## Extending Trojan notation to finer resolution

Dave Keenan
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### Extending Trojan notation to finer resolution

This began with the following new thread post by Cmloegcmluin Xenharmonic Feisbeuk on facebook.

I've moved it here so that (a) it doesn't get lost, and (b) in subsequent posts I can show the actual symbols by using the forum "smilies".
Cmloegcmluin Xenharmonic Feisbeuk
Can anyone recommend a notation system to me like Maneri-Sims, but with higher resolution? 16.67¢ doesn't approximate my pitches closely enough.

I am comfortable with a couple JI notation systems, but the piece I'm working on is not JI. If I were to notate it with a JI notation system, there'd be a couple problems:
1) it would connote JI, which is not how I'm thinking about it, and I don't want the performers to interpret it that way, and
2) it's unnecessarily complex to represent my pitches with the JI symbols.

My pitch system is octave-repeating, so all I'm looking for really is a (12n)edo where 12n > 72, and an associated set of accidentals. 12n is because I'm still notating this on standard Western sheet music so the accidentals will be deviations from 12edo. I suppose it'd be ideal if one step of this (12n)edo was < 5¢.

If it doesn't exist, I would be surprised, but I can design it, if someone might suggest an ideal (12n)edo to use? For example, I know that savarts are precedented (300edo). Of course 1200edo might be a good choice, with accidentals for +/- 50, +/- 25, +/- 10, +/- 5, and +/- 1 cent.
Paul Erlich
You can use the Sagittal notation system for say 612-tET.

15 replies
• Cmloegcmluin Xenharmonic Feisbeuk
I feel like Sagittal connotes JI, since it is based on comma alterations

Paul Erlich
Well it often deliberately conflates ratios, useful for temperaments.

Cmloegcmluin Xenharmonic Feisbeuk
I'll have to look into Sagittal more closely now. thanks for the suggestion.

Cmloegcmluin Xenharmonic Feisbeuk
whoa! I really put off delving into Sagittal for too long.

I *still kinda* feel like it connotes JI, but it doesn't bother me enough not to use it. I know it's popular. I really do not want to proliferate standards.

do you know where I'd find symbol sequences up to the apotome for ed's like 612 and 624? I don't see those in the main pdf... only seems to go up to 224 there.

Paul Erlich
Perhaps Dave Keenan or George Secor could help.

Dave Keenan
Hi Cmloegcmluin. The Sagittal system has a notation intended for the purpose you describe. It's called the Trojan notation. It is a superset notation for all 12n-edo notations up to 240-edo. So it has a resolution of 5 cents. This resolution can be improved by using accent marks to raise and lower by 2 cents.

See the first 2 paragraphs and the footnote on page 18, and see Figure 10 on page 19 of http://sagittal.org/sagittal.pdf

Sagittal has notations for 612-edo and 624-edo, but these are not based on a 700¢ fifth, but rather the near-just fifth of those edos. They are notated using essentially the same symbols as the Herculean JI notation, which can be seen here:
http://sagittal.org/SagittalJI.gif

Dave Keenan
Cmloegcmluin, Figure 10 shows only the pure-Sagittal version. The double-shaft symbols on the right-hand side can be replaced with a combination of conventional sharp symbol and the downward version of the symmetrically-opposite sagittal symbol.

Also, Trojan as described, is one-sagittal-symbol per alteration. But there's nothing to stop you making a multi-sagittal version of it, by using only say the 50¢, 20¢, 10¢ and 5¢ symbols and the 2¢ accent.

Dave Keenan
You're also welcome to use the Stein-Zimmerman semi-sharp and semi-flat symbols for your 50¢ symbols, in combination with Sagittals. See Figure 1 on page 3 of http://sagittal.org/sagittal.pdf

Dave Keenan
Instead of symbols for 50¢, 20¢, 10¢, 5¢, it might be better to use the symbols for 8, 4, 2, 1 degrees of 192-edo, and use the accent as 0.5 degrees. i.e. quarter-tone, eighth-tone, sixteenth tone etc. That way you never have to double-up a symbol, and you never need symbols pointing in opposite directions.

