Dave Keenan wrote: ↑Tue Sep 01, 2020 11:07 am
That is correct. And perhaps instead of looking at tina error we should ensure that any tina comma we choose is consistent with the 809-EDA/8539-EDO obvious mapping (patent val).

Or perhaps both?

I'd need to review the original motivation for considering tina error. Probably we talked about it already, earlier in this thread. It seems obvious that you'd rather the primary commas were close to the middles of their capture zones rather than on their edges, but I don't at present have a lucid sense of just how important that is and why.

Update: I've reviewed the thread. I found stuff about us requiring tina error be less than a quarter-tina, which makes sense because beyond that point you'd be crossing into the territory of the next half-tina. We were considering the dot to represent a third-tina at some point though, in which case we'd have to prevent tina error from being anything more than a sixth-tina.

I did find this thought shared in a private communication back in March:

We want the exact sizes of the symbols (the size of their primary commas) to be reasonably evenly spaced in any given precision level, so that we minimise the maximum error in any ratio that must be notated approximately.

So, that makes sense. That's probably the main reason actually.

Re: consistency, is there a handy-dandy tool you're aware of for calculating the consistency of some system? Something like Graham Breed's tools maybe?

Here are three resources you shared with me about it:

http://www.tonalsoft.com/enc/c/consistent.aspx
https://en.xen.wiki/w/Consistent
http://www.huygens-fokker.org/docs/consist_limits.html
But they don't lead me to such a tool nor am I able to turn one up with my web searching skills.

I'd have enough information to build one myself, but I'd rather not, of course, if someone else has already built out a robust solution.

Or is it as simple as this: each tina must individually get mapped to unison by the patent val for 809-EDA? Or did we for sure

decide on 8539.00834-EDO? Probably we should just check both...

And yes, "comma usefulness" would have been a good name for that thing I think we no longer need.

An unuseful usefulness metric it is, then.

You would also need to only show symbols smaller than the half-apotome

This makes sense to me, yes. As you said before, the commas above the half-apotome mirror are dependent on those below it.

Think of it purely as a way of finding good tina definitions, or at most an alternative kind of JI notation, as an extension of the prime factor notation to a prime-factor-combination notation.

These statements were instrumental for me in getting my head around your ask.

So we're not actually going to think about re-allocations for the symbols stricken from the table. We're just getting them out of the way so we can isolate the symbol for each 2,3-free class whose notating comma has the best absolute apotome slope.

A subtlety, then: mightn't it be handy to actually have two options: one with the smallest

*positive *apotome slope, and one with the smallest

*negative *apotome slope? Because if we're going to use them to find tinas by subtracting ones which are close in cents to each other, it would be nice if their 3-exponents were really similar. And wouldn't it be a shame if we had to subtract two commas from each other and one had a negative 3-exponent and the other had a positive 3-exponent, when we might have found a better meta-comma as the difference of two commas with 3-exponents with the same sign?

We know the apotome slope will be really close to the abs3exp for tinas since their fraction of the apotome is so small. So basically it'll just be the difference of the 3-exponents of the two diffed commas.

Actually, if I'm just going to write us a script to comb over these things and find the ones with 1-to-14-tina-sized differences, mightn't I as well just include all (up to 4 or 5 I think we found) of the notating commas per 2,3-class?

I can still print out a new version of that N2D3P9-sorted 136 most popular 2,3-free class table but with the single best apotome sloped notating comma per row, if you want. To be clear, you'd want that for every row — not just the rows which have exactly notating Sagittal symbols, right?

Here are two reasons why that would be too extreme a restriction for general symbol definition (as opposed to merely finding good tina definitions): 1. it would disallow

as 5s (although we could make an exception for accents because they would be defined as "meta-commas"), 2. it would disallow a symbol for the optional comma for 17/1 that some people prefer.

To be clear: it would disallow

as 5s because the 1/5C (Pythagorean comma, 81/80) has a better apotome slope than it (2.676 beats 7.880).

And that 17/1 bit is related to Dave Ryan's preference covered

here. It's the choice between the 17k w/ apotome slope 6.462 and the 17C w/ apotome slope 4.093. Sagittal uses the 17C to represent prime 17 in its Prime Factor notation, eschewing the 17k (which Dave Ryan prefers). Sagittal's (precision level) JI notation, on the other hand, allows either one. A prime-factor-combination notation would have the same limitation as the Prime Factor notation, though.

But this is all not terribly relevant to the current work at hand. It seems we agree that we should not actually pursue release of such a "prime-factor-combination" or "meta-comma" notation. However, we can use it to help make the best decisions about the tina commas. In other words, such a notation can be worked into the JI precision level notation and become part of it at this precision level (and perhaps use this strategy when reassigning those couple of bad apple commas in the Extreme notation which we recently found w/ N2D3P9).

If we did that, there would be no need to apportion the "votes" for a given 2,3-free ratio among the multiple commas that could be used to notate it, as George and I did in the past.

Perhaps this is a subtlety contained in the Xenharmonikôn article, but I don't think I was aware of such apportioning.

No. Not mentioned in the XH article. But it's done in one of the spreadsheets posted in this forum, and I think I described how the apportioning was done, somewhere in this thread.

Let me know if you find it. I just reviewed the thread and couldn't find it.

Are you saying that when multiple symbols were assigned the same comma, this was in part justified because there were just so many votes for that comma that it warranted multiple symbols?

We didn't think of it in that way, but I suppose that is an indirect consequence of what we did. It was more that when assigning a comma to a symbol, a comma for a more popular ratio could lose out to a less popular one because the comma for the more popular ratio had too large a 3-exponent and so got only a small proportion of the votes.

Are you saying that had apotome slope / abs3exp not knocked some commas out of the running, it would have been even more skewed toward the most popular ratios getting all the symbols?

I agree this sounds like an interesting experiment. I can get to it soon.

Thanks!

I still owe us some tina smileys for the forum, too, apparently!

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I thought I should resurface a thought we were working with earlier in the thread: that we should choose tina commas with 3-exponents close to +8 since when subtracted from the 5-schisma they would lead to a low 3-exponent. That was only for tinas 1 through 7, though, and it was because we were planning to set tinas 8 though 14 as the difference between the 5-schisma and the corresponding tina.

I expect that this thought would be outmoded (that's right — I just used outmoded as a past-tense verb, not an adjective!) if this experiment works out. In some sense this experiment is an extension of it. We may find a meta-comma difference between two commas which is in the zone for being assigned to the 13-tina symbol, by which we really mean, since there is no 13 tina, that the difference between it and the 5-schisma would be the 1 tina.

Or should I only look for ones within 9 tinas, which is the range we directly need commas for? Okay, yes, I see that you specifically said 9 tinas earlier. Got it.

It looks like we have not already 100% locked down a half-tina, so I should group possibilities by half-tina.

And the reason I brought up this 5-schisma-difference bit was to confirm that we no longer need the 7.5-, 8-, 8.5-, and 9- tina symbols to be half-5-schisma-mirrors of the 6.5-, 6-, 5.5-, and 5- tina symbols. That actually it might be a feature for them to not be complements of each other, if that allows us to exactly notate more as-of-yet not exactly notated 2,3-free classes.