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

[Dave Keenan]

To follow this stuff, it really helps to memorise the chain of fifths. Its a part of conventional musical education. Repeat after me :
EF
see-GEE
dee-AY
ee-BEE

EF see-GEE dee-AY ee-BEE

F C G D A E B

This chunk is a godsend for me. That's so cool that it works out that way for G (I worked it out and I assume what you mean is that none of 3/2, 5/4, 7/4, 9/8, 11/8, or 13/8 require accidentals when G is 1/1, ...

You should include 1/1 in the list, and I think you mean they don't require sharps or flats. They certainly require comma accidentals — for 5/4, 7/4, 11/8 and 13/8.

but you wouldn't necessarily get that for other nominals as 1/1... you don't get it for the couple others I checked).

It's fairly easy to prove it doesn't work for any other 1/1 nominal (with that particular choice of notational commas for the primes 5, 7, 11, 13) by induction on the chain of fifths. It uses every bare nominal, F C G D A E B. So every shift of one position on the chain-of-fifths adds another sharp as you go fifthward, or another flat as you go fourthward.

Where can I learn more of this? I'm embarrassed to say I haven't read Doty's Just Intonation Primer yet.

It depends what you mean by "this". In any case, it wouldn't hurt to read Doty.

I've heard about setting A to something else than 440 Hz, but usually it's pretty close to 440 Hz. Is the choice of letter really just all about the accidentals? Meaning I could set it to A if that gave the nicest accidentals, but if I actually wanted my piece to be in a different register than that I could just say that A is something crazy like 311.13 Hz, and most performers would be okay with that?

No. If you're willing to choose any of C G D A E to notate your 1/1, there's no reason to go outside of the range 400 Hz to 480 Hz for 1/1. That's ±10%. That's because the largest step between those nominals is a Pythagorean minor third (A:C or E:G), which is 27:32 = 1.185, and 480/400 = 1.2.

Whether performers are OK with an A that differs from 440 Hz depends on the nature of their instrument — whether it is made for that reference, or how easily it can be retuned (and possibly whether the performer is cursed with absolute pitch perception ).

Cool idea [using a 3-comma accidental]. Why not just keep adding conventional sharp signs though? Wouldn't it be fairly intuitive to indicate a triple sharp like ?

Sure. That's what's usually done.

I feel like I'd rather see that than myself.

You've got it wrong. is a shift of 7 fifths, while is a shift of 12 fifths. So F (3×7-3 = 18 fifths from D) would not be F (2×12+7 = 31 fifths from D) but G (12 + 7 - 1 = 18 fifths from D). I chose D as my zero reference here because it is the point of symmetry of the chain of fifths (as conventionally-notated). With 3-comma accidentals the chain could be notated:
E B F C G D A E B F C G     E B F C G D A E B F C G     E B F C G D A E B F C G

But it's probably not such a great idea, since it uses an accented symbol. And the symbol is based on the 5-comma symbol. The 5-nesses of the core and the accent cancel each other out.

The Sagittal JI calculator spreadsheet does allow setting the double-flats and double-sharps as 1/1, and seems to work alright. Is that something I should fix then, when I do the work to have it help you out with choosing the most conventional options it suggests? Or do you think I may as well leave it that way?

If it aint broke, don't fix it. Leave it.

Sorry, I should have given an example of what I mean. is the multi-sagittal for 13. If I go to the Sagittal JI Calculator and put in 13, leaving C as the default 1/1, indeed I get as my result (on an A, and technically it's but let's just ignore the diacritics for now). G and D as 1/1 also agree with . But if you use A as your 1/1, is not one of the options; the only option with a bare nominal is F, and the accidental is . E and B also give this result. (F agrees with C, G, and D). So it's split roughly evenly, 4 against 3. Was preferred because 4 > 3? Or is there something deeper I'm not getting yet?

Are you confusing the prime factor notation (which is the topic of this thread) with the Olympian notation (which is what you get from George's spreadsheet)? The Olympian notation has multiple symbols for notating 13, the prime factor notation has only one.

Or are you just asking (slightly off-topic) why George's spreadsheet offers 8:13 as D : B but not as A : F. And conversely why it offers 8:13 as A : F and not D : B?

The reason is, that George believed that one should not combine an apotome symbol with another symbol pointing in the opposite direction if that symbol is larger than a half-apotome. It also stems from the primacy of the pure Sagittal (at least in George's mind). It enforces only one mixed combination for every pure sagittal symbol.

