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### Re: Magrathean diacritics

Posted: Tue Jun 23, 2020 12:24 am
Dave Keenan wrote: Mon Jun 22, 2020 8:39 pm
volleo6144 wrote: Mon Jun 22, 2020 10:15 am The "u" term heavily penalizes commas with many primes on one side but only one or two, or zero, on the other, such as our yellow 2 [5831:5832], 4 [3024:3025], and 5 [2400:2401] tinas.
So "u" is the only thing differentiating 7/5 from 35 etc, but not very well. I'd like to understand why "t" isn't doing that. But maybe if you take the log of "u" it will work even better.
No, "t" is doing that, but in a different way: 77:125 (for example) is (18-15)/20 = 0.15 for t and (125-77)/200 = 0.24 for u. It's just that t does this to a greater extent (because of its higher scaling of 5 per 100 instead of 5 per 1000) for the typical case of distinguishing 5:7 from 1:35, while u heavily penalizes things with a lot of primes on one side (49:9765625n for the half-tina is immediately out because its u value is over 40,000, while its t value is a more tolerable 1.8).

And then we have the issue of how to prevent Pythagorean schisminas like [-1054 665> (665edo3n, which has a respectably low error of 0.037806 tinas from the half-tina) from being all the primary commas for all the tinas.

### Re: Magrathean diacritics

Posted: Tue Jun 23, 2020 6:02 am
I have spun out the subtopic about the comma complexity metric into its own topic here: viewtopic.php?f=4&t=493

### Re: Magrathean diacritics

Posted: Tue Jun 23, 2020 6:04 am
Re: the tina smileys, I do plan to add those to the forum eventually. I have added Sagittal smileys before so I know how. Would it be helpful to have them sooner than later? I think our discussion at the moment is not being hindered dramatically by lacking them.

### Re: Magrathean diacritics

Posted: Tue Jun 23, 2020 6:25 am
cmloegcmluin wrote: Tue Jun 23, 2020 6:04 am Re: the tina smileys, I do plan to add those to the forum eventually. I have added Sagittal smileys before so I know how. Would it be helpful to have them sooner than later? I think our discussion at the moment is not being hindered dramatically by lacking them.
See, I believe that is in fact the case, but it's just ... nice to have. I certainly wasn't here for when the , , , and smilies were added in November of 2016 (as indicated by some caching-related response headers), so I don't know exactly when in the discussion of Olympian they were added, and wouldn't know when that would be in the Magrathean discussion.

### Re: Magrathean diacritics

Posted: Tue Sep 01, 2020 8:23 am

At last I can come back to this thread, after a digression that resulted in the longest thread in the Sagittal forum, by far:
Developing a notational comma popularity metric, which resulted in the discovery/development of the beautiful function N2D3P9 as a replacement for sum-of-prime-factors-with-repeats, as a popularity metric for the equivalence classes of rational pitches that can be notated with a given microtonal accidental, each such pitch class being represented by a 2,3-free superunison ratio, as listed here in order of their N2D3P9 value.

At first I thought we would then have to create a comma popularity metric (as per the name of the thread), by combining N2D3P9 with the 3-exponent and/or the apotome-slope of the 2 or 3 commas that could be used to notate each 2,3-free ratio. But some things that @cmloegcmluin wrote in that thread have apparently been working away in my brain subconsciously, and finally made me realise that may not be necessary.

In this post, I noted that the first 2,3-free ratio in the above list that does not yet have a sagittal symbol, 25/11, is a candidate to be notated by a 1 tina accent applied to the existing symbol for 49/11  . That makes 1 tina = 1225/121n which is what George Secor suggested for it. There may be other un-notated but moderately popular ratios that could be notated with tina accents. This might be the best way to assign commas to the accents for 1 thru 9 tinas.

But the realisation I had this morning, goes further than making me wonder if there may be other un-notated but moderately popular ratios that could be notated with tina accents. It makes me wonder if perhaps we should not have given multiple symbols below the half-apotome to any 2,3-free ratio, no matter how popular it is. And should only have given a symbol to the comma with the lowest absolute value of apotome slope.

apotome_slope = exponent_of_3 - 7 × untempered_size_in_cents/113.685

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.

