## consistent Sagittal 37-Limit

cmloegcmluin
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### consistent Sagittal 37-Limit

I apologize to @Dave Keenan for posting this while he is still preoccupied, but my drafts are starting to pile up out of control, so I figured I'd start posting questions I have that are for the Sagittal community, not necessarily for him.

In the Sagittal-SMuFL-Map, we provide information about each sagittal's membership in different notations, and in the column for the Prime Factor notation, we do not report primes higher than 37. In this case, it is because:
• The Sagittal-SMuFL-Map concerns itself with font characters. A character may represent a sagittal or it may represent a diacritic, but not both; symbols which are combinations of sagittals and diacritics do not get their own font characters.
• 41 is the next prime after 37, and it is the first prime which requires a diacritic to distinguish it from a simpler prime (see here). In other words, it is the first Prime Factor notation symbol which does not consist only of a sagittal.
• Therefore, the ability to report membership in the Prime Factor notation in a table about characters breaks down past 37.
37 is also where Dave capped off his clever prime-to-keyboard-key mapping for a proposed WinCompose sequence set for Prime Factor notation (see here).

And a few posts later, 37 is also where he capped his enumeration of common combinations of primes a WinCompose sequence set could handle (see here).

Dave also gives 37 as the prime limit for his scraping of the Scala usage statistics (see here).

Dave also wrote me some time ago in an email that the extreme precision standard JI notation includes primes up to 37.

Given these few pieces of evidence, it feels like if we did want to propose a consistent prime limit for situations in Sagittal that may call for one, 37 would be our prime candidate (sorry, I had to do it). Any prime would be fairly arbitrary, but the fact that 41 needs a diacritic in Prime Factor notation is because it is the first one which is almost the same as an earlier prime (5), so that's a pretty good signal to cut ourselves off.

The reason I write all this today is because, as taken from the original JI Notation Spreadsheet that George produced (latest version of that document here), the extreme precision JI notation does actually include one prime above 37 (perhaps I misunderstood the point Dave was trying to make in that email). In fact, it skips right over 41 and 43, going straight to 47. It is the 47-small-diesis, represented by  . So I was thinking that it might be preferable to set the default value of this symbol to a 37-limit comma instead, unless anyone knows a compelling reason for it to be so high limit.

As for suggestions, I'm sure I could come up with an algorithm for finding 37-limit commas between 36.326¢ and 36.588¢ and then choose one which is simple and ranks not too badly on the Scala usage statistics. However, I expect that Dave may already have such infrastructure for comma determination ready-to-go, if he agrees this is a worthwhile endeavor. I guess we could just sum the monzos of and and be done with it though.

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

I simply failed to notice (or remember) that George had included a ratio of a prime greater than 37 among the primary commas (definitions) of the Olympians. Well spotted. I expect he tried to find a comma for that symbol, that would notate a ratio with a lower prime limit, but did not find any that would notate any ratio more popular than 47/1. Popularity doesn't tend to obey prime limits. Popularity is more like a sum-of-prime-factors limit. But feel free to check for 37 limit commas for that symbol, to make sure George didn't miss anything. I did have a way of generating such commas. I could eventually dig up my old spreadsheet. But perhaps it's best if you do it independently, in case my algorithm missed some possibilty too.

I wonder if Wolfram Alpha could help.

The single-key-per-prime WinCompose idea could be extended to 47 in the following kludgey way:

 1 2 3 4 5 6 7 8 9
Q W E R T Y U I O
A S D F G H J K L
Z X C V B N M
becomes
         5   7
11  13      17  19
41  23      47  29
31  43      37

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

or perhaps not quite so kludgey, and allowing for future expansion up to prime 73:
  41  43 5   747
11  13      17  19
23          29
31          37

cmloegcmluin
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### Re: consistent Sagittal 37-Limit

Dave Keenan wrote: Wed May 13, 2020 10:20 am but did not find any that would notate any ratio more popular than 47/1.
Do you have a link to the full popularity info mentioned here: viewtopic.php?p=258#p258

Calculating commas I can figure out no problem. But I’m less sure about statistical popularity.

I have to admit I am a fan of the ingenuity and intuitively of your less kludgey keyboard mapping here. But it wouldn’t sway me from wanting to limit to 37 where we can, given it can be justified.

herman.miller
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Joined: Sun Sep 06, 2015 8:27 am

### Re: consistent Sagittal 37-Limit

I found 529/518 (2^-1 * 7^-1 * 23^2 * 37^-1), which is around 36.37875 cents,

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

herman.miller wrote: Thu May 14, 2020 10:00 am I found 529/518 (2^-1 * 7^-1 * 23^2 * 37^-1), which is around 36.37875 cents,
Hey! Great to hear from you Herman.

By ignoring powers of 2 and 3 we obtain the simplest ratio that could be notated with an accidental for that comma, which is 529/259. And we can calculate its "estimated popularity rank" as the sum of all its prime factors, which is 7+23+23+37 = 90. So it's likely to be much less popular than 47/1 whose estimated popularity rank is of course just 47.

Here are the spreadsheets with the actual popularity stats from the Scala archive:
PopularityOfRatios.xlsx
popularityOfCommas.xlsx

cmloegcmluin
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### Re: consistent Sagittal 37-Limit

I found ~36.529¢ = | 1 -4 1 1 -1 1 > = 910/891, AKA the 455/11C.

It doesn't show up in the Scala stats, but by the sum of primes heuristic, we get 5+7+11+13=36 which beats 47. Also it's cute that it uses 455 like the mina. And it's only 13-limit.

cmloegcmluin
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### Re: consistent Sagittal 37-Limit

...so it's basically just , were that a valid symbol. The 11S minus the 455n.

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

cmloegcmluin wrote: Thu May 14, 2020 1:42 pm ...so it's basically just , were that a valid symbol. The 11S minus the 455n.
11S minus 455n is which is a valid symbol. That symbol's definition is the more popular 11:23S.

The main reference here is the Sagittal Standard JI Notation Calculator Spreadsheet. In particular its "Commas" and "Boundaries" sub-sheets.

The boundary between and is one of the few that "split a mina" — in this case the 75th. I think that what your discovery of the comma 11:455S = [ 1 -4 1 1 -1 1 > = 910/891 is telling us, is that this boundary should be lowered to include this comma in the capture zone for rather than 11:455S being a replacement for 47S = 48/47 as the definition of  .

What commas do you get for based on summing its components — either sum of flags and diacritics, or sum of core and diacritics, or sum of any kinds of subsets? Is 529/518 one if these, @herman.miller?

cmloegcmluin
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### Re: consistent Sagittal 37-Limit

Oh, right. I still get the mina and tina confused sometimes, clearly.

I agree we should move that boundary to accommodate the comma I suggested.

Well, I can clean this up later, but here’s a dump of reasonable-ish possibilities I found. Maybe one of these will work.

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