But in the prime-factor notation you have no choice. There is only one symbol for prime 13 so 8:13 has to be D : B and A : F.

I definitely don't understand the statement "We're just minimising the absolute value of the offset from 1/1 along the chain of fifths".

Let's use the example of prime 13. Why did we choose instead of to notate 13 in the prime factor notation? With C as 1/1, 13 could be A or A. On the chain of fifths they are Ab Eb Bb F C G D A. We chose A because A is closer to C on the chain of fifths. The thinking behind that was that this would minimise the number of sharps and flats for the "average" choice of 1/1, which we took to be D, the symmetry point.

But there is definitely an argument to be had, that the "average" choice of 1/1 is closer to G, in which case there would be fewer sharps and flats on average if we allow the 3 exponents of the prime commas to range from -4 to +7 instead of -5 to +6. This doesn't affect the choice for 13 (or any primes below it), but it does affect 17. It would lead us to prefer the 17-kleisma (2187/2176) to the 17-comma (4131/4096). Dave Ryan prefers the 17-kleisma for other reasons. I actually think it's a good choice too, because it is a visually-simpler symbol. But George preferred to stick to the symmetrical -6 to +6 (which is defacto -5 to +6 for the primes we've looked at so far (I think)).

I think it would be great if the choice between 17C and 17k was a parameter for the prime factor notation calculator, and hence the choice between -5 to +6 and -4 to +7.

[Regarding factors of 3] So... I do think there may be some fundamental inner workings of Sagittal (and extended H-E systems in general) that I don't quite intuit yet. I'll try to take you through my experience so far. Keep in mind that I don't have a strong musical background -- I've been writing music as long as I can remember, but I've never mastered an instrument or studied it academically.

That's good to know. I'm sure there are many others like you. And it puts you in a good position to explain the Sagittal system to others, once you feel you understand it sufficiently yourself.

• Ah, interesting. Commatic alterations. That makes sense. Because each of the primes has a different deviation from standard tuning.
• Okay, each comma can have a bunch of powers of 2 in it. Fine, I understand octave equivalency.
• ...Wait, what...? These commas have a bunch of powers of 3 in them, too! I don't like that. I almost never use fifths. They also aren't pitch class equivalent, so now I'll be limited by that nature. And I'll probably have to memorize the circle of fifths, etc...
• (Ignores JI notation systems for many years, focusing on writing music for computers that don't need to deal with all this nonsense)
• (Decides he'd like to have some of his music performed by humans, starts trying to figure out JI notation systems again)
• I still don't really understand the powers of 3. They seem to magically work out a lot of the time

That's priceless, for helping me understand where you're coming from.

You only have to memorise FCGDAEB. There's nearly always a scrap of paper on my desk somewhere with a chain of at least 35 fifths written on it for counting purposes.

Even though Yer tuning doesn't have any pure fifths, it has some near-misses, of 692c, 711c and 718c. Pure fifths are such strong attractors, it could be useful, in effect, to have the notation tell you where they are so you can avoid them.

Part of the reason I included that snippet of me working out 11*13, if only semi-consciously, is that I expected you might say something like this: that I should have just known that right off the bat. But I don't think I get yet why those powers of 3 just work themselves out. I still feel like I have to do it manually. And that's why I'm so uncomfortable with the idea of anything other than -5 to +6 fifths and just following instructions someone who knows what they're doing has told me will work. Either that or I really do just need to learn this by working out enough examples until things click (or by implementing it in code, although I'd rather work out the bugs in my understanding before I try programming them... there'll be room enough for bugs later in just the implementation errors!).

Oh sure. We've got to get you up to understanding the reasons behind the instructions. Not just following them blindly. I'm sure it will click soon and you'll realise it's simpler than you thought.

One thing you've made me realise is that we really should have the exponent of 3 (and hence the offset along the chain of fifths) shown for each of those prime commas in the first post of this topic.

I'm trying to figure out all the thoughts behind the sentences: "It's not a convention to go from -5 to 6. It's not even a convention to limit it to 12 Pythagorean notes. It just seemed like a good idea to George and I." The chart above seems to have been provided by a guy named Dave Ryan.

Dave Ryan told me what to do, but I made the chart. But the interesting thing is that he doesn't agree with -5 to 6 or -6 to 6. In fact he doesn't place any a priori limits like that, but has a complicated algorithm for deciding the best offset for any prime. As mentioned, he prefers +7 for prime 17. That's the first prime where we disagree. (Although I could easily be convinced to go with +7 too).

