## Magrathean diacritics

cmloegmcluin
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### Re: Magrathean diacritics

Dave Keenan wrote: Thu Sep 03, 2020 2:00 pm Thanks for that table. It is certainly of interest, but I don't see how it lets me verify which metacomma occurs first for each tina.
What do you mean by "first"? I imagine there is some pattern to the cells you chose to give a thick black border, among the ones highlighed in red for being within 9 tinas in size. And I expect that this pattern is "first" for some definition of first. But I can't figure out what it is.

And I don't understand what it means for a metacomma to "occur" for a tina.
I was not taking any notice of whether commas were exactly notated (without using tina accents), as that is something that could change.
What do you mean by "change"? Do you mean changing some of the comma definitions for the Extreme level, as previously discussed elsewhere?

I can't quite parse this sentence. But I thought that taking notice of whether commas are exactly notated is central to our present work in at least some sense.
It would make more sense to me if you did it for commas that are not yet exactly notated without using minas (or tinas), since minas are essentially triple-tinas.
I think this "without using tinas" concept must be key, but I don't understand what it means.

What do you mean by "did it"?
For example, the table above fails to suggest the obvious definitions for 3 tinas and 6 tinas.
Do you mean the commas we already have assigned to and  , the 455n and 65:77n, or 4096/4095 and 2080/2079, respectively? I see them both in the table...

It tells me that the 455n would help us exactly notate 35/13, which is the 5th most popular 2,3-free-class which Sagittal does not yet exactly notate. It tells us that it would allow exact notation of 35/13 since it is the case that we already have a symbol for 169/1. It also tells us that it allows the notation of 3 additional as-of-yet not-exactly-notated commas in this upper slice of popularity. It also tells us that if we want to notate 35/13, we have only one other choice: 1716/1715, a candidate for the 7 tina, which would notate 35/13 relative to 49/11.
Dave Keenan wrote: Thu Sep 03, 2020 11:01 pm I think you've run with what I wrote here:
Dave Keenan wrote: Tue Sep 01, 2020 8:23 am In this post, I noted that the first 2,3-free ratio in the above list that does not yet have a sagittal symbol, 25/11, is a candidate to be notated by a 1 tina accent applied to the existing symbol for 49/11  . That makes 1 tina = 1225/121n which is what George Secor suggested for it. There may be other un-notated but moderately popular ratios that could be notated with tina accents. This might be the best way to assign commas to the accents for 1 thru 9 tinas.
and that's fine. The more approaches the better.

But I'm going with what I wrote later in that post:
... it would be interesting to see that single lowest-slope comma, and its size in cents, listed against each ratio in the above list. Then search for cases where two such commas differ by 9 tinas or less, and consider assigning each tina accent to that correctly-sized difference whose pair of 2,3-free ratios have the lowest maximum N2D3P9.
I don't see the difference between these two things. I thought they were the same thing. I thought the table I provided addresses that thing.
I've checked my results above, and have attached the spreadsheet I used, with bold outlines on the cells corresponding to the meta-commas I listed.
I think your table is just a different look at the same data. Is it not? You have the LAAS-per-2,3-free-class on the vertical and other LAAS-per-2,3-free-class on the horizontal, with the tinas in the cells. I just pivoted the table so that the tinas are on the vertical and the other LAAS-per-2,3-free-class are in the cells.
I planned to give that list of tinas and their lowest-N2D3P9 metacommas again here, but with the pairs of ranks converted to meta-comma names, so we could compare them to the yellow-highlighted rows in the tables you mentioned from way back. But I ran out of time tonight.
No worries.
I'd be pleased if you can generate another list of 131 2,3-free ratios, this time with the notational comma for each, that has lowest abs3exp (LATE) (and smallest size, in the case of a tie), instead of lowest absolute apotome slope (LAAS).
I think I'll need to get my head around the problem a little better before proceeding. Maybe a video call would be good once we're both up.

Dave Keenan
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### Re: Magrathean diacritics

Thank you for asking these questions.
cmloegcmluin wrote: Fri Sep 04, 2020 6:30 am What do you mean by "first"? I imagine there is some pattern to the cells you chose to give a thick black border, among the ones highlighed in red for being within 9 tinas in size. And I expect that this pattern is "first" for some definition of first. But I can't figure out what it is.
I mean first in order of increasing N2D3P9, which corresponds to first in reading order (left to right, top to bottom) in my spreadsheet, provided we stay within the lower-left triangle of my difference/metacomma matrix/2D-table. The upper-right triangle is redundant but I couldn't be bothered eliminating it.

If we ignore the second example I gave for 7 tinas, and the one I gave for 7.5 tinas, then each of the black bordered cells is the first time that whole number of tinas occurs (as the rounded value of some metacomma) as we progressively add commas in N2D3P9 order.

In the case of 7 tinas, two such metacommas appear at the same time (same N2D3P9 level), in which case the N2D3P9 rank of the earlier comma of each pair can be used as a tie-breaker in deciding which metacomma should become the rational definition of 7 tinas. Which is the same as being first in reading order.
And I don't understand what it means for a metacomma to "occur" for a tina.
Each rounded tina value corresponds to multiple different metacomma ratios in the matrix. A cell whose value rounds to n tinas, I'm calling "an occurrence of a metacomma for the nth tina".
I was not taking any notice of whether commas were exactly notated (without using tina accents), as that is something that could change.
What do you mean by "change"? Do you mean changing some of the comma definitions for the Extreme level, as previously discussed elsewhere?
Yes. That's what I mean.
I can't quite parse this sentence. But I thought that taking notice of whether commas are exactly notated is central to our present work in at least some sense.
I don't see it that way. Of interest? Certainly. But central? I don't think so.

If there was ever a complete insane-JI-precision-level, with a symbol for every one of the 405 tinas in the half-apotome, some of the ratios that presently have a symbol that includes a 1-mina accent, would need to instead use an accent for 2 or 4 tinas. Similarly some ratios that presently have a symbol that includes a 2-mina accent, would need to instead use an accent for 5 or 7 tinas.