Cmloegcmluin Xenharmonic Feisbeuk
Dave Keenan thank you so much for your guidance (and of course for the work of developing Sagittal in the first place!)

I feel a bit embarrassed because on page 18 the Sagittal paper basically starts hand-feeding me exactly what I was looking for... I think I must have run out of steam or gotten distracted by the diagrams on page 16 and 17. Sorry!

I think I will use Stein-Zimmerman just for sharps and flats, and Sagittal for anything that deviates from 12tet.

I also think I'll use the conventional sharps and flats and the downward or upward versions of the smaller absolute value Sagittal accidentals, just so my performers have half as many symbols to learn.

192edo will suffice for me for now, since with the accent mark for a 0.5 degree I get 3.125¢ precision. But as long as I'm investing in a notation system, I'd like to see how much more precision I can get for what cost. If I'm understanding correctly, if 612edo and 624edo are not based on a 700¢ fifth then they won't work for Trojan, i.e. won't be in terms of deviations from 12edo. So I wouldn't want that. Is there any higher edo than 192 that still uses the 700¢ fifth, and if so, can you refer me to the symbol sequence for it? On another thread I think I understand you're saying Trojan is extensible to 720-edo, but I'm not sure how to extend it myself.

Dave Keenan
Cmloegcmluin 720-edo can be notated based on 12-edo by using a Trojan notation for 240-edo and using the accent to raise or lower by 1/3 of a degree of 240-edo (1.67c). You can think of the various options in terms of number systems with possibly-mixed bases.

The 384-edo (192×2) notation described above is conceptually simple because it is strictly binary, because 192-edo is 12×16 where 16 = 2×2×2×2, requiring a maximum of 4 arrow-like "digits". Then we use the accent as an additional binary digit. So we have 2×2×2×2×2. But it only gets you to 3.125¢ resolution.

720-edo gets you the best 12-edo-based resolution possible in sagittal (without using an additional type of accent mark) but it's a lot more messy as it involves 240 = 12×20 where 20 = 5×2×2 with the accent as a balanced-trinary digit (-1, 0, +1) giving 5×2×2×3.

The 5 can be factored into 2×2.5 to give the familiar 1:2:5 decimal currency basis. And the accent can be used as a base-2.5 digit corresponding to 2¢, giving us 600-edo resolution. I note that the U.S. 25¢ coin is something of an aberration among decimal currency systems. Most have 20¢ coins instead. Even the US banknotes go 1 2 5 10 20 50 100. So for the 600-edo system we'd have symbols for 2 5 10 20 50 cents.

Another option would be 648-edo which maximises the use of balanced-trinary digits. This would be 3×216-edo where 216 = 12×18 and 18 = 2x3x3 then the accent would be another ×3. But it can be difficult to mentally compute the result of a bunch of different sized things pulling in opposite directions.

Dave Keenan
We could notate 612-edo and 624-edo relative to 12-edo, as 17×3 and 13×2×2. But since you're not going for JI approximations, there seems little point.

Cmloegcmluin Xenharmonic Feisbeuk

Dave How about the next power of 2, 768-edo? Has that symbol sequence been documented?

Dave Keenan
768-edo could be notated relative to 12-edo by adding the mina diacritic to the 384-edo notation which, as described above, adds the schisma diacritic to the 192-edo notation. The mina (pronounced "meena") diacritic (Olympian extension) is not yet implemented in the Bravura font. However you could simply use a scaled down version of the schisma diacritic (acute and grave). Say 1/2 or 2/3 of the point size. Whatever seems the best compromise between being distinct in size and being visible.

You can see what the real mina diacritic is intended to look like, here: viewtopic.php?f=10&t=254&p=587...

