## consistent Sagittal 37-Limit

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### Re: consistent Sagittal 37-Limit

...I mean the mina and schisma. *facepalm*

Dave Keenan
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### Re: consistent Sagittal 37-Limit

cmloegcmluin wrote: Thu May 14, 2020 3:39 pm Oh, right. I still get the mina and tina confused sometimes, clearly.
"Clearly. Particularly since there are no tinas involved here. We're only dealing with minas and schismas. "

Cool list. Can you easily add a sum-of-prime-factors column? And whack [pre]...[/pre] around the whole table so they line up (assuming tab-delimited)?

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### Re: consistent Sagittal 37-Limit

Here's that again, cleaned up:

cents			monzo					ratio			comma		"rank"
36.53357970860179	| -13 15 -1 -3 >			14348907/14049280	1/1715C		26
36.5888086148658	| 22 -12 -1 0 1 0 -1 >			46137344/45172485	11/85C		33
36.35693218545814	| -10 11 1 -1 -2 >			885735/867328		5/847C		34
36.542244404772084	| 20 -5 -2 0 0 -2 >			1048576/1026675		1/4225C		36
36.52933627696398	| 1 -4 1 1 -1 1 >			910/891			455/11C		36
36.5259897559322	| -21 11 0 1 -1 0 0 1 >			23560551/23068672	133/11C		37
36.39991273195046	| 7 -5 1 -2 >				12160/11907		95/49C		38
36.36675894371364	| -6 6 -1 0 0 1 0 0 0 -1 >		9477/9280		13/145C		47
36.568970302424745	| -4 -5 0 0 1 0 0 2 >			3971/3888		3971/1C		49
36.52575261336313	| 16 -13 0 0 0 -1 1 1 >			21168128/20726199	323/13C		49
36.3958168885402	| -23 12 -2 0 0 1 0 0 0 0 1 >		214170723/209715200	403/25C		54
36.42993229295411	| -8 9 0 1 0 0 -1 0 0 0 -1 >		137781/134912		7/527C		55
36.429695150384504	| 29 -15 0 0 1 -1 0 0 0 0 -1 >		5905580032/5782609521	11/403C		55
36.51580850736912	| 20 -10 0 -2 -1 0 0 0 0 0 1 >		32505856/31827411	31/539C		56
36.45875309521124	| 12 -9 -1 0 1 -1 0 0 0 1 >		1306624/1279395		319/65C		58
36.458990237780995	| -25 15 -1 1 0 0 -1 0 0 1 >		2912828121/2852126720	203/85C		58
36.38937629765928	| -14 13 0 1 0 0 0 0 -1 -1 >		11160261/10928128	7/667C		59
36.466364671724484	| 15 -2 -1 0 0 0 0 0 -1 0 -1 >		32768/32085		1/3565C		59
36.50671892301813	| 9 -11 0 0 -1 2 0 0 1 >		1990144/1948617		3887/11C	60
36.38176472114606	| -17 6 0 1 1 -1 0 0 0 0 1 >		1740123/1703936		2387/13C	62
36.469881961561335	| -24 12 1 0 1 0 1 0 0 -1 >		496897335/486539264	935/29C		62
36.49542261655127	| -2 4 -2 0 0 0 0 0 -1 1 >		2349/2300		29/575C		62
36.33008942237331	| 8 -7 0 0 1 0 0 0 1 -1 >		64768/63423		253/29C		63
36.56576865573409	| 24 -15 0 0 -1 1 1 0 -1 >		3707764736/3630273471	221/253C	64
36.49083601109001	| -26 7 0 1 2 0 0 0 0 0 0 1 >		68538393/67108864	31339/1C	66
36.54687033562636	| 5 -3 0 -1 1 0 1 0 0 0 -1 >		5984/5859		187/217C	66
36.359147367200194	| -9 -1 -1 0 1 0 0 0 1 0 1 >		7843/7680		7843/5C		70
36.51043795685604	| -18 8 1 0 1 0 0 0 1 0 -1 >		8299665/8126464		1265/31C	70
36.567854565336546	| 24 -7 0 -1 0 0 0 0 0 -1 0 -1 >	16777216/16426557	1/7511C		73
36.437799045682624	| 14 -4 0 -1 0 -1 1 0 0 0 0 -1 >	278528/272727		17/3367C	74
36.57951194405419	| -27 12 0 0 0 2 0 -1 0 1 >		2604592341/2550136832	4901/19C	74
36.37010546474512	| 16 -9 0 0 0 2 0 -1 0 -1 >		11075584/10845333	169/551C	74
36.328003512771055	| 8 -15 0 1 0 1 1 0 0 0 0 1 >		14653184/14348907	57239/1C	74
36.48101673474406	| -6 4 0 -1 1 0 0 1 0 0 0 -1 >		16929/16576		209/259C	74
36.48972027400201	| 2 5 0 0 1 0 0 -2 0 -1 >		10692/10469		11/10469C	78
36.5038897891344	| -12 13 0 0 -1 0 1 -1 0 0 -1 >		27103491/26537984	17/6479C	78
36.401366666912025	| -9 7 1 0 0 -1 0 0 1 0 0 -1 >		251505/246272		115/481C	78
36.403857591873766	| 26 -10 0 0 0 1 -1 0 -1 0 0 -1 >	872415232/854261883	13/14467C	90
36.3778862349868	| -12 6 0 0 -1 0 -1 0 0 1 0 1 >		782217/765952		1073/187C	94
36.58577773698017	| -11 7 0 0 0 0 1 -1 0 -1 1 >		1152549/1128448		527/551C	96
36.33733023969181	| -18 10 0 0 -1 0 0 0 -1 0 1 1 >	67729203/66322432	1147/253C	102
36.560480131393085	| -16 10 0 0 0 0 -1 0 1 0 1 -1 >	42101937/41222144	713/629C	108