So it doesn't make sense to exclude from consideration, metacommas between commas that are both presently notated, unless they both have symbols that are free of mina accents. And I think it so unlikely that any two mina-free symbols would have a sub-9.5-tina metacomma between their comma definitions, that I choose to entirely ignore, in this search for tina definitions, whether a comma presently has a symbol or not.
It would make more sense to me if you did it for commas that are not yet exactly notated without using minas (or tinas), since minas are essentially triple-tinas.
I think this "without using tinas" concept must be key, but I don't understand what it means.
I was suggesting that you should only exclude from consideration, already-notated commas whose symbol does not include any mina accents. I probably should have left off the "(or tinas)" here. I was really just including the obvious, since anything that is already notated, has a symbol that does not include any tina accents, except in so far as a 1-mina accent is also a 3-tina accent and a 2-mina accent is also a 6-tina accent.
What do you mean by "did it"?
For example, the table above fails to suggest the obvious definitions for 3 tinas and 6 tinas.
Do you mean the commas we already have assigned to and  , the 455n and 65:77n, or 4096/4095 and 2080/2079, respectively?
Yes.
I see them both in the table...

It tells me that the 455n would help us exactly notate 35/13, which is the 5th most popular 2,3-free-class which Sagittal does not yet exactly notate. It tells us that it would allow exact notation of 35/13 since it is the case that we already have a symbol for 169/1. It also tells us that it allows the notation of 3 additional as-of-yet not-exactly-notated commas in this upper slice of popularity. It also tells us that if we want to notate 35/13, we have only one other choice: 1716/1715, a candidate for the 7 tina, which would notate 35/13 relative to 49/11.
Maybe I don't understand how you plan to use this table. Why don't you go ahead and give us your proposal for the metacomma definition for each whole tina.

The way I read it (past tense), you would give it to the leftmost metacomma in your matrix, which means you would define 3 tinas as 4225/4224 and 6 tinas as 2176/2175.
I don't see the difference between these two things. I thought they were the same thing. I thought the table I provided addresses that thing.
The first is only considering metacommas between commas that are not already both notated. The second is considering all metacommas. Of course in both cases the commas being considered are only the single "best" comma for each 2,3-equivalence-class. So they share that recent innovation.
I think your table is just a different look at the same data. Is it not? You have the LAAS-per-2,3-free-class on the vertical and other LAAS-per-2,3-free-class on the horizontal, with the tinas in the cells. I just pivoted the table so that the tinas are on the vertical and the other LAAS-per-2,3-free-class are in the cells.
I agree about the pivoting, but the difference is in which LAAS-per-2,3-free-classes are included on the horizontal. I include them all. You only include those which are not already notated.
I think I'll need to get my head around the problem a little better before proceeding. Maybe a video call would be good once we're both up.
I felt I could be clearer in text. I hope it worked.

Dave Keenan
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### Re: Magrathean diacritics

LASS-based first metacommas:

1 tina as 121:1225n = 121:49·25n
2 tinas as 29:275n = 29:25·11
3 tinas as 455n = 35·13n
4 tinas as 7:3025n = 7:121·25n
5 tinas as 25:2401n = 25:74n
6 tinas as 65:77n
7 tinas as 7:2125n = 7:125·17n
7 tinas as 7:425n = 7:25·17n (a close second, the 5s-complement of the first)
7.5 tinas as 2875n = 125·23n (This could give us a ratio for the dot, as a meta-meta-comma with 7 or 8 tinas)
8 tinas as 5:253n = 5:23·11n
9 tinas as 539n = 49·11n

Some previous lists for comparison:
viewtopic.php?p=1613#p1613
viewtopic.php?p=1826#p1826
viewtopic.php?p=1832#p1832

There's a lot of agreement there. The most difficult to agree on are 7, 2, 1 and 0.5.

cmloegmcluin
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### Re: Magrathean diacritics

Dave Keenan wrote: Fri Sep 04, 2020 9:33 am
cmloegcmluin wrote: Fri Sep 04, 2020 6:30 am What do you mean by "first"?
I mean first in order of increasing N2D3P9, which corresponds to first in reading order
ohhhhhhhhhhhh. Okay, yes. I see it now.
the upper triangle is redundant but I couldn't be bothered eliminating it).
I've made enough internal interval matrices in my day to have recognized at least that part myself (and also the feeling of not bothering to clean it up. They have a perfect word for that in Japanese: めんどくさい).
And I don't understand what it means for a metacomma to "occur" for a tina.
Each rounded tina value corresponds to multiple different metacomma ratios in the matrix. A cell whose value rounds to n tinas, I'm calling "an occurrence of a metacomma for the nth tina".
ohhhhhhhhhhhh. Okay, yes I see that now too.

I guess I'd been stuck thinking like this: how can a metacomma occur for a different tina other than the tina it is? I didn't get that you meant any metacomma for any tina, starting from some N2D3P9-based "beginning" and moving forward. My table's structure and approach was more like this: okay for each row, I've got this specific metacomma which is for this specific tina... and now what can I do with it?
I thought that taking notice of whether commas are exactly notated is central to our present work in at least some sense.
I don't see it that way. Of interest? Certainly. But central? I don't think so.
Okay. I think I was just majorly thrown off by the inclusion of the Sagittal symbol, if present, for each LAAS notating comma. And got fixated on that. Or at least it deeply colored my processing of the problem space. It's not your fault. You did at some point say it was merely a nice-to-have.

I am reprocessing your most recent comment back on the N2D3P9 thread about this list forming a new basis for Sagittal. I can see that now, now that I understand what you're trying to do here.