Cmloegcmluin Xenharmonic Feisbeuk
Dave Keenan thanks! I don’t understand the system well enough yet to figure what the final result of that insertion of schisma and mina would look like (I don’t know where to add them). Sorry to be a dunce but it’d be awesome if we just had these somewhere with their corresponding cents values. I’m happy to help make it presentable for the benefit of others if someone can provide the raw data.

Dave Keenan
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### Re: Extending Trojan notation to finer resolution

Here's my proposed multi-Trojan 12-relative 768-edo notation.

Symbol	Degrees	Cents	Tones	Unicode in
up down	of 768			SMuFL fonts
---------------------------------------------
64	100	1/2	U+E262 U+E260
32	50	1/4	U+E30A U+E30B
16	25	1/8	U+E370 U+E371
8	12	1/16	U+E302 U+E303
4	6.2	1/32	U+E300 U+E301 (substituted for the official Trojan  )
2	3.1	1/64	U+E3F2 U+E3F3
1	1.6	1/128	U+E3F4 U+E3F5 (not in Bravura font. Use U+E3F2 U+E3F3 at 2/3 point size)


Symbols should be arranged leftward from the notehead in order of decreasing alteration.
o

If there are no sagittal symbols other than diacritics, then the relevant bare shaft symbol should be used to the right of but not touching them. U+E34F0 (up) or U+E3F1 (down)
o

Dave Keenan
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### Re: Extending Trojan notation to finer resolution

Dave Keenan
Cmloegcmluin, I have no problem with cents. You could even do both. i.e. Sagittal Trojan symbols /on/ the staff, with the cents for any such symbols stacked /above/ the staff.

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### Re: Extending Trojan notation to finer resolution

This is awesome Dave. Thank you so much.

Oy, it's been quite a whirlwind the past few months for me.

I taught myself Johnston notation in a single day in a last-ditch stunt for a competition. I think I started with the absolute hardest notation system, because looking back on it now I find Johnston so unnecessarily complex, at least for the type of music I'm writing.

So I started using HEWM because I'd heard a lot about it too and folks recommended it on Facebook. And that worked fine but not so well for some really precisely tuned stuff of mine that either a) isn't JI or b) spirals so far from the original pitch that the numerators and denominators are in the trillions and the JI symbols become impractical. For that I need something else. And I really wanted to settle on a single notation system for all of my scores.

This Trojan style will work for my stuff that isn't simpler JI. And the simpler JI stuff will be served by other Sagittal sets.

I've known about Sagittal since 2011 when I attended the first Xenharmonic Praxis Summer Camp. But I don't perform myself, and I didn't start trying to get my xenharmonic performed until very recently. Having gone over the paper several times in the past couple days, I have to say that the theory behind this xenharmonic notation system resonates with me the most, and that of course is way more important!!!

Dave Keenan
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### Re: Extending Trojan notation to finer resolution

@cmloegcmluin, You certainly have been prolific on topics xenharmonic recently.

I am glad that you like the theory behind the Sagittal system. As you have probably read, it was designed to notate both EDOs and JI, right from the beginning, and was designed with the help of an online community of xenharmonists.

Like you, we always wanted to see how far we could push it, in all kinds of directions. And every time we did, we would extend or optimise some aspect of the system. We might change the choice of which commas to symbolise, or what flag combinations to use, or what the approximate untempered size of each flag was in cents. But we had a strict rule that whenever we changed the system to deal with some more complex class of tunings, say larger EDOs or higher prime limits, these changes could never "reach back" and make it more complicated to notate the simpler tunings we had already dealt with. It's not that we couldn't change how they were dealt with, only that the cognitive load must remain the same or become less. There was a ratchet that could only go down.

Eventually the system stabilised, and only then did we publish.

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### Re: Extending Trojan notation to finer resolution

@Dave Keenan do you happen to have available higher resolution images of mi and mo, the up and down versions of the 1\768edo step symbol, ~1.6 cents?

