Magrathean diacritics

[Dave Keenan]

I just found this in email from George, subject heading: "Re: Primary commas still undecided", dated: 7-Dec-2006 (so it's in between the two sets linked above). This is not the whole email. It's only the part relating to 25/11C and 11/49C.

>> Dave wrote:
>>> I understand there is still some uncertainty about whether ~~| is
>>> 11:25C or 11:49C at the super-olympian level. Is that correct?
>>
>> Yes, but I intend to show why we should prefer 11:49C (later).
>
> Hmm. How much _later_? 

Ummm -- okay, now (see below).

> 11:49 does have wonderfully low slope and so
> will probably be better for notating ETs.
>
> I guess that means poor old 11:25C gets the double right accented
> symbol |~'' ?

Yep.  It boils down to this:  Assuming that we're going to distinguish
between the two (at the olympian level), then one of them gets ~~| and
the other gets something else.  Given a choice between |~'' and .)|~.
for "something else", I think we would prefer the former.

So which comma gets ~~| ?  This is answered by looking at this several
ways.

1) Compare the symbol & comma sizes:

SoCA for |~'' is 17.377c
11:25C is 17.399c
SoF for ~~| is 17.459c
11:49C is 17.576c

Both symbols are closest to 11:25C, but |~'' is closer than ~~| to
11:25C, and 11:49C is closest to ~~| ; so ~~| gets 11:49C.

2) Use monotonic symbol cores as a tie-breaker:

Assuming that the 36th mina will be split halfway between 11:25C and
11:49C (at 17.488c), then |~'' must be either above or below that
boundary, and ~~| will be opposite.  Putting |~'' below the boundary
maintains a monotonic symbol core sequence with |~, so |~'' gets
11:25C.

3) Determine what's most needed for ET notations:

Many divisions (e.g., 388, 742) have 11:25C and 11:49C as the same
number of steps, so ~~| can represent both commas in those divisions.

However, where the number of steps differs, the number of steps for
11:25C will usually agree with 23C (for 311, 441, 525, 653, 665, 935,
but not 795; also cf. 207, 301), so if ~~| is 11:25C, then there will
be very few opportunities to use the ~~| symbol, since we would
probably prefer |~ .  Using ~~| as 11:49C would not only allow more
opportunities to use that symbol, but would also make it possible to
associate |~ with both 23C and 11:25C (remembering that this amounts to
dropping the double-right accent from the 11:25C symbol) in those
instances.

---

My response the next day had subject heading: "Forget the justification, just give me super-olympian", which pretty much says it all. I accepted George's recommendation on the matter.

But as we see from the 2007 email linked from the previous post, nearly a year later George decided to unsplit the 36th mina, so only one of 25/11C and 11/49C required a symbol.

[cmloegcmluin]

Before we get into it, I just wanted to drop a few miscellaneous notes.

• This is something I feel like a doofus for caring about, but it is kind of cute how the term "occam" is almost a scrambled version of "comma".
• When, re: the 7/425n, I said "I can't figure out what I would have meant by "most occurrences in Extreme Precision"", it popped back in my thoughts, and I thought maybe by occurrences I meant what we came to mostly refer to as "votes", i.e. the Scala usage stats. But {425/7}₂,₃ does not appear in the Scala stats, so I'm still at a loss.
• I just wanted to acknowledge that by finding the best comma for each semitina zone, we've made a big dent in the process of assigning commas for each tina of the Insane notation. It's certainly not the final list (e.g. 59/7n is the winner by direct badness for the horn and bare shaft slot, but we'd probably want to install the 10241/5n there, since it's what we're saying the horn glyph primarily represents; similarly 1/205n wins by direct badness for the wedge and bare shaft slot, though we're going with 1/5831n for the wedge's 2-tina value, and 253/5n wins by direct badness for the wedgebird and bare shaft slot, though we're going with 187/175n for the wedgebird's 8-tina value). Even if this was the final list of commas, we'd still need to choose which core each of them belong to. If anyone ever mentions tina-splitting, I will destroy them. Anyway, I'm not proposing it to be our next project. Honestly I'd rather not think about Insane again for a longy long time.

The following (post-mortem of a miscommunication) .... Is that correct?

Yes, everything you say about what I was thinking and meaning is correct. Even if I didn't fully understand the ambiguities myself at the time.

Lets leave list-A as best Insane commas, and now we have list-B as best Ultra commas (which only differs from existing Ultra commas in one place, and may become actual Ultra commas in future). Such mathematical purity.

Haha. Yes.

I suspect it was only the Scala archive stats that favoured {49/11}_2,3 over {25/11}_2,3.