That's probably going too far, and it might be weakened to notating the commas having either the lowest slope or the lowest 3-exponent of the available commas, or to having absolute slope below some threshold. But it would be interesting to see that single lowest-slope comma, and its size in cents, listed against each ratio in the above list. Then search for cases where two such commas differ by 9 tinas or less, and consider assigning each tina accent to that correctly-sized difference whose pair of 2,3-free ratios have the lowest maximum N2D3P9.

But we'd still need to consider how close to a whole number of tinas the difference was. I'm using the term "difference" here for kind of meta-commas between commas.

### Re: Magrathean diacritics

Posted: Tue Sep 01, 2020 9:37 am
Dave Keenan wrote: Tue Sep 01, 2020 8:23 am At last I can come back to this thread, after a digression that resulted in the longest thread in the Sagittal forum, by far:
Developing a notational comma popularity metric, which resulted in the discovery/development of the beautiful function N2D3P9 as a replacement for sum-of-prime-factors-with-repeats, as a popularity metric for the equivalence classes of rational pitches that can be notated with a given microtonal accidental, each such pitch class being represented by a 2,3-free superunison ratio, as listed here in order of their N2D3P9 value.
I suppose we should credit phpBB, then, with an assist in the development of N2D3P9: we never would have been able to pull it off had the forum not supported the "Split topic" capability.

On a serious note, if you haven't checked out the Xen wiki post for N2D3P9, please do! I'm quite happy with what we came up with.
At first I thought we would then have to create a comma popularity metric (as per the name of the thread), by combining N2D3P9 with the 3-exponent and/or the apotome-slope of the 2 or 3 commas that could be used to notate each 2,3-free ratio. But some things that @cmloegcmluin wrote in that thread have apparently been working away in my brain subconsciously, and finally made me realise that may not be necessary.
Interesting!

So we had discussed the development of a comma popularity metric (well, I recently had suggested we call it a comma "usefulness" metric which incorporated the notated ratio popularity metric N2D3P9 along with abs3exp and/or apotome slope to measure its usefulness for notating EDOs, but haven't heard back from you on that suggestion) as an intermediate step to the development of a badness metric which incorporates a final component of (tina) error. It looks like you are suggesting that we no longer need either the comma popularity metric nor the badness metric, because you may have a better method for choosing tina commas. Is that correct?
In this post, I noted that the first 2,3-free ratio in the above list that does not yet have a sagittal symbol, is a candidate to be notated by a 1 tina accent applied to the existing symbol for 49/11  . That makes 1 tina = 1225/121n which is what George Secor suggested for it. There may be other un-notated but moderately popular ratios that could be notated with tina accents. This might be the best way to assign commas to the accents for 1 thru 9 tinas.

But the realisation I had this morning, goes further than making me wonder if there may be other un-notated but moderately popular ratios that could be notated with tina accents.
Yes I remember you mentioning this. I think this is a cool approach.
It makes me wonder if perhaps we should not have given multiple symbols below the half-apotome to any 2,3-free ratio, no matter how popular it is. And should only have given a symbol to the comma with the lowest absolute value of apotome slope.
Let me make sure I follow correctly. In the extreme you might be suggesting that the linked table above ideally would only have a single symbol for each row, so that the duplicate symbols could be reallocated to exactly notate ratios which are not exactly notated. And that the symbol remaining in each row after the duplicates were reallocated would be the one with the lowest absolute apotome slope.

If so, that sounds interesting. But I'm afraid it might have to be more complicated than that. I use the word "afraid" because I don't yet fully understand the problem space enough to imagine this. But I'm concerned that one wouldn't simply be able to do such a reallocation without fundamentally redoing each precision level.

You spoke of the difference between assigning symbols to commas, and assigning commas to symbols. Assigning symbols to commas is how Sagittal started out, of course. And there may not be a hard break from one mode to the other. In some sense it's always about bringing them together. Just that in the beginning, the set of commas that need symbols is the somewhat obvious bit, and then as you press into deeper precision levels, its the symbols which get more obvious and the commas which are less certain.

Wasn't the creation of the precision levels something like a hugely multidimensional game, with comma popularity as one dimension, the assemblage of a set of tiers commas with each tier roughly equally spaced throughout the half-apotome as some other dimensions, element arithmetic so that the chosen commas would be recombinable in logical visual ways as another dimension, etc.

