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

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

Post by Dave Keenan »

Correction: "any chord and its inversion" -> "any chord and its inverse". e.g. D:F#:A and G:Bb:D.

Not really sure what you're after. You could directly show on a staff, each of the scales and chords I described.

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

Post by cmloegmcluin »

I suppose it's the G version which in particular I'd like to see, how the harmonic series happens to fit most closely to the sharp- and flat- less nominals on a chain of Pythagorean fifths when you base it on G. Something like: a ring, corresponding to the octave, with the fifths and the harmonic series points labeled on it.

I guess what would be even better, though, what I really want, rather, is just the feature I have planned for the web app notation calculator, where you can at the click of a button change the 1/1, and see how that affects every pitch in your tuning, so at a glance you can eyeball which 1/1 is best for your piece. I remind you that a favorite tuning of mine worked out so that B was the ideal 1/1, even though you decided to only do 4 out of the 7 nominals in your chart here: viewtopic.php?p=662#p662
cmloegcmluin wrote: Mon Apr 06, 2020 7:28 am One thing you may notice is that most of my examples are based on a 1/1 of B. I know this is not the most common choice, but again, Yer is not your average scale. B was chosen for Yer for a similar reason that C, G, D, or A are often chosen for other scales: because it results in the fewest sharps and flats.
And I expect it won't be the first time I have some bizarre tuning and would want to rapidly determine which 1/1 lines it up with the chain of fifths most nicely. Without such a tool, figuring that is -- for me, anyway -- really tedious and nerve-racking. In fact, it's less about the work to do it, and more the fact that the prospect of doing it is so alien and intimidating to me that I just say "forget it" and go back to writing music for computers instead of humans.

I would prefer if that chart just had the other nominals provided, honestly. You've already done more than half of them, I mean. I suppose, though, that leaving out E, B, and F from that chart might have been not so much a "good enough" type of decision, so much as it was a way of intentionally implicitly dismissing them, i.e. even if those 1/1's might give you less sharps and flats for a given tuning, it wouldn't be worth the cost of using them, that cost being the lower level of familiarity with the layout of the circle of fifths relative to them as 1/1 which the average performer is likely to have. Would you say that's true?

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

Post by Dave Keenan »

cmloegcmluin wrote: Mon Mar 01, 2021 11:53 am I suppose it's the G version which in particular I'd like to see, how the harmonic series happens to fit most closely to the sharp- and flat- less nominals on a chain of Pythagorean fifths when you base it on G. Something like: a ring, corresponding to the octave, with the fifths and the harmonic series points labeled on it.
That would be good, although it's only the first 13 harmonics that are sharp-and-flat-less. But I thought it was essentially shown by the image and table in the first post of this thread. The 5-comma requires the nominal to be +4 on the chain of fiths and the 7-comma requires -2. The only way they can both be sharp-and-flat-less is if G is 1/1 as follows:

 F  C  G  D  A  E  B
-2 -1  0 +1 +2 +3 +4

It then happens that the nominals for 1, 3, 9, 11 and 13 all fit within the same range of fifths. But for 15 you must introduce F#.

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

Post by Dave Keenan »

I guess what would be even better, though, what I really want, rather, is just the feature I have planned for the web app notation calculator, where you can at the click of a button change the 1/1, and see how that affects every pitch in your tuning, so at a glance you can eyeball which 1/1 is best for your piece. I remind you that a favorite tuning of mine worked out so that B was the ideal 1/1, even though you decided to only do 4 out of the 7 nominals in your chart here: viewtopic.php?p=662#p662
Sure. That will be good.
And I expect it won't be the first time I have some bizarre tuning and would want to rapidly determine which 1/1 lines it up with the chain of fifths most nicely. Without such a tool, figuring that is -- for me, anyway -- really tedious and nerve-racking. In fact, it's less about the work to do it, and more the fact that the prospect of doing it is so alien and intimidating to me that I just say "forget it" and go back to writing music for computers instead of humans.
That suggests to me that you don't keep a chain-of-fifths ruler from Fbb to Bx on a handy scrap of paper to refer to. A change of 1/1 is just a slide along that slide-rule.
I would prefer if that chart just had the other nominals provided, honestly. You've already done more than half of them, I mean. I suppose, though, that leaving out E, B, and F from that chart might have been not so much a "good enough" type of decision, so much as it was a way of intentionally implicitly dismissing them, i.e. even if those 1/1's might give you less sharps and flats for a given tuning, it wouldn't be worth the cost of using them, that cost being the lower level of familiarity with the layout of the circle of fifths relative to them as 1/1 which the average performer is likely to have. Would you say that's true?
Short answer: Yes. Longer answer: Almost. But I'm pretty sure I said it's all about minimising sharps and flats. So if B does that for your scale, you should use B.

