Page 5 of 16

Re: Magrathean diacritics

Posted: Wed May 27, 2020 12:50 pm
by Dave Keenan
cmloegcmluin wrote:
Wed May 27, 2020 11:39 am
So consider :`::)|: . Its mina represents 3136/3135 which is 0.552, about a tina higher than a mina. So you're saying that in the Insane precision level, the comma :`::)|: represents, the 49/55s, would get assigned to :@4::)|: (maybe one day that'll turn into a Magrathean smilie for 4-tinas!)? But wouldn't that mean that we'd have to change the comma for :`::)|: to something else?

This feels kind of ass-backwards. Surely the other precision levels weren't developed this way. I thought lower levels, once locked in, were locked in. Unless the Extreme precision layer is different because it's the first layer where we don't really care enough about the commas we found for it and don't mind if they get booted to a later (higher) precision level. Or am I still not getting this?

I just thought that we were stuck with :`::)|: being the 49/55s, and even if that was more like 4 tinas than 3 tinas off from :)|: 's 19s, that's how it was, and we'd still be using it as our basis from which we'd deviate by tinas up and down.
Oh man! You are so right. What was I thinking? And yet I believe Ash's idea still survives, despite my wrong explanation.

However, there is something different about the extreme to insane step. We don't keep the accents from the lower level and merely add new types of accents. We replace the mina accents with tina accents. So we could be seen to be skipping the extreme level and jumping straight from very high to insane. But that's a conundrum for another day.
Back on the consistent 37 limit thread, you asked for me to "add commas that correspond to zero minas (preferably without requiring horizontal scroll bars)" to my table of attested values for minas. I can't figure out what you would mean by this.
That idea doesn't survive. Complete nonsense, sorry.
Or maybe "suboptimal" is basically the call of Chthulu in this context. Turn away!
Maybe. :)

Re: Magrathean diacritics

Posted: Wed May 27, 2020 1:08 pm
by Dave Keenan
Thanks for the list. But the horizontal scrolling is driving me nuts, and I miss not being able to see how far a comma is from the exact tina.

Could you please add two decimal places to your "tinas" column, and shift the ratio and monzo columns to the far right (as they have very little bearing on the decision).

I'll be looking to come up with a single badness measure that is a weighted sum of the absolute error in tinas, the absolute 3-exponent and the SoPF>3.

And maybe you could include commas with absolute 3-exponents up to 13.

Re: Magrathean diacritics

Posted: Wed May 27, 2020 1:14 pm
by Dave Keenan
I see you have a tina error column. That's good. But I'd still like to see the decimal places of tinas in the first column. It's easy enough to mentally round them, and you've grouped them by nearest tina anyway.

Re: Magrathean diacritics

Posted: Wed May 27, 2020 1:29 pm
by Dave Keenan
You apparently missed where I said:
Dave Keenan wrote:
Tue May 26, 2020 11:40 pm
I retract my earlier rejection of 7:3025n ...
just before I said:
So we've nailed down 3, 4, 6, 8 and 9 tinas.
i.e. the 4 tina comma that I claim is nailed down, is 7:3025n, which I didn't know was George's at the time.

Re: Magrathean diacritics

Posted: Wed May 27, 2020 2:06 pm
by Dave Keenan
I agree that:
2 tinas should be Ash's 5831n.
5 tinas should be George's 25:2401n

Since I'm accepting that the 5 and 9 tina commas do not have to be 5-schisma complements, then I guess I'm accepting that no pair that add to 14 have to be complements.

I don't understand why you like 1729n for 7 tinas. I'd prefer one of the two that are currently represented by a 2-mina accent in Olympian.

I'm aesthetically-pleased when any of these turn out to be superparticular, even though that isn't a consideration. :)

In the case of 8 tinas, I might be willing to pull out the nail from 5:253n and go with the one that has 3 occurrences in Olympian, namely 13:77n.

0.5 tina and 1 tina are complete bastards.

One thing we haven't looked at is whether the commas follow a consistent prime mapping, of either whole tinas or whole half-tinas. That might be a way to chose 1-tina and 0.5-tina.

