140th mina

Dave Keenan
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140th mina

[Moderator note: this thread was broken out from the JI Notation Spreadsheet thread. The last post there is reproduced as a quote below.]
cmloegcmluin wrote: Tue Apr 07, 2020 1:56 am Sagittal Standard JI Notation Calculator Spreadsheet.xlsx
(attachment last edited: May 2nd, 2020)

Alright, I’ve updated the JI calculator spreadsheet to use the new Olympian diacritics (linked above). While I was at it I also updated it to:
1. include pure notation
2. include the Unicode characters so if you have the Bravura font installed you can see the symbols (pending the Olympian diacritics getting implemented in SMuFL)
3. report the error between your pitch and the default pitch of suggested symbols
4. report the count of fifths each pitch is away from 1/1
5. hide the guts of the calculator on other tabs to clean up the UI
6. patch what I suspect was a bug where the calculator would suggest instead of simply *
I made a few other tweaks here and there too, such as switching the names of each level to be based on their precision instead of the Greek proper names for their maximum symbol set.

These changes are not a strict win. Perhaps to some people the pure version, the unicode, or the accuracy measurements are just clutter. So please let me know what you think or if there any revisions you would like to see made. Otherwise I hope it works for you and you enjoy notating some JI in Sagittal!

* You can test this for yourself by entering C as your 1/1, -4 as your count of 3's, and 1 as your count of 5's. I found that you can fix the problem by increasing the upper bound of symbol suggestion from 68.08453082 to 68.8, but I was not sure if that might cause other problems, so instead I simply added a layer at the end where if it ever suggests it swaps it out for (and the equivalent problems for flat, double sharp, and double flat).

Wow! Thanks for all that.

That bug is interesting. Well spotted! It relates to something I noticed a week or so ago:

The upper boundary of the largest single-shaft symbol in Olympian is currently the nearest odd half-tina to 139.5 minas, which is 68.08453 ¢, as you noted.

But the upperbound for single-shaft sagittals should be the same as the upperbound for large-dieses. This is the apotome-complement of the lower-bound for medium-dieses, and is given in footnote 7 on page 8 of http://sagittal.org/sagittal.pdf as sqrt([-30 19⟩) = 68.57250822 ¢.

That's one mina higher than the upperbound of the current largest single-shaft symbol.

That suggests there ought to be one more single-shaft Olympian, at the 140th mina, namely . Its primary comma (default interpretation) should be 25:77L, which is the apotome-complement of the 93rd mina, 25:77M. That is so that every medium diesis has a single-shaft complement. Currently 25:77M is the only one that doesn't. Or equivalently, it is so that we have single-shaft symbols for all large-dieses.

I think its upper bound should be the aforementioned 68.57250822 ¢ even though that is not on a half-tina. But that's not entirely clear to me. Perhaps it should be symmetrical with the upperbound [Edit: Oops! That should be "lowerbound". Thanks Douglas] of the 93rd mina 25:77M, about the half-apotome.

In any case, this would require changing the symbol for 25:77M (the 93rd mina) from to to agree with the apotome-complement rules.

Perhaps George thought that the former was a better symbol for 25:77M based on sum-of-flags. Or perhaps it is the effect it might have on the related 45.6 cent Herculean or Athenian boundary that led to his choice. This deserves investigating. But I can't justify the time at present.

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Re: 140th mina

I agree that this issue deserves investigation. Clearly it taps into a number of competing and profound technico-aesthetic considerations, which is exciting to me. I have so many questions! (There are "apotome-complement rules"? Where can I find them?! And what's the "related 45.6 cent Herculean or Athenian boundary"?) But I also agree that it may not be the rabbit hole we want to go down at the moment. Whatever we decide, it'll be a cinch to fix in the web calculator (and we can come back and update the spreadsheet calculator too).

Dave Keenan
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Re: 140th mina

cmloegcmluin wrote: Sat Apr 11, 2020 11:58 am I agree that this issue deserves investigation. Clearly it taps into a number of competing and profound technico-aesthetic considerations, which is exciting to me. I have so many questions! (There are "apotome-complement rules"? Where can I find them?! And what's the "related 45.6 cent Herculean or Athenian boundary"?) But I also agree that it may not be the rabbit hole we want to go down at the moment. Whatever we decide, it'll be a cinch to fix in the web calculator (and we can come back and update the spreadsheet calculator too).
I suspect the complete apotome-complement rules only exist in email so far, but here they are: In the pure notation, the apotome-complement of a core symbol is given by Figure 13 on page 24 of http://sagittal.org/sagittal.pdf, with the exception that the apotome-complement of is  . The apotome complement of a diacritic'd symbol requires inverting all the diacritics in addition to taking the apotome complement of the core as above, except in the cases of
and  where the apotome-complement of is  .

