called "strictly superunison", by anology with Rothenberg propriety's "proper" vs "strictly proper" dichotomy, and probably other examples. Amusingly, the first hit I get for a web search on "strictly proper" is this, defining it "of a transfer function, where the degree of the numerator is less than that of the denominator" (my italics). Not that I think that's a dealbreaker for using "strictly" in contexts either where n>d or n<d (the latter of course being "strictly subunison"); I just found it jarringly unfortunate. Maybe there's a better word than "strictly", such as "exclusively", drawing on readers' familiarity with "inclusive" and "exclusive" in the context of number ranges.

Re: unitary and unital: I've looked into these out of curiosity. As for unital, I can't make heads or tails of it — a bit too hifalunital for me. *pause for applause* Unitary, on the other hand, is easy enough for me to understand at least in the context of "unitary divisors", which I'm surprised I've never heard of before or wondered about. Interesting stuff. Thanks for the reference.

And a thumbs up to each the rest of the talking points you made in your previous post! No rush on the edits to Development.
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Re: developing a notational comma popularity metric

Post by Dave Keenan »

cmloegcmluin wrote: Tue Aug 18, 2020 2:46 pm Whoa, I really like "superunison", yes! We should add that, and "subunison", to our topic in the just-for-fun subforum about multiplicative equivalents of additive concepts.
Please do.
Actually, I think I like these even more than the more succinct "super" and "sub", which may be effective in circles where they've become accepted, sure, but until then, may be a bit too succinct... I expect they would beg some "huh?"s from folks. "Superunison", five syllables though it may be, just rolls right off the tongue, and is fairly memorable.
Agreed.
Unless you meant you thought "superunison" was only suitable in musical contexts, and not general numeric ones, because "unison" is not a standard mathematical term for 1/1.
Right. "Unison" is not a mathematical term.
I wasn't actually suggesting we put "super" in the N2D3P9 write-up, by the way. Now that you've upgraded it to "superunison", though, if we did decide to put it in, I think we'd still want to include n≥d in a parenthetical.
I have added "superunison", twice. It is explained the first time it is used. The explanation was already there. See what you think.
Oh crap, though... now that begs the question: how should we distinguish between n≥d and n>d? I suppose it only matters if n≥d can't be called "superunison". Perhaps it can, though, while the latter gets called "strictly superunison", by anology with Rothenberg propriety's "proper" vs "strictly proper" dichotomy, and probably other examples.
That works for me. That's the standard usage of "strictly". https://en.wikipedia.org/wiki/Strict
Amusingly, the first hit I get for a web search on "strictly proper" is this, defining it "of a transfer function, where the degree of the numerator is less than that of the denominator" (my italics). Not that I think that's a dealbreaker for using "strictly" in contexts either where n>d or n<d (the latter of course being "strictly subunison"); I just found it jarringly unfortunate. Maybe there's a better word than "strictly", such as "exclusively", drawing on readers' familiarity with "inclusive" and "exclusive" in the context of number ranges.
No. Strictly is fine. It's the "proper" part that means n ≤ d, not the "strictly" part which merely excludes n = d, leaving n < d. That's what I mentioned earlier, about an "improper fraction" being one where n ≥ d.
No rush on the edits to Development.
I will not have them finished before your bedtime. But hopefully I can have them waiting for you in the morning.
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Re: developing a notational comma popularity metric

Post by cmloegcmluin »

Dave Keenan wrote: Tue Aug 18, 2020 3:36 pm
cmloegcmluin wrote: Tue Aug 18, 2020 2:46 pm We should add that, and "subunison", to our topic in the just-for-fun subforum about multiplicative equivalents of additive concepts.
Please do.
Done.
I have added "superunison", twice. It is explained the first time it is used. The explanation was already there. See what you think.
It works. Thanks.
That's the standard usage of "strictly". https://en.wikipedia.org/wiki/Strict
Ah, so it is. You may go ahead and LMGTFY me if the need ever arises again.
It's the "proper" part that means n ≤ d... That's what I mentioned earlier, about an "improper fraction" being one where n ≥ d.
Ha. Wow. I think "improper fractions" has been such a familiar term to me that I hardly even register the word "proper" in it! Now it's all completely clear.
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Re: developing a notational comma popularity metric

Post by cmloegcmluin »

Phew, okay. So I've finished reviewing the entire topic. Interesting experience.

Knowing all that I know now, it's clear sometimes when you're trying to get a key point across, though at the time it didn't connect with me, and so you had to try two or three times. Thanks for being patient with me.