You're probably not actually suggesting something that extreme, though. Perhaps the isolation of the single symbol per 2,3-free ratio is just meant as a prerequisite for your proposed experiment.

I think this is along the lines of this thread we had going on:
Dave Keenan wrote:
cmloegcmluin wrote:
Dave Keenan wrote:
cmloegcmluin wrote:
Dave Keenan wrote:
cmloegcmluin wrote: I still have no idea how we're going to weight abs3exp and apotome slope. What objective data do we have to model that after?
Bugger all. Although we did throw away some data on 3-exponents from the Scala archive data, because initially it was the full pitch ratios with their individual counts, before I crunched them down to 5-rough. I think the full data is in a spreadsheet somewhere in this forum. That's data on what 3-exponents people use in pitch ratios. But I assume the average (signed) 3 exponent will be close to zero.
I know you shared it because I have a copy of it in my Google Sheets.

Yes, I've found it here: viewtopic.php?p=1331#p1331
We're more interested in: When choosing what comma to give to some symbol, how much are we willing to allow the 5-rough-no-pop-rank to increase in order to have a comma that doesn't require so many sharps or flats to go along with it.
Or also perhaps how much are we willing to allow the 5-rough-no-pop-rank to increase in order to have a comma which allows notation of ratios which are not yet exactly notatable?
Hmm. I don't thing George and I ever did that. But it doesn't seem unreasonable.
If one goal is to enable as many ratios as possible to be exactly notatable, it seems reasonable. But I'm thinking this could turn into quite the rabbit hole of a thing to optimize for.
Yes. Let's set the whole revisit of the Extreme JI notation to one side until we've assigned the tina ratios and updated SMuFL and Bravura. I'm pretty sure there are no unaccented symbols or unsymbolled accents in danger of being redefined, and hence no effect on the definitions already in SMuFL (except the errors you found some months back).
Am I correct in connecting these two threads, and if so, does resurfacing any of these thoughts affect things at all for you?
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. 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?
That's probably going too far, and it might be weakened to notating the commas having either the lowest slope or the lowest 3-exponent of the available commas, or to having absolute slope below some threshold. But it would be interesting to see that single lowest-slope comma, and its size in cents, listed against each ratio in the above list. Then search for cases where two such commas differ by 9 tinas or less, and consider assigning each tina accent to that correctly-sized difference whose pair of 2,3-free ratios have the lowest maximum N2D3P9.
I agree this sounds like an interesting experiment. I can get to it soon.

### Re: Magrathean diacritics

Posted: Tue Sep 01, 2020 11:07 am
cmloegcmluin wrote: Wed Aug 26, 2020 4:33 pm So I know the final layer is a "badness" metric, which will incorporate (tina) error and be used to choose commas for new symbols. And I know the intermediate layer is what we've been calling a comma-no-pop-rank, which you've said we could use for revisiting existing comma assignments. Is it really even a "popularity rank"? It will heavily leverage a popularity rank, but by including apotome slope (and possibly abs3exp) doesn't it sufficiently depart from that definition? It's not a badness rank, but perhaps it's a notational "usefulness rank" or "utility rank", since the apotome slope is there to help tell us how useful the comma is for notating EDOs as the size of the fifth varies?
cmloegcmluin wrote: Tue Sep 01, 2020 9:37 am So we had discussed the development of a comma popularity metric (well, I recently had suggested we call it a comma "usefulness" metric which incorporated the notated ratio popularity metric N2D3P9 along with abs3exp and/or apotome slope to measure its usefulness for notating EDOs, but haven't heard back from you on that suggestion) as an intermediate step to the development of a badness metric which incorporates a final component of (tina) error. It looks like you are suggesting that we no longer need either the comma popularity metric nor the badness metric, because you may have a better method for choosing tina commas. Is that correct?
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).

