## developing a notational comma popularity metric

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

Dave Keenan wrote: Sat Aug 15, 2020 10:53 am OK. I agree to using the capitalisation "N2D3P9" in math expressions too.
Great.
Good work finding those earlier agreements re the topic of this thread. I think you'll likely find somewhere where I also agreed to not changing the title at such a late stage, despite it being not quite appropriate, because the forum software doesn't change the title of all the posts. You'd have to do that one post at a time.
I remember saying that myself! I'll haven't come across the evidence yet, but in any case, this is true.
But I note that there are two steps to get from a 5-rough-no-pop-rank to a badness metric. The intermediate step is the comma-no-pop-rank [just trying out possible terms here] whose job is to distinguish between commas that notate the same 5-rough pitch ratio (as well as those for different 5-rough pitch ratios). This includes a consideration of abs3exp and/or apotome-slope. This is useful independent of a badness measure, in determining which commas deserve symbols.
Okay, interesting.

What else gets included in a "badness measure", then? I found this: "A "badness" measure is typically a combination of a complexity measure and an error measure. But we have two kinds of complexity measure in this case." I wasn't aware that a badness measure was an established concept, let alone was I aware that it had conventions, and I also don't know what an error measure is in general or what it would be in our case.

What use would the intermediate no-comma-pop-rank have? You say it would be useful independent of a badness measure, but if the badness measure is what we would use to determine which commas deserve symbols (Magrathean accents), then I now know of zero uses for the no-comma-pop-rank-or-whatever in and of itself.
So we could choose to make the title of this thread come true by continuing on to that — then start a new thread for the badness, which includes a consideration of the error (in cents or fractions of a tina, or other ina).

If we start a new thread for comma-no-pop-rank, what should we call it?
The use of the adjective "notational" is messy. Sometimes it refers to the comma and sometimes the popularity. A notational comma is a comma used for notation (i.e. a comma that is used as the value of an accidental or diacritic symbol) as opposed to say a vanishing comma or tempered-out comma. It doesn't make sense (to me) to talk of a notational pitch ratio or notational 5-rough ratio. They aren't used for notation, they need to be notated. They are notated ratios, not notational ratios. But it does make sense to me to talk of the "notational popularity" of such (non-comma) ratios, i.e. how often they turn up needing to be notated, as opposed to say how often they occur as intervals between notes, or how much people like to hear them in chords.
Indeed, messy.

Agreed that ratios aren't "notational". I used "notatable" here, though I think I prefer your "notated". And even more so I prefer "notational popularity of ratios".
Hence "comma-no-pop-rank" could also be "no-comma-pop-rank", where "no" is short for "notational". But "5-rough-no-pop-rank" could not be "no-5-rough-pop-rank". Hence using the form "comma-no-pop-rank" to make the similarities and differences clear relative to "5-rough-no-pop-rank".
Agreed. I will try to be consistent henceforth in using "no-comma-pop-rank" and "5-rough-no-pop-rank".
BTW, Can you think of an easy way to generate a list of all the 5-rough ratios with an N2D3P9 of 130 or less? The idea would be to sort them on N2D3P9 and see which ones we have comma-symbols for (in primary roles only). Some would have more than one symbol, but are there any early in the list that have no symbol?
Well, first thought is: since N2D3P9 includes prime-limit, I could just grab the N2D3P9 of all 127-limit ratios. And then the 5-exponent would have to between -9 and 5 because \frac{5}{3}^9 is ~99.23 and \frac{5}{2}^5 is ~97.65 and if you go any bigger than 9 or 5 respectively then you'll exceed 130. I could repeat this logic for each prime. That's still pretty dumb; I'll still filter a bunch of >130 results in the end, and it'll be millions of N2D3P9's to check, but I think (?) I've crunched harder numbers on this topic ...

Where'd you get the 130 number from? Is that the N2D3P9 of your least favorite comma in Sagittal?
Last edited by Dave Keenan on Sun Aug 16, 2020 6:55 am, edited 1 time in total.
Reason: Corrected N2D3P5 typo to N2D3P9

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

Here's another fun graph.

