Where each piece of wire was cut it is rough. What is the best way
to polish and smooth this piece?
Pain in the neck, isn't it? If you're doing it by hand, the answer
is filing (coarse to fine) followed by sandpapers or sanding sticks,
buffers, and polishers - but be very careful about machine polishing
chains, as they are easily grabbed and turned into dangerous flails
or missiles. The whole process is easier if you think ahead and
minimize the roughness as you work. Also, if you solder the joints,
a little of the roughness may be covered. Any way you do it (by
hand) it's not easy. I'll defer to others about tumbling.