Re: Magrathean diacritics

Posted: Wed May 27, 2020 4:19 pm
by cmloegcmluin
And yet I believe Ash's idea still survives, despite my wrong explanation.
Meaning that even if :`::)|: was stuck being the 49/55s, it could still be useful to have a tina representing the value that, when applied to 49/55s, would get a comma which was very close to being 3 tinas away from :)|: , which could be helpful in preventing stuff like OOBSOFLS and DEFLL exceptions. That makes sense to me.
You apparently missed where I said:
Dave Keenan wrote: ↑Tue May 26, 2020 6:40 am
I retract my earlier rejection of 7:3025n ...
just before I said:
So we've nailed down 3, 4, 6, 8 and 9 tinas.
i.e. the 4 tina comma that I claim is nailed down, is 7:3025n, which I didn't know was George's at the time.
Yes, sorry. Updated in the original post.
I don't understand why you like 1729n for 7 tinas. I'd prefer one of the two that are currently represented by a 2-mina accent in Olympian.
I'll be looking to come up with a single badness measure that is a weighted sum of the absolute error in tinas, the absolute 3-exponent and the SoPF>3.
Looking at that list now, I don't see why I said it, either.

I look forward to your consolidated tina badness metric.
I'm aesthetically-pleased when any of these turn out to be superparticular, even though that isn't a consideration. :)
You're not alone!!
One thing we haven't looked at is whether the commas follow a consistent prime mapping, of either whole tinas or whole half-tinas. That might be a way to chose 1-tina and 0.5-tina.
Do you mean that iff the Extreme precision level as it is now is consistent when mapped to 809- (or perhaps 1618- ) EDA, then it would be a helpful filter that candidates for the 1- and 0.5- tina also be consistent?
Thanks for the list. But the horizontal scrolling is driving me nuts, and I miss not being able to see how far a comma is from the exact tina.

Could you please add two decimal places to your "tinas" column, and shift the ratio and monzo columns to the far right (as they have very little bearing on the decision).
And maybe you could include commas with absolute 3-exponents up to 13.
I see you have a tina error column. That's good. But I'd still like to see the decimal places of tinas in the first column. It's easy enough to mentally round them, and you've grouped them by nearest tina anyway.
I will have to get this to you tomorrow.

Re: Magrathean diacritics

Posted: Wed May 27, 2020 6:06 pm
by Dave Keenan
cmloegcmluin wrote:
Wed May 27, 2020 4:19 pm
I look forward to your consolidated tina badness metric.
Don't hold your breath.
One thing we haven't looked at is whether the commas follow a consistent prime mapping, of either whole tinas or whole half-tinas. That might be a way to chose 1-tina and 0.5-tina.
Do you mean that iff the Extreme precision level as it is now is consistent when mapped to 809- (or perhaps 1618- ) EDA, then it would be a helpful filter that candidates for the 1- and 0.5- tina also be consistent?
No. I just meant calculate the obvious mapping ("patent val")* from primes to degrees of 809-EDA. which starts off
⟨ 8539 13534 19828 ... ]
then take its inner product with the monzo for each candidate, and publish the number of tinas it comes to. Then we can compare that with the number of tinas obtained by rounding the comma's untempered size.

For the sub-tinas represented by the dot, I would not use a 1618-EDA mapping, as that is likely to be highly inconsistent. I would try each of the EDAs I listed in a previous post in this thread. And compare the tempered number of sub-tinas with the rounded untempered number for each candidate comma.

Or perhaps we should adopt @cmloegcmluin's suggestion from an admin-only topic:
Seriously, though, what if... just what if... the dot represented "full stop". Like, there is nothing beyond this level of insanity. What I'm suggesting is that we just say the dot represents anything you want, so long as its less than a tina.
*Actually, the 809-EDA mapping of prime 3 is not so obvious ("not so patent") because a simple calculation of how many tinas are in prime 3 gives 13534.60451 which rounds up to 13535. But if you then calculate how many tinas are in an apotome, as 7 × 13535 - 11 × 8539 you get 816, not 809. It has to be 13534 to get 7 × 13534 - 11 × 8539 = 809.

I get the right answer for prime 3 straight off, if I use the optimal tina given here:
https://en.xen.wiki/w/The_Riemann_Zeta_ ... and_Tuning
i.e. 1200/8539.00834 = 0.140531541 ¢
This corresponds to neither an exact EDO nor an exact EDA. But we should use the obvious mapping for this tina.