Did I miss somewhere where this concept of -5 to +6 fifths is codified for Sagittal in general?

It is absolutely not codified in Sagittal in general. It is mentioned for the first time in the first post of this topic (actually as officially -6 to +6, but defacto -5 to +6).

It seems like it's only relevant to figuring out these multi-sagittals.

Yes! You got it.

Otherwise you're free to use whichever sharped and flatted nominals you want.

Yes.

My sense is that these multi-sagittal might only "work out" with respect to the count of 3's in their monzos as long as you followed this exact -5 to +6 chain.

Where are you getting these counts of three? 3⁻¹ for 11, 3³ for 13? They don't seem to correspond to the monzos for or , which have powers of 1 and -5, respectively, not -1 and 3.

No. They should work out any way. They are exactly the exponent of 3 in their monzos. You just seem to have made an arithmetic error for the case of 13 above.

The symbol is for the 11-medium diesis up = 33/32 = [ -5 1 0 0 1 >. So to notate 11 you have to cancel out that 3¹ in the comma accidental, by having 3⁻¹ in the combination of nominal and sharp or flat, meaning that this must be one place to the left of the nominal for 1/1, on the chain of fifths. So if 1/1 is C then 11 (or 11/8 if you prefer) is F.

The symbol is for the 13-large-diesis down = 26/27 = [ 1 -3 0 0 0 1 >. So to notate 13 you have to cancel out that 3⁻³ in the comma accidental, by having 3³ in the combination of nominal and sharp or flat, meaning that this must be 3 places to the right of the nominal for 1/1, on the chain of fifths. So if 1/1 is C then 13 (or 13/8 if you prefer) is [thinking "CGDA"] A.

For homework I suggest you work out (or look up) the monzos for all the prime commas listed in the first post, and post them here.

[cmloegcmluin]

No. If you're willing to choose any of C G D A E to notate your 1/1, there's no reason to go outside of the range 400 Hz to 480 Hz for 1/1. That's ±10%. That's because the largest step between those nominals is a Pythagorean minor third (A:C or E:G), which is 27:32 = 1.185, and 480/400 = 1.2.

You say there's "no reason", but as far as I can understand, if the cleanest notation for a scale in a piece of mine was when I set A to 1/1, but I just didn't like as much how the piece sounded when played that high, so I really wanted it to shift the whole thing down to 311 Hz, while of course keeping the same notation, it looks like I would not be able to achieve this.

I think it would be great if the choice between 17C and 17k was a parameter for the prime factor notation calculator, and hence the choice between -5 to +6 and -4 to +7.

Yes, in my mindmap for this project I have proposed that the calculator should let you adjust the range of fifths like that. This feature would be relevant for the JI calculator too. Perhaps others.

And I think it would be a good thing to alert users that they were diverging from the standard if they get different results as a consequence of changing the range to something other than -6 to +6.

It is absolutely not codified in Sagittal in general. It is mentioned for the first time in the first post of this topic (actually as officially -6 to +6, but defacto -5 to +6).

Holy moly. Okay, I just re-read the original post and it makes a lot more sense now. I promise I read it at some point, but it was just so beyond me at that point, I was overwhelmed!

You just seem to have made an arithmetic error for the case of 13 above.

No, I see what I did though. I grabbed the power of 3 from the monzo for instead of ... I wasn't doing any arithmetic.

The symbol is for the 11-medium diesis up = 33/32 = [ -5 1 0 0 1 >. So to notate 11 you have to cancel out that 3¹ in the comma accidental, by having 3⁻¹ in the combination of nominal and sharp or flat, meaning that this must be one place to the left of the nominal for 1/1, on the chain of fifths. So if 1/1 is C then 11 (or 11/8 if you prefer) is F.

The symbol is for the 13-large-diesis down = 26/27 = [ 1 -3 0 0 0 1 >. So to notate 13 you have to cancel out that 3⁻³ in the comma accidental, by having 3³ in the combination of nominal and sharp or flat, meaning that this must be 3 places to the right of the nominal for 1/1, on the chain of fifths. So if 1/1 is C then 13 (or 13/8 if you prefer) is [thinking "CGDA"] A.

Phew..... thank you for holding my hand on that one. My sense of utter relief is far outweighing my frustration (feeling like I should have been able to get that myself).

