## Extending Trojan notation to finer resolution

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

cmloegcmluin wrote: Tue Mar 31, 2020 3:57 am 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).
That is correct. And yes "style" seems like a reasonable term for these. But I note that JI, Trojan and EDO are, in a sense, one style. That's the style where, for fifths in a small range of sizes (2 to 4 cents wide) we define a set of symbols, and capture zones for them, based on the tempered size of the symbols' commas. Although we haven't literally done that (yet) for fifth sizes other than Pythagorean (JI) or 700 cents (Trojan).
See viewtopic.php?p=843#p843 and viewtopic.php?p=807#p807.
So, first of all, thank you for going to that length to find a Sagittal that worked for my use case!
You're welcome.
I personally think that this new style is worth keeping and maintaining.
Agreed.
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.
Definitely worth a try.
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!
You need to re-read page 13 of http://sagittal.org/sagittal.pdf from the section heading, and the last paragraph on page 15, and the first paragraph on page 23, and the third paragraph of http://sagittal.org/whatpitch.txt.

When the fifths are not pure (Pythagorean), the untempered size of the default comma should be irrelevant. It's the tempered size that matters. The 700c fifth is 1.955c narrower than a pure fifth, so the tempered size in cents, of a comma with a 3-exponent of n differs from its untempered (JI) size by n × 1.955c. For example, the tempered apotome is 113.685 - 7×1.955 = 100 cents. So that's the size of in 700c land.

I did violate this for the diacritics, in my binary 12-relative proposal, because I could see no other way to get fine enough resolution. But this principle should not be violated unless there is a very good reason. But when you are forced to violate it, It makes sense to fall back to the untempered size of symbols or diacritics. For that reason, I find your use of the diacritics preferable to mine, provided we can justify the use of a symbol for 3.125 cents.

I also violated it for my 6.25c symbol, because I thought the standard Trojan for that was visually too big and complicated. I expect George would not have approved of that, unless we could come up with a secondary comma for that symbol whose tempered value was in the right ballpark. In any case, because I've done it for 6.25 cents, I can't complain that you've done it for 3.125 cents.

But I see no reason not to use symbols with approximately the correct tempered size (the standard Trojans), for 100 cents down to 12.5 cents. So it should at worst be:

cents	symbol	tempered size	untempered size
100		100		113.685
50				53.273
25
12.5
6.25				5.758
3.125				3.378
1.5625				1.954
0.78125				0.833
0.390625 			0.423


You might fill in the blanks in the above, for homework.
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.
"12R binary" or "binary 12R" is good. But I note that any mixed notation could be considered "polysymbolic" as it may have a sharp or flat and a sagittal.

cmloegmcluin
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### Re: Extending Trojan notation to finer resolution

When the fifths are not pure (Pythagorean), the untempered size of the default comma should be irrelevant. It's the tempered size that matters.
This was all I needed to read and I immediately knew what I had done wrong. Of course, in working on notating 137edo, I did just this sort of thing — consider the tempered values of Sagittal symbols' commas instead of their just values. Thank you for correcting me.

It's interesting because as far as I can tell, we're only tempering the 3. For 137edo I didn't notice a problem because it's a big enough edo that the commas were in a 0-8 step range. But for 12edo a good number of them were tempered out completely to 0 steps and then you can't compare which of them are more accurate representations of apotome fractions. It makes sense to only temper the 3, but it hadn't occurred to me.
"12R binary" or "binary 12R" is good. But I note that any mixed notation could be considered "polysymbolic" as it may have a sharp or flat and a sagittal.
Well I noticed on another thread recently you threw out the word "monosagittal". So how about "monosagittal" and "polysaggital"?
JI, Trojan and EDO are, in a sense, one style.
Fair. Upon further review, I don't feel like I need a word anymore for something like "style". "Notation" is fine. Sagittal is a notation system with many notations.
I expect George would not have approved of that, unless we could come up with a secondary comma for that symbol whose tempered value was in the right ballpark.
gets tempered to 3.802 cents, if that's close enough to 3.125?
You might fill in the blanks in the above, for homework.
cents	symbol	tempered size	untempered size
100		100.000          113.685
50	        51.318           53.273
25	        28.274           16.544
12.5	        13.686           21.506
6.25		 17.488		  5.758
3.125		 -2.487            3.378
1.5625	       -13.686           1.954
0.78125	        6.698             0.833
0.390625 	 4.333             0.423


There seem to be some commas that are even closer to the 100*2^-n cents points in this 3-tempered world, however, I suppose symbol simplicity and flag arithmetic are still important considerations, even if this is kind of a notation for dummies.