There are only a few commas that beat 47 by the sum-of-primes popularity heuristic, and only a few of those which are also smaller in cents than the 455/11C that we decided should be included in the capture zone for the next higher symbol and therefore the boundary will be moved down.

I realized my method was wrong for searching the sheets Dave shared re: comma popularity. At least one of these commas does appear in the list: the 5/847C. It appears once It'll be a contender then.

At the point of only a single appearance, though, the actual usage stats don't count for much. The 5/847C is only 11-limit, sure, but the 95/49C beats it at only 7-limit. 95/49C also has a lower count of fifths (5 vs 11). So 95/49C would be my vote.

13/145C is the only other possibility. It doesn't beat 47, only matches it. And it's 29-limit.

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### Re: consistent Sagittal 37-Limit

I think they are all small dieses not commas, so "S" not "C".

95/49S = 12160/11907 is 19-limit and its monzo is [ 7 -5 1 -2 0 0 1 >, but you have its "rank" correct at 5+7+7+19 = 38. So it is still a contender.

Could you filter to those less than 910/891 and calculate their apotome-slopes.

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### Re: consistent Sagittal 37-Limit

Dave Keenan wrote: Fri May 15, 2020 3:43 am I think they are all small dieses not commas, so "S" not "C".
Ah, yes you are right! Cutoff for C to S being 33.382¢ = L/2 = | 13.5 -8.5 >.
95/49S = 12160/11907 is 19-limit and its monzo is [ 7 -5 1 -2 0 0 1 >, but you have its "rank" correct at 5+7+7+19 = 38. So it is still a contender.
Let that be a lesson to you, kids: this is what happens when you decide to manually remove the trailing zeroes when you notice them after getting the tabs just right, instead of just going back and regenerating things the way you want programmatically. See I'm going to have to do it anyway now.
Could you filter to those less than 910/891 and calculate their apotome-slopes.
Yes, apotome slope was what I was hinting at when calling out the smaller count of fifths for the 95/49S. Here you go.

cents			monzo				ratio			comma		limit	apotome slope		"rank"
36.35693218545814	| -10 11 1 -1 -2 >		885735/867328		5/847S		11	1.2792165082625901	34
36.5259897559322	| -21 11 0 1 -1 0 0 1 >		23560551/23068672	133/11S		19	1.285164788879173	37
36.39991273195046	| 7 -5 1 -2 0 0 0 1 >		12160/11907		95/49S		19	-3.8421863287452656	38
36.36675894371364	| -6 6 -1 0 0 1 0 0 0 -1 >	9477/9280		13/145S		29	-0.3198905655426278	47


As you can see I also filtered out those with rank higher than 47. I missed the 133/11S before; it's just below the 455/11S in size. I also added the limit in.

