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

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

Post by ᴄᴍʟᴏᴇɢᴄᴍʟᴜɪɴ »

Dave Keenan wrote: Mon Mar 01, 2021 4:43 pm
cmloegcmluin wrote: Thu Feb 25, 2021 10:00 am 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.

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

Post by Dave Keenan »

cmloegcmluin wrote: Tue Mar 02, 2021 5:20 am
Dave Keenan wrote: Mon Mar 01, 2021 4:43 pm
cmloegcmluin wrote: Thu Feb 25, 2021 10:00 am 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:B
   17 :~|(::b: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.

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

Post by ᴄᴍʟᴏᴇɢᴄᴍʟᴜɪɴ »

Dave Keenan wrote: Tue Mar 02, 2021 8:24 am 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.

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

Post by 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::~|(::b: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 (:~|(::b: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 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.

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