Prime-factor Sagittal JI notation (one symbol per prime)

[cmloegcmluin]

If you prefer the vectorized version I put together for the upcoming tutorial video, I can snapshot that instead.

Yes, please replace it with your new version.

Done.

Re: the proposition for a related image which goes up to prime 23, I think it gets kind of messy past 16 because the harmonics stop being in order. I certainly agree we shouldn't replace what we've got (up to 13) with it. If you think it would be handy to our higher-limit brethren, I won't stop you from making it. But I also agree it would be low priority.

Also, I thought that maybe until we have a full-fledged interactive calculator ready for this notation — like we have for the precision level notation, in spreadsheet form — we might want to add a note explaining what to do with the bolded 3-exponents. I had totally forgotten about those, and also the bolding isn't exactly enough to help them jump off the page atcha to remind you they're there.

Done. Let me know what you think of the new edit, between horizontal rules?

Oh, that's excellente!

One nitpick, and this is my fault: I was wrong to suggest that it goes up to 23 unbroken. It skips 21. Oh and I see you observed and reported this yourself in the next post. But the text in the intro post still suggests otherwise.

Those instructions are quite helpful, yes. They give you an instant answer for the nominal+sharp/flat for any single otonal pitch up to 23. The generalized instruction would explain that, say you're notating 7/5, then you'd need to sum the 3-exponents for those two. I.e. the 7-comma's 3-exponent is 2, and the 5-comma's 3-exponent is -4, but the 5 is in the denominator so you flip the sign to 4, so the sum is 6, so you'd notate that off of the nominal -6 away from your 1/1, which in the case of G would be Db.

But don't agonize over it. The calculator will be immensely helpful for this sort of thing.

Oh, another thing: you determined the Prime Factor notation, as you say in the intro, "The symbols were obtained as follows: As we came to each new prime, we determined its Olympian symbol." To be clear, you mean you determined its Olympian symbol with your 1/1 set to G, right? It might be nice to mention that, if so.

[Dave Keenan]

If you prefer the vectorized version I put together for the upcoming tutorial video, I can snapshot that instead.

Yes, please replace it with your new version.

Done.

Can I please have two tweaks to that image?
1. Nudge the inverted "1" and "3" of the 13th harmonic mnemonic up just enough to open up a gap between the 1 and the staff-line, as is already the case for the first "1" of the 11th harmonic mnemonic.
2. Change the word "Partial" to "Harmonic". I used the word "Partial" for the Wikipedia image because that's what existing images had, so I thought that would maximise its chances of being accepted. But it's not technically correct. A square wave has only odd harmonics, so its 3rd partial is its 5th harmonic. And struck metallophones have inharmonic partials.

Re: the proposition for a related image which goes up to prime 23, I think it gets kind of messy past 16 because the harmonics stop being in order. I certainly agree we shouldn't replace what we've got (up to 13) with it. If you think it would be handy to our higher-limit brethren, I won't stop you from making it. But I also agree it would be low priority.

Yeah. It might be better to just accept a fourth and two seconds, with:

   23 C
   19 B
   17 A
   13 E
   11 C
    9  A
    7 F
    6  D
    5 B
    4  G

But there would be a major accidental pileup. Even worse if we include 15 and 21.

One nitpick, and this is my fault: I was wrong to suggest that it goes up to 23 unbroken. It skips 21. Oh and I see you observed and reported this yourself in the next post. But the text in the intro post still suggests otherwise.

Fixed. Thanks.

Those instructions are quite helpful, yes. They give you an instant answer for the nominal+sharp/flat for any single otonal pitch up to 23. The generalized instruction would explain that, say you're notating 7/5, then you'd need to sum the 3-exponents for those two. I.e. the 7-comma's 3-exponent is 2, and the 5-comma's 3-exponent is -4, but the 5 is in the denominator so you flip the sign to 4, so the sum is 6, so you'd notate that off of the nominal -6 away from your 1/1, which in the case of G would be Db.

But don't agonize over it. The calculator will be immensely helpful for this sort of thing.

Yes. I think I'm agreeing with you when I say that would be too much information for that first post, particularly for an edit 5 years after the fact.

Oh, another thing: you determined the Prime Factor notation, as you say in the intro, "The symbols were obtained as follows: As we came to each new prime, we determined its Olympian symbol." To be clear, you mean you determined its Olympian symbol with your 1/1 set to G, right? It might be nice to mention that, if so.

No. I mean we determined the Olympian symbol for its comma (1/1 irrelevant). Fixed. Thanks.