After this clicked for me, I ended up deleting 80% of my response here. A hellscape of confusion, cleared away!

So in the case of my 11*13, I have a total (1) + (-3) = -2 powers of 3 I need to cancel out, so that's why I move up two positions in the chain of fifths from C to D.

Why do we even need this table? Isn't this technique a million times easier? Maybe not for everyone. I'm a big believer in different learning styles. Well, now we have a couple different ways to attack the problem.

Re: the table, though, if it's officially -6 to +6, then wouldn't you need to divide the octave up into 13 ranges in that table, not 12?

One thing you've made me realise is that we really should have the exponent of 3 (and hence the offset along the chain of fifths) shown for each of those prime commas in the first post of this topic.

For homework I suggest you work out (or look up) the monzos for all the prime commas listed in the first post, and post them here.

That would probably have been a huge help to me, yes. I can't seem to get the table formatting to work out for me on the forum yet. And I think the exponents of 3 are really the salient information. But here you go:

 
5	80/81		-4
7	63/64		 2
11	33/32		 1
13	26/27		-3
17 	4131/4096	 5
19 	513/512		 3
23 	736/729		-6
29 	261/256		 2
31 	31/32		 0
37 	37/36		-2
41 	82/81		-4
43 	129/128		 1
47 	47/48		-1
53 	53/54		-3
59 	531/512		 2
61 	244/243		-5

Indeed, all within -5 to +6.

Upon further reflection maybe the full monzos would be nice to have since actually some of these commas are not even present in the standard extreme precision level JI notation.

[Dave Keenan]

No. If you're willing to choose any of C G D A E to notate your 1/1, there's no reason to go outside of the range 400 Hz to 480 Hz for 1/1. That's ±10%. That's because the largest step between those nominals is a Pythagorean minor third (A:C or E:G), which is 27:32 = 1.185, and 480/400 = 1.2.

You say there's "no reason", but as far as I can understand, if the cleanest notation for a scale in a piece of mine was when I set A to 1/1, but I just didn't like as much how the piece sounded when played that high, so I really wanted it to shift the whole thing down to 311 Hz, while of course keeping the same notation, it looks like I would not be able to achieve this.

There's nothing stopping you from doing it. My "No" was in response to your question whether most performers would be OK with that. But if the instrument does the transposing, it shouldn't be a problem.

I think it would be great if the choice between 17C and 17k was a parameter for the prime factor notation calculator, and hence the choice between -5 to +6 and -4 to +7.

Yes, in my mindmap for this project I have proposed that the calculator should let you adjust the range of fifths like that. This feature would be relevant for the JI calculator too. Perhaps others.

And I think it would be a good thing to alert users that they were diverging from the standard if they get different results as a consequence of changing the range to something other than -6 to +6.

Definitely alert them. I would prefer only that one option. But you might check what the range of 3 exponents is for the commas used by EHEJIPN.

Holy moly. Okay, I just re-read the original post and it makes a lot more sense now. I promise I read it at some point, but it was just so beyond me at that point, I was overwhelmed!

That's perfectly understandable.

Phew..... thank you for holding my hand on that one. My sense of utter relief is far outweighing my frustration (feeling like I should have been able to get that myself).

After this clicked for me, I ended up deleting 80% of my response here. A hellscape of confusion, cleared away!

That's good to hear.

So in the case of my 11*13, I have a total (1) + (-3) = -2 powers of 3 I need to cancel out, so that's why I move up two positions in the chain of fifths from C to D.

Indeed.

Why do we even need this table? Isn't this technique a million times easier? Maybe not for everyone. I'm a big believer in different learning styles. Well, now we have a couple different ways to attack the problem.

Re: the table, though, if it's officially -6 to +6, then wouldn't you need to divide the octave up into 13 ranges in that table, not 12?

You're confusing two levels. That Pythagoraen[12] lookup table is meta to the thing you just did above. It is a way to determine what symbol and what exponent of 3 the prime-factor notation should use for each prime. The thing you did above was (once you have the symbols and 3-exponents for each prime) how you should notate a given ratio (which may have a combination of primes).

When starting out, it made sense not to favour negative over positive, hence -6 to +6. But ultimately, it's far better to limit to 12 Pythagoreans than 13, because 12 is distributionally even, a MOS. And 13 has the awkwardness that e.g. G# is higher than Ab. So once we looked at all the primes to 61, and saw that prime 23 was the only one that might have wanted -6, but in fact it had a smaller comma for +6, then we were justified in contracting to -5 to 6.