So the 13/145S actually has the smallest apotome slope. It's the 95/49S which has the worst apotome slope since I forgot to take into account that the -5 is negative and apotome slope is minimized when you have close to 7 fifths. In any case, all four of these are below the average (absolute value of) apotome slope for commas, which is 1.773 (the max being 6.690). Does apotome slope even matter much here since this is a diacriticized symbol, which is unlikely to be used for EDOs?

So my vote is for the 5/847S then. It's the 2nd best apotome slope, lowest limit, actually has at least one usage in the Scala stats, and has the lowest sum-of-primes popularity heuristic rank. Let me know what your votes are.

I will write a script sometime soon to verify that we don't have any other secondary commas akin to the 455/11S which warrant doing a boundary shift.

Dave Keenan
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### Re: consistent Sagittal 37-Limit

Great work. Thanks. Could you please edit-in the values for the existing primary comma, 47S = 48/47 for comparison above.

I agree that the absolute value of the 3-exponent is probably a better metric than the slope. But slope is still worth considering. Are you sure those slopes are correct?

apotome_slope = exponent_of_3 - 7 × untempered_size_in_cents/113.685

A comment you made earlier makes me worry that you might have misread this as:

apotome_slope = (exponent_of_3 - 7) × untempered_size_in_cents/113.685

Can someone please generate a list of the commas that are sum-of-elements or sum-of-subsets for the symbol we are considering redefining  . i.e.
+ + +
+ +
+ +
+ +
+ +
+
+
+
+
Did I miss any?

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### Re: consistent Sagittal 37-Limit

In the above sum-of-subsets, we should also consider the common secondary commas for and  .

I'm pretty sure they exist — one for each diacritic — but I can't readily find what they are.

They can be found by subtraction. i.e. take the primary comma for a symbol with a mina diacritic, and subtract the primary comma the same symbol with the mina diacritic removed, and see if you get something other than 455n or 65:77n.

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### Re: consistent Sagittal 37-Limit

Dave Keenan wrote: Fri May 15, 2020 11:58 am Great work. Thanks. Could you please edit-in the values for the existing primary comma, 47S = 48/47 for comparison above.
I will do this soon.
I agree that the absolute value of the 3-exponent is probably a better metric than the slope. But slope is still worth considering. Are you sure those slopes are correct?

apotome_slope = exponent_of_3 - 7 × untempered_size_in_cents/113.685

A comment you made earlier makes me worry that you might have misread this as:

apotome_slope = (exponent_of_3 - 7) × untempered_size_in_cents/113.685
Dang! They were wrong.

They may teach us how to round numbers a bit differently here in the USA than they do in Australia, but I'm pretty sure they teach us the same order of operations.

Actually the first script I ran this morning generated the correct values. But when I compared them against the ones I've got in my Everything Sagittal sheet, they didn't match, size-wise. So I checked my formula in the sheet, and it corresponded to the incorrect order of operations in your second example. I suppose I shouldn't assume my past self had something more correct about Sagittal than my present self!

I will fix those soon.
cmloegcmluin wrote: Fri May 15, 2020 6:25 am I will write a script sometime soon to verify that we don't have any other secondary commas akin to the 455/11S which warrant doing a boundary shift.
I found one further example! The case of  . Other than and  , everything seems to be hunky-dory.

The problem with is that its sum-of-elements is greater than its upper bound. It's the opposite problem as whose sum-of-elements are lower than its lower bound.

