## 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.

cmloegcmluin
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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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Joined: Tue Sep 01, 2015 2:59 pm
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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.