Attachments
ArchiveVsN2D3P9.png

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

Dave Keenan wrote: Sat Aug 15, 2020 11:53 am Here's another fun graph.
Fun indeed! Which axis is which?

Here's the evidence that I talked about the possibility of having to update every reply's title individually: viewtopic.php?p=1927#p1927
And that was way back on page 6!
Ain't nobody updating >300 reply titles one by one. No way, no how.
So... you may have talked about it, too, of course, but I just haven't come across it yet.

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### Re: developing a notational comma popularity metric

cmloegcmluin wrote: Sat Aug 15, 2020 11:56 am Fun indeed! Which axis is which?
Archive vertical. I think you're always supposed to say "<vertical> versus <horizontal>". These are all the ratios in the archive that have N2D3P9_rank ≤ 130. Note several ratios having archive_rank = 604.5 (count = 1) and several others having archive_rank = 329 (count = 2). Archive rank 123.5 corresponds to count = 9. So it doesn't take much noise in the count to make a ratio's archive rank jump from 134.5 to 329 or 604.5.

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### Re: developing a notational comma popularity metric

cmloegcmluin wrote: Sat Aug 15, 2020 8:38 am Haha. Yes N2P9 is my second favorite character, after R2D2!
Ah Sorry. I was mistaken. The third droid whose parts were used was NR-N99, the Persuader-class droid enforcer.

You will believe this metric or else ....
I also think "N2D3P9" just has a certain charm, a certain je ne sais quoi. Though I stress it secondary-un-secondary-un-primary-un (w/ the same stress pattern as "counterdemonstration") ...
Or you could pronounce "N2D3P9" with the same stress pattern as (and rhyming with) "enter the free P mine" (a phosphate mine).

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### Re: developing a notational comma popularity metric

cmloegcmluin wrote: Sat Aug 15, 2020 11:28 am Well, first thought is: since N2D3P9 includes prime-limit, I could just grab the N2D3P9 of all 127-limit ratios. And then the 5-exponent would have to between -9 and 5 because \frac{5}{3}^9 is ~99.23 and \frac{5}{2}^5 is ~97.65 and if you go any bigger than 9 or 5 respectively then you'll exceed 130. I could repeat this logic for each prime. That's still pretty dumb; I'll still filter a bunch of >130 results in the end, and it'll be millions of N2D3P9's to check, but I think (?) I've crunched harder numbers on this topic ...
Good approach, thanks, but far wider bounds than necessary.

For ratios of the form p/1 where p is prime, N2D3P9(p/1) = p/2 × 1/1 × p/9 = p2/18. So the largest p for which N2D3P9(p/1) ≤ 130 is the largest prime less than √130 × 18. Namely p = 47.
Where'd you get the 130 number from? Is that the N2D3P9 of your least favorite comma in Sagittal?
No. I just pulled it out of my arsenal. I guess that joke doesn't work for Americans, due to a difference in the spelling and pronunciation of a certain profanity.

130 was the limit I chose for the two graphs above, because it seemed likely, based on the first graph beginning to curve away from the straight line, above 130, that the archive was missing ratios above that.

It also seemed appropriate because it was a little larger than the number of minas in a half-apotome, which is the number of Olympian symbols with independent comma definitions.

And now, its prime limit of 47 also confirms it as a good choice. Maybe we'll find that George was justified in giving 47/1 an Olympian symbol after all. [Edit: No. 47/1 was competing with 23/11 for the 75th mina. I get N2D3P9(47) ≈ 123 and N2D3P9(23/11) ≈ 108. So 23/11 still wins.]

For ratios of the form 5n5/1, N2D3P9(5n5/1) = (5/2)n5 × 1/1 × 5/9 = (5/2)n5 × 5/9. So the largest n5 for which N2D3P9(5n5/1) ≤ 130 is the largest integer less than log5/2(130 x 9/5) = log5/2(234). Namely n5 = 5. So max is 55 = 3125.