That [3-exponents or full monzos] would probably have been a huge help to me, yes. I can't seem to get the table formatting to work out for me on the forum yet. And I think the exponents of 3 are really the salient information. But here you go:
...
Indeed, all within -5 to +6.

Upon further reflection maybe the full monzos would be nice to have since actually some of these commas are not even present in the standard extreme precision level JI notation.

You seem to have used the [ pre ] tags admirably, to produce that table. Currently there is no other way. But you might want to investigate "BBCodes for tables". If you find a good set of definitions for them, I could implement them for this forum. Yes, it's annoying that you can't directly type a tab character, as the browser just eats the tab as a command to go to the next field or something. So I either prepare such tables in a text-editor like Notepad++, using a monospaced font, or I copy a tab from somewhere and then Ctrl+V (paste) becomes my effective tab-character key.

I suggest giving the full monzo, but bolding the 3-exponent, and eliding strings of more than 3 zeros. i.e. every prime after 13 looks like [ n m ... 1 >.

[cmloegcmluin]

I suggest giving the full monzo, but bolding the 3-exponent, and eliding strings of more than 3 zeros. i.e. every prime after 13 looks like [ n m ... 1 >.

Something like this then:

5	80/81		[   4  -4   1   0   0 >
7	63/64		[  -6   2   0   1   0 >
11	33/32		[  -5   1   0   0   1 >
13	26/27		[   1  -3            ... 1 >
17 	4131/4096	[ -12   5            ... 1 >
19 	513/512		[  -9   3            ... 1 >
23 	736/729		[   5  -6            ... 1 >
29 	261/256		[  -8   2            ... 1 >
31 	31/32		[  -5   0            ... 1 >
37 	37/36		[  -2  -2            ... 1 >
41 	82/81		[   1  -4            ... 1 >
43 	129/128		[  -7   1            ... 1 >
47 	47/48		[  -4  -1            ... 1 >
53 	53/54		[  -1  -3            ... 1 >
59 	531/512		[  -9   2            ... 1 >
61 	244/243		[   2  -5            ... 1 >

[Dave Keenan]

Thanks for those. I've used them to update the first post of this topic.

[cmloegcmluin]

Quick question re: style:

JI Notation prescribes that the visual elements representing alterations should be sorted from right to left, largest to smallest. However, the same principle would not be as effective for Prime Factor Notation. The natural thing to do seems rather to be to sort by the size of the primes, from right to left, smallest to largest. This would conform Sagittal to the pattern used by EHEJIPN. I assume this should be the standard, but I noticed this pattern was not followed in the Tonality Diamonds attached in earlier posts (and I am not sure where to find other examples).

Please let me know if anyone disagrees.

[cmloegcmluin]

Ooh, here's another fun question that has arisen in the course of putting what I've learned on this thread into practice:

What should we do for Plain Text Longform when using Prime Factor Sagittal?

I don't think it's a good idea to simply concatenate the symbols, even though ambiguity doesn't look like it should arise regularly, if at all; in fact, the first example I've been able to find is:

(!/|~|(

which could be read as either

(a subharmonic 29, subharmonic 5, and a 17) or

(a 13, a 23, and a subharmonic 7).
But these are weird high-limit pitches, and they don't even respect the suggested ordering-by-size-of-prime principle in my previous post.

Simply adding spaces between the pitches is subpar because they may be desired to delimit separate pitches.

We want to reserve the pairs <>[]{}+- as delimeters, operators, or to represent pairs of non-Sagittal, but Sagittal-compatible, accidentals.

That's something Dave said to me on another thread, so perhaps we don't want to use + or - ...

Any issue with underscores?
In which case we'd have either (!_/|_~|( or (!/_|~_|(

And perhaps a more reasonable example of a 143/128 which would be (!/_/|\

I am definitely open to alternative suggestions.

[Dave Keenan]

Quick question re: style:

JI Notation prescribes that the visual elements representing alterations should be sorted from right to left, largest to smallest. However, the same principle would not be as effective for Prime Factor Notation. The natural thing to do seems rather to be to sort by the size of the primes, from right to left, smallest to largest. This would conform Sagittal to the pattern used by EHEJIPN. I assume this should be the standard, but I noticed this pattern was not followed in the Tonality Diamonds attached in earlier posts (and I am not sure where to find other examples).