= 17C = | -12 5 0 0 0 0 1 > = 4131/4096 = 14.730¢
= 65:77n down = | -5 3 -1 1 1 -1 > = 2079/2080 = -0.833¢
so together they are:
14.730¢ - 0.833¢ = 13.897¢
but the upper bound for is 13.842¢.

The next question is: where exactly to move these boundaries? It occurred to me that we must be careful how far we move them, because if me move them too far then we'll end up with the opposite problem for the neighboring symbol.

I'll answer for first since it's simpler. We can't move the upper bound of any higher than 14.191¢, because the next symbol up,  , has a default value of (Note: edited later from 14.307 to be correct) 14.191¢ coming from its primary comma, the 245C. It's sum-of-flags value is 14.307¢ (14.730¢ - 0.423¢), even higher, so we don't need to worry about it. In other words, we need to land it somewhere between 13.897¢ and 14.191¢. It looks like if you move it up by half a mina you get 14.123¢. Good enough?

Here's where it gets gnarly though. So we've established that we need to move the lower bound of lower than 36.529¢ so that it includes its own sum-of-elements. But we must be respectful of the next symbol down, which is  . We can't move it lower than its primary comma or its sum-of-elements, which are 36.326¢ and 36.529¢, respectively. So the higher of the two is the sum-of-elements, so it's the one that matters. So we can't move the boundary lower than 36.529¢. But wait... by the other criteria, we must move it lower than 36.529¢! We can't do both!

Those values are so close that I'll get more exact figures for them. Sum-of-elements for  is 36.529336277. Sum-of-elements for  is 36.529336277. So by sum-of-elements they're exactly the same thing!

Indeed:
 | -2 2 1 0 -1 >
| 15 -8 -1 >
| -12 2 1 1 0 1 >
| 1 -4 1 1 -1 1 >

And:
 | -4 -1 0 2 >
| 5 -3 1 -1 -1 1 >
| 1 -4 1 1 -1 1 >

So what does that mean? I think it means that it was a mistake to make both and valid symbols, since their sum-of-elements values are identical. I think we need to retire one or the other of them, and replace it with an alternative.

The first option I saw was to replace with  . But then I observed that there are no other valid symbols which combine a double mina with a schisma in the same direction. So I tabled that.

We may try replacing with  , which has sum-of-elements at 36.520¢, which would give us a tiny sliver of space between it and 36.529¢ to put its upper bound. It could still accomplish the original goal of giving it a primary comma of any of these four 37-limit commas I suggested which have cent values of 36.356¢, 36.525¢, 36.399¢, 36.366¢ since all of those are less than 36.529¢.

We can't replace with because its sum-of-elements is 36.930¢ which is greater than that zone's upper bound which is 36.888¢.
Can someone please generate a list of the commas that are sum-of-elements or sum-of-subsets for the symbol we are considering redefining  . i.e.
+ + +
+ +
+ +
+ +
+ +
+
+
+
+
Did I miss any?
I want to get this soon.
In the above sum-of-subsets, we should also consider the common secondary commas for and  .

I'm pretty sure they exist — one for each diacritic — but I can't readily find what they are.

They can be found by subtraction. i.e. take the primary comma for a symbol with a mina diacritic, and subtract the primary comma the same symbol with the mina diacritic removed, and see if you get something other than 455n or 65:77n.
I also want to get this soon.

But I'm spent from investigating the other stuff.

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### Re: consistent Sagittal 37-Limit

Commas on the point of an arrow.
Angels on the head of a pin?

We need to understand why George split the 75th mina.

Here's a result of some Sagittal archaeology (on my disk drive). It's a spreadsheet George used when designing the various JI precision-levels, including Olympian. It may shed some light. It is dated 9-Oct-2007 which is about a week later than the emails below, so it's slightly updated from what was being discussed.

Note that it has information on both Sheet 1 and Sheet 2.
Attachments
NotDeriv.xlsx

Dave Keenan
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### Re: consistent Sagittal 37-Limit

I searched email from George for "75th mina" and hit paydirt.

I include the entire HUGE email below, split over two posts. It addresses many fine points of Olympian symbol assignment, not to mention religion and philosophy.

