## Extending Trojan notation to finer resolution

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### Re: Extending Trojan notation to finer resolution

An interesting issue arose on an email thread between Dave and me, and I thought I'd surface it here:
The other strange thing I noticed while looking at your Yer binary-12R notation, is that you often have components (diacritics or symbols) pointing in opposite directions. This happens on notes 2, 3 8, 9, 10, 14, 15, 16, 17. This shouldn't happen with the binary notation. The fact that it doesn't need to happen, is one of the advantages of a binary notation over say a balanced-trinary notation (which would require fewer symbols per note for similar resolution).
The Yer notation study can be found here.

Information on balanced-trinary notation can be found in the initial post on this thread, here.

I hadn't thought about orienting all the symbols and diacritics in the same direction when trying my hand at this binary notation. I was only striving to minimize the total number of symbols. These aspirations are not always the same.

For example, to notate my pitch 13⋅17⋅19 or 4199/4096 which is a ≈42.996¢ deviation from 12edo, I used three symbols: +50, -6.25, -0.78125, which brings it to 42.96875, which — at only 0.02725 more than just — is within half of the smallest available resolution step 0.390625, so the notation cannot do any better. However, if you prefer all the symbols to be oriented in the same direction then you should instead select +25, +12.5, +3.125, +1.5625, and +0.78125 to arrive at the same approximation, using five total symbols.

Using fewer symbols is simpler, of course, but so is using symbols that are all oriented in the same direction. Should we prescribe one approach over another?

At the moment, I am feeling like I actually prefer orienting the symbols all in the same direction, now that it has been brought to my attention (i.e. even though it wasn't my own instinct when experimenting with the notation). But I'm not sure yet if it is better enough to be worth standardizing.

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### Re: Extending Trojan notation to finer resolution

cmloegcmluin wrote: Tue Apr 14, 2020 1:12 pm I hadn't thought about orienting all the symbols and diacritics in the same direction when trying my hand at this binary notation. I was only striving to minimize the total number of symbols. These aspirations are not always the same.
An excellent point, which I had completely failed to appreciate. Thanks! And I really should have, since it's analogous to what I've done many times in assembly-language (and what optimising compilers do) when multiplying by small constants using shifts and adds or subtracts. e.g. to multiply n by 7 you don't use n<<2 + n<<1 + n, you use n<<3 - n.
Using fewer symbols is simpler, of course, but so is using symbols that are all oriented in the same direction. Should we prescribe one approach over another?
No, I don't think so. I'm happy with just pointing out the advantages and disadvantages of each.

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### Re: Extending Trojan notation to finer resolution

In your initial post here (originally posted to Facebook) you brought up "balanced-trinary" notation:
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.
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.
You also mentioned it in an email thread on the JI Notation Spreadsheet:
The other strange thing I noticed while looking at your Yer binary-12R notation, is that you often have components (diacritics or symbols) pointing in opposite directions. This happens on notes 2, 3 8, 9, 10, 14, 15, 16, 17. This shouldn't happen with the binary notation. The fact that it doesn't need to happen, is one of the advantages of a binary notation over say a balanced-trinary notation (which would require fewer symbols per note for similar resolution).
This technique keeps returning to my thoughts. I think it may be valuable to offer, in addition to Binary 12R, a Trinary 12R:

cents		symbols	tempered size	untempered size	EDO step
±100		 	100.000		113.685 	12
±33.33333333	 	31.174		27.264		36
±11.11111111	 	11.199		24.884		108
±3.703703704	 	2.487		20.082		324
±1.234567901	 	6.698		0.833		972
±0.4115226337	 	4.333		0.423		2916


Points of order:
1. 12 is already composed of 2's and 3's. It therefore feels just as reasonable to break the octave down into smaller parts by 2's or by 3's.
2. 2 and 3 both hold a special place in Sagittal already, being the primes which form the foundations of its commatic intervals.
3. I do not think it is worthwhile to explore notations such as quaternary, quinary, or beyond. I have a hard time believing anyone would crave such things.
4. Between the apotome symbol and the diacritics, all of the symbols are different from Binary 12R. I think that's probably a good thing.
5. I could see an argument for as the symbol for one step of 324EDO. It has tempered size 4.955 and untempered size 14.730.
6. Compared with the Binary 12R notation, more of the symbols' tempered sizes are close to the alteration (as opposed to the untempered size). The one which is closer is indicated in bold, per your advice for the earlier Binary 12R table.
7. I suggest that for simplicity's sake, rather than allowing mixing of binary and trinary divisions such as in your 648EDO example above, we restrict things w/r/t these polysagittal notations to either entirely binary or entirely trinary (i.e., just this notation above, and the one Binary 12R proposed earlier).
8. If you like this suggestion, should we find a name for this pair of notations? The "-ary 12R's"? Oof. I love the word "arity" but I'm not sure it helps us here. I don't think either of us were gung-ho about "Binary 12R" in the first place, so maybe questioning that is on the table.

