I think that most symptoms described in the first
message were due to