But it's so close to including 56 that I'd like to increase the range to N2D3P9 ≤ 136, so 56 = 15625 is included. Max p is still 47.

Numerator must be greater than denominator, so the greatest power of 5 in the denominator is 52, because 29/25 just squeaks in with N2D3P9(29/25) = 129.784.
Last edited by cmloegcmluin on Sun Aug 16, 2020 7:59 am, edited 1 time in total.
Reason: Correcting occurrences of "N2D2P9" to "N2D3P9". It would appear you accidentally swapped a C3PO part for an R2D2 part :P

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

To find the greatest prime in the denominator, we look at ratios of the form pi+1 / pi, where N2D3P9 = pi+1/2 × pi/3 × pi+1/9. So we want pi+12 × pi ≤ 136×2×3×9 = 7344. The cube-root of 7344 is 19.438, and the maximum prime in the denominator is 17.

Max power of any prime p in the numerator is $$\lfloor \text{log}_{p/2}(136×9/p)\rfloor = \lfloor \text{log}_{p/2}(68×9)\rfloor - 1$$.

Floor-function brackets, ⌊ and ⌋, are typed as ⎄⇧L[ and ⎄⇧L] using WinCompose. \lfloor and \rfloor in $$\LaTeX$$.

PrimeMax exponentMax exponentNumber of
in numeratorin denominatorexponent values
5629
7427
11214
13214
17 thru 23113 (for 3 primes)
29 thru 47102 (for 6 primes)
>47001

[Edit: Corrected for 19 and 23 in denominator as 25/19 and 25/23.]
[Edit 2: Corrected for 49 in the denominator as 125/49.]

This is just the kind of pattern I have noticed in the Scala archive data over the years. What a great metric! I'm sure glad you didn't give up on your code. We never would have found that metric otherwise.

So a brute-force generate-and-test (to find all 5-rough ratios with N2D3P9 ≤ 136) would generate 9×7×4×4×33×26 = 1 741 824 monzos before whittling them down to approximately 136 monzos. That's the beauty of the -P9 version, as opposed to the -P1 version — you know there will be approximately n ratios having N2D3P9 ≤ n.
Last edited by cmloegcmluin on Sun Aug 16, 2020 3:39 pm, edited 2 times in total.

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### Re: developing a notational comma popularity metric

cmloegcmluin wrote: Sat Aug 15, 2020 11:28 am What else gets included in a "badness measure", then? I found this: "A "badness" measure is typically a combination of a complexity measure and an error measure. But we have two kinds of complexity measure in this case." I wasn't aware that a badness measure was an established concept, let alone was I aware that it had conventions, and I also don't know what an error measure is in general or what it would be in our case.
Yes. That's relevant.

The error, in our case, is the difference, either in cents, or in decimal fractions of a tina, between a whole number of tinas and a comma proposed as its definition.
What use would the intermediate no-comma-pop-rank have? You say it would be useful independent of a badness measure, but if the badness measure is what we would use to determine which commas deserve symbols (Magrathean accents), then I now know of zero uses for the no-comma-pop-rank-or-whatever in and of itself.
I may have overstated its usefulness on its own. It is not useful on its own for giving commas to tina symbols. But note that I was talking about giving symbols to commas, which is subtly different from giving commas to symbols. That's kind of how we started out, with a list of the most popular commas and then we designed symbols for them. So it would only really be of use in revisiting the assignment of single-shaft symbols to see how good a job we did.
So we could choose to make the title of this thread come true by continuing on to that — then start a new thread for the badness, which includes a consideration of the error (in cents or fractions of a tina, or other -ina).

If we start a new thread for comma-no-pop-rank, what should we call it?
That was pretty much a rhetorical question, intended to make you realise that this thread already has that name.
This is fairly useless, but at least it's something: https://en.xen.wiki/w/Badness

Badness is about establishing how much accuracy you are willing to give up to reduce complexity or vice versa. Another approach is to not to define a badness measure, but just plot complexity against error, for each candidate ratio for some number of tinas, on a scatter plot, and decide on an acceptable region on the plot.