I just checked this, and found I was wrong... it may have been the lower slope that caused {49/11}_2,3 to win when using the closer Scala ranks or the closer SoPFRs (21 vs 25) as the unpopularity metric.

That makes total sense to me. Yeah, that's probably the explanation.

So, in theory 11/49C is better for notating EDOs, but the only EDO I can find being used for is 198-edo. It's used for 3 degrees. But I just checked and 25/11C is also 3 degrees of 198-edo. So that's no barrier to redefining as 25/11C.

If only there was a way to capture this occurrence rate across EDOs of a given symbol, so instead of manually checking such things we could factor it into a single mathematical formula for uselessness...

Just kidding. That'd be insane!

Such a redefinition would involve an update to the SMuFL notation, but it needn't delay the current update re Magrathean.

Yeah, that should be a totally separate effort. We'd previously expressed some trepidation about changing SMuFL class names, in case that would break points of integration with various other software such as MuseScore or VexFlow. Descriptions seem completely safe to alter, but it would seem weird to have a symbol with className "accSagittal11v49CommaUp" yet a description all about the 25/11C.

Even if say we searched George/Dave email and found some really good reason to keep as 11/49C, there are other reasons given, earlier in this thread, to break the tie for 7 tinas in favour of 7/425n.

So we go with the list given here:
viewtopic.php?p=2633#p2633

I will draft an email to Daniel Spreadbury and send it to you for comment/completion.

Yes, let's go with that list.

I searched George/Dave email re 11:49C versus 11:25C.

I just found this in email from George, subject heading: "Re: Primary commas still undecided"

...

My response the next day had subject heading: "Forget the justification, just give me super-olympian", which pretty much says it all. I accepted George's recommendation on the matter.

...
nearly a year later George decided to unsplit the 36th mina, so only one of 25/11C and 11/49C required a symbol.

What do you take away from these as the reason(s) why 11:49C was chosen over 11:25C for ~~| ( )? And are they still valid? I have my own reading/theory but I don't want to influence yours.

Overall, my head is just spinning from trying to form an opinion on the matter.

I like this idea of monotonic core sequences. Looks like it came up in other emails posted on that consistent 37 thread, where it also gets referred to as MTC or "core-crossovers". In this one the rule is laid out where an accented core can't cross over an unaccented one. I believe that is the case but I'll add a test to ensure it always stays the case.

As best as I can tell, "super-olympian" was an earlier name for what became the "Magrathean" symbol set, so when you and George refer to super-olympian you're talking about the Insane JI precision level notation. It's weird though because I see things like "A super-olympian symbol was assigned" or "Thus olympian is much better for DAFLL than super-olympian. I would reserve super-olympian only for emergencies, where someone absolutely insists on separate symbols for distinguishing between two complicated commas" but as far as I knew no major effort had yet been made to assemble the 282 (404-122) new sagittals and primary commas required for Insane. Also, I remember asking about "Herculean-X" in the past and getting an answer, but it's hard to search for in my email, mind-map, and on the forum; it just wants to turn up all results for "Herculean". So I don't remember the difference. Next time you explain it, I'll have to figure out a better way not to lose it! Sorry.

I note that the issue of the 49/11 and 25/11 also came up here, in the first post coming back to this topic after developing N2D3P9; the fact that we're not going with the 1225/121n for the 1 tina after all does not mean that whichever one we choose between 49/11 and 25/11, the other one won't be exactly notated. We could still choose the other one of the two as the primary comma for the symbol which is the other comma's core up or down a 1-tina horn accent. The fact that the symbol which is a bare shaft with a 1-tina horn is not the 1225/121n does not preclude this. We've established that there are numerous mina-accented commas in the Extreme notation which are not separated by exactly the 1/455n. Similarly, the 1225/121n is the difference between the 1/1225k and 1/121k, which we have a dedicated topic going for already, but I don't think it makes any difference that we didn't choose the 1225/121n as the 1-tina comma, or whether is 49/11 or 25/11. By the way, the list of 809 best commas per semitina zone gives 25/11C as the 124th tina and 11/49C as the 125th tina (it also gives 1/121k as the 51st tina and 1/1225k as the 52nd).

This post also mentions that vote apportioning process you did when choosing commas for Extreme, and initiated the LATE comma subproject we pursued as an alternative to it. We decided to eschew redundancy from our considerations when choosing Insane commas, but if we're going to reevaluate any sane commas — in this case a High comma — we probably do still need to consider redundancy. In the cyan-hilited text of your second link from May 2005, you lament losing both the spreadsheet where you implemented your process and the memory of how to do it, but here (in May 2020, fifteen years later ) you managed to reverse-engineer it. So think this process you detail could be one objective approach to applying factor #6 from that list I compiled. Another approach might be punishing non-LATE commas, either on a binary or gradated basis; i.e. our expanded ATE submetric which measures uselessness is one thing but comparing ATEs against other commas in the same 2,3-free class is a way of scoring redundancy.