And in a parallel thread at this time, we were planning the first Sagittal prank.

You can skip to the part about the 75th mina, but before you go, you will need to know that "right accents" are mina accents, and SoCA stands for Sum of Core and Accents.

You also need to know that DAFLL stands for Drop Accents For Lower Levels, sometimes referred to as DAFLR for Drop Accents For Lower Resolution. This is considered a desirable property for an Olympian (or Herculean) boundary, namely that if you find the Olympian symbol for some ratio, and drop the accents from it, you should obtain the same symbol you would have got by just finding the Athenian or Promethean symbol (whichever is the lowest level at which the core symbol appears) for the original ratio. "Accents" here refers to both mina and schisma diacritics.

This assumes that lower precision levels are designed first. And higher precision levels are not allowed to "reach back" and complexify the lower levels. But clearly we do allow the Olympian level to reach back and tweak the lower level boundaries by the smallest possible amount so they match its own.

George Secor in email, 2-Oct-2007 wrote:
--- Dave Keenan <d.keenan@...> wrote:

> Hi George,

Hi Dave,

Here's my reply to your last long e-mail, which I completed on Friday
and reviewed over the weekend. It's a good thing I took the time to
review it, because after going over all the latest things we said, a
"what-if" idea occurred to me:

<< As I've worked it out, dropping right accents from olympian symbols
will almost always result in the correct maximum-split herculean (which
I'll subsequently refer to as herculean-X) symbol: out of the 116 minas
in the half-apotome, only 5.5 minas do not follow DAFLL [drop accents
for lower limits/levels/whatever] (so few that they could easily be
memorized -- and most of these exceptions could be eliminated fairly
easily; I'll leave the details for later). >>

Rather than leaving the details for later, on Saturday I investigated
exactly what it would take to eliminate *all* of the exceptions and
found that there was no good reason not to do it! (Robert Walker will
be delighted, to say the least!) Thus there are two levels of Sagittal
which we can highly recommend for JI: one with right accents and the
other without. I've attached a new zip file (JI-Nota.zip) to replace
the one I previously sent.

I've gone through this message to update those changes (a search for
the word "update" will find all of these). I was thinking of
reordering parts of it to flow more logically (and editing out the
parts that are repetitive), but that would take a lot more time &
effort, and it's really not worth the trouble. I suggest that you read
the entire message (25 pages!) to get everything in context before

> At 11:22 PM 4/06/2007, you wrote:
>> Anyway, I now agree that it [Assignment of 49M to (/| ] shouldn't be
> changed.
>
> That's good. If your plan was to do a reductio ad absurdum on
> whatever rules you were using before, then showing that they assigned
> 49M to .)/|\ was a good way to go about it.
>
> And it still seems to me that mina boundaries and DAFLR were causing
> the problem.

After all this time I don't even remember what the "problem" was, and
as much as I'm tempted to want to go back and figure it out in order to
argue that mina boundaries probably weren't causing the "problem", I'm
biting my tongue and forcing myself to forget it and go on, because
there's no point in pursuing that any further.

In the rest of this discussion I'll continue to refer to minas, because
I still think that they're the solution to, rather than the cause of,
problems of notating the less popular commas. (I'll be making a case
for this as I go along.) For one thing, when I needed to identify
those regions where there are only very unpopular commas, I found it
very useful to use mina boundaries in order to locate them. Therefore,
I'll be referring to them by number of minas.

>> This isn't merely about detail, but rather that it's an example that
>> points up the question concerning how far we should go in defining
>> commas. I was making the point that, just because it's *possible* to
>> define a very unpopular and complex comma (rank 322, complexity 36.780)
>> with a relatively complicated symbol, I don't think that's a sufficient
>> reason to do it, since the comma could always be notated in a secondary
>> role using a simpler symbol.
>
> Agreed. But IF there are other commas less popular that have a symbol
> OR there is not much of a gap between it and the previous more
> popular comma that has a symbol THEN it should get a symbol if
> possible.