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

cmloegcmluin wrote: Wed Apr 01, 2020 6:17 am It's interesting because as far as I can tell, we're only tempering the 3. ... It makes sense to only temper the 3, but it hadn't occurred to me.
Right. In this case we aren't dealing with any particular EDO, we just know we have 700c fifths, so 3's are all we can temper. I note that if we were coming up with a notation for Bohlen Peirce, we might choose to temper the 2's.
Well I noticed on another thread recently you threw out the word "monosagittal". So how about "monosagittal" and "polysaggital"?
Sure. So both the prime factor notation and the binary 12R notation are polysagittal. And "monosymbolic", or "monoaccidental", might be a synonym for "pure".
I expect George would not have approved of that, unless we could come up with a secondary comma for that symbol whose tempered value was in the right ballpark.
gets tempered to 3.802 cents, if that's close enough to 3.125?
I see is 7:25k. At first I wrote: "That assuages my guilt nicely. Thanks. ". But then I went to add the note: "(as 7:25k)" to the appropriate row of (a copy of) the table above, and realised that we're using as 6.25c, not 3.125c.
There seem to be some commas that are even closer to the 100*2^-n cents points in this 3-tempered world, however, I suppose symbol simplicity and flag arithmetic are still important considerations, even if this is kind of a notation for dummies.
With the above set, we're lucky that flag arithmetic doesn't enter into it, because there is no symbol that can be made by combining flags from other symbols. Most are mono-flagular. I mean single-flag.

I don't understand what you mean when you say it's a notation for dummies.

cmloegmcluin
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### Re: Extending Trojan notation to finer resolution

Right. In this case we aren't dealing with any particular EDO, we just know we have 700c fifths, so 3's are all we can temper.
Ah, okay. So when I calculated the notation for 137, I was correct to temper all the commas by its patent val. Got it.
I note that if we were coming up with a notation for Bohlen Peirce, we might choose to temper the 2's.
I'll have that one on the backburner. I've only got a couple remaining posts planned for this forum in "learning / prepping for building the calculator mode", and the very last one is about non-octave tunings, especially BP. Before you wrote that, in fact, Bohlen-Pierce had not come up once on this forum, and very few references to BP in Sagittal occur anywhere online.
Sure. So both the prime factor notation and the binary 12R notation are polysagittal. And "monosymbolic", or "monoaccidental", might be a synonym for "pure".
I like monosymbolic. So we have:

Notation			-sagittal	-symbolic
JI, Trojan, EDO (Pure)		mono		mono
JI, Trojan, EDO (Mixed)		mono		poly
Prime Factor, Binary 12R	poly		poly


I think it goes without saying that we're not considering monosymbolic and polysymbolic as replacements for pure and mixed.

(aside: having discovered I can paste a tab character in and then copy and paste it within this window, I feel I've leveled up a belt color in my table-fu! Especially with the smilies disrespecting and thus disrupting their monospaced environs...)
we're using as 6.25c, not 3.125c

Sorry for mistakes due to rushing.
Indeed there is no secondary comma for in the correct ballpark. Amusingly, gets sent to 6.458c, so we could be justified in switching the two. I know you seek to minimize size-order reversals, but when it's on the table, what are the criteria for deciding whether to do so? If it helps, I don't see showing up in other EDO or Trojan notations, so it wouldn't confuse anyone with respect to that.

cents		symbol	tempered size	untempered size		justifying 2ary comma
100			100.000		113.685
50			51.318		53.273
25			28.274		16.544
12.5			13.686		21.506
6.25			-2.487		3.378			 (6.458)
3.125			17.488		5.758			 (3.802)
1.5625			-13.686		1.954
0.78125			6.698		0.833
0.390625		4.333		0.423

mono-flagular
Careful, if you make linguistic jests I'm going to take them as at least partly seriously...
"Monosimaiac"? "Monostocheic"?
...Single-flag it is.
I don't understand what you mean when you say it's a notation for dummies.
I was only thinking of myself a couple months ago knowing basically nothing about Sagittal. I knew I could throw a on a note to give it +100 cents. I was just looking for other things I pile onto the notes to zero them in on some rather arbitrary cents deviations that bore no relation to JI, circles of fifths, EDOs, etc. Binary 12R is certainly less elegant than JI, EDO, and Trojan notations, but "for dummies" is not something I'm compelled to propagate or sell is as.