But I wouldn't want to add any other 1/1's to that post because I wouldn't want novices to be misled about what was likely to be useful.

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

Post by Dave Keenan »

A chain-of-fifths slide-rule is a solution to many problems.

FbbCbbGbbDbbAbbEbbBbbFb Cb Gb Db Ab Eb Bb F  C  G  D  A  E  B  F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
offsets from 1/1                -5 -4 -3 -2 -1  0 +1 +2 +3 +4 +5 +6(+7)
odd harmonics                   17    19  7 11  1  3  9 13  5 15 23(17-alternative)

Just insert or delete spaces in the bottom two rows, to slide the slide-rule and change the 1/1.

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

Post by cmloegmcluin »

Dave Keenan wrote: Mon Mar 01, 2021 1:48 pm ...it's only the first 13 harmonics that are sharp-and-flat-less.
Sure, understood.
The 5-comma requires the nominal to be +4 on the chain of fiths and the 7-comma requires -2.
Oh, how ridiculous of me. I'm writing the tutorial for that very concept as we speak. Of course. Seeing them line up in terms of their position in cents relative to each other is meaningless. All that matters is their addresses in terms of 3-exponent.

And so I reiterate my ask here, which you may have missed while you were away. I think it would help make this concept clearer.
Dave Keenan wrote: Mon Mar 01, 2021 2:00 pm That suggests to me that you don't keep a chain-of-fifths ruler from Fbb to Bx on a handy scrap of paper to refer to. A change of 1/1 is just a slide along that slide-rule.
Oh wow. That gives me a really helpful way to think about it. I see that you explained that back there as well. But at the time I couldn't quite put the full picture together in my head.

Maybe a good way for me to approach notating my various bizarre JI scales is to just start with D as my 1/1. Then mark all the places where my pitches are.
                                        1        2     1 1     3  1        2  1  1  1         
Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx

Then I would just slide my top row side to side until it minimized the sharps or flats. Eye-balling the above random example, I think I would want to shift it a little to the left, to be on C instead.
Short answer: Yes. Longer answer: Almost. But I'm pretty sure I said it's all about minimising sharps and flats. So if B does that for your scale, you should use B.

But I wouldn't want to add any other 1/1's to that post because I wouldn't want novices to be misled about what was likely to be useful.
Thanks for explaining. That's really good to know.

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

Post by cmloegmcluin »

Whoa. That's quite impressive to me how those odds go all the way up to 23 before repeating, and only leave one slot unused. Isn't that remarkable?

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

Post by Dave Keenan »

cmloegcmluin wrote: Mon Mar 01, 2021 2:47 pm Maybe a good way for me to approach notating my various bizarre JI scales is to just start with D as my 1/1. Then mark all the places where my pitches are.
                                        1        2     1 1     3  1        2  1  1  1         
Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx

Then I would just slide my top row side to side until it minimized the sharps or flats. Eye-balling the above random example, I think I would want to shift it a little to the left, to be on C instead.
You got it! I agree, C looks optimal.
cmloegcmluin wrote: Mon Mar 01, 2021 2:53 pm Whoa. That's quite impressive to me how those odds go all the way up to 23 before repeating, and only leave one slot unused. Isn't that remarkable?
Yes, it is. God got that that right. Or maybe she had no choice in the matter. ;)

Edit: Oops, no. I omitted 21 which must share C with 11.

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

Post by Dave Keenan »

cmloegcmluin wrote: Thu Feb 25, 2021 10:00 am @Dave Keenan I noticed that in the first post of this topic, the diagram is different than the one that appears on the home page. For some reason the mnemonics and actual symbols are flipped. That's fine, just curious. Except that the mnemonic for pao is... well, just a pao, not the Roman numeral mnemonic. 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. I had to make a new one for the JI Wikipedia article (according to my understanding of their rules). At that time I decided it was better to put the true notation before the mnemonics, and I added the Roman 5.
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?

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

Post by Dave Keenan »

I worked out a way to extend that 13-limit stack-of-thirds staff image to prime 23 (but so far without additional mnemonics). It has odd harmonics to 25 except it omits 21 because this would be notated :!):C and so competes with 23 for the top position. That top "third" (17:23) is a doozy (really a fourth).

2x 23 :|~::#:C
2x 17 :~|(::b:A
2x 15 :\!::#:F
   25 :\ \!::#:D
   19 :)|::b:B
   16  G
   13 :(!/:E
   11 :/|\:C
    9  A
    7 :!):F
    6  D
    5 :\!:B
    4  G

I wouldn't want to replace the 13-limit image with this, as it's so much more messy and scary-looking. But it might be useful for some high-limit JI users, particularly if we can come up with some mnemonics. But it's low priority.

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