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### Re: Extending Trojan notation to finer resolution

But I don't like the for 1/324 octave due to the size reversal. And that it's so far from its untempered size.

I considered 1/324 octave as as secondary comma 11:35k (2816:2835)? Tempered size 3.822 ¢. Trouble is, it's visually way too similar to  .

It would be useful to include a column in your tables that says what comma the tempered and untempered sizes are based on, and whether this is the symbol's primary definition.

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### Re: Extending Trojan notation to finer resolution

Dave Keenan wrote: Sun Apr 26, 2020 1:45 pm Not a bad idea.

But I don't like the for 1/324 octave due to the size reversal. And that it's so far from its untempered size.
When you say size reversal, what did you mean exactly? It looks like those symbols are both decreases from in both tempered size and untempered size. But maybe you meant the appearance of the symbol, how its flags take up more space.
I considered 1/324 octave as as secondary comma 11:35k (2816:2835)? Tempered size 3.822 ¢. Trouble is, it's visually way too similar to  .

might be a good candidate, if we switch to using untempered sizes one step earlier, before going into the diacritics; its untempered size is 3.378 (tempered -2.487). Visually it works since it looks smaller than the other symbols.
It would be useful to include a column in your tables that says what comma the tempered and untempered sizes are based on, and whether this is the symbol's primary definition.
Interesting. I had not yet even considered commas other than the primary commas of each symbol. Here's what the original table would have looked like (also correcting my completely wrong information about the schima diacritic):

cents		symbols	tempered size	untempered size	EDO step	comma	primary?
±100		 	100.000		113.685 	12		apotome	yes
±33.33333333	 	31.174		27.264		36		7C	yes
±11.11111111	 	11.199		24.884		108		5:19C	yes
±3.703703704	 	2.487		20.082		324		19C	yes
±1.234567901	 	-13.686		1.954		972		5s	yes
±0.4115226337	 	4.333		0.423		2916		455n	yes


And here's another suggestion, including that insight:

cents		symbols	tempered size	untempered size	EDO step	comma	primary?
±100		 	100.000		113.685 	12		apotome	yes
±33.33333333	 	31.569		29.614		36		3125C	no
±11.11111111	 	11.199		24.884		108		5:19C	yes
±3.703703704	 	3.802		7.712		324		7:25k	no
±1.234567901	 	-13.686		1.954		972		5s	yes
±0.4115226337	 	4.333		0.423		2916		455n	yes


Even though there were symbols with secondary commas that came closer to 33.333 and 11.111 cents, I decided to keep and  , because they are actually used in 36edo and 108edo and it would seem strange not to use them for that same step size here.

You'll see that I went with over my earlier suggestion of  . One thing has going for it is that it symbol-element-wise it is a strict reduction from which is the smallest defined 12R symbol cutting off just about here, at around 4¢.

Finally, re: the name issue:

When I was on that video call with Stephen the other morning, he expressed some confusing about the name "12R". He didn't know when to use "12N" and when to use "12R". He knew that "N" was just a variable/multiple and that "R" stood for "relative", but still found it confusing that both were used. Actually "12N" or "12-n" or any such variation does not even appear in the Xenharmonikon article (although maybe the title of Figure 10 should be "Trojan Symbol Sequences for 12N Equal Divisions"?). But people I think are generally quite familiar with this 12N concept, and so I think that for Stephen, the name "12R" clouded his ability to see it as being a notation should use for anything other than a 12-N EDO (I had to explain to him how I used 12R for a couple different proprietary tunings of mine that were not EDOs, but either also weren't JI or didn't really use fifths, and also weren't going to be performed by specifically microtonal-obsessed players, and that therefore the best notation for me was going to be one relative to 100 cent intervals).

So I wonder if maybe we could consider a name other than 12R. I mean, I think it is important that we encourage people to recognize that "this is the notation where everything is relative to the familiar 12 equal divisions of an octave into 100¢ intervals", but maybe instead of focusing on the 12 equal divisions part, we could focus on the 100¢ part? Like call it "100 Cent Notation" or "100 Cent Relative Notation"? I like that because:
1. it emphasizes how the capture zone diagram for it ranges from 0 to 100.*
2. it simplifies the description of the system from being based on two values (12 divisions of the interval 2/1, to simply 100¢ repeating. Would it be accurate to say it reduces it from a rank-2 to a rank-1 name, then? In any case, it occurs to me that if we ever get around to notating Bohlen-Pierce, it would probably want a stack of 13R EDOs in the same manner as Trojan's 12R EDO stack. But then in one place the R means EDOR and in the other it means EDTR. I'd rather say 100¢R than 12EDOR myself.
How about instead of Binary and Trinary we call it "Bisect" and "Trisect"? The latter, in a single word, also captures the equal-sectioning up of the interval. Unfortunately, they lose the implication of being polysagittal, because "binary" and "trinary" suggest digital systems, which each next digit being one order of magnitude greater. So maybe we should stick to our guns on Binary and Trinary.