In the very first post (actually yanked from the "consistent 37 limit" topic), you gave as part of the original ask:
Even more complicated: In order of decreasing frequency of occurrence, we have the ratios 11:35, 7:55. 5:77 and 1:385, which are all treated equally by SoPF>3. 
I had totally forgotten that you had sought this type of balance of primes across the vinculum from the outset. I went back over the spreadsheets I was using before I started turning to code to get my answers, and indeed I was playing around with these two prime against one combos for a while. It's clear from looking at the formula for N2D3P9 that the only aspect which affects balance is the count of primes, not what those primes are, which is why these three all come out the same. But I didn't need my automatic solver code to tell me that — the inconsistency of this type of balance should have been evident to us from a few cells of the data. Oops!
Dave Keenan wrote: Sun Jun 21, 2020 8:31 pm ...Are your n and d, that are used in computing Tenney height (log product complexity)...
That one's technically from the topic for finding tina commas, but I drop it here because apparently I had heard of product complexity before, despite later claiming otherwise. :oops:
Dave Keenan wrote: Mon Jun 29, 2020 10:39 am What is the argument for why a composer would prefer ratios with lower sopf, as opposed to merely lower sopfr?
I agree the metric should with little contention represent things composers are likely to make their decisions on. For me, sopf seems pretty reasonable (as reasonable as π did, anyway), because many scales are based on just generators which are iterated repeatedly. Does it not seem reasonable to you that we should give ratios like 625/1 even the littlest bit of a pass for using only 5's? And wouldn't we want to punish e.g. 5.11/7.13 vs. 13.13/5.5?

I feel like sopfr, sopf, and gpf are each capturing a different aspect of the harmonic content which is important to capture: sopfr the overall harmonic content, sopf the variety of it, and gpf the extremeness of it. My stats vocab is lacking here; perhaps someone else knows better terms for these abstract aspects of data than "variety" and "extremeness".
You make a good case. I read this as, "When you've used a given prime once, the cost of repeating it is less than the cost of the initial use. Sopfr treats repeats as having the same cost as the initial use. Sopf treats repeats as having zero cost.

I wonder if it would make sense to instead use a single intermediate function between sopaf() and sopafr() that costed repeats at some fractional power of the repeat number (which is the absolute value of the prime exponent). I suggest, in the interests of keeping it pronounceable, we call it sopafry().
I replay this clip from the topic because toward the very end of our selection process for a final metric we decided between wbl1 and wybl1. When I found wybl1 with SoS(-1) 0.003240125 I about blew a gasket. But you preferred wbl1 due to its simplicity and how its better SoS(1) also indicated better fitability. However, since I think its clear that using the parameter `y` is psychoacoustically plausible (per the above exchange) and also improves SoS(-1) by roughly 15% between metrics which are differentiated only by its inclusion, I wonder if we might reconsider including it.

I know it's late in the game to suggest such a change, so feel free to dismiss it on those grounds alone. I missed the boat. That ship has sailed. Etc.

The first place where they disagree is actually in the top 10 5-rough ratios: wbl1 puts 13/1 more popular than 49/1, while wybl1 puts 49/1 more popular 13/1, which agrees with the Scala stats. Though it's only a matter of 463 vs 447 votes. And pretty much starting with rank 12 my ability to compare them breaks down; they're both pretty close to the real ranking, of course, since they both have extremely low SoS, but they are close in really different ways. Around rank 12, being one or two ranks off costs about 0.0001 SoS so it's not making too much a difference anymore.

I'm not sure if I'm really suggesting we should reconsider wybl1 anymore. I just thought I should at least put these facts out there.
Dave Keenan wrote: Fri Jul 03, 2020 2:28 am I think maybe we should make w positive and describe the numinator weighting function as log2(p) - w.
I don't remember ever seeing this! Sorry! We just plunged forward with w remaining negative.
cmloegcmluin wrote: Sat Jul 04, 2020 3:33 am But I doubt we'll have significant disagreement on what constitutes a chunk of complexity.
lulz
Dave Keenan wrote: Sun Jul 12, 2020 2:45 pm soa1pfar(n) + soa2pfar(d), where a1p = p → lb(p)+w, a2p = p → lb(p)+b, ar = r → ry
I totally didn't pick up on this a1p vs. a2p thing. I like it a lot better than ap vs. mp and would have used it had I noticed. Sorry! Good idea!
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