Magrathean diacritics

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Dave Keenan
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Re: Magrathean diacritics

Post by Dave Keenan »

[Moved from the Developing a notational comma popularity metric thread]

Here's a badness function with rounded parameters approximately in the middle of the ranges that give maximum matches for the existing extreme-level commas.

badness = lb(N2D3P9) + (AAS/8.5)^1.5 + 2^(ATE-10) + 0.5×AERR

You could call it a member of the LPEI family of notational comma badness functions, where "I" stands for identity function.

When I apply that to the 0.5-tina candidates, it doesn't change the result I described above. In this case AERR is the number of half-tinas that the comma is away from the nearest whole-multiple of a half-tina. In order for my favourite (0.58 tina) comma to win, I need to use 1.5×AERR instead of 0.5×AERR.

When using 1.5×AERR with the extreme-level commas, it is still possible to match 101 commas, but not 102.

It's all so messy and subjective, and ultimately unimportant, that I think if we still can't agree on a tina comma, after all the work we did deriving N2D3P9 (the only metric with any claim to objectivity (or at least intersubjectivity)), then we should simply list our favourites and toss a coin or roll a dice to choose one.
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Re: Magrathean diacritics

Post by Dave Keenan »

It seems that the comma for the half-tina dot is the only one that remains unclear, based on the above extreme-derived complexity and badness metrics.

It could be:
0.5 tina 5/1517n (wins with <1.5×AERR)
or
0.5 tina 77/185n (wins with ≥1.5×AERR)

Here are those I think are clear winners for the whole tinas, based on the above extreme-derived complexity and badness metrics:
1 tina 59/7n
2 tinas 1/205n
3 tinas 1/455n
4 tinas 3025/7n
5 tinas 2401/25n
6 tinas 65/77n
7 tinas 7/425n
8 tinas 253/5n
9 tinas 1/539n

I searched on these comma names in this (Magrathean diacritics) thread.

They are all long-standing favourites except for 0.5, 1, 2 and 8 tinas. Those for 1 and 8 tinas are recent favourites (since N2D3P9). That for 2 tinas (1/205n) has been proposed before, by you (cmloegcmluin).

77/185n for the 0.5 tina has been proposed before, by me. 5/1517n has never been mentioned before by anyone, anywhere on this forum. 1517 = 37×41.

Do you want to argue for one or the other 0.5 tina comma?

I've attached your wonderful spreadsheet, updated with the complexity and badness measures, with the winners highlited.
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tinaCommaCandidatesWithN2D3P9LessThan5298.xlsx
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Re: Magrathean diacritics

Post by cmloegcmluin »

Exciting stuff. I can smell victory.

I had a couple hours this morning so I decided to try out my recently described metacommas technique.

I used the LPEI formula from your spreadsheet and divided the half apotome into half tina buckets. For each one I found its best comma by LPEI. I then found every metacomma between one of these commas and an Extreme comma that was < 9.5 tinas and sorted them into buckets for each candidate. I then looked at which metacomma was most common in each bucket.

The results are pretty interesting. Unsurprisingly, the commas which are direct LPEI winners for the <9.5 tina buckets appear high in these lists, but they are not always the clear winner. What I'm proposing is that what matters too, in addition to whether the tinas are good as commas themselves is: as accents applied to existing commas, do they help us exactly notate commas that are the most useful. I'm pretty sure this theme has come up on the forum before, either here or back on the Magrathean thread, but I don't have the time right now to review how that panned out. I might have time tonight.

For now, here's my raw results, attached. They agree with the direct LPEI winners for tinas 3, 4, 5, 6, 7, and 9, but give different answers for 0.5, 1, 2, and 8 (the problem children). One thing I notice right away is that this process results in 121/1225n being the clear winner for tina 1, which was George's proposed 1-tina.

I'm also interested in modifying the badness metric (whether or not we insert this metacomma layer to our decision process) to include punishment for commas who have already been exactly notated in Extreme, But that will have to wait for another time.
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Re: Magrathean diacritics

Post by cmloegcmluin »

Found it. Here’s where I brought it up: viewtopic.php?p=1508#p1508

Then you said it was a good idea: viewtopic.php?p=1526#p1526

But then I think we soon likened the approach to throwing open the gates to admit some Lovecraftian horror. Well, I guess with the tools we’ve developed by this point, it wasn’t that bad to compute.
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Re: Magrathean diacritics

Post by cmloegcmluin »

Let me know if you want me to re-run this but only gathering the best comma every whole tina, i.e. that we should only care about counting metacommas for commas that would actually appear in the Insane notation.
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Re: Magrathean diacritics

Post by Dave Keenan »

Thanks for the most common metacommas. I'm leaning towards using them.
cmloegcmluin wrote: Mon Nov 02, 2020 7:03 am Let me know if you want me to re-run this but only gathering the best comma every whole tina, i.e. that we should only care about counting metacommas for commas that would actually appear in the Insane notation.
Sure. If it's easy. [Edit: On second thoughts, see viewtopic.php?p=2619#p2619]

It would also be good to look for the most common meta-meta-comma for the half-tina dot. i.e. find the most common metacommas for the odd half-tina multiples as well as the whole tinas, then find the most common difference between metacommas for consecutive half tinas.
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Re: Magrathean diacritics

Post by Dave Keenan »

To summarise the result of your recent most-common-metacomma search, I list below, the most common, and any with a count within 20% of the most common. Those with more than 1 are hilited.