A new "basis" doesn't necessarily mean dramatic change. I believe you may mean something more like: an improved justification for some of the design decisions. You have probably noticed that, among the 9-tina-and-below-sized intervals between pairs of LAAS notating commas for the 2,3-free pitch classes sorted by N2D3P9 estimated popularity, the sequence of their tina sizes in reading order (AKA decreasing combined popularity of the pair of 2,3-free pitch classes) begins 3, 3, 6, 3, 9, 6, 9, 3, 5... The first EIGHT of them are all multiples of 3! And on top of that, all eight of those are within not only a sixth of a tina from a whole tina, but within a TENTH! If that's not a compelling argument for the mina being the defining interval of the Extreme precision level, I don't know what is!

(This kind of "click" moment is exactly the kind of fulfillment I was referring to my email this morning, by the way... victories in designing Sagittal notations really feel good cuz you have to work so hard for them)
I was suggesting that you should only exclude from consideration, already-notated commas whose symbol does not include any mina accents.
I get this idea now too.
Maybe I don't understand how you plan to use this table. Why don't you go ahead and give us your proposal for the metacomma definition for each whole tina.
No need. Your table is awesome.

I could flip a couple switches in my code and regenerate my table but with all LAAS commas across the horizontal, not only the not-yet-exactly notated ones. But then it would still leave you with a ton of work to do. When I posted those tables, as I may have said, I didn't have much time to explain or introduce them. But what I meant to imply was that the next step would be to start highlighting rows and columns, like a big crazy logic puzzle, trying to optimize for the most, most-popular ratios getting newly exacted notated. It wasn't going to be easy or systematic. It would have sucked. I just didn't see the solution to the problem you'd already seen in this more objective reading-order approach. I suppose mine might have been able to notate more new intervals exactly, but your approach feels more in line with the emergent/bottom-up design principle precedents in Sagittal. Whodathunk the one of us who is its co-creator who has been working on it for decades already woulda come up with it

And also, we're just trying to assign the primary commas to the symbols which consist of these tina accents with a bare shaft, right? When we get to the project of assigning commas to every tina in the Insane precision level (let's tentatively schedule that project for late 2025 or so, yeah?) we can worry about choosing commas that optimize for that sort of coverage. I remind you that the actual values of the mina accent elements in the Extreme precision level were not nearly as consistent as you thought they were, taking on something like 25 different values. So it'd definitely be precedented for the tina accent elements in Insane precision symbols to represent other ratios of about that tina size.

Yeah, actually, so why would we even agonize over exact notation at this stage at all?!

Maybe you were already there. That only just registered for me.
The way I read it (past tense), you would give it to the leftmost metacomma in your matrix, which means you would define 3 tinas as 4225/4224 and 6 tinas as 2176/2175.
So, no, that's not how I was planning to go about it. But my way would be electing the way of pain.

Said another way: you're like a hardened vet in this domain. I walk into a jungle like this and all I can manage to do is turn one puzzle into a new puzzle. But you've turned it into a solution! I didn't even realize we were this close.
I felt I could be clearer in text. I hope it worked.
It did!

Okay, I can get you that LATE-based list soon. I assume that smaller size is the tie-breaker because a matrix of overall smaller commas is more likely to net us some tina-sized metacommas.

Dave Keenan
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### Re: Magrathean diacritics

I hope you didn't miss my latest, because it was submitted seconds before your latest.
cmloegcmluin wrote: Fri Sep 04, 2020 11:12 am I've made enough internal interval matrices in my day to have recognized at least that part myself (and also the feeling of not bothering to clean it up. They have a perfect word for that in Japanese: めんどくさい).
Ha!
A new "basis" doesn't necessarily mean dramatic change. I believe you may mean something more like: an improved justification for some of the design decisions.
Yes! Exactly!
You have probably noticed that, among the 9-tina-and-below-sized intervals between pairs of LAAS notating commas for the 2,3-free pitch classes sorted by N2D3P9 estimated popularity, the sequence of their tina sizes in reading order (AKA decreasing combined popularity of the pair of 2,3-free pitch classes) begins 3, 3, 6, 3, 9, 6, 9, 3, 5... The first EIGHT of them are all multiples of 3! And on top of that, all eight of those are within not only a sixth of a tina from a whole tina, but within a TENTH! If that's not a compelling argument for the mina being the defining interval of the Extreme precision level, I don't know what is!
Yes! I'm glad you were impressed by that too.
Whodathunk the one of us who is its co-creator who has been working on it for decades already woulda come up with it
Ha. Thanks. But don't sell yourself short. Just keep reading.
And also, we're just trying to assign the primary commas to the symbols which consist of these tina accents with a bare shaft, right? When we get to the project of assigning commas to every tina in the Insane precision level (let's tentatively schedule that project for late 2025 or so, yeah?) we can worry about choosing commas that optimize for that sort of coverage. I remind you that the actual values of the mina accent elements in the Extreme precision level were not nearly as consistent as you thought they were, taking on something like 25 different values. So it'd definitely be precedented for the tina accent elements in Insane precision symbols to represent other ratios of about that tina size.
And with that, you've blown my current approach clean out of the water.

Since my approach is based on metacommas, it is totally irrelevant to assigning the primary commas to the symbols which consist of these tina accents with a bare shaft. And you're right that that's what we put into the SMuFL documentation.

So we shouldn't be looking at metacommas at all! We should just go up the N2D3P9 list until we find actual commas of the right size, i.e. as the single best comma for some 2,3-equivalence class. And we can just use the LATE comma for each class since LATE is indistinguishable from LAAS at those sizes. And as you pointed out, they are unlikely to ever be used for EDO notation (except the 8539-EDO notation that you get for free).

We'll probably need the 307 list (maybe more) to get all the tinas. But it's a much simpler problem. Thanks!
Yeah, actually, so why would we even agonize over exact notation at this stage at all?!