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

cmloegcmluin wrote: Thu Apr 02, 2020 1:58 am I like monosymbolic. So we have:

Notation			-sagittal	-symbolic
JI, Trojan, EDO (Pure)		mono		mono
JI, Trojan, EDO (Mixed)		mono		poly
Prime Factor, Binary 12R	poly		poly

Agreed.
I think it goes without saying that we're not considering monosymbolic and polysymbolic as replacements for pure and mixed.
Agreed.
(aside: having discovered I can paste a tab character in and then copy and paste it within this window, I feel I've leveled up a belt color in my table-fu! Especially with the smilies disrespecting and thus disrupting their monospaced environs...)
Well done.
Indeed there is no secondary comma for in the correct ballpark. Amusingly, gets sent to 6.458c, so we could be justified in switching the two. I know you seek to minimize size-order reversals, but when it's on the table, what are the criteria for deciding whether to do so? If it helps, I don't see showing up in other EDO or Trojan notations, so it wouldn't confuse anyone with respect to that.
I really don't like it. It wouldn't be so bad if they weren't both scrolls, and therefore near-mirror-images, and therefore easily confused.

I was thinking what a great mnemonic it was that the single-flag symbols alternated left/right in size order.
mono-flagular
Careful, if you make linguistic jests I'm going to take them as at least partly seriously...
"Monosimaiac"? "Monostocheic"?
...Single-flag it is.
The etymology of the above escapes me. But "monovexillar" ...

cmloegmcluin
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### Re: Extending Trojan notation to finer resolution

Good points. We won't switch those two's size-order then.

Another random idea: what if, in order to reduce the count of symbols required to use this notation, we allow more symbols for the three-quarters-of-the-way positions. These symbols are valid flag combinations:

cents				symbol	tempered size	untempered size
37.5   = 25.0  + 12.5			41.961		38.051
9.375 =  6.25 +  3.125			17.508		9.688


The only other valid combination of these flags is which would be 53.125c; untempered it's 56.482c and tempered it's 48.662c, which is not a three-quarter position but a 17/32nd which is really weird.

Of course if we wanted to make a new glyph for /|( that could be handy. In fact, without it, I'm not sure this proposal is worth considering.
I was thinking what a great mnemonic it was that the single-flag symbols alternated left/right in size order.
I had noticed that too, yes. Worth preserving.
monovexillar
I hope your wink emoji is because it bothers you as much as it does me to mix Greek and Latin roots. In Greek "simaia" is flag and and "stoicheia" is element (I missed an "i" in the previous post, oops) but I definitely didn't know that before looking it up and I wouldn't expect any non-Greeks to recognize those either.

I am winding up for a big post on the forum about terminology, by the way, where we can have a more practical discussion of glyphs, elements, symbols, accidentals, etc.

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

cmloegcmluin wrote: Fri Apr 03, 2020 1:49 am Good points. We won't switch those two's size-order then.

Another random idea: what if, in order to reduce the count of symbols required to use this notation, we allow more symbols for the three-quarters-of-the-way positions. These symbols are valid flag combinations:

cents				symbol	tempered size	untempered size
37.5   = 25.0  + 12.5			41.961		38.051
9.375 =  6.25 +  3.125			17.508		9.688

Yes. I had noticed that too.
The only other valid combination of these flags is which would be 53.125c; untempered it's 56.482c and tempered it's 48.662c, which is not a three-quarter position but a 17/32nd which is really weird.
Yeah. Forget that.
Of course if we wanted to make a new glyph for /|( that could be handy. In fact, without it, I'm not sure this proposal is worth considering.
That could really make the system untidy. I'd have to see some serious interest in this notation before I'd even consider it. When you think about what comma it is, based on sum-of-flags, it's the 7-comma, same as . Notation software usually allows fine positioning of accidentals, so you could overlay the shafts of and .
monovexillar
I hope your wink emoji is because it bothers you as much as it does me to mix Greek and Latin roots.
Goodness me no. I hadn't noticed. Oh the shame of it.

I could weasel and point out that "vexillar" is English (but from the Latin, as you say). But no, you are right. It must be "univexillar".
I am winding up for a big post on the forum about terminology, by the way, where we can have a more practical discussion of glyphs, elements, symbols, accidentals, etc.
Cool!

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### Re: Extending Trojan notation to finer resolution

Right then! So our final proposal for Binary 12R is:

cents		symbol	tempered size	untempered size		EDO step
100			100.000		113.685 		12
50			51.318		53.273			24
25			28.274		16.544			48
12.5			13.686		21.506			96
6.25			17.488		5.758			192
3.125			-2.487		3.378			384
1.5625			-13.686		1.954			768
0.78125			6.698		0.833			1536
0.390625		4.333		0.423			3072


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

Right. But I suggest you edit the table above, to embolden the tempered size for the first four, and the untempered size for the last 5.

cmloegmcluin
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Good idea. Done.