By "the related 45.6 cent Herculean or Athenian boundary" I mean the boundary between and in Herculean, and the boundary between and in Athenian, which can be seen here: http://sagittal.org/SagittalJI.gif

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Re: 140th mina

Dave Keenan wrote: Tue Apr 07, 2020 12:32 pm That bug is interesting. Well spotted! It relates to something I noticed a week or so ago:

The upper boundary of the largest single-shaft symbol in Olympian is currently the nearest odd half-tina to 139.5 minas, which is 68.08453 ¢, as you noted.

But the upperbound for single-shaft sagittals should be the same as the upperbound for large-dieses. This is the apotome-complement of the lower-bound for medium-dieses, and is given in footnote 7 on page 8 of http://sagittal.org/sagittal.pdf as sqrt([-30 19>) = 68.57250822 ¢.

That's one mina higher than the upperbound of the current largest single-shaft symbol.

That suggests there ought to be one more single-shaft Olympian, at the 140th mina, namely .
Agreed. Let's add that to the JI Notation Spreadsheet, as well as to the JI Precision Levels Diagram (and to my Everything Sagittal sheet as a valid symbol).
Its primary comma (default interpretation) should be 25:77L, which is the apotome-complement of the 93rd mina, 25:77M. That is so that every medium diesis has a single-shaft complement. Currently 25:77M is the only one that doesn't. Or equivalently, it is so that we have single-shaft symbols for all large-dieses.
I agree that it being the only M without a L is a strong reason to add it.
I think its upper bound should be the aforementioned 68.57250822 ¢ even though that is not on a half-tina. But that's not entirely clear to me. Perhaps it should be symmetrical with the upperbound of the 93rd mina 25:77M, about the half-apotome.

An interesting dilemma. On one hand we have an opportunity to say that any interval up through the size category of large diesis can be represented by a single-shaft symbol (68.57250822¢). On the other hand we can preserve symmetry of the capture zones at the extreme JI precision level so that apotome complements stay paired in that middle zone (68.084530821955500¢ + 0.421576042241213¢ = 68.5061068642¢).

Why can't we accomplish both by changing the upperbound of the 93rd mina to be the mirror of the large diesis upperbound about the half-apotome? In other words, we set the upperbound of at 68.57250822¢, and then move the 93rd mina upperbound from 45.178899193516400¢ down by 68.57250822¢ - 68.5061068642¢ = 0.0664013558¢ (about a half-tina) to 45.1124978377¢. This does not cross either of the neighboring commas, which are at 45.561¢ and 44.970¢. It wouldn't be the first capture zone boundary with an irregular definition.
In any case, this would require changing the symbol for 25:77M (the 93rd mina) from to to agree with the apotome-complement rules.
Agreed. I will update that to the three places listed above, too.
Perhaps George thought that the former was a better symbol for 25:77M based on sum-of-flags.

Sum of flags information:

25:77M actual value 45.561¢.

sum of flags:
is 43.013
is 1.954
is 0.423
43.013 + 1.954 + 0.423 = 45.39

sum of flags:
is 46.394
is -0.833
46.394 - 0.833 = 45.561

So actually the latter is much closer!
Or perhaps it is the effect it might have on the related 45.6 cent Herculean or Athenian boundary that led to his choice.
By "the related 45.6 cent Herculean or Athenian boundary" I mean the boundary between and in Herculean, and the boundary between and in Athenian, which can be seen here: http://sagittal.org/SagittalJI.gif
See above where I show it doesn't cause them to fall outside their zones.
I suspect the complete apotome-complement rules only exist in email so far, but here they are: In the pure notation, the apotome-complement of a core symbol is given by Figure 13 on page 24 of http://sagittal.org/sagittal.pdf, with the exceptions that the apotome-complement of is  . The apotome complement of a diacritic'd symbol requires inverting all the diacritics in addition to taking the apotome complement of the core as above, except in the cases of
and  where the apotome-complement of is  .
Whoa! Totally hadn't noticed that before.

In my "Anatomy of an Apotome" sheet I give as the only one which is its own apotome complement. When you checked it before you didn't call that out as wrong. Since in that diagram I am using the Very High precision level to illustrate points about Mixed (Evo) vs. Pure (Revo) flavors and symbol element arithmetic and such, I believe that is still an accurate statement. It would only be upon reaching the Extreme precision level that it would cease to be its own apotome complement, because that's where it gets broken down into an even number of chunks.

Were it to get broken down into five chunks instead then we wouldn't have to make any exceptions (like  ,  ,  ,  , and  ). Probably there was a reason that wasn't done though. I guess the problem is that is already toward the bottom of that section. In that case I would actually recommend reducing from four chunks to three. Oh, but the problem with that is that the boundaries have to be symmetrical and does not fall inside the middle of three chunks there. Well, you all probably already went through this before... alas...