I couldn't find it in here:
http://sagittal.org/Sagittal2_character_map.pdf

I installed the Sagittal font but can't figure out if there's instructions for how to use it anywhere, i.e. which keys map to which symbols.

The "smilies" on this site have it, but it's not very detailed - only 4 pixels.

I want to use it in my score, but the workaround you suggested given its absence from the Bravura font is not desirable, since the workaround is to shrink to 2/3 size the symbol for 2\768, but I am also using the symbol for 2\768, and I think it's not distinct enough from it.

Thanks for your attention to this!

Dave Keenan
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### Re: Extending Trojan notation to finer resolution

cmloegcmluin wrote: Fri Mar 06, 2020 1:10 pm @Dave Keenan do you happen to have available higher resolution images of mi and mo, the up and down versions of the 1\768edo step symbol, ~1.6 cents?
They aren't in the Sagittal font either (which is now obsolete). It's way past time we should have made an outline font version of them. I assume you've seen, in the SMuFL document, that there are codepoints reserved for them. I don't suppose you'd be interested in adding them (and maybe the magrathean diacritics) to a copy of the Bravura font? The free font editor FontForge, by George Williams is a masterpiece of good user-interface design. I once described it to a colleague as follows:
FontForge is just a joy to use. Everything works just as you expect it to. The program's model is intuitive. The help is excellent. And it costs — nothing. When using FontForge, one has the sense that a single mind finely crafted the design before finely crafting the implementation. And it looks beautiful. George Williams is my hero.
One way to understand the intended form of these symbols is that conceptually they are miniature versions of without their shafts. But to see their intended alignment relative to the staff lines, see this post:
viewtopic.php?p=814#p814
The 1 mina symbols are identical to the 3 tina symbols. Unfortunately it shows only a slightly higher resolution (1.33 times that of the smiley version). This time they consist of 5 pixels  — that's when the vertical distance between staff-line centres is 8 pixels.

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### Re: Extending Trojan notation to finer resolution

Thanks so much Dave. Using FontForge, along your explanation of the symbol (assuming I identified which symbol you said it was a shaftless version of correctly... it does seem like there are very many Sagittal symbols which look quite similar to each other), I was able to get the mark working in my score! I had a harder time figuring out how to get MuseScore to just find my new font (turned out I had to "install for all users"... sheesh) than I did with everything else combined; FontForge indeed was a pleasure to work with.

As for the magrathean diacritics, that might be a bit out of scope for me, especially without a bit more knowledge about Sagittal and associated domain knowledge. I still feel very much like I've stumbled across some ancient civilization when I try to make sense of posts on this forum.

Dave Keenan
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### Re: Extending Trojan notation to finer resolution

cmloegcmluin wrote: Sat Mar 07, 2020 3:22 pm Thanks so much Dave. Using FontForge, along your explanation of the symbol ..., I was able to get the mark working in my score!
That was quick. Well done! Can you send me a copy of your modified Bravura font. I'd like to check your work, and if satisfactory, submit it to Steinberg for inclusion in the next version of Bravura. I sent you a private message on this forum, with my email address. I realise the font file is quite large even when zipped (83 MiB or 87 MB). I have temporarily increased the attachment size limit for this forum, so you could upload it as an attachment to your next post in this topic, if you wish.
assuming I identified which symbol you said it was a shaftless version of correctly...
Sorry. I should have said that the referenced symbols are at U+E308 and U+E309, and the mi and mo diacritics should be placed at U+E3F4 (upward) and U+E3F5 (downward). I hope you found those.
it does seem like there are very many Sagittal symbols which look quite similar to each other
Yes. I should have warned you. Only the bravest warriors should be allowed to gaze upon the entire Sagittal symbol set. Lesser men and women have been known to run screaming from the sight. It's almost inevitable that a system capable of notating 52 different pitches within a single chromatic semitone without using diacritics (1618 with diacritics), will have symbols which are similar to each other. But within any one tuning, the small set of symbols required can be sufficiently distinct. It's a bit like having a palette of 52 colours. Some are bound to look very similar, but if you need less than 8 colours in any particular application, they can be quite distinct.