I think we should eventually (after the writeup of the 5-rough-no-pop-rank) continue on to the comma-no-pop-rank in this thread (to make its title become true), then start a new thread for the badness (or complexity vs error plot region).

Hence "comma-no-pop-rank" could also be "no-comma-pop-rank", where "no" is short for "notational". But "5-rough-no-pop-rank" could not be "no-5-rough-pop-rank". Hence using the form "comma-no-pop-rank" to make the similarities and differences clear relative to "5-rough-no-pop-rank".
Agreed. I will try to be consistent henceforth in using "no-comma-pop-rank" and "5-rough-no-pop-rank".
That's not what I suggested above (now hilited).

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### Re: developing a notational comma popularity metric

Dave Keenan wrote: Sat Aug 15, 2020 1:16 pm Archive vertical. I think you're always supposed to say "<vertical> versus <horizontal>".
I wasn't aware there was a convention for that, but I do see evidence for it.
Dave Keenan wrote: Sat Aug 15, 2020 1:47 pm Ah Sorry. I was mistaken. The third droid whose parts were used was NR-N99, the Persuader-class droid enforcer.
Ah yes, the "snail droid". How could I have forgotten....

Lauren and I do, actually, share a strange fascination with Wat Tambor, the lead engineer for the Techno Union (which manufactures the NR-N99), so I'm amused to share this odd affiliation with him now.
Or you could pronounce "N2D3P9" with the same stress pattern as (and rhyming with) "enter the free P mine" (a phosphate mine).
Another option, yes! Nothing binding us to single-word stress-patterns. Thanks as always for thinking outside the box.
Good approach, thanks, but far wider bounds than necessary.
Thanks for the correction from your original "but wrong results". I was planning to quibble that my results were okay, just quicker and dirtier than yours.

Thanks for crunching the numbers and preparing that table of min and max prime exponents for me. That's exactly everything I need to get the answer to your question without really having to think about anything... just hammer out a little code.
No. I just pulled it out of my arsenal. I guess that joke doesn't work for Americans, due to a difference in the spelling and pronunciation of a certain profanity.
You could've at least given the joke a chance!
130 was the limit I chose for the two graphs above, because it seemed likely, based on the first graph beginning to curve away from the straight line, above 130, that the archive was missing ratios above that.
Got it.
It also seemed appropriate because it was a little larger than the number of minas in a half-apotome, which is the number of Olympian symbols with independent comma definitions.
Yes, that makes sense. Oy, I'm wincing at how much I had to think about it to verify my understanding of that statement. I suppose it's a good thing — by the time I get back to working on educational materials for Sagittal I will have forgotten everything about it and so that'll help me approach it as a newcomer again
And now, its prime limit of 47 also confirms it as a good choice. Maybe we'll find that George was justified in giving 47/1 an Olympian symbol after all. [Edit: No. 47/1 was competing with 23/11 for the 75th mina. I get N2D3P9(47) ≈ 123 and N2D3P9(23/11) ≈ 108. So 23/11 still wins.]
That's great news.

Out of curiosity I asked what N2D3P9 had to say about the 121k vs 1225k issue. Interestingly, while 243/242 has lower SoPF>3 than 19683/19600 (22 vs. 24), it has higher N2D3P9 (≈16 vs. ≈12). We'll wait on the comma-no-pop-rank metric though to make a call on that front.
Dave Keenan wrote: Sat Aug 15, 2020 3:49 pm This is just the kind of pattern I have noticed in the Scala archive data over the years. What a great metric! I'm sure glad you didn't give up on your code. We never would have found that metric otherwise.
I'm glad, too! Thanks for saying so.