Maybe you were already there. That only just registered for me.
I've often encouraged less agonising over these defs, but not exactly for that reason.
Said another way: you're like a hardened vet in this domain. I walk into a jungle like this and all I can manage to do is turn one puzzle into a new puzzle. But you've turned it into a solution! I didn't even realize we were this close.
Ha! And now you've made it even closer.
Okay, I can get you that LATE-based list soon. I assume that smaller size is the tie-breaker because a matrix of overall smaller commas is more likely to net us some tina-sized metacommas.
Make that laté a double-shot. i.e. Do the whole 307 list.

Originally, I suggested that tie-breaker because that's what we did in deciding the Prime Factor notation. But now it's obvious. We're more likely to net us some tina-sized commas. Forget meta-commas.

cmloegmcluin
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### Re: Magrathean diacritics

2,3-equivalentbestbestbest
pitchnotatingnotatingnotating
ratioN2D3P9commacommacomma
classranktinasmonzosymbol
1/11166.945[ -19 12 ⟩
5/12153.042[ -4 4 -1 ⟩
7/13194.015[ 6 -2 0 -1 ⟩
25/14139.139[ 11 -4 -2 ⟩
7/55 40.973[ 10 -6 1 -1 ⟩
11/16379.098[ -5 1 0 0 1 ⟩
35/17347.058[ 2 2 -1 -1 ⟩
125/18292.181[ 7 0 -3 ⟩
13/19344.049[ -10 4 0 0 0 1 ⟩
49/110254.024[ -4 -1 0 2 ⟩
11/511276.859[ -2 2 1 0 -1 ⟩
25/712 54.876[ -5 2 2 -1 ⟩
13/513311.909[ 3 -1 1 0 0 -1 ⟩
11/714 68.941[ 7 -4 0 1 -1 ⟩
49/515401.934[ -3 4 1 -2 ⟩
17/116104.824[ -12 5 0 0 0 0 1 ⟩
55/117226.056[ -1 -3 1 0 1 ⟩
175/118308.900[ -9 1 2 1 ⟩
19/119 24.039[ -9 3 0 0 0 0 0 1 ⟩
625/120363.777[ -14 3 4 ⟩
13/721270.935[ -7 5 0 1 0 -1 ⟩
65/122191.007[ -6 0 1 0 0 1 ⟩
77/123185.083[ -11 3 0 1 1 ⟩
245/124100.982[ 0 -5 1 2 ⟩
49/2525248.892[ 1 0 2 -2 ⟩
17/526 90.921[ 3 -3 -1 0 0 0 1 ⟩
25/1127123.817[ 2 -2 2 0 -1 ⟩
125/728 98.166[ 1 2 -3 1 ⟩
23/129117.732[ 5 -6 0 0 0 0 0 0 1 ⟩
91/130 16.911[ -3 6 0 -1 0 -1 ⟩
343/131 60.009[ -10 1 0 3 ⟩
19/532 10.136[ 6 -5 -1 0 0 0 0 1 ⟩
13/1133 35.049[ 5 -3 0 0 1 -1 ⟩
121/134 50.803[ -1 5 0 0 -2 ⟩
17/735298.839[ -6 3 0 -1 0 0 1 ⟩
25/1336158.866[ 7 -5 2 0 0 -1 ⟩
35/1138 82.844[ -8 4 1 1 -1 ⟩
55/738221.983[ 3 0 -1 1 -1 ⟩
77/538171.180[ 4 -5 -1 1 1 ⟩
85/140 48.218[ 8 -1 -1 0 0 0 -1 ⟩
275/141239.959[ -16 5 2 0 1 ⟩
875/142155.858[ -5 -3 3 1 ⟩
29/143238.300[ -8 2 0 0 0 0 0 0 0 1 ⟩
19/744218.054[ -3 1 0 -1 0 0 0 1 ⟩
23/545270.774[ 1 -2 -1 0 0 0 0 0 1 ⟩
95/146129.004[ 5 1 -1 0 0 0 0 -1 ⟩
143/147 85.852[ 4 2 0 0 -1 -1 ⟩
31/148391.135[ 5 0 0 0 0 0 0 0 0 0 -1 ⟩
3125/149210.735[ -10 -1 5 ⟩
35/1351117.893[ -3 1 1 1 0 -1 ⟩
65/751385.023[ 0 -2 1 -1 0 1 ⟩
91/551136.131[ -1 -2 -1 1 0 1 ⟩
49/1153125.074[ -1 2 0 -2 1 ⟩
343/554213.051[ -14 5 -1 3 ⟩
119/155256.137[ -1 5 0 -1 0 0 -1 ⟩
325/156 37.965[ -2 -4 2 0 0 1 ⟩
385/157 32.041[ -7 -1 1 1 1 ⟩
17/1158367.780[ 1 -1 0 0 -1 0 1 ⟩
1225/159114.885[ -15 3 2 2 ⟩
169/160120.901[ 9 -1 0 0 0 -2 ⟩
121/561102.239[ -3 -1 -1 0 2 ⟩
77/2562324.222[ 0 -1 -2 1 1 ⟩
125/4963 95.850[ 5 -4 3 -2 ⟩
25/1764243.963[ -1 1 -2 0 0 0 1 ⟩
23/765330.307[ -3 3 0 1 0 0 0 0 -1 ⟩