FontForge indeed was a pleasure to work with.
George Williams should get some kind of award for it.
As for the magrathean diacritics, that might be a bit out of scope for me, especially without a bit more knowledge about Sagittal and associated domain knowledge.
Fair enough. I may complete them myself, following your lead, before submission to Steinberg.
I still feel very much like I've stumbled across some ancient civilization when I try to make sense of posts on this forum.
We do like our ancient Greek mythology, and there is a strange similarity between many of the Sagittal symbols and the symbols on the Anaphorian Tablet of Celudore,

http://anaphoria.com/musinst.html

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### Re: Extending Trojan notation to finer resolution

I spent the weekend diving deep into Trojan notation and EDOs. Just 12 days ago I still believed they were the same thing. Now I look upon the periodic table and it actually maybe makes at least a little bit of sense!

As I was studying, it occurred to me -- wait a second, that one piece of mine I put into some form of Sagittal per Dave's easy instructions -- I think I ended up with multiple sagittals per note -- isn't that not how Trojan works -- isn't that only how Prime Factor Sagittal works? So I came back to this thread to review.

Indeed, you proposed this as a "multi-Trojan" notation! (And of course on the Facebook thread you say "Also, Trojan as described, is one-sagittal-symbol per alteration. But there's nothing to stop you making a multi-sagittal version of it, by using only say the 50¢, 20¢, 10¢ and 5¢ symbols and the 2¢ accent.")

I must say it was delightful for me to reread this thread. It's hard to believe that was 7 weeks ago. Feels both really short and really long. At the time I had no idea what you were talking about for the most part, and clearly my attitude was "please just tell me what to do". But now I totally get it.

As far as I can find, this is the only place where such a notation has been proposed or worked with. And it's distinct from the previously existing four Sagittal notation styles* (JI, Trojan, EDO, Prime Factor).

So, first of all, thank you for going to that length to find a Sagittal that worked for my use case!

I personally think that this new style is worth keeping and maintaining.

I'm also excited to say that I think I know what I'm doing enough now to hold some opinions, and maybe even suggest modifying your original proposal.

My motivation for this suggestion is that I'd like this notation to support even better accuracy; I want it to compare to Olympian. Your original suggestion capped out at 1.6 cent resolution. But I think with some tweaks we can make a logical notation that is accurate to a half cent.

Here's what I'm thinking: if we're devising a new distinct style, must we bind ourselves to the Trojan capture zones? We're using to represent a 25 cent alteration, while the default value of in JI notation is 16.544 cents. Why not just use the symbols which are a good balance of close to the alteration, and simple? Don't worry about the flag arithmetic or anything like that... this is the notation for dummies!

cents     symbol    default value
100                113.685
50                  53.273
25                   27.264
12.5                 9.688
6.25                 5.758
3.125                3.378
1.5625               1.954
0.78125              0.833
0.390625             0.423


This way you also get a kind of tiered effect where all those with barbs are together, then arcs, then scrolls, then accents, then breves. From here we could even dig into the Magratheans.

Just because its also 12R doesn't mean we have to call it Trojan either. And maybe since we just renamed Prime Factor Sagittal away from Multi-Sagittal (to avoid potential confusion with multi-shaft symbols) we should question the "Multi-" part too. I'd like the name to encode both its relationship to Trojan and its relationship to Prime Factor, if possible. The best name that occurs to me so far is "12R Binary Notation", which doesn't associate it well with Prime Factor. However, maybe we could make a point of referring to both Prime Factor and 12R Binary notations as "polysymbolic" notations.

* Would "style" be the best term here?