developing a notational comma popularity metric

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Dave Keenan
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Re: developing a notational comma popularity metric

Post by Dave Keenan »

cmloegcmluin wrote: Fri Nov 20, 2020 3:32 am Perhaps I should pick myself up a copy of OTSOT, then. Or at least stop publicly admitting that I haven't read it. ;)
I suggest just using the Internet Archive version:
https://archive.org/stream/onsensations ... 5/mode/2up
Sometime you might read the table of contents and skim the index and follow up anything that piques your interest.
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cmloegcmluin
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Re: developing a notational comma popularity metric

Post by cmloegcmluin »

cmloegcmluin wrote: Tue Sep 29, 2020 1:40 am That 2.920050977... number in particular seems exciting, but I think your description of it and Mill's number as mere "compressions" for the primes is on point.
Numberphile recently released a new video on this constant:

Around 5:45 James notes that the constant is not "predictive", AKA it won't be able to find primes you don't know about yet. Which is another way of capturing its uninterestingness. Or as Brady puts it, it's the primes generating the constant, not the other way around.

It would be cool if a particular prime-to-constant compression algorithm resulted in a constant which had other properties of interest, but I don't have any reason to believe such a thing should exist.

Ha. I typed that sentence before I finished watching the video. There is in fact, a pleasant property of this constant, which they start going into at the 10:00 mark.
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Re: developing a notational comma popularity metric

Post by Dave Keenan »

cmloegcmluin wrote: Thu Dec 03, 2020 5:46 am It would be cool if a particular prime-to-constant compression algorithm resulted in a constant which had other properties of interest, but I don't have any reason to believe such a thing should exist.

Ha. I typed that sentence before I finished watching the video. There is in fact, a pleasant property of this constant, which they start going into at the 10:00 mark.
Fascinating. Thanks.
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Re: developing a notational comma popularity metric

Post by Dave Keenan »

I note that the subject of this thread ultimately became:
Developing a notational comma badness metric.

Things hopped back and forth a bit between this thread and the Magrathean diacritics thread. The final badness metric (that was used indirectly in obtaining the final list of tina commas) was:

From: viewtopic.php?p=2636#p2636

LPEI_badness = lb(N2D3P9) + (AAS/9.65)^1.7 + 2^(ATE-9.65) + 0.8×AERR

lb() is the base-2 logarithm.

N2D3P9 is as described here: https://en.xen.wiki/w/N2D3P9

AAS is the absolute value of the apotome slope
= comma_3_exponent - apotome_3_exponent × untempered_comma_cents / untempered_apotome_cents
= comma_3_exponent - 7 × untempered_comma_cents / 113.685c

ATE is the absolute value of the 3-exponent of the comma.

AERR is the absolute value of the error which is the number of inas the comma is away from the nearest whole ina. The ina is the width of the capture zone at the relevant precision level. So AERR always ranges from 0 to 0.5.

"LPEI" stands for "log, power, exponential, identity" which lists the functions applied to the four input values before scaling and adding them.

The LPE notational comma complexity metric is obtained from the above by omitting the 0.8×AERR term.
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