17/1366239.226[ 2 -1 0 0 0 1 -1 ⟩
125/1167 29.225[ -6 6 -3 0 1 ⟩
133/168169.977[ 15 -5 0 -1 0 0 0 -1 ⟩
625/769 84.263[ 16 -6 -4 1 ⟩
115/170131.635[ -10 2 1 0 0 0 0 0 1 ⟩
19/1171355.060[ 4 -2 0 0 1 0 0 -1 ⟩
37/172337.548[ -2 -2 0 0 0 0 0 0 0 0 0 1 ⟩
49/1373 90.025[ -6 5 0 -2 0 1 ⟩
29/574224.397[ 7 -6 -1 0 0 0 0 0 0 1 ⟩
455/175 3.008[ 12 -2 -1 -1 0 -1 ⟩
539/176 8.932[ 17 -5 0 -2 -1 ⟩
1715/177 93.034[ 6 3 -1 -3 ⟩
25/1978163.178[ 2 -1 -2 0 0 0 0 1 ⟩
55/1380 48.952[ -10 5 1 0 1 -1 ⟩
65/1180188.091[ 1 1 -1 0 1 -1 ⟩
143/580 67.190[ -8 2 -1 0 1 1 ⟩
121/782310.157[ -12 5 0 -1 2 ⟩
19/1383320.011[ -1 1 0 0 0 1 0 -1 ⟩
187/184325.078[ 6 1 0 0 -1 0 -1 ⟩
31/585403.962[ -1 -1 -1 0 0 0 0 0 0 0 1 ⟩
91/2586289.173[ -5 2 -2 1 0 1 ⟩
55/4987 27.968[ -3 2 -1 2 -1 ⟩
605/188 36.900[ 14 -3 -1 0 -2 ⟩
343/2589199.148[ 1 -3 -2 3 ⟩
41/190151.164[ 1 -4 0 0 0 0 0 0 0 0 0 0 1 ⟩
35/1792357.119[ -1 0 1 1 0 0 -1 ⟩
85/792145.797[ -2 -1 1 -1 0 0 1 ⟩
119/592103.095[ 3 1 1 -1 0 0 -1 ⟩
125/1394172.769[ -8 3 3 0 0 -1 ⟩
175/1195.5 70.198[ 4 0 -2 -1 1 ⟩
275/795.5375.025[ -1 4 -2 1 -1 ⟩
425/197201.260[ 4 3 -2 0 0 0 -1 ⟩
169/598 32.141[ -13 5 -1 0 0 2 ⟩
121/2599255.281[ -7 3 -2 0 2 ⟩
43/1100 95.874[ -7 1 0 0 0 0 0 0 0 0 0 0 0 1 ⟩
161/1101 76.283[ 1 4 0 -1 0 0 0 0 -1 ⟩
221/1102360.127[ 11 -2 0 0 0 -1 -1 ⟩
1375/1103 86.917[ -12 1 3 0 1 ⟩
4375/1104169.761[ -20 5 4 1 ⟩
23/11105 94.421[ 9 -5 0 0 1 0 0 0 -1 ⟩
29/7106209.739[ 10 -5 0 1 0 0 0 0 0 -1 ⟩
209/1107403.137[ -14 4 0 0 1 0 0 1 ⟩
19/17108 80.785[ -3 2 0 0 0 0 1 -1 ⟩
637/1109210.927[ 3 4 0 -2 0 -1 ⟩
2401/1110134.007[ 16 -3 0 -4 ⟩
145/1111 85.258[ -4 -2 1 0 0 0 0 0 0 1 ⟩
35/19113371.096[ -7 5 -1 -1 0 0 0 1 ⟩
95/7113 65.012[ 1 -3 1 -1 0 0 0 1 ⟩
133/5113183.880[ 0 3 1 -1 0 0 0 -1 ⟩
77/13116158.966[ 1 1 0 -1 -1 1 ⟩
91/11116229.065[ 11 -5 0 -1 1 -1 ⟩
143/7116108.163[ 2 -4 0 -1 1 1 ⟩
25/23118385.184[ -8 5 2 0 0 0 0 0 -1 ⟩
47/1119259.372[ 4 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 ⟩
31/7120197.120[ -1 2 0 1 0 0 0 0 0 0 -1 ⟩
475/1121282.046[ 1 5 -2 0 0 0 0 -1 ⟩
37/5122318.410[ -5 5 1 0 0 0 0 0 0 0 0 -1 ⟩
23/13123 59.372[ 4 -2 0 0 0 1 0 0 -1 ⟩
65/49124 63.017[ 2 -1 -1 2 0 -1 ⟩
715/1125 71.949[ 19 -6 -1 0 -1 -1 ⟩
847/1126244.819[ 5 3 0 -1 -2 ⟩
29/25127377.439[ 3 -2 -2 0 0 0 0 0 0 1 ⟩
247/1128201.143[ 0 -5 0 0 0 1 0 1 ⟩
49/17129149.200[ 8 -6 0 2 0 0 -1 ⟩
155/1130264.823[ -12 3 1 0 0 0 0 0 0 0 1 ⟩
15625/1131 57.693[ -6 -5 6 ⟩
289/1132 42.703[ -5 -2 0 0 0 0 2 ⟩
175/13133.5 35.149[ -1 3 -2 -1 0 1 ⟩
325/7133.5231.981[ 4 -6 2 -1 0 1 ⟩
245/11135.5111.171[ 14 -6 -1 -2 1 ⟩
539/5135.5 22.835[ 2 3 1 -2 -1 ⟩
595/1137399.821[ -6 -2 1 1 0 0 1 ⟩
169/7138 73.114[ -3 -1 0 -1 0 2 ⟩
143/25139 53.287[ 7 -6 -2 0 1 1 ⟩
121/35140345.800[ 5 -2 1 1 -2 ⟩
1625/1141 51.868[ -17 4 3 0 0 1 ⟩
1925/1142121.001[ 3 5 -2 -1 -1 ⟩
55/17144288.178[ -8 4 1 0 1 0 -1 ⟩
85/11144214.738[ 5 -5 1 0 -1 0 1 ⟩
187/5144172.036[ 10 -3 1 0 -1 0 -1 ⟩
31/25146251.996[ -6 4 2 0 0 0 0 0 0 0 -1 ⟩
6125/1147 38.157[ 11 1 -3 -2 ⟩
845/1148273.943[ 5 3 -1 0 0 -2 ⟩
343/125149352.190[ -3 1 -3 3 ⟩
41/5150304.206[ -3 0 -1 0 0 0 0 0 0 0 0 0 1 ⟩
