Dave Keenan wrote: ↑Fri Aug 14, 2020 10:01 am

It's disappointing that while this metric distinguishes 35/1 from 7/5, and distinguishes 55/1 from 11/5, and 77/1 from 11/7, it does not distinguish 77/5 from 55/7 or 35/11.

Which have 55, 61, and 92 votes, respectively. Yeah, that is a bummer, huh?

Ooh, well this is interesting... I almost can't believe that with w and b set to such weird numbers as -0.944786887715889 and -1.561335378 that these three ratios can come out with identical antivotes, so I looked them up in my detailed logs, and actually I rank them differently... but not for a good reason. 35/11 gets rank 37 with 8.597237100787783 antivotes; 55/7 and 77/5 get rank 38.5, being tied with 8.597237100787785. Yup: that's one of them famous JavaScript floating point arithmetic discrepancies.

Fun factlet: I actually had to revise one of my tests yesterday to verify values with significantly less precision, because a SoS calculation started failing on CI even though it was passing locally. I was able to reproduce the failure on my work laptop, which shares Linux architecture with the container the tests are running on in my free CI solution. So it's that finicky!

Anyway, probably what I should do is stop dealing with antivote values with any greater precision than the billionths with which we've been exchanging final SoS values. This will apparently make the difference between values tying or not, which as we've already seen, particularly on a large scale, can have nontrivial impacts on the final SoS.

You're kidding yourself if you think I'm going to wait for my scripts to re-run for days to get slightly fixed values, though

I can at least get them for our selected contenders though.

All that's cool and all, but it doesn't address the main issue which is the failure of WBL-1 to differentiate between 77/5, 55/7, and 35/11. Though I note that this differentiation was not among

those enumerated by myself a couple days ago. It's similar to the fourth task, balance, except it's not "prime balance", but just plain old "balance", and thus it would be the only one of the five tasks which is not with respect to primes (i.e. it would be "prime repetitions", "prime content", "prime limit", "prime balance", "balance") but simplify magnitude of the numbers in the numerator and denominator. That makes me a tiny bit suspicious of it. But let's see if it holds for any other permutations of 3 primes:

35/13: 25 votes

65/7: 11 votes

91/5: 11 votes

55/13: 4 votes

65/11: 6 votes

143/5: 13 votes

77/13: 8 votes

95/11: 1 votes

143/7: 9 votes

I'm not seeing enough of a pattern to think this is something our metric must capture. What do you think?

Rounding W and B to 1/2 and 1/3 increases the SoS over all ratios from 0.010786256 to 0.011129715. These are approximate values due to my way of dealing with primes above 97, but that's a 3% increase. It increases the SoS over the first 80 ratios from 0.003886596 to 0.004205825 (an 8% increase).

Okay, my code is set up to verify that, as long as I can figure out what the equivalent w and b values would be for those W and B... seems like this should be easy enough... just reverse engineer your transformations... Right okay, so if W = 2

^{w} and B = 2

^{b}, then I just need to take lb of each to get back. And if W = 1/2 and B = 1/3, then w = -1 and b = -1.58496250072, which is indeed extremely close to their original values of -0.944786887715889 and -1.561335378, respectively. I confirm this gives SoS(-1) of 0.00420582488763467, SoS(-1)-all of 0.010749073335214025 (well that one's actually even lower than your non-rounded SoS(-1)-all... but we've implicitly established that your tool will struggle with SoS(-1)-all), and SoS(1) of 12566.5.

I don't like "wobble" at all, because it tells the first-time reader (or hearer) absolutely nothing useful about the mathematical nature of the function, or its purpose. "unpop" is slightly better.

I agree with not liking wobble for that reason. Okay, let's strike it from consideration.

Yeah, "unpopularity" appears to be an actual word, but it just sounds wrong to me for some reason. It's a mouthful in any case.

Alright, we'll keep thinking on it, then.

Oh — so have you for certain decided that even though wybl beat wbl on SoS(-1), wbl beat wybl on SoS(1) by enough that it's better? I'm just a bit surprised by how much you seem to care about SoS(1) now, when everything I'd spent days running scripts for was to desperately minimize SoS(-1) (why wasn't I running that script for 3 days on SoS(1) and then in the end checking on SoS(-1) to see how that looked? Why was it this way and not the other?)

"Bennedetti height" is a ridiculously-obscure name for something that can be (and often has been) *completely described* in the same number of syllables, as "product complexity". And for goodness sake, why write "BH(n/d)" for what can be written as "n·d", and then have to explain what BH is?

Ah ok. I'd never heard the phrase product complexity before (and web searches don't readily turn up evidence of it being an accepted or popular mathematic term). From an anecdotal standpoint, Bennedetti height seems to come up all the time when I find myself in discussions of xenharmonics (and not always brought up by myself). Those things both said, if you do not think the appeal to xenharmonicists of seeing BH in our metric will be worth the cost of some people potentially not knowing or caring about it, then I am certainly fine sticking with n·d.

Thanks for reminding me that "copfr" is a (very sensible) name of our own making. Again, "big omega" tells you nothing. It has zero mnemonic value relative to any of its

many other uses. But we should mention it in passing, in any exposition on this metric, as we should mention "gpf" as an alternative to "prime-limit".

Yeah, speaking of exposition, who's writing our findings up for Xenharmonikôn?!

I thought of asking that earlier, but I wasn't sure you'd think there'd be wider interest in what is essentially an inner mechanic to Sagittal.

I'm not sure I've finished with potentially-more-memorable transformations of this metric, so it might be a bit premature to name it. But I'm hoping we can come up with an acronym or abbreviation based on its components, possibly including product-complexity and prime-limit.

I hope so too. Perhaps my list of 4 (or 3, or 5) tasks our metric has could be a source of inspiration, too. "Simplicity" and "complexity" are another pair of words which have come up when talking about the purpose of this metric. Interestingly, the word "prime" itself contains an essence of fundamentality, simplicity, or importance. How about "primacy"? We could take the "primacy" of a ratio? A problem is that this metric gives bigger numbers for the opposite thing. So it could be ratio obscurity. I kind of like that.