Was there something of particular importance to the "balanced" word you've oft included along with "trinary"?

So... a bit weird, but "CCR Notation" (C, Roman numeral for 100; C, cents; R, relative)? Okay I suppose we don't folks to confuse it with Creedence Clearwater Revival notation, though

How about: 100cr Notations, including all 12n EDOs.
Then also we have Binary 100cr Notation and Trinary 100cr Notation.

* Although there is an argument that to be more similar to the capture zone diagram for JI precision levels it should only go up to 50 cents, and provide the Revo flavor of the upper half as a supplement soon thereafter (perhaps same thing for the rest of the EDOs in the paper... that quite confused Stephen as well). But it should probably still be called 100 Cent Notation over 50 Cent Notation, because 50 Cent would not work for this trinary notation.

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### Re: Extending Trojan notation to finer resolution

Sorry I'm not responding to everything in this post. Only the stuff I don't have to think much about.

Re: 12R. One can't expect to pack everything into a short name, but I agree "12R" can easily be read as equivalent to "12N", merely using a different variable, R instead of N, for some unknown reason. So perhaps expand it to "12-rel" or "12-relative".

I'm afraid I don't like any of your other suggested names for this, but I choose not to take the time to explain, at this time. Sorry.

Yes, "trinary" is perhaps better than "ternary". The "balanced" part merely refers to using -1, 0, 1 as the values of the "digits" of the number system, as opposed to using 0, 1, 2. In this context, "balanced" will be assumed, so the word can be omitted.

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### Re: Extending Trojan notation to finer resolution

OK. You win.

I can't come at using "100 cents" in the name because I think it would be even less obvious what we were on about. Almost no one calls (or thinks of) our common scale as "the 100 cent scale". They think of it as the division of the octave into 12 equal parts.

And what is special about a particular apotome-fraction notation, like those implied by the different colours in the periodic table, is not the size of their steps, but the size of their fifths, since the nominals are in a chain of those fifths. Although I suppose you could say it is the size of their apotome/chromatic-semitone. I suppose you could correctly refer to it as the "100 cent apotome notation". But that's getting even further from common parlance. The "100 cent chromatic-semitone scale" is a very long name.

But you could notate some scales relative to a chain of 700 cent fifths in such a way that no 100 cent interval appeared or was implied, because it only required a chain of 6 fifths.

Would you even be tempted to refer to a similar notation, based on say 22-EDO/superpythagorean fifths, or 31-EDO/meantone fifths, by the size of its apotome?

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### Re: Extending Trojan notation to finer resolution

Dave Keenan wrote: Thu May 28, 2020 4:54 pmOK. You win.
"Tell him what he's won!"
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Dave Keenan wrote: Thu May 28, 2020 9:40 am The "balanced" part merely refers to using -1, 0, 1 as the values of the "digits" of the number system, as opposed to using 0, 1, 2.
That makes total sense. I see that I could have simply Google'd it. I must have subconsciously assumed it was a you-ism.
Re: 12R. One can't expect to pack everything into a short name, but I agree "12R" can easily be read as equivalent to "12N", merely using a different variable, R instead of N, for some unknown reason. So perhaps expand it to "12-rel" or "12-relative".
I like 12-Relative. I think that solves the problem. Not that I would discourage anyone from referring to it as 12R for short. But perhaps the Xenharmonikon article, on p. 18, could simply introduce it as 12-Relative, without the abbreviation and parenthetical.
Dave Keenan wrote: Thu May 28, 2020 4:54 pm But you could notate some scales relative to a chain of 700 cent fifths in such a way that no 100 cent interval appeared or were implied, because it only required a chain of 6 fifths.
You're right. I see now that while once you iterate the chain of fifths past 12 generations the structure can be collapsed to rank-1, before that point, it's still rank-2, and thus 12-Relative is the superior name.

Thanks for keeping your finger on the pulse of composers. I think you are probably right that if we held a survey "do you compose on a 100 cent lattice" or "do you compose on 12 equal divisions of the octave"? the EDOs would have it. They're the ones with an acronym after all.

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### Re: Extending Trojan notation to finer resolution

And do let me know what you think of the symbol choices for the Trinary 12-Relative someday, but no rush at all.

Dave Keenan