53/1151230.282[ 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ⟩
119/25152 49.948[ -7 3 -2 1 0 0 1 ⟩
253/1153145.224[ 8 0 0 0 -1 0 0 0 -1 ⟩
125/77154331.736[ -7 4 3 -1 -1 ⟩
203/1155 44.285[ -14 4 0 1 0 0 0 0 0 1 ⟩
49/19156229.986[ 5 -4 0 2 0 0 0 -1 ⟩
625/49157109.753[ -10 4 4 -2 ⟩
23/17158179.853[ -2 1 0 0 0 0 -1 0 1 ⟩
125/17159397.005[ -5 5 -3 0 0 0 1 ⟩
169/25160 18.238[ 2 -3 -2 0 0 2 ⟩
323/1161 38.083[ 2 4 0 0 0 0 -1 -1 ⟩
43/5162248.916[ -11 5 -1 0 0 0 0 0 0 0 0 0 0 1 ⟩
29/11163140.798[ 3 -1 0 0 1 0 0 0 0 -1 ⟩
35/23165177.265[ 1 -1 1 1 0 0 0 0 -1 ⟩
115/7165325.651[ -4 0 1 -1 0 0 0 0 1 ⟩
161/5165 76.759[ -5 0 -1 1 0 0 0 0 1 ⟩
65/17168 86.183[ 6 -5 1 0 0 1 -1 ⟩
85/13168392.268[ -2 3 -1 0 0 1 -1 ⟩
221/5168207.085[ 15 -6 1 0 0 -1 -1 ⟩
625/11170 15.322[ 9 -2 -4 0 1 ⟩
665/1171323.019[ 11 -1 -1 -1 0 0 0 -1 ⟩
3125/7172237.305[ 12 -2 -5 1 ⟩
37/7173110.492[ 4 -1 0 1 0 0 0 0 0 0 0 -1 ⟩
1001/1174279.868[ 10 0 0 -1 -1 -1 ⟩
575/1175 21.407[ 6 2 -2 0 0 0 0 0 -1 ⟩
55/19177202.018[ 8 -6 1 0 1 0 0 -1 ⟩
95/11177300.898[ -11 5 1 0 -1 0 0 1 ⟩
209/5177252.821[ 7 -1 1 0 -1 0 0 -1 ⟩
23/19179260.639[ -5 3 0 0 0 0 0 -1 1 ⟩
217/1180 56.904[ -3 -3 0 1 0 0 0 0 0 0 1 ⟩
121/13181227.907[ 8 -3 0 0 -2 1 ⟩
185/1182184.506[ 2 -6 1 0 0 0 0 0 0 0 0 1 ⟩
361/1183 48.077[ -18 6 0 0 0 0 0 2 ⟩
299/1184180.273[ 13 -3 0 0 0 -1 0 0 -1 ⟩
245/13185.5 76.122[ 9 -3 -1 -2 0 1 ⟩
637/5185.5 57.884[ 7 0 1 -2 0 -1 ⟩
343/11187319.090[ 5 0 0 -3 1 ⟩
59/1188360.101[ -2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ⟩
2401/5189147.910[ 1 5 1 -4 ⟩
133/25190 30.838[ 4 -1 2 -1 0 0 0 -1 ⟩
31/11191 38.766[ -11 6 0 0 -1 0 0 0 0 0 1 ⟩
833/1192358.848[ -16 4 0 2 0 0 1 ⟩
77/65194 5.924[ 5 -3 1 -1 -1 1 ⟩
91/55194242.967[ -4 3 1 -1 1 -1 ⟩
143/35194261.205[ -2 0 -1 -1 1 1 ⟩
121/49196304.827[ -5 4 0 2 -2 ⟩
29/13197105.749[ -2 2 0 0 0 1 0 0 0 -1 ⟩
1331/1198313.760[ 12 -1 0 0 -3 ⟩
47/5199106.330[ 8 -3 1 0 0 0 0 0 0 0 0 0 0 0 -1 ⟩
2275/1200156.050[ 8 2 -2 -1 0 -1 ⟩
2695/1201161.975[ 13 -1 -1 -2 -1 ⟩
77/17203 80.259[ 1 -2 0 1 1 0 -1 ⟩
119/11203173.765[ -5 1 0 1 -1 0 1 ⟩
187/7203131.062[ 0 3 0 1 -1 0 -1 ⟩
61/1205 50.594[ 2 -5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⟩
8575/1206 79.131[ 21 -5 -2 -3 ⟩
125/19207316.220[ -2 3 -3 0 0 0 0 1 ⟩
37/25208165.368[ -1 1 2 0 0 0 0 0 0 0 0 -1 ⟩
1183/1209327.138[ -7 -2 0 1 0 2 ⟩
275/13210.5104.090[ 6 -1 -2 0 -1 1 ⟩
325/11210.5341.133[ -3 5 -2 0 1 -1 ⟩
605/7212.5157.115[ -8 1 1 -1 2 ⟩
847/5212.5 91.776[ 9 -1 1 -1 -2 ⟩
65/19215166.969[ 3 -3 1 0 0 1 0 -1 ⟩
95/13215335.947[ -6 2 1 0 0 -1 0 1 ⟩
247/5215354.185[ -4 -1 -1 0 0 1 0 1 ⟩
41/7217296.875[ 1 1 0 1 0 0 0 0 0 0 0 0 -1 ⟩
85/49218302.242[ 4 -2 -1 2 0 0 -1 ⟩
935/1219330.880[ -13 2 1 0 1 0 1 ⟩
169/11220142.055[ 4 -5 0 0 -1 2 ⟩
289/5221195.745[ -9 2 -1 0 0 0 2 ⟩
125/91222366.785[ -2 1 3 -1 0 -1 ⟩
275/49223.5 14.065[ 12 -6 -2 2 -1 ⟩
539/25223.5130.207[ -6 1 -2 2 1 ⟩
3025/1225189.942[ 10 1 -2 0 -2 ⟩
31/13226 73.815[ -6 3 0 0 0 -1 0 0 0 0 1 ⟩
205/1227165.067[ -14 4 1 0 0 0 0 0 0 0 0 0 1 ⟩
175/17228.5204.077[ 3 -4 2 1 0 0 -1 ⟩
425/7228.5 7.245[ -2 5 -2 1 0 0 -1 ⟩