Dave Keenan
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Re: 140th mina

cmloegcmluin wrote: Thu May 14, 2020 1:49 am Why can't we accomplish both by changing the upperbound of the 93rd mina to be the mirror of the large diesis upperbound about the half-apotome?
That seems to be an excellent solution. Thanks! You have addressed in advance all of the concerns I might have had about it. Please go ahead and implement it, and all the other things you discuss above, in the JI Notation Spreadsheet, if you haven't already.

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Re: 140th mina

I'm realizing there's one key piece of information we haven't determined yet, though, which is the primary comma for the new symbol .

It needs to be between 68.0845308219555¢ (the boundary between the 139th and 140th mina) and 68.57250822¢ (the apotome-complement of the lower-bound for medium-dieses = √| -30 19>).

It's sum-of-elements is

| -16 11 1 0 0 -1 >
67.29106161¢

+

| 5 -3 1 -1 -1 1 >
0.8325242041¢

=

| -11 8 2 -1 -1 >
68.12358581888735¢

which is inside that range, so that's good, given our recent observations on this thread.

Here are some contenders:

												apotome
cents	monzo					ratio			comma		limit	slope		"rank"	Scala stat uses
68.323	|  13  -2 -3 -1                 >	8192/7875		1/875L		7	-5.409		22	21
68.124	| -11   8  2 -1 -1              >	164025/157696		25/77L		11	0.599		28	0
68.296	|   0  -7  2  1  0  1           >	2275/2187		2275/1L		13	-8.410		30	0
68.293	| -22   8  1  1  0  0  0  1     >	4363065/4194304		665/1L		19	0.601		31	0
68.430	| -20  13  0  0  0  1  0 -1     >	20726199/19922944	13/19L		19	3.612		32	44
68.355	|  21 -15  0  0  2  0 -1        >	253755392/243931419	121/17L		17	-13.228		39	1
68.312	|   4   1 0   1  0  0 -1 -1     >	336/323			7/323L		19	-3.605		43	1
68.486	|  20 -12 -1  0 -1  0  0  0 0 1 >	30408704/29229255	29/55L		29	-11.446		45	4
68.390	| -28  15  0  0  1 -1  0  0 1   >	3630273471/3489660928	253/13L		23	4.813		47	1


I'll abstain from submitting my preference for now.

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Re: 140th mina

FACEPALM AGAIN

Right, it's the 25/77L, being the apotome complement of the 25/77M.

Well, you know what, that exercise was fun anyway. No regrets.

Dave Keenan
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Re: 140th mina

That's OK. I think you've been working way too hard, and way too far into the night on this stuff. I really appreciate all the work you're doing. But please enjoy some well-earned rest.

I just reread an earlier post of yours in this thread. I haven't had time to think it out fully. But you talk about changing the upperbound of mina 93 to match the upperbound of the new mina 140 symbol. But isn't it the lowerbound of mina 93 that is its mirror-image about the half-apocalypsetome.

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Re: 140th mina

Dave Keenan wrote: Sat May 16, 2020 4:14 pm That's OK. I think you've been working way too hard, and way too far into the night on this stuff. I really appreciate all the work you're doing. But please enjoy some well-earned rest.
Yeah, I don't excel at doing too many things at once. Which is what I am doing.
I just reread an earlier post of yours in this thread. I haven't had time to think it out fully. But you talk about changing the upperbound of mina 93 to match the upperbound of the new mina 140 symbol. But isn't it the lowerbound of mina 93 that is its mirror-image about the half-apocalypse.
You are correct. That is what I meant. Though if you look back at the first post in this thread, you may find that you started it

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Re: 140th mina

Here's a question though. So the comma for the 140th mina should be the apotome complement of the comma for the 93rd mina, sure. But are the 77/25M and 25/77L the ideal pair?

Maybe the 77/25M was ideal for the 93rd mina before the 140th mina was introduced, but now that it has a paired comma, we should reconsider it.

As you can see, in the work I did to find the ideal primary comma for this new symbol for the 140th mina, 25:77(M/L) is not a common comma. It has a low sum-of-primes heuristic rank, but zero actual usages in the Scala stats. Whereas the 875(M/L) has an even better sum-of-primes heuristic rank while also having 21 usages in the Scala stats. And also it's lower limit. Only drawback is its relatively high apotome slope.

Or are the Scala stats supposed to dominate these decisions, out of deference to the wider community of composers rather than our own whims as crafters of the notation? In that case the 13:19(M/L) would be our winner.

I would like to know more about the priority of the criteria used to make these calls.