Please let me know if anyone disagrees.

@cam.taylor, can you explain what rule you used, if any, for the diamonds you gave earlier in this thread?

In the few examples I give above, I seem to be assuming greater-alteration-closer-to-the-notehead, the same as for the mono-sagittal JI notations.

Is this another user-settable parameter for the calculator, like the choice of 17-comma versus 17-kleisma?
Greater alteration closer to the notehead
versus
Smaller prime closer to the notehead.

There is little reason for EHEJIPN or Johnston notation to consider ordering symbols by size of alteration, because, unlike Sagittal, their symbols give little indication of how big their alteration is.

HEWM however, appears to use "greater alteration closer to the notehead (or nominal)".

We need more images of Sagittal on the staff, not merely in text. Something that would be worth investigating, is how to get Bravura characters, on this forum, to show in everyone's browser, without them having to install the font on their computer. It should then be possible, possibly using the "Bravura Text" font, in conjunction with suitable WinCompose definitions, to type staves and type notes and accidentals onto them at specified positions. It's built into the font somehow. There is info about this somewhere on the SMuFL site. I think there are "combining characters" which, when they follow a note or accidental character, raise or lower it to a specified staff position.

[Dave Keenan]

Ooh, here's another fun question that has arisen in the course of putting what I've learned on this thread into practice:

What should we do for Plain Text Longform when using Prime Factor Sagittal?

I don't think it's a good idea to simply concatenate the symbols, even though ambiguity doesn't look like it should arise regularly, if at all; ...

That's a very good point, that I had not previously considered.

Incidentally, I just learned that, thanks to UTF-8, "plain text" no longer means ASCII (or the 7-bit subset of Unicode). See https://en.wikipedia.org/wiki/Plain_text.

I found it interesting to ask myself how one would characterise that subset without using computer-geek terms like "ASCII" or "7-bit". What is it about that subset that makes us want to limit ourselves to it, in this application. I decided it was mostly because it's the subset that can be typed directly on almost all keyboards. By "directly" I mean with one key press (with or without a shift key). But it's too long to say "the long and short keyboard-character representations" or "the long and short keycaps-character-set representations". I think I'll go back to calling them "the long and short ASCII representations".

Simply adding spaces between the pitches is subpar because they may be desired to delimit separate pitches.

I don't understand what's wrong with using spaces between long ASCII representations of accidentals.

Did you mean to write: "Simply adding spaces between the accidentals is subpar because they may be desired to delimit separate pitches."?

What's wrong with (! /| ~|( E (!/ |~ |( A? It's clear where a pitch ends because of the letter nominal. One could even use a comma without it being confusable with a mo, (! /| ~|( E, (!/ |~ |( A.

Any issue with underscores?

I don't like how it makes them look connected. And it is the short ASCII for .

[cmloegcmluin]

In the few examples I give above, I seem to be assuming greater-alteration-closer-to-the-notehead, the same as for the mono-sagittal JI notations.

Ah, so you are! I should have noticed that.

Is this another user-settable parameter for the calculator, like the choice of 17-comma versus 17-kleisma?
Greater alteration closer to the notehead
versus
Smaller prime closer to the notehead.

Yes, noted.

There is little reason for EHEJIPN or Johnston notation to consider ordering symbols by size of alteration, because, unlike Sagittal, their symbols give little indication of how big their alteration is.

True. And there may be a good benefit to the standard for Sagittal, across all its notations, to be ordering by alteration size, even if it "would not be as effective" (retroactive italicization) for Prime-Factor Sagittal. I may have been a bit hasty to "assume [ordering by prime] should be the standard", dismissing ordering by alteration size as not the "natural thing to do".

HEWM however, appears to use "greater alteration closer to the notehead (or nominal)".

Your link to HEWM is broken in the above. Did you mean to link to http://www.tonalsoft.com/enc/h/hewm.aspx, or somewhere more specific? I've reviewed that page and while indeed the commas are listed in the first table in descending order of size rather than by the prime order, I do not find an example of a pitch altered by more than one symbol at a time such that we could know what HEWM prescribes for this.

We need more images of Sagittal on the staff, not merely in text.