That first condition isn't a good reason for requiring that a symbol be
assigned. Since we've already agreed that *all* minas must have at
least one symbol assigned (so as not to have half-cent gaps in the
notation), there will be situations where the only possible assignments
will be very unpopular and/or high-prime-limit commas, and for many of
these it will not even be clear which of several competing commas
should be assigned.

Here's my list of the half-dozen minas (in order of size) that are
regions of lowest popularity. For each I give the simplest symbol,
followed by several comma-candidates for defining that symbol. For
each candidate I give (in parentheses) the following information:
popularity rank, prime limit (if high), and weighted complexity. I
also indicate the ones that are SoCA or alternate SoCA. (If you need
more data, such as distance from SoCA and alt. SoCA of a comma, you can
consult NotDeriv.xlsx, by which I determined these figures.)

)|. at 6 minas: 23:43s (393 pop. rank, 43 limit, 69.193 cplx.) vs.
29:61s (509 pop. rank, >47 limit, 90.852 cplx.) vs. 35:187s (677 pop.
rank, 40.039 cplx.) vs. 13:625s (705 pop. rank, 52.842 cplx.) vs.
19:4375s (Alt. SoCA, 1618 pop. rank, 103.390 cplx.)

/|.. at 42 minas: 19:73C (396 pop. rank, >47 limit, 93.365 cplx.) vs.
19:169C (454 pop. rank, 54.060 cplx.) vs. 625:2401C (662 pop. rank,
49.167 cplx.) vs. 19:4375 (674 pop. rank, 82.861 cplx.) vs. 253C (763
pop. rank, 47.634 cplx.)

)/|. at 50 minas: 5:187C (384 pop. rank, 39.899 cplx.) vs. 420175C (537
pop. rank, 108.230 cplx.) vs. 65:77C (552 pop. rank, 42.968 cplx.) vs.
13:125C (782 pop. rank, 39.302 cplx.)

(|. at 67 minas: 42875C (344 pop. rank, 79.301 cplx.) vs. 17:37C (411
pop. rank, 37 limit, 61.170 cplx.) vs. 4235C (426 pop. rank, 62.782
cplx.) vs. 23:125C (630 pop. rank, 53.241 cplx.)

'//|. at 91 minas: 499S (546 pop. rank, >47 limit, 598.932 cplx.) vs.
17:49S (562 pop. rank, 37.911 cplx.)

/|).. at 98 minas: 83M (289 pop. rank, >47 limit, 99.815 cplx.) vs.
5:187M (438 pop. rank, 45.286 cplx.)vs. 49:85M (650 pop. rank, 36.697
cplx.) vs. 11:325M (SoCA, not in pop. lists, 47.958 cplx.)

Of these, the 91st mina, with '//|. as symbol, is the one with the
least popular most-popular comma, 499S (I hope you understood that!).
Considering its >47 prime limit and huge complexity, 17:49S would seem
to be a better choice. Whichever is assigned, the conclusion follows
that it would be mandatory to assign all commas with better than 556 or
562 popularity rank. This amounts to no cutoff point at all, because
it would include so many commas as to require that nearly all possible
symbols be assigned commas.

Consideration of how frequently the gaps of unassigned commas occur is
a much better criterion.

> So the question is "How big do we let the gaps get (or the sum total
> of gaps) before we stop". Lets just see if something naturally
> suggests itself. I think I'd want to stop if we got to the point
> where there were as many unsymbolised commas as symbolised. So if
> we're assigning 150 or 200 symbols then we wouldn't want to look
> further than about the 300 or 400 most popular commas. Except maybe
> for a few Geneisms.
>
> And for "popular" above you can substitute "simple" if you prefer.
>
> A few that have occurred to me in the past as natural stopping places
> are:
> Rank
> 283 just includes 455n
> 346 just includes 65:77n
> 454 just includes 7:121C

I found your stopping places to be overly generous; the gaps start
occurring way before that (see below).