And yes, "comma usefulness" would have been a good name for that thing I think we no longer need.
It makes me wonder if perhaps we should not have given multiple symbols below the half-apotome to any 2,3-free ratio, no matter how popular it is. And should only have given a symbol to the comma with the lowest absolute value of apotome slope.
Let me make sure I follow correctly. In the extreme you might be suggesting that the linked table above ideally would only have a single symbol for each row, so that the duplicate symbols could be reallocated to exactly notate ratios which are not exactly notated. And that the symbol remaining in each row after the duplicates were reallocated would be the one with the lowest absolute apotome slope.
You would also need to only show symbols smaller than the half-apotome, in order for there to only be one symbol per 2,3-free ratio. But with that proviso, yes, that is exactly what I'm suggesting in the extreme case, which is the case I'm suggesting we use to find tina definitions.

It would be similar to the one-symbol-per-prime notation. It would be extending that to a one-symbol-per-2,3-free-ratio notation, and we would be considering a comma's usefulness in notating EDOs.

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.
If so, that sounds interesting. But I'm afraid it might have to be more complicated than that. I use the word "afraid" because I don't yet fully understand the problem space enough to imagine this. But I'm concerned that one wouldn't simply be able to do such a reallocation without fundamentally redoing each precision level.
Yeah. We're not going to mess with the existing JI precision levels (except perhaps in minor ways). 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.
You spoke of the difference between assigning symbols to commas, and assigning commas to symbols. Assigning symbols to commas is how Sagittal started out, of course. And there may not be a hard break from one mode to the other. In some sense it's always about bringing them together. Just that in the beginning, the set of commas that need symbols is the somewhat obvious bit, and then as you press into deeper precision levels, its the symbols which get more obvious and the commas which are less certain.
Yes. That's a good way of thinking about it.
Wasn't the creation of the precision levels something like a hugely multidimensional game, with comma popularity as one dimension, the assemblage of a set of tiers commas with each tier roughly equally spaced throughout the half-apotome as some other dimensions, element arithmetic so that the chosen commas would be recombinable in logical visual ways as another dimension, etc.
Indeed it was.
You're probably not actually suggesting something that extreme, though. Perhaps the isolation of the single symbol per 2,3-free ratio is just meant as a prerequisite for your proposed experiment.
You got it.
I think this is along the lines of this thread we had going on:
Dave Keenan wrote: ...
Am I correct in connecting these two threads, and if so, does resurfacing any of these thoughts affect things at all for you?
You are correct in connecting them. Resurfacing them here doesn't change my thoughts on this. They are part of what surfaced from my subconscious this morning.
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.
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.
I agree this sounds like an interesting experiment. I can get to it soon.
Thanks!

### Re: Magrathean diacritics

Posted: Wed Sep 02, 2020 8:49 am
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.
It would be similar to the one-symbol-per-prime notation. It would be extending that to a one-symbol-per-2,3-free-ratio notation
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.

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!

------

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.

### Re: Magrathean diacritics

Posted: Wed Sep 02, 2020 1:47 pm
cmloegcmluin wrote: Wed Sep 02, 2020 8:49 am
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?
Sure. And I should have said 8539.00834-EDO, which is 808.964346-EDA. And that reminds me that we need to use this number (not 809, it gives the wrong mapping) for the boundaries of the JI precision-level notations, if we haven't already, with the usual mirroring about the half-apotome.
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 think we should not specify any ratio for the dot, just call it a "fractional tina". I think you suggested elsewhere, something like:

We could think of the dot as a full-stop that says, "Let the insanity stop here". Let it represent anything you want, so long as it is less than a tina.

[joke] The 0.003347 ¢ meta-comma between 5:19C and 13C would be a valid definition for Gene Ward Smith's "neutrina" (one degree of 30723-EDA, 0.003700 ¢, ≈1/38th of a tina). If the dots were used to represent neutrinas, 13C and 5:19C would have the same core, schisma accent and tina accent, but could differ in one of them having 18 dots up and the other having 19 dots up (or something like that). [/joke]