I'm sad because I donated my copy of The Sagittal Songbook to San Francisco's Center for New Music, but I can't get to it now due to the COVID-19 lockdown. It occurred to me this morning that I might have scanned versions of the documents I brought home from Xenharmonic Praxis Summer Camp 2011; unfortunately all the physical versions were lost by the USPS when I moved back from Japan a couple years ago, but I did find some stuff that may be of interest:

XPSC 2011 scanned docs containing Sagittal.pdf

(722.56 KiB) Downloaded 333 times

Of course if one needs to know how to position the Sagittal accidentals, I think there are enough illustrative examples in the Xenharmonikon article to work with.

Something that would be worth investigating, is how to get Bravura characters, on this forum, to show in everyone's browser, without them having to install the font on their computer.

Perhaps if we cross your experience with phpBB and mine with web development, we could figure out a solution. If you have access to a CSS file that gets loaded on the forum, you could add a `@font-face` rule with the `src` property pointing to a `url` on the server for the main site, where we'd be hosting a webfont format of Bravura.

It should then be possible, possibly using the "Bravura Text" font, in conjunction with suitable WinCompose definitions, to type staves and type notes and accidentals onto them at specified positions. It's built into the font somehow. There is info about this somewhere on the SMuFL site. I think there are "combining characters" which, when they follow a note or accidental character, raise or lower it to a specified staff position.

Awesome! This is indeed a thing: https://www.w3.org/2019/03/smufl13/tables/combining-staff-positions.html

(Edit: some of the below statements I since discovered to be wrong. The combining characters and staff positioning seems to work fine in any program as long as you make sure to use the font Bravura Text, not Bravura.)

However, it doesn't work just anywhere. Try those in Open Office or Word and they'll have no effect. I tried them within a text box in my music notation software MuseScore, however, and they worked like a charm. It looks like we're not the only ones struggling with this outside of the appropriate environments. So... it's doubtful even if we did the font available here if we'd be able to make it do such fancy things.

If you're curious, though, if you want to type a )~||E on a staff you can use WinCompose to input by Unicode, in this order:

1. a stave,
2. the combining character to shift upwards by 3 staff positions,
3. the accidental,
4. again the same combining character,
5. and then the notehead

The notehead defaults to centered on B, which does not meet the SMuFL spec: "Noteheads should be positioned as if on the bottom line of the staff", which is funny because that's one thing Word and OpenOffice got right!

So to be absolutely clear that'd be:

1. RightAlt ue014 Enter
2. RightAlt ueb92 Enter
3. RightAlt ue3b0 Enter
4. RightAlt ueb92 Enter
5. RightAlt ue0A4 Enter

Incidentally, I just learned that, thanks to UTF-8, "plain text" no longer means ASCII (or the 7-bit subset of Unicode). See https://en.wikipedia.org/wiki/Plain_text.

I found it interesting to ask myself how one would characterise that subset without using computer-geek terms like "ASCII" or "7-bit". What is it about that subset that makes us want to limit ourselves to it, in this application. I decided it was mostly because it's the subset that can be typed directly on almost all keyboards. By "directly" I mean with one key press (with or without a shift key). But it's too long to say "the long and short keyboard-character representations" or "the long and short keycaps-character-set representations". I think I'll go back to calling them "the long and short ASCII representations".

I hadn't even thought that geekily about it yet. Reflecting on it, I'm realizing that "ASCII", for me, has the strongest associations with "ASCII art" (example here), and that's the only reason why I thought these were called ASCII in the first place: because they were attempts to represent images using only the shapes available in typable characters. So in case anyone else is coming to the table with mainly those associations, it's another reason to prefer ASCII?

Simply adding spaces between the pitches is subpar because they may be desired to delimit separate pitches.

I don't understand what's wrong with using spaces between long ASCII representations of accidentals.

Did you mean to write: "Simply adding spaces between the accidentals is subpar because they may be desired to delimit separate pitches."?

Yes, that was a mistake, sorry.

What's wrong with (! /| ~|( E (!/ |~ |( A? It's clear where a pitch ends because of the letter nominal. One could even use a comma without it being confusable with a mo, (! /| ~|( E, (!/ |~ |( A.

Any issue with underscores?

I don't like how it makes them look connected. And it is the short ASCII for .

Good points. Okay.

In these examples, the first accidental is separated from the nominal by a space too. For consistency, when notating Prime-Factor Sagittal, should this space always be present, even when a note has only a single accidental, e.g. ~|( E? I could see a case for letting the conventional sharp or flat sign stick right to the nominal still, though, e.g. ~|( #E.