So as you know I'm currently reading my way back over this topic from page 1 and taking notes about the different threads we pursued (including the blue threads of death... ). So far we've each only almost given up once apiece (I seem to recall almost giving up three times myself, you maybe only twice). It's occasionally agonizing how close you or I come to wbl1 before my code gave us some confidence that we couldn't do better.
So a brute-force generate-and-test (to find all 5-rough ratios with N2D3P9 ≤ 136) would generate 9×6×4×4×3×28 = 663552 monzos before whittling them down to approximately 136 monzos. That's the beauty of the -P9 version, as opposed to the -P1 version — you know there will be approximately n ratios having N2D3P9 ≤ n.
I agree, the 9 is paying off.
Dave Keenan wrote: Sat Aug 15, 2020 10:07 pm The error, in our case, is the difference, either in cents, or in decimal fractions of a tina, between a whole number of tinas and a comma proposed as its definition.
That makes total sense. 'Spose I coulda come up with that myself, sorry.
I may have overstated [the intermediate comma-no-pop-rank's] usefulness on its own. It is not useful on its own for giving commas to tina symbols. But note that I was talking about giving symbols to commas, which is subtly different from giving commas to symbols. That's kind of how we started out, with a list of the most popular commas and then we designed symbols for them. So it would only really be of use in revisiting the assignment of single-shaft symbols to see how good a job we did.
That makes total sense, too. As I mentioned earlier in this topic I have already set up a test in my code to do just this revisiting. Its description is "checks that every symbol's primary comma is its best-ranked comma in its secondary comma zone according to our metric". I worried for a moment that SoPF>3 didn't pass this test at all, but then I realized that other factors were considered including 3-exponent stuff. You followed up saying that not much else besides abs3exp and apotome slope was involved. According to that, then, we would expect the test to pass when our comma-no-pop-rank function is used.

I still have no idea how we're going to weight abs3exp and apotome slope. What objective data do we have to model that after? Besides of course Sagittal commas, which if — and this is a big "if" — we no longer cared about our reputations, we could just fit it to those and call it day.

I could imagine it might be something to the effect of: for ratios of a given N2D3P9 rank, what does the distribution of apotome slopes look like? Does this comma under consideration fall on the better or worse side of this distribution?
I think we should eventually (after the writeup of the 5-rough-no-pop-rank) continue on to the comma-no-pop-rank in this thread (to make its title become true), then start a new thread for the badness (or complexity vs error plot region).
Understood re: badness metric vs complexity-vs-error-plot, and agreed on the plans for a new topic.
Hence "comma-no-pop-rank" could also be "no-comma-pop-rank", where "no" is short for "notational". But "5-rough-no-pop-rank" could not be "no-5-rough-pop-rank". Hence using the form "comma-no-pop-rank" to make the similarities and differences clear relative to "5-rough-no-pop-rank".
Agreed. I will try to be consistent henceforth in using "no-comma-pop-rank" and "5-rough-no-pop-rank".
That's not what I suggested above (now hilited).
Bah! I'm a doofus. FWIW, I understood 90% of your intentions w/r/t this issue; I just misread a couple things at the very end. I totally get it now and I agree with your reasoning. I started correcting the previous post but I honestly think it'll just make things more confusing; I'll get it right moving forward.

Man, I look forward to having more free time and thus being able to take my time responding here rather than just sort of frantically responding whenever I get a minute and making these sorts of mistakes.

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### Re: developing a notational comma popularity metric

I checked all 1492992 ratios in that range. 128 [Edit: 131] of them had N2D3P9 < 136, which I think qualifies as approximately the same as 136.

[Edit: Changed "0.11" to "0.33" for 1/1.]
[Edit: Changed "0.33" to "1.00" for 1/1.]
[Edit: Changed ratios to directed form and zero-indexed levels.]
[Edit: Added actual ranks and estimate ranks.]