>> I've come to this conclusion as a result of the following:
>> 1) 25:49M (rank 62, complexity 27.332) *requires* '/|\ in a secondary
>> role, and
>
> Yes I see that is so, because it is only 0.02 cents from 55M.
>
>> 2) We've agreed to exclude 13:17M (rank 117, complexity 31.869) from a
>> separate symbol definition because it's closer to 1/2-apotome than
>> )/|\, which is defined as 5:49M.
>
> Yes I see the denial of a symbol to 13:17M as the result of a (new)
> rule whereby accents cannot be used where they cause a core to cross
> the half-apotome (or M/L diesis) boundary, except in the single
> special case of )/|\'

I chose )/|\. as the simplest symbol for 595M (or 17:37M, if you prefer
that one; it's not very clear which comma is simpler) over '(/|. and
'/|\'', the only other possibilities. Its apotome complement would
then have to be )/|\'' for 595L (or 17:37L), as opposed to .|\)' or
.(|).. . (Another reason for choosing .)/|\ is that this promotes
DAFLL.) Since this double-right-accented symbol crosses the M/L
boundary, I would restate your rule in a way that would not require any
exceptions: "A left-accent cannot be used where it causes a core to
cross the half-apotome (or M/L diesis) boundary."

> At least 13:17S gets to be symbolised by (|' along with 29S.

Yes, and smart defaults would easily distinguish 13:17M from 5:49M.

>> If we're excluding the above from symbol assignments, then why bother
>> to assign significantly less popular, more complex commas with more
>> complicated symbols?
>
> But those two presumably just result in small gaps in the list.

Yes, with 13:17M we're only out to pop. rank 117. (At this point, I'm
repeating 3 paragraphs from my message of 31 July, because it directly
that the gaps are no longer small nor infrequent once you pass rank 133
(where the next 3 out of 4 commas are >23 limit). After that point, I
ended up with more gaps than assignments, even though there were still
plenty of regions [i.e., minas] where commas still needed to be
assigned. Above rank 150, no more than 20% of the commas were assigned
to symbols. By this point the weighted complexity varies all over the
place (e.g., consider these two 11-limit commas: 25:77S is pop. rank
143 & wc 29.162, while 385C is rank 144 & wc 60.529). In giving
priority to one obscure comma over another I found it very difficult to
attach much significance to differences in pop. rankings.

The less popular comma-symbol assignments therefore seem a bit
arbitrary, which is not a good thing if boundaries between symbols are
to be based on the commas assigned. I concluded, therefore, that it
would be safer (and much more straightforward) to stick with strict
233-EDA boundaries, which would guarantee that the yet-unassigned
regions would be treated equitably.

While I did find it necessary to split some minas, there were not very
many (only 6 up to 1/2-apotome in olympian, where there were two fairly
popular commas within the same mina boundaries). For super-olympian, I
split 38 more minas, in most cases to notate either the next most
popular commas (through rank 133) or as many remaining 7-limit commas
as possible.

Mina 6 is an excellent example of a region in which assignment of )|.
to a comma is not at all clear. Popularity, weighted complexity, prime
limit, and proximity to SoCA of the symbol each suggest different
orders of preference, so it's largely a judgment call.. I settled on
19:4375s for two reasons: it's the alternate SoCA, and it's only 0.055c
from the midpoint of this mina (hence is a good nominal value for the
symbol).

To summarize, once popularity drops beyond a certain point in the list,
it becomes much more important to fill out the remaining gaps by comma
*size* than by popularity. Using minas to set symbol boundaries has
1) The boundaries are much more objective in that they do not depend on
the (sometimes unclear and therefore somewhat arbitrary) choice of
comma to define a given symbol; and
2) It's a straightforward (objective) way to determine how many (and
where) symbols remain to be defined (for the olympian level, i.e., the
finest one that will be required for most computer-based applications).

Super-olympian is a superset of olympian that distinguishes a few dozen
more (very obscure) commas with additional symbols (by splitting minas
halfway between the symbol definitions; these are held in reserve, in
the event someone requires separate symbols for some of these.