Alright, all that out of the way, let me even attempt to confront the question directly...

So if I've got the history down correctly, in May 2005 y'all are figuring out the High notation. The 9°58 comma needs to be decided. You're deciding between 125C, 49/11C, 11/25C, and 95C. While 125C would be the most popular (well, someone thought it shouldn't be more popular, but it definitely is), its bad 3-exp renders it useless. The playing field is then reduced to just the 49/11C vs the 11/25C. I don't really understand what's going on with the 125C in these discussions, but maybe I don't need to worry too much about that.

Then in 2006 the mina is split, and it is decided that should get the 49/11C on account of proximity in cents and that should get 25/11C on account of avoiding core crossovers. I haven't double-checked these assertions, but in principle they make sense. I guess it's not super clear to me why you couldn't split a mina and have both sides share a core, in which case I would think should just go to the best comma, which is 25/11C by LPEI today.

I find the messaging on the 2007 email really confusing; it's like... it's split, it's not split, it's added, it's removed, it's not distinguished, it's assigned... bleaugh. Anyway, it's clear that in the end, that mina was unsplit "so that the boundary between the |~ and ~~| cores coincides in olympian and herculean-X". I want to understand what that means better. It looks like there are several other places where the bound splitting a mina in Extreme extends down into Ultra or even High. As y'all said, since it almost always involves a split between two different cores, it has to extend down or it'd create a DAfLL exception. So I just don't understand why the mina wouldn't have been kept split while keeping the bound beween and coinciding in lower levels, or what else could have been meant by this statement.

Alright, I look forward to your reading/theory on this now that I've had my chance to share my thoughts uncolored by yours.

[cmloegcmluin]

I've just made a recommendation for the issue of the 1/1225k vs 1/121k using our new LPEI badness metric: viewtopic.php?p=2656#p2656 I recommend the 1/121k because it has lower badness, AKA it's "better". Almost feels wild to be able to objectively state such an encompassing thing, but I guess that's what a lot of this work has been driving toward, and that's the feeling of it paying off!

For the record, the JI notation as realized in my code which we used in finding tina commas (the topic of the linked post above) inherited its configuration from George's calculator, so it used the 1/1225k, the one I do *not* recommend. I just re-ran the script using the 1/1225k in place of the 1/121k (still with the 25/11C in place of the 11/49C), and it did not change anything that would affect our tina comma choices.

In other news, I know we've been using semitina on this topic lately, and I praised the term when you first introduced it, but I then when I noticed that to be consistent I should start referring to the half-apotome as the semiapotome, it occurred to me to question the etymology. And yeah, unfortunately, semi- is the Latin prefix, not the Greek. We would want hemiapotome, and so then I think I'd prefer hemitina, again, in the name of consistency. Of course I'll keep the "semi-" in semisharp, etc., since those are popular conventional terms, but I don't think semiapotome would be well established. Actually a web search turns up a few hits for it, including on this forum. Let me know if you prefer semiapotome to hemiapotome, or if you prefer hyphenating or not.

[cmloegcmluin]

I just went through my old notes about what to do after developing an unpopularity metric. The only thing I noticed was that we never factored in 5-schisma slope and/or having 3-exponents close to 8, as described here. Maybe somewhere we dismissed it; I had brought it up again here.

[cmloegcmluin]

You could call it a member of the LPEI family of notational comma badness functions, where "I" stands for identity function.

Why is the tina/mina error submetric considered an "identity function"?

[Dave Keenan]

Before we get into it, I just wanted to drop a few miscellaneous notes.

• This is something I feel like a doofus for caring about, but it is kind of cute how the term "occam" is almost a scrambled version of "comma".

 o ↔ c
m     c
 m   a

Just thought I'd join you in doofus-land for a moment.

• When, re: the 7/425n, I said "I can't figure out what I would have meant by "most occurrences in Extreme Precision"", it popped back in my thoughts, and I thought maybe by occurrences I meant what we came to mostly refer to as "votes", i.e. the Scala usage stats. But {425/7}₂,₃ does not appear in the Scala stats, so I'm still at a loss.

I think that was when we looked at the occurrences of metacommas between extreme commas. I think that was Ash's idea.