Re: Magrathean diacritics

Posted: Thu May 28, 2020 2:57 am
by cmloegcmluin
Dave Keenan wrote:
Wed May 27, 2020 6:06 pm
cmloegcmluin wrote:
Wed May 27, 2020 4:19 pm
I look forward to your consolidated tina badness metric.
Don't hold your breath.
I think what you're really looking for, when you asked me above to expand the scope to 13 for the absolute 3 exponent, is not just to permit George's 49:9765625n, but to keep removing constraints on our searches for commas until we are no longer finding any commas better than the ones we've already found. And for that I think we need a consolidated mina badness metric. I can come up with something. 37 is a reasonably hard prime limit, but I think we could even go above 51 SoPF if everything else was absolutely beautiful.

I can come up with something.
calculate the obvious mapping ("patent val")* from primes to degrees of 809-EDA. which starts off
⟨ 8539 13534 19828 ... ]
then take its inner product with the monzo for each candidate, and publish the number of tinas it comes to. Then we can compare that with the number of tinas obtained by rounding the comma's untempered size.
*Actually, the 809-EDA mapping of prime 3 is not so obvious ("not so patent") because a simple calculation of how many tinas are in prime 3 gives 13534.60451 which rounds up to 13535. But if you then calculate how many tinas are in an apotome, as 7 × 13535 - 11 × 8539 you get 816, not 809. It has to be 13534 to get 7 × 13534 - 11 × 8539 = 809.

I get the right answer for prime 3 straight off, if I use the optimal tina given here:
https://en.xen.wiki/w/The_Riemann_Zeta_ ... and_Tuning
i.e. 1200/8539.00834 = 0.140531541 ¢
This corresponds to neither an exact EDO nor an exact EDA. But we should use the obvious mapping for this tina.
With the patent val, the mappings were all over the place. With the optimal val, every comma in our list mapped to its rounded size.
For the sub-tinas represented by the dot, I would not use a 1618-EDA mapping, as that is likely to be highly inconsistent. I would try each of the EDAs I listed in a previous post in this thread. And compare the tempered number of sub-tinas with the rounded untempered number for each candidate comma.

Or perhaps we should adopt @cmloegcmluin's suggestion from another thread:
Seriously, though, what if... just what if... the dot represented "full stop". Like, there is nothing beyond this level of insanity. What I'm suggesting is that we just say the dot represents anything you want, so long as its less than a tina.
Well I was thinking some more about that.... and that's essentially what the secondary comma zone means anyway, yeah?

Perhaps what I should have said was: what if the primary comma of the half-tina is the unison. Meaning that for purposes of digital playback, it has no effect.

Re: Magrathean diacritics

Posted: Thu May 28, 2020 3:22 am
by Ash9903b4
Dave Keenan wrote:
Wed May 27, 2020 12:19 pm
Thanks @Ash9903b4 and @cmloegcmluin.

Unfortunately, a good EDO doesn't necessarily mean a good EDA. The two happen to be almost indistinguishable in the case of 8539edo and 809-EDA. But that is not the case for 16808edo and 1592-EDA. 1592-EDA is more like 16804edo and is nothing special.

I dusted off my old EDA search spreadsheet (attached) and pushed it out to 3000-EDA. It's somewhat quirky since it only looks at errors in the commas commonly represented by Sagittal elements (flags and accents).
In that case, we might want to look at divisions that make for both a good EDO and a good EDA, where the apotome doesn't deviate too much from its established size. 5587EDA/58973EDO is still a solid choice, as well as 1914EDA/20203EDO, and in particular 3436EDA/36269EDO has both a low average error (0.1263) and a low max relative error (0.3764) on your spreadsheet, as well as being a zeta integral EDO (and probably a zeta peak EDO beyond those listed on the wiki page, but I admit that I haven't checked all EDOs between 34691 and 36269 to confirm this).

The idea to make dots map to the unison sounds interesting, but I don't think it would be that much of a problem if we skipped JI entirely and just made them stand for 1 degree of either 20203 or 36269 by default.

Re: Magrathean diacritics

Posted: Thu May 28, 2020 3:27 am
by cmloegcmluin
I updated my earlier chart: latest highlights, re-ordering columns, adding tina values.