0.5 tinas 4675/13n
0.5 tinas 77/185n
1 tina 121/1225n
2 tinas 319/7n
2 tinas 1/5831n
2 tinas 625/1001n
2 tinas 35/1573n
2 tinas 2023/5n
3 tinas 1/455n
4 tinas 3025/7n
5 tinas 2401/25n
5 tinas 2431/5n
6 tinas 65/77n
7 tinas 7/425n
8 tinas 1/2875n
8 tinas 253/5n
8 tinas 77/13n
8 tinas 187/175n
9 tinas 1/539n
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Re: Magrathean diacritics

Post by Dave Keenan »

Here's the result of an earlier metacomma search of a different kind:
viewtopic.php?f=10&t=430&p=2365&hilit=25%2F11n#p2365

1 tina as 121/1225n
2 tinas as 275/29n [This comma is rubbish. It's closer to 1.5 tinas.]
3 tinas as 1/455n
4 tinas as 3025/7n
5 tinas as 2401/25n
6 tinas as 65/77n
7 tinas as 2125/7n
7 tinas as 7/425n (a close second, the 5s-complement of the first)
8 tinas as 253/5n
9 tinas as 1/539n

Regarding the suggestion that 7.5 tinas as 1/2875n could give us a ratio for the dot, as a meta-meta-comma with 7 or 8 tinas. It doesn't. The only such is 17/805n, but it is rubbish because it is too close to 1/4-tina.

Those for 1, 2 and 4 tinas above, help notate low-N2D3P9 ratios that are not yet exactly notated.
Some information about how this list was derived, is in this earlier post:
viewtopic.php?p=2353#p2353

Here's where I argued for 77/185n for the 0.5 tina because it was the metacomma between the two half-tina-multiple commas (below 10 tinas) having the lowest N2D3P9, namely 6 and 6.5 tinas. And you've just shown that 77/185n is also a fairly common metacomma.
viewtopic.php?p=2393#p2393

We agreed long ago, that 8 tinas should be 77/13n because it is the schisma-complement of 6 tinas as 65/77n. And you've just shown that 77/13n is also a fairly common metacomma.
viewtopic.php?f=10&t=430&p=1493&hilit=s ... ment#p1493

So, for me, only the 2 tina comma remains a conundrum.
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Re: Magrathean diacritics

Post by Dave Keenan »

Of your 2 tina metacommas that are within 20% of the max count., most can be rejected because they are outside of the range 1.75 to 2.25 tinas.

2.41 tinas 319/7n badness ∞
2.11 tinas 1/5831n badness 10.3
1.59 tinas 625/1001n badness ∞
1.7506 tinas 35/1573n badness 11.4
1.73 tinas 2023/5n badness ∞

I declare 1/5831n the winner. I note that has been the favourite for a long time.

So. The list:

0.5 tina 77/185n
1 tina 121/1225n
2 tinas 1/5831n
3 tinas 1/455n
4 tinas 3025/7n
5 tinas 2401/25n
6 tinas 65/77n
7 tinas 7/425n
8 tinas 77/13n
9 tinas 1/539n

I have checked that these all map to the correct number of tinas, using the zeta-peak mapping, including the comma for the 0.5 tina dot mapping to zero.

Overall, this list has a greatest prime factor of 37. The 0.5 tina comma is the only one with a prime greater than 17.
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Re: Magrathean diacritics

Post by Dave Keenan »

Oops!

I just noticed that 121/1225n is 1.26 tinas, and so is outside the ±0.25 range that I used to eliminate 2 tina candidates. i.e. it is slightly closer to an odd multiple of a half-tina and so could be notated with a dot.

The most common 1 tina metacommas on your list are:

1.26 tinas, 121/1225n, 27 times, badness ∞
1.42 tinas, 11/3125n, 19 times, badness ∞
1.20 tinas, 10241/5n, 18 times, badness 11.75, N2D3P9 2252, ATE 0
0.83 tinas, 221/7n, 17 times, badness 13.7, N2D3P 243, ATE 12
? tinas, 1547/1375n, 15 times, badness high, N2D3P9 >5298 or ATE > 15
1.07 tinas, 196625n, 12 times, badness 12.4, N2D3P9 4437, ATE 1
0.85 tinas, 25/14399n, 12 times, badness 12.7, N2D3P9 4722, ATE 2
0.90 tinas, 833/65n, 11 times, badness 11.75, N2D3P9 1420, ATE 8

They all zeta-map to 1 tina.

Although I've given them above, I just realised that the N2D3P9 of a metacomma doesn't matter one whit. Same with its AAS and ATE. And if those metrics don't matter for a metacomma, then neither does the badness measure we're using here. But I think the error still matters. What really matters is that when added or subtracted, the metacomma converts one low-badness comma into another low-badness comma. An example of this can be seen here:
viewtopic.php?p=2393#p2393

So I'm voting for 10241/5n (19-limit), here on the basis that it's the most common metacomma in the range 0.75 to 1.25 tinas.

Maybe you want to argue why 121/1225n (11-limt) should be used for 1 tina despite it being 1.26 tinas.
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