[cmloegcmluin]

Can I please have two tweaks to that image?
1. Nudge the inverted "1" and "3" of the 13th harmonic mnemonic up just enough to open up a gap between the 1 and the staff-line, as is already the case for the first "1" of the 11th harmonic mnemonic.
2. Change the word "Partial" to "Harmonic". I used the word "Partial" for the Wikipedia image because that's what existing images had, so I thought that would maximise its chances of being accepted. But it's not technically correct. A square wave has only odd harmonics, so its 3rd partial is its 5th harmonic. And struck metallophones have inharmonic partials.

That all makes sense. Thanks for explaining. Done.

No. I mean we determined the Olympian symbol for its comma (1/1 irrelevant). Fixed. Thanks.

Got it. Yes, that makes more sense as well.

[Dave Keenan]

I'd like to record here, something I learned about the Sagittal choice of 17-comma (4131/4096) for the prime-factor notation where 17 is notated G:A versus the common alternative, which is the 17-kleisma (2187/2176) where 17 is notated G:G. I once suggested to George that we might go with the pack here, despite the kleisma having a greater absolute 3-exponent, and so requiring a greater offset from 1/1 on the chain of fifths (+7 instead of -5). One reason I gave was that the symbol was much simpler and easier to remember. George was adamant that this would be a mistake.

I learned the following recently, which confirmed his wisdom, while trying to find a good way of extending the harmonic-13th-chord's stack of thirds to include primes 17, 19 and 23.

I decided that to qualify, a stacked interval had to both be notated as some kind of a third (i.e. the nominals had to be 2 letters apart), and it had to have a size that was thirdish. I limited the size to the rather generous range of 205 to 497 cents, i.e. bigger than 9/8 (an obvious second) and smaller than 4/3 (an obvious fourth).

I set up a spreadsheet to generate all 27-integer-limit ratios (23-prime limit) and test them against these criteria. Then I repeated the exercise with the 17-kleisma (G) instead of the 17-comma (A).

There are six such ratios of 17 that are thirdish in size, listed in order of size below. Here are their notations when 1/1=G (omitting the comma symbols). The notations that spell them as thirds are hilited.

              Sagittal  Other
Ratio  Cents  notation  notations
---------------------------------
15:17  217¢   F#:Ab     F#:G#
17:20  281¢   Ab:B      G#:B
14:17  336¢   F:Ab      F:G#
17:21  366¢   Ab:C      G#:C
17:22  446¢   Ab:C      G#:C
13:17  464¢   E:Ab      E:G#

The most thirdish are the the middle four, which are between 250 and 450 cents in size. The standard Sagittal choice of nominal for prime-17 notates 3 of those 4 as thirds. The other choice notates only 1 of those 4 as a third, and completely fails the most thirdish pair in the middle.

What's special about thirds? I think there's something about the "critical band" in human psychoacoustics. People just love to make chords out of stacked thirds, at least in the middle registers (with the odd fourth thrown in sometimes).

I expect there are other properties of the choice of nominal (and hence comma) for 17, that could be weighed against this. But I was not previously aware of this data point in favour of the standard Sagittal choice. Well done, George.

[volleo6144]

So with the thread on removing the 47S from the Extreme precision level (and replacing it with 11:23S), we found that the 53C might be less far out than we'd hoped, and I also noticed that some of the commas we considered had N2D3P9 above 73²/18; my latest post there (which probably came off as more condescending than I'd like, but I can't figure out how to make it not sound like that without looking like I've left again) necessitated figuring out what the next symbols in the prime-factor notation would be after 61:

prime	symbol	comma	ratio	error	notes
67		67k	16281:16384	1407:1408
71		71C	71:72	5:19.71n
73		73C	72:73	73n
79		79S	79:81	6399:6400
83		83M	249:256	17:7.83n	some discussion below was as a result of me thinking it would mirror the 79S and just assuming this was 83S = 81:83
89		89C	64881:65536	49:11.89n	similar situation to 23, where +6 and -6 are both in range*
97		97C	96:97	4752:4753	represents 89.97n, 2097152:2097819, 0.55¢
101		101s	8181:8192	505n
103	?	103C	8192:8343	none yet	see below

The repair strategy of keeping the mina accents breaks down after 101, because for the 103C is already used as () for the 53C, and the 103C falls into the capture zone of itself and not that of .

* With 23, we have two options that are both within the 68¢ limit: the 23C (729:736, 16.54¢), and the 23S (16384:16767, 40.00¢) which is exactly a 3C larger; since these both have a 3-exponent of ±6, we choose the smaller one: the 23C. Mirroring this, we have the 89S (712:729, 40.85¢), and the 89C (64881:65536, 17.39¢) which is also a 3C away, except that the smaller odd limit is the larger comma this time around.