More realistically, the dot might be used for either 1/2-tina or 1/3-tina. We don't need to specify.
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.
Yeah. That's still valid. But only for whole tinas.
Re: consistency, ...
I just mean you should calculate the obvious mapping from primes to degrees of 8539.00834-EDO (yes, definitely that one), which starts off
⟨ 8539 13534 19827 ... ]
then take its inner product with the monzo for each candidate comma, and ensure it's the same as the number of tinas obtained by rounding the comma's untempered size.
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.
Exactly.
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?
No, I think not, because the abs3exp (or slope) of a meta-comma (for an accent mark) is irrelevant, except (possibly) if it becomes the definition or primary comma for the symbol consisting of that accent mark beside a bare shaft.
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?
No. That just reintroduces the need to rank commas according to some combination of their abs3exp and their N2D3P9, in order to decide which is the most deserving meta-comma for a given number of tinas. When you only use the comma with the lowest absSlope (or alternatively the lowest abs3exp) for each 2,3-equivalence-class, then you don't have that problem.
I can still print out a new version of that N2D3P9-sorted 136 most popular 2,3-free class table
I think we need all symbols with N2D3P9 < 307. viewtopic.php?f=4&t=493&p=2252#unpopularSymbols
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?
Yes, every row. In fact the symbols are irrelevant to this process, although it would still be interesting to see them in the table.

### Re: Magrathean diacritics

Posted: Wed Sep 02, 2020 3:07 pm
Dave Keenan wrote: Wed Sep 02, 2020 1:47 pm I should have said 8539.00834-EDO, which is 808.964346-EDA. And that reminds me that we need to use this number (not 809, it gives the wrong mapping) for the boundaries of the JI precision-level notations, if we haven't already, with the usual mirroring about the half-apotome.
Cool. This is just the sort of thing I can build into the code. So it'll be documented, tested, feed the web calculator, be the foundation for regenerating assets such as the SMuFL map, precisions diagram, spreadsheet calculator, etc., and catch regressions if we ever try to tweak things.

Just this morning I added tests for half-apotome mirroring, and I soon plan to add tests for protected secondary commas, etc.
More realistically, the dot might be used for either 1/2-tina or 1/3-tina. We don't need to specify.
Elsewhere I had suggested that for purposes of playback, it would alter pitch by nothing (AKA its primary comma would be unison). But I suppose it could be a 1/3-tina.

But more importantly, yes, I do think I'm settling on the half-tina dot as this "full-stop... end the insanity here". We may lose the support of some sentient extraterrestrials who are physiologically capable of distinguishing pitches that are hundredths of cents apart, but they've all been fairly quiet on the forum recently...
Re: consistency, ...
I just mean you should calculate the obvious mapping from primes to degrees of 8539.00834-EDO (yes, definitely that one), which starts off
⟨ 8539 13534 19827 ... ]
then take its inner product with the monzo for each candidate comma, and ensure it's the same as the number of tinas obtained by rounding the comma's untempered size.
Oh right, that makes sense.

Sorry, I'm just laughing/crying for a moment in remembering that we're dealing with an EDO for which the tiny intervals we're concerning ourselves with map to multiple degrees...
the abs3exp (or slope) of a meta-comma (for an accent mark) is irrelevant, except (possibly) if it becomes the definition or primary comma for the symbol consisting of that accent mark beside a bare shaft.
...and then that symbol consisting of an accent beside a bare shaft gets used in an EDO.
And you don't see too many or on the Periodic table, do you...? So yeah, that makes complete sense.
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?
No. That just reintroduces the need to rank commas according to some combination of their abs3exp and their N2D3P9, in order to decide which is the most deserving meta-comma for a given number of tinas. When you only use the comma with the lowest absSlope (or alternatively the lowest abs3exp) for each 2,3-equivalence-class, then you don't have that problem.
Alright, I trust you on this one. My assumption was that including all these notating commas was that we'd find some new (tina-sized) cents-differences. It sounds like you have already convinced yourself that the additional notating commas per 2,3-class would be redundant, i.e. introduce no additional cents-differences. I could probably convince myself of this with a little bit of ... what do you call it... "reasoning"... but I'm afraid I'm spent for the day already.
I can still print out a new version of that N2D3P9-sorted 136 most popular 2,3-free class table
I think we need all symbols with N2D3P9 < 307. viewtopic.php?f=4&t=493&p=2252#unpopularSymbols
I'll get that soon. But tomorrow I've allocated to another project so you shouldn't plan to wake up to it. Sorry for the delay.
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?
Yes, every row. In fact the symbols are irrelevant to this process, although it would still be interesting to see them in the table.
I can include them. Shouldn't be tough.