• I just wanted to acknowledge that by finding the best comma for each semitina zone, we've made a big dent in the process of assigning commas for each tina of the Insane notation. It's certainly not the final list (e.g. 59/7n is the winner by direct badness for the horn and bare shaft slot, but we'd probably want to install the 10241/5n there, since it's what we're saying the horn glyph primarily represents; similarly 1/205n wins by direct badness for the wedge and bare shaft slot, though we're going with 1/5831n for the wedge's 2-tina value, and 253/5n wins by direct badness for the wedgebird and bare shaft slot, though we're going with 187/175n for the wedgebird's 8-tina value). Even if this was the final list of commas, we'd still need to choose which core each of them belong to. If anyone ever mentions tina-splitting, I will destroy them. Anyway, I'm not proposing it to be our next project. Honestly I'd rather not think about Insane again for a longy long time.

You may be right about those comma choices. Or not. An actual insane precision level notation is not something I have any interest in either, at least not at this time, or in the foreseeable future. In fact, someone else can do it after I'm dead.

I suspect it was only the Scala archive stats that favoured {49/11}_2,3 over {25/11}_2,3.

I just checked this, and found I was wrong... it may have been the lower slope that caused {49/11}_2,3 to win when using the closer Scala ranks or the closer SoPFRs (21 vs 25) as the unpopularity metric.

That makes total sense to me. Yeah, that's probably the explanation.

I no longer think it is the explanation. More below.

If only there was a way to capture this occurrence rate across EDOs of a given symbol, so instead of manually checking such things we could factor it into a single mathematical formula for uselessness...

Aaaaaargh.

Just kidding. That'd be insane!

Phew.

Descriptions seem completely safe to alter, but it would seem weird to have a symbol with className "accSagittal11v49CommaUp" yet a description all about the 25/11C.

Yes. We'll have to decide which is the lesser of the two weevils eventually — having such a mismatch, or changing the names.

Overall, my head is just spinning from trying to form an opinion on the matter.
...

Super-olympian and Herculean-X are explained here:
viewtopic.php?p=2659#p2659
starting in the 4th paragraph.

Alright, I look forward to your reading/theory on this now that I've had my chance to share my thoughts uncolored by yours.

I believe the explanation lies in Super-olympian. In Super-olympian it made sense to assign 11/49C to  , and 25/11C to the other symbol in the split mina, because of their sizes in cents. But when Super-Olympian was no longer a thing, and the mina was no longer split, it made sense for to be the Olympian symbol, to maintain monotonic cores. And the association between 11/49C and was so strong (and George still thought Super-olympian might be a thing again some day) that he simply deleted 25/11C along with the unwanted Super-olympian symbol, despite it having the lower complexity (including by Georges own "weighted complexity" metric). He did not consider reassigning to 25/11C because if Super-olympian was ever resurrected, it would not make sense to assign 11/49C to the other symbol, because they would then be out of order of the symbols' SoFLS (sum-of-fewest-lower-subsets). Or at least that's my theory.

I suspect Super-olympian may explain other not-lowest-badness assignments in Olympian too. I don't think it is a valid reason any more.

[cmloegcmluin]

 o ↔ c
m     c
 m   a

You and your hexagons again!

• When, re: the 7/425n, I said "I can't figure out what I would have meant by "most occurrences in Extreme Precision"", it popped back in my thoughts, and I thought maybe by occurrences I meant what we came to mostly refer to as "votes", i.e. the Scala usage stats. But {425/7}₂,₃ does not appear in the Scala stats, so I'm still at a loss.

I think that was when we looked at the occurrences of metacommas between extreme commas. I think that was Ash's idea.

Perchance, mayhaps. I can't be bothered to verify.

Honestly I'd rather not think about Insane again for a longy long time.

An actual insane precision level notation is not something I have any interest in either, at least not at this time, or in the foreseeable future. In fact, someone else can do it after I'm dead.

Perhaps in our lifetimes one of these newfangled neural nets will be intelligent enough to figure it out for us.

I believe the explanation lies in Super-olympian. In Super-olympian it made sense to assign 11/49C to  , and 25/11C to the other symbol in the split mina, because of their sizes in cents. But when Super-Olympian was no longer a thing, and the mina was no longer split, it made sense for to be the Olympian symbol, to maintain monotonic cores. And the association between 11/49C and was so strong (and George still thought Super-olympian might be a thing again some day) that he simply deleted 25/11C along with the unwanted Super-olympian symbol, despite it having the lower complexity (including by Georges own "weighted complexity" metric). He did not consider reassigning to 25/11C because if Super-olympian was ever resurrected, it would not make sense to assign 11/49C to the other symbol, because they would then be out of order of the symbols' SoFLS (sum-of-fewest-lower-subsets). Or at least that's my theory.

