FloraC wrote: ↑Thu Nov 12, 2020 3:37 am
@cmloegcmluin Thanks! You answered a lot of my questions.

Good to hear.

Dave has emailed me the solution in ascii which is really difficult to read. Nice to see the graphical form.

Yeah, the graphical form, or "smilies" as we call them, are a game-changer. That's why we're always trying to get people to move Sagittal-related conversations over from Facebook or wherever to the fourm.

I totally empathize with the ASCII form of the Sagittal symbols being just One More Thing to work through. Once you get used to them though, you start to like them. It's pretty remarkable just how well Sagittal symbols can be evoked using what anyone can type on a keyboard already (as opposed to some other microtonal notations). If you haven't found it yet, the

Sagittal-SMuFL-Map is the canonical reference at this time for the ASCII form (and much more!).

Yeah I checked

the diagram, but some commas are too complex to approach, such as the

= 45927/45056. I didn't have any idea how I could find that ratio. Now I see it's the apotome-complement of 22/21.

Correction:

is 45/44, which is far less intimidating. It's actually the

which is 45927/45056. It's the difference of the presence or absence of a right scroll. In ASCII they're (|( and (| respectively.

But yeah, some of those ratios look surprisingly complex at first. The ratios for the Athenian symbols go:

5120/5103
896/891
4131/4096
81/80
64/63
45927/45056
45/44
6561/6400
36/35
33/32

But if you remove all the 2's and 3's from them, they look less lopsided:

7/5
11/7
17/1
5/1
7/1
11/7
11/5
25/1
35/1
11/1

And for notation purposes, those're what matter more, and those are the quotients you'll find in the names of the commas, which are also in that chart. Let's go through a concrete example. I'm doing this as much for myself as I am doing it for you, honestly, since I was actually just yesterday resolving my own lack of clarity regarding this concept which is pretty fundamental to Sagittal.

So all that 45927/45056 really is is the pitch you'd get if you applied the

symbol to the Pythagorean nominal (A,B,C,D,E,F,G) which was your 1/1. Let's go with D for this example. So

D is 45927/45056. But that's only 1 out of 56 different ratios in the octave which this accent gives you the power to notate! Why 56? Let's break that number down.

We've got 35 different pitches we can get to already using only 2's and 3's:

F through

B. and for each of those pitches we can apply the accent upwards

or downwards

, so that's 35×2 = 70 pitches. Well, it

*would* be 70, were it not for the fact that Sagittal doesn't go beyond two sharps/flats/apotomes (chromatic semitones) away from the nominal. I've referred to it fondly as "the edge of the world". So actually 14 of those 35 you can only notate in one direction from, namely, back toward the natural, whichever direction that is (up for double-flats, down for double-sharps). So 70-14 = 56.

Okay. So let's look at some of the other 55 pitches besides the up-from-D one we already looked at. As you already pointed out, 45927/45056 is the apotome complement of 22/21, so that's another way of saying that if we went up an apotome (or sharp), it would take one move by 45927/45056 back down to get us to 22/21. So

D is 22/21. And here're all the remaining pitches notated by

I find which are simple enough such that no number in their quotient has more than 2 decimal digits:

G

88/63
A

11/7
E

33/28
B

99/56
A

63/44
D

21/11
G

14/11
C

56/33
There are still a bunch of really nasty ones in there. Take

B which is 2711943423/1476395008. You're probably not going to need that one. Or

F which is 2952790016/2711943423. I grabbed these two pitches in particular because it doesn't get any worse than them; they're the furthest pitches away from 1/1 on the chain of fifths which still apply the Sagittal alteration away from the natural in the same direction. It's a way to explain that "edge of the world": the next step away from D past

B is

F, and we say that it no longer makes sense to move

past that point, because you'll only get worse junk than 2711943423/1476395008.

And to complete the point I'm making, if you use one of the simpler-looking symbols, like 64/63, it's not that the pitches it notates are overall simpler. It simply displaces which addresses in the chain of fifths are the ones with the big nasty looking ratios with all the 2's and 3's in them and which are the ones with the really useful pitches. It all just depends on how many each of the 2's and 3's are required to pull the part of the ratio which includes 5's and beyond back down to commatic size. At a certain point, you can find ratios which have the combination of 5's, 7's, 11's, etc. that you need but it just takes too many darn 2's and 3's to get it to be comma sized, and then it ceases to be of much use for notating, because nowhere even in that chain of 35 Pythagorean fifths (ranging from

F [27 -17⟩ to

B [-26 17⟩ ) can you find any position where the final pitches you get are worth trying to play (or as you put it, can you "find" them). That's why when choosing ratios for Sagittal, Dave and George looked at the ATE, or absolute three-exponent, of the pitches, to determine — even if the 2,3-free pitch classes the commas notate are popular ones which many musicians would likely want exactly notated — is the comma actually notationally

*useful*.

Anyway, I encourage you to do your own investigations to convince yourself how this works out, if my explanation doesn't do the trick. Hopefully it helps, though!

Plus, I thought

was

with a schisma accent.

Whoops, nope. That little curl on the left is a "scroll", meant to be the same flag as you see on the

symbol, or )| in ASCII.

Accents will never be contiguous, or attached, to the symbol core.

with a schisma accent looks like this:

, or '/|\ in ASCII.

The left scroll which represents the flag with the smallest pitch alteration value of any Sagittal flag. You can see that because it's the furthest symbol to the left in the JI precision levels diagram. If you go any smaller than

you're in the realm of the accents. And since the flags were designed so that their visual size corresponds to the size of their pitch alteration, so it's the smallest flag in appearance too. So it's not surprising that if you were to confuse any flag with an accent, it'd be this left scroll.

Dave and George did their best to make the symbols as distinct as possible at a glance, for ease of sight-reading. Some of those symbols with left scrolls, though, do get a big tough, as Dave himself will attest. Worst offenders probably being the

vs. the

.

Is it reasonable to use

instead of

for 2\198?

Short answer: even if it is, prefer using the standard notation, to reduce the burden on yourself and the community. Unless you have an important reason for your exception.

One consideration when deciding an EDO notation is accounting for the tempered fifth. Some symbols' values change a ton in terms of their position within the apotome when you vary the size of the fifth. We say they have high "apotome slope". This is another way in which symbols can be notationally unuseful.

The primary comma for

has two 3's in its monzo, [4 2 0 0 -1 -1⟩, and since 198-EDO's fifth at 703.030¢ (116°198) is sharp by 1.075¢, we have to adjust by 2×1.075¢. Since [4 2 0 0 -1 -1⟩ = 12.064¢, we end up with 14.215¢ (some rounding happened in there). And now how about

? Its monzo is [5 -6 0 0 0 0 0 0 1⟩. It's going to change more, by -6×1.075¢, from 16.544 to 10.093. What's the exact size of 2\198, then? 12.121. It's a close call.

is 2.094¢ off, while

is 2.028¢ off. On this basis,

would be slightly preferred. But I'm sure there are other probably more important reasons. I'll defer to

@Dave Keenan for those. So is it reasonable? I certainly wouldn't say it's unreasonable. But maybe not recommended.

Both 45927/45056 and 45/44 (they indeed differ by 5120/5103

) map to 7\198. Should I still bother with

instead of simply

?

Let me know if this isn't addressed by the earlier part of my response here.