>> Who will ever use them?
>
> Probably only Gene, but who can say?
>
>> I would assign relatively unpopular, complex commas to symbols only if:
>> 1) They're 7-limit, or
>> 2) They're defined as a higher-prime-limit comma (generally as SoFA) so
>> that they can be used in a secondary role by a 7-limit comma, in order
>> to distinguish that comma from another 7-limit comma having the same
>> number of minas.
>
> I had trouble with the referents of your two "they"s above (in the
> two "they're"s. I'll assume the first refers to the commas and the
> second to the symbols.

Yes (sorry!).

> I find the phrase "having the same number of minas" to be redundant
> here. Couldn't it simply be "to distinguish that comma from another
> 7-limit comma <period>".

Yes, you're right.

> Now I certainly think that is a good thing to do. And a good way to
> approach it. But I wouldn't rule out assigning a symbol to a comma
> where the symbol has neither primary nor secondary 7-limit role, if
> the (non-7-limit) comma was more popular than the least popular of
> the other primary roles.

As I pointed out above, that would justify assigning anything with a
popularity rank better than 556 or 562 to some complicated symbol for
no reason other than that we have a bunch of obsure symbols that would
otherwise go unused. For example, do we really need '|(.. for 13:55k
(SoCA, pop. rank 385) and .)|(.. for 13:35k (pop. rank 347), when there
are many *more popular* commas that are unassigned? I expect that
there would be instances where, because of an unfavorable size order,
this might result in assignment of a more complicated symbol to a much
more popular comma in order that a simple symbol could be reassigned to
a highly unpopular comma (the alternative would be to dig deeper down
the popularity list to come up with something for the more complicated
symbol).

>> I looked at many of these less-popular 7-limit commas last night, and I
>> was delighted when I observed that most of them will get symbols that
>> are only moderately complicated, e.g.:
>>
>> 625S ~|\..
>> 3125C '|)'
>> 15625k '|('
>> 15625C .(|'
>> 49:625k ~|..
>>
>> Some of the above are in secondary roles.

As it turns out, all of above commas have been assigned the above
symbols in *primary* roles, but there are other, more complicated
7-limit ones that get secondary roles. For example, in the 27th mina
78125C is a secondary role for .~|(' (primary role of 11:23C), while
)~|''. is defined as 1715C.

>> It's not required that each
>> one be defined in a primary role, only that 7-limit commas be capable
>> of being notated uniquely up to a certain exponent level, e.g., all
>> 7-limit commas having the sum of the 5 and 7 exponents equal to 6 (or
>> possibly 7) or less be distinguishable from one another by different
>> symbols.
>
> This is a _great_ thing to do. I've argued for it before. To see how
> far you can push notation of the 5^n*7^m plane before it starts
> getting too frayed around the edges. Clearly higher powers of 5 are
> of more interest than higher powers of 7 (in about a 7:5 ratio I'd
> guess).
>
> We should publish this as a separate list. This would be "the maximal
> Sagittal 7-limit notation". No precision level need be mentioned. No
> boundaries need be defined. If you need to notate a 7-limit comma
> that doesn't have a symbol you just reuse the symbol whose 7-limit
> default is the closest.

FYI, the following are the simplest (most popular, or most notable)
pairs of commas *not distinguished* from one another within the 7 limit
in super-olympian:

1715k (2^25:3^9*5*7^3), 54.537 cplx, 206 pop. rank (vs. 78125k 151 pop. rank)
7:3125M (7:5^5), 68.457 cplx., 581 pop. rank (vs. 5:49M, 19 pop. rank)
40353607s (7^9), 188.689 cplx., 611 pop. rank (vs. 5s, 4 pop. rank)
7s (3^14*7:2^25), 224.798 cplx., no pop. rank (vs. 49:15625s, 1446 pop. rank)

We could distinguish 7s using .|(, which is SoCA, if you think anyone
might want that. (The symbol is certainly simple enough.) I've seen
the 7-schisma mentioned once or twice before, and this would
distinguish it from another 7-limit comma, 49:15625s in a theoretical
discussion. It's not really needed for notating 7/4 in JI until you
take F# as 1/1, and even then it's needed only for Fb'!(, which is the
*2nd alternate* spelling.
[Continued in next post]