I suspect Super-olympian may explain other not-lowest-badness assignments in Olympian too. I don't think it is a valid reason any more.

Wow. Yeah, I should have ever sussed out the story behind the 11/49C vs 25/11C had to do with this Super-olympian. That story adds up, from my perspective anyway.

So, for purposes of having chosen primary commas for the tinas, we are comfortably justified in temporarily substituting the 25/11C for the 11/49C to break a tie for the 7-tina. But the bigger issue of fixing the 15th ( from 1/1225k -> 1/121k) and 36th minas will be part of an audit of the other 18 non-best Extreme commas, which we now suspect may be largely explained by fall-out from the elimination of the super-olympian notation.

I know I'm keeping you busy this week. I await your thoughts on the material in the other posts I piled on this topic (hemitina & hemiapotome, 5-schisma-slope, identity function).

[Dave Keenan]

For the record, the JI notation as realized in my code which we used in finding tina commas (the topic of the linked post above) inherited its configuration from George's calculator, so it used the 1/1225k, the one I do *not* recommend. I just re-ran the script using the 1/1225k in place of the 1/121k (still with the 25/11C in place of the 11/49C), and it did not change anything that would affect our tina comma choices.

That's great to know. Thanks for thinking to check it.

In other news, I know we've been using semitina on this topic lately, and I praised the term when you first introduced it, but I then when I noticed that to be consistent I should start referring to the half-apotome as the semiapotome, it occurred to me to question the etymology. And yeah, unfortunately, semi- is the Latin prefix, not the Greek. We would want hemiapotome, and so then I think I'd prefer hemitina, again, in the name of consistency. Of course I'll keep the "semi-" in semisharp, etc., since those are popular conventional terms, but I don't think semiapotome would be well established. Actually a web search turns up a few hits for it, including on this forum. Let me know if you prefer semiapotome to hemiapotome, or if you prefer hyphenating or not.

In general I prefer "half tina" (or "fractional tina") and "half apotome". My reason for using "semitina" here was specifically in the context of your metacomma searches, because I wanted you to see it as a unit in its own right, not as a half of something else, as I wanted you to think in terms of integers only. I guess "hemitina" would have done just as well. I don't think semi or hemi should be hyphenated as they are not really words in their own right. I think semi is more English. Apart from technical medical or mathematical terms, there's really only "hemisphere".
https://www.thefreedictionary.com/words ... -with-hemi
But there are dozens of English words beginning with "semi" where it means half.
https://www.thefreedictionary.com/words ... -with-semi

[Dave Keenan]

I just went through my old notes about what to do after developing an unpopularity metric. The only thing I noticed was that we never factored in 5-schisma slope and/or having 3-exponents close to 8, as described here. Maybe somewhere we dismissed it; I had brought it up again here.

Those now seem irrelevant in the context of finding the most common metacomma definitions.

You could call it a member of the LPEI family of notational comma badness functions, where "I" stands for identity function.

Why is the tina/mina error submetric considered an "identity function"?

Not "an" identity function, "the" identity function. The identity function is the Clayton's function — the function you have when you're not having a function.
https://www.youtube.com/watch?v=ylH43Tcaj60
The output of the identity function is identical to its input. The error component was neither compressed nor expanded, but left exactly as it was. Sure it was multiplied by a weighting factor, but so were the functions of N2D3P9, ATE and AAS.

[cmloegcmluin]

In general I prefer "half tina" (or "fractional tina") and "half apotome". My reason for using "semitina" here was specifically in the context of your metacomma searches, because I wanted you to see it as a unit in its own right, not as a half of something else, as I wanted you to think in terms of integers only. I guess "hemitina" would have done just as well. I don't think semi or hemi should be hyphenated as they are not really words in their own right. I think semi is more English. Apart from technical medical or mathematical terms, there's really only "hemisphere".
https://www.thefreedictionary.com/words ... -with-hemi
But there are dozens of English words beginning with "semi" where it means half.
https://www.thefreedictionary.com/words ... -with-semi

Alrighty then. And indeed the word "semitina" did the trick for helping to see it as a unit in its own right.

I'll stick with "half apotome" (unhyphenated) in the code and in the educational materials then. And if we're not going to establish hemiapotome as a thing, then I'll stick with semitina, because of its higher popularity in English.

Not "an" identity function, "the" identity function.

Ahhhhhhh okay. That totally makes sense now. I had lost hold of what E and P stood for, to contextualize the choice of I. So it has nothing to do with error, and everything to do with growth.