169/35230226.156[ -7 3 -1 -1 0 2 ⟩
143/125231206.329[ 3 -2 -3 0 1 1 ⟩
43/7232289.889[ -1 -1 0 -1 0 0 0 0 0 0 0 0 0 1 ⟩
49/23233136.292[ -9 5 0 2 0 0 0 0 -1 ⟩
91/17235 45.210[ -4 1 0 1 0 1 -1 ⟩
119/13235208.814[ 0 -2 0 1 0 -1 1 ⟩
221/7235166.111[ 5 0 0 1 0 -1 -1 ⟩
625/13237 19.727[ -4 -1 4 0 0 -1 ⟩
875/11238.5223.240[ 0 4 -3 -1 1 ⟩
1375/7238.5280.933[ -6 -1 3 -1 1 ⟩
187/25240185.939[ -5 5 2 0 -1 0 -1 ⟩
931/1241278.063[ -13 2 0 2 0 0 0 1 ⟩
67/1242 77.694[ 14 -5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ⟩
391/1243222.556[ -7 -1 0 0 0 0 1 0 1 ⟩
2125/1244187.357[ 19 -5 -3 0 0 0 -1 ⟩
125/121245400.677[ 0 0 3 0 -2 ⟩
215/1246 57.168[ 3 3 -1 0 0 0 0 0 0 0 0 0 0 -1 ⟩
319/1247191.601[ 2 4 0 0 -1 0 0 0 0 -1 ⟩
805/1248 62.380[ 16 -4 -1 -1 0 0 0 0 -1 ⟩
77/19250161.044[ -2 0 0 1 1 0 0 -1 ⟩
133/11250 92.980[ -2 -1 0 1 -1 0 0 1 ⟩
209/7250211.848[ -3 5 0 1 -1 0 0 -1 ⟩
41/25252351.752[ -4 3 2 0 0 0 0 0 0 0 0 0 -1 ⟩
53/5253 77.240[ 5 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 ⟩
1105/1254295.831[ -18 5 1 0 0 1 1 ⟩
6875/1255 66.125[ 8 3 -4 0 -1 ⟩
29/17256133.476[ 4 -3 0 0 0 0 -1 0 0 1 ⟩
21875/1257 16.719[ -16 1 5 1 ⟩
259/1258143.532[ -8 0 0 1 0 0 0 0 0 0 0 1 ⟩
343/13259284.041[ 0 3 0 -3 0 1 ⟩
55/23261108.324[ -6 3 1 0 1 0 0 0 -1 ⟩
115/11261247.463[ 5 -1 -1 0 1 0 0 0 -1 ⟩
253/5261 7.818[ -12 4 -1 0 1 0 0 0 1 ⟩
35/29264223.642[ -5 3 1 1 0 0 0 0 0 -1 ⟩
145/7264362.782[ 6 -1 -1 1 0 0 0 0 0 -1 ⟩
203/5264 30.382[ 1 -4 -1 1 0 0 0 0 0 1 ⟩
95/49266383.028[ 1 0 -1 2 0 0 0 -1 ⟩
1045/1267250.095[ -10 0 1 0 1 0 0 1 ⟩
37/11268 41.551[ -3 3 0 0 1 0 0 0 0 0 0 -1 ⟩
437/1269141.771[ -4 -3 0 0 0 0 0 1 1 ⟩
71/1270172.307[ 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ⟩
143/49271339.876[ 0 1 0 2 -1 -1 ⟩
169/125272171.280[ -2 1 -3 0 0 2 ⟩
1573/1273293.246[ -9 -1 0 0 2 1 ⟩
85/19275 72.257[ -1 2 -1 0 0 0 -1 1 ⟩
95/17275 66.882[ 12 -6 -1 0 0 0 1 -1 ⟩
323/5275114.959[ -6 0 -1 0 0 0 1 1 ⟩
43/25277235.013[ 4 -3 -2 0 0 0 0 0 0 0 0 0 0 1 ⟩
161/25278229.801[ -9 4 -2 1 0 0 0 0 1 ⟩
47/7279 65.357[ -2 3 0 1 0 0 0 0 0 0 0 0 0 0 -1 ⟩
3185/1280197.024[ 18 -4 -1 -2 0 -1 ⟩
3773/1281369.893[ 4 5 0 -3 -1 ⟩
221/25282220.988[ 0 2 2 0 0 -1 -1 ⟩
12005/1283287.049[ 12 1 -1 -4 ⟩
725/1284 67.784[ 0 6 -2 0 0 0 0 0 0 -1 ⟩
175/19285.5284.862[ 0 -2 2 1 0 0 0 -1 ⟩
475/7285.5 78.915[ -14 5 2 -1 0 0 0 1 ⟩
341/1287 12.037[ 10 -1 0 0 -1 0 0 0 0 0 -1 ⟩
37/35288 42.550[ -8 5 -1 -1 0 0 0 0 0 0 0 1 ⟩
29/19289214.262[ 1 -1 0 0 0 0 0 -1 0 1 ⟩
73/1290169.930[ -3 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⟩
385/13292.5312.009[ -3 5 -1 -1 -1 1 ⟩
455/11292.5382.107[ 7 -1 -1 -1 1 -1 ⟩
715/7292.5122.066[ -13 4 1 -1 1 1 ⟩
1001/5292.5126.826[ 14 -4 1 -1 -1 -1 ⟩
31/17295313.041[ -4 2 0 0 0 0 -1 0 0 0 1 ⟩
377/1296226.651[ 7 1 0 0 0 -1 0 0 0 -1 ⟩
91/19298125.995[ -7 3 0 1 0 1 0 -1 ⟩
133/13298128.029[ 3 -4 0 1 0 -1 0 1 ⟩
247/7298246.897[ 2 2 0 1 0 -1 0 -1 ⟩
125/23300232.142[ -4 1 3 0 0 0 0 0 -1 ⟩
209/25301 99.779[ 11 -5 2 0 -1 0 0 -1 ⟩
235/1302396.586[ -11 2 1 0 0 0 0 0 0 0 0 0 0 0 1 ⟩
Sorry for the brevity: gotta go make dinner.