2,3-equivalentexactlyintroducing
pitchnotatingsymbolScalaScala
ratioJIsubsetN2D3P9archivearchive
classN2D3P9symbolsindicesrankrankoccurrences
1/110117624
5/11.39 3, 0225371
7/12.72 0, 3, 3333016
25/13.47 3, 0441610
7/54.54 0, 3551318
11/16.72 3, 0, 0661002
35/16.81 3, 4, 0, 077875
125/18.68 3, 4, 488492
13/19.39 4, 4, 4910447
49/19.53 2, 2, 2109463
11/511.2 1, 3, 31111339
25/711.34 3, 41214312
13/515.65 4, 2, 21316205
11/715.69 1, 11412324
49/515.88 2, 41515246
17/116.06 2, 11613318
55/116.81 1, 3, 31724119
175/117.01 4, 4, 41817168
19/120.06 2, 21918166
625/121.7 4, 4, 42021143
13/721.9142120145
65/123.47 4, 4, 4225040
77/123.5342325111
245/123.82 4, 32419165
49/2526.4732523134
17/526.76 3, 42626108
25/1128.01274742
125/728.364283362
23/129.39 2, 22922136
91/132.86 4, 4305730
343/133.354313170
19/533.43 3, 2322797
13/1134.43 4, 4332989
121/136.973442.546
17/737.46 4, 4354050
25/1339.12 4, 4, 43652.534
35/1139.213382892
55/739.213383461
77/539.2143835.555
85/140.143407820
275/142.014411477
875/142.534427621
29/146.72433267
19/746.824437.552
23/548.98 2, 4, 4454445
95/150.14467223
143/151.642476626
31/153.39 4, 4, 4483080
3125/154.2544952.534
35/1354.77516825
65/754.77 4, 451102.511
91/554.77 4, 451102.511
49/1154.92535433
343/555.5845455.531
119/156.1955252.53
325/158.68456604.51
385/158.8245737.552
17/1158.87 4, 4, 45835.555
1225/159.55 4, 4594147
169/161.034608614
121/561.62611477
77/2565.35 4, 4626327
125/4966.17636327
25/1766.9464134.58
23/768.57654742
17/1369.574664742
125/1170.02671477
133/170.19683292
625/770.896911310
115/173.4770604.51
19/1173.54 2, 27155.531
37/176.06 4, 47242.546
49/1376.684731477
29/577.877459.528
455/182.15475186.55
539/182.3576186.55
1715/183.37777621
25/1983.5637879.519
55/1386.0680217.54
65/1186.064801646
143/586.06809013
121/786.27821646
19/1386.913834544
187/188.3184217.54
31/588.98856825
91/2591.284863292
55/4991.54873951
605/192.438894.512
343/2592.63896825
41/193.39907223
35/1793.6692123.59
85/793.6692-0
119/593.6692217.54
125/1397.8494604.51
175/1198.0395.5102.511
275/798.03 4, 495.51477
425/1100.35973292
169/5101.71983292
121/25102.799217.54
43/1102.721005829
161/1102.86101604.51
221/1104.36102252.53
1375/1105.03103-0
4375/1106.3441041477
23/11107.76 4, 41056327
29/7109.021068315
209/1110.31107604.51
19/17113.651087223
637/1115.01109604.51
2401/1116.7211079.519
145/1116.81111-0
35/19116.991131646
95/7116.99113604.51
133/5116.99113604.51
77/13120.49116134.58
91/11120.49 4, 4, 4116217.54
143/7120.494116123.59
25/23122.451181646
47/1122.721195137
31/7124.5712011310
475/1125.3512194.512
37/5126.761227024
23/13127.351236327
65/49127.791243292
715/1129.1125186.55
847/1129.412611310
29/25129.7812711310
247/1130.36128604.51
49/17131.1241293292
155/1133.47130604.51
15625/1135.631316327

I have to run to help make dinner, so I haven't had time to check this over carefully, but I thought I'd get some results to you ASAP.

By the way, if you take the N2D3P9 of the primary commas for every Sagittal symbol in the JI notation and then sort them, this is what it looks like:

So eleven of them are N2D3P9 > 136. There's one comma that's particularly bad, the 14641k; with four elevens, its SoPF>3 is 44, but its N2D3P9 comes out ≈1118. Oof. Seems likely that any comma-no-pop-rank incorporating N2D3P9 will be able to find something better than that.
Attachments
sorted n2d3p9 of sagittal JI notation primary commas.png