[Dave Keenan]

That's cool that you worked out those higher primes, although I don't think there will be many customers for them. Interesting problem with 103. I think your solution (using a tina accent) is the right one.

[volleo6144]

I think your solution (using a tina accent) is the right one.

Come to think of it, have we even finalized how tina accents work if they're not against a bare shaft? Is the 226th tina because the 55C is 226.056 (809) or 226.046 (8539) tinas, or because 65 minas is 225.687 (809/233) or 225.624 (8539/2460), or for some other reason? Sorry, I was away...

[Dave Keenan]

Come to think of it, have we even finalized how tina accents work if they're not against a bare shaft? Is the 226th tina because the 55C is 226.056 (809) or 226.046 (8539) tinas, or because 65 minas is 225.687 (809/233) or 225.624 (8539/2460), or for some other reason? Sorry, I was away...

is the 226th tina because the 55C is 226.056 (809).

That's consistent with what happens in taking from high to ultra to extreme precision (Promethean to Herculean to Olympian). As shown in this (somewhat out-of-date) chart:
https://sagittal.org/SagittalJI.gif
i.e. a symbol has the same default/primary comma at every precision level.

There may however be some problems in going from extreme to insane (Olympian to Magrathean) where an existing mina accent turns out to not be 3 tinas (or a 2-mina accent not 6 tinas). We haven't figured out what to do about those yet. I'm not even sure if there are any. I have a vague memory of someone looking into that in the Magrathean thread.

There's a design rule that Sagittal follows, whereby a later-developed, less-commonly-used feature (like Magrathean) cannot "reach back" and make a more-commonly-used feature (like Olympian) more complicated. That's not to say it can't cause us to revise an earlier-developed feature, only that we can't revise it in a way that makes it harder to understand or harder to use.

We also haven't assigned symbols or default/primary commas to every Magrathean capture zone. But I have stated elsewhere my lack of interest in doing so. My interest in Magrathean pretty much ceased once we got font outlines and SMuFL descriptions for the tina accents and made the deadline and got them into SMuFL/Bravura.

[volleo6144]

While checking the accuracy of my other post again, I noticed something... interesting: since the 75th mina isn't split anymore, 47:48 is now represented by and not , and the value the dropped accent represents is changed from 7²·47n = 2303:2304 to 11:47n = 24057:24064.

On to the... 31, 61, 127-limit!

prime	symbol	comma	ratio	notes
103		103C	8192:8343
107		107C	107:108	is 23·107n, 19683:19688, 0.44¢
109		109C	108:109
113		113k	1017:1024
127	? ?	127C	127:128	43·127n = 16383:16384 ~ 0.11¢ ~ ¾ tina

And with that, we get two more required tina accents, this time together with a mina accent—just as the 41C is the first comma to require a mina accent as 5·41n = 80·82:81² = 6560:6561 away from the existing 5C, the 109C and 127C also require tina accents as 107·109:108² and 127·129:128² away from the 107C and 43C, and the 43C itself requires a 2-mina accent to distinguish it from the 17C. What happens here?

[Dave Keenan]

While checking the accuracy of my other post again, I noticed something... interesting: since the 75th mina isn't split anymore, 47:48 is now represented by and not , and the value the dropped accent represents is changed from 7²·47n = 2303:2304 to 11:47n = 24057:24064.

Thanks for pointing that out. I've updated the tables in the first post of this thread, with the new symbol for 47. But I've left the old one there too, in case we decide George had the right idea after-all, in splitting the 75th mina so that 47 could be exactly notated in Olympian.

I think you have the wrong symbol for 83. I think it should be DropMinas() = because whatever comma that is, has a smaller 3-exponent according to the JI notation calculator spreadsheet.

And for 109 I get so it will need a tina accent to distinguish it from 107. So 109 will be That's ":l4:"

On to the... 31, 61, 127-limit!

Given that you went beyond 61, I think it was good to go to 127, to complete the octave. But please, no more.

127

? ?

127C

127:128

43·127n = 16383:16384 ~ 0.11¢ ~ ¾ tina

And with that, we get the second required tina accent, this time together with a mina accent—just as the 41C is the first comma to require a mina accent as 5·41n = 80·82:81² = 6560:6561 away from the existing 5C, the 127C is the second to require a tina accent, being 127·129:128² away from the 43C, which itself requires a 2-mina accent to distinguish it from the 17C. What happens here?

I think 127 should be That's ":@5:"

You needn't worry. I haven't felt you were being condescending. But given your obvious math and CS skills, I'd be perfectly willing to make allowances for any social difficulties. I suspect we're all a bit aspie around here.