Hi, Karen, Here's what I would say, though there are always other
ways to deal with things.
To get invisible joins,
1) make sure the parts mate as perfectly as possible. My students
tear their hair out trying to do this, but there's no way around it.
Then use minimal solder.
2) use the hardest solder you can manage. The harder the solder, the
closer the color match.
3) Assuming this is silver, after all heat-work is done, "bring up
the fine silver". This means gentle heating and pickling as many
times as it takes (up to 10 sometimes) to deplete the surface, and
especially the solder, of copper. Heat the piece without flux with
a soft flame until it just turns yellow, then plunge it into hot
pickle. It will pickle very quickly. Rinse, dry, do it again. It
will gradually take more heat to get discoloration, and when it
stays white, you are done.
The last creates a disguise for solder and for firescale, but it is
just a thin layer of soft fine silver, so don't depend on it to last
on high-wear items like rings. Concentrate on #1 & 2. But any piece
(IMHO) worth spending more than an hour making is worth #3, since
this will also substantially decrease or delay tarnish. It will also
create that beautiful soft white finish that is so popular these
days, especially in Europe.
I just looked at your post again, and I may see another issue. If
you file correctly to begin with, the sawing-through is unnecessary,
and possibly counter-productive. Get the ends flat and square, then
bend them together. Don't worry about the shape of the ring: they'll
meet best if they meet straight, so your ring can be quite squashed
in shape. Then, instead of binding, displace the ends to the side so
that you can force them past each other (as though they came from a
coil), then snap them back. Then do the same to the other side,
and/or to the top and bottom. This can be quite difficult on a heavy
ring, but it work-hardens the shank and creates spring, so that
ultimately the ends are pressing hard on each other. This works
way better than binding to create a tight fit. Also, I like to
solder from the inside, so any mess ends up there where cleanup is
less obvious. I stand the ring up, against a pin or block if
necessary, with the joint at the bottom. Easy enough to round it out
after it is soldered.