Up to N2D3P9 of 307 gives us commas only for tinas 3, 6, 7, 8, and 9.

So far they all agree with the ones you listed using the LASS-based metacomma technique (yes I did see the post you got in just before mine — thanks for those comma name translations and extra info).

Dave Keenan
Posts: 2044
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: Magrathean diacritics

Brevity can be good.

That's awesome! Thanks. It's particularly good to have 7 tinas nailed down.

If you calculate the N2D3P9 of the (2,3-free ratios notated by the) remaining metacommas on my list, and take the maximum, that's an upper bound on how high you will need to go in N2D3P9 to get all the whole tinas as commas.

cmloegmcluin
Posts: 1681
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer (he/him/his)
Contact:

### Re: Magrathean diacritics

Just another quick note: I can just use my find-commas script to find the remaining tinas, after dinner. For example, running (these min and max cents values are 4.5 and 5.5 tinas):

Code: Select all

npm run find-commas -- --max-n2d3p9 Infinity --min-cents 0.63236406334 --max-cents 0.77288941075

Turns up this:
symbol nameratiomonzocentsapotome slopelimit5-rough sopfrN2D3P9
2695n4296700485/4294967296[ -32 13 1 2 1 ⟩ 0.698 12.957 11 30 205.868
2401/25n2401/2400[ -5 -1 -2 4 ⟩ 0.721 -1.044 7 38 324.209
2431/5n2431/2430[ -1 -5 -1 0 1 1 1 ⟩ 0.712 -5.044 17 46 956.644
11/2375n2376/2375[ 3 3 -3 0 1 0 0 -1 ⟩ 0.729 2.955 19 45 1149.02
1/2303n2304/2303[ 8 2 0 -2 0 0 0 0 0 0 0 0 0 0 -1 ⟩ 0.752 1.954 47 61 1503.35
625/833n2500/2499[ 2 -1 4 -2 0 0 -1 ⟩ 0.693 -1.043 17 51 1517.60
1/3689n59049/59024[ -4 10 0 -1 0 0 -1 0 0 0 -1 ⟩ 0.733 9.955 31 55 1588.32
289/95n151519232/151460685[ 19 -13 -1 0 0 0 2 -1 ⟩ 0.669-13.041 19 58 1610.02
1/20449n134217728/134165889[ 27 -8 0 0 -2 -2 ⟩ 0.669 -8.041 13 48 1846.09
1/4669n4782969/4781056[ -10 14 0 -1 0 0 0 0 -1 -1 ⟩ 0.693 13.957 29 59 1880.57
1331/875n1395654656/1395032625[ 20 -13 -3 -1 3 ⟩ 0.772-13.048 11 55 2196.65
289/325n2601/2600[ -3 2 -2 0 0 -1 2 ⟩ 0.666 1.959 17 57 2464.08
3553/25n209801097/209715200[ -23 10 -2 0 1 0 1 1 ⟩ 0.709 9.956 19 57 2604.44
5/41503n2657205/2656192[ -6 12 1 -3 -2 ⟩ 0.660 11.959 11 48 2641.97
1309/23n11781/11776[ -9 2 0 1 1 0 1 0 -1 ⟩ 0.735 1.955 23 58 3205.84
95/2401n194560/194481[ 11 -4 1 -4 0 0 0 1 ⟩ 0.703 -4.043 19 52 3343.99
23023n16783767/16777216[ -24 6 0 1 1 1 0 0 1 ⟩ 0.676 5.958 23 54 3677.28
19/2431n2432/2431[ 7 0 0 0 -1 -1 -1 1 ⟩ 0.712 -0.044 19 60 4062.92
9025/11n72200/72171[ 3 -8 2 0 -1 0 0 2 ⟩ 0.696 -8.043 19 59 4366.26
7/9425n15032385536/15026494275[ 31 -13 -2 1 0 -1 0 0 0 -1 ⟩ 0.679-13.042 29 59 4428.88
1/232925n2097152/2096325[ 21 -2 -2 -1 -3 ⟩ 0.683 -2.042 11 50 4448.22
125/7889n4194304000/4192538049[ 25 -12 3 -3 0 0 0 0 -1 ⟩ 0.729-12.045 23 59 5833.56
13/7105n28431/28420[ -2 7 -1 -2 0 1 0 0 0 -1 ⟩ 0.670 6.959 29 61 6200.43
199375n1595000/1594323[ 3 -13 4 0 1 0 0 0 0 1 ⟩ 0.735-13.045 29 60 10038.0
265837n531674/531441[ 1 -12 0 0 2 3 ⟩ 0.759-12.047 13 61 11999.6
So we can see that the first entry in the table didn't turn up in the previous table because the slot for 2695/1 got beaten out by the better LATE comma [ 13 -1 -1 -2 -1 ⟩ which is 161.975 tinas.

And while the previous table cuts off at N2D3P9 around 307, had it gone just a little bit longer to 325 then we would have nabbed 2401/25n, which was our yellow-highlighted comma for the 5 tina back on the page 6 table.

So don't worry 0.5, 1, 2, 4 -- I'll find the others after dinner.

Dave Keenan
Posts: 2044
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: Magrathean diacritics

[Written before reading your previous:] Yikes! The worst one is the metacomma for 1 tina. It has N2D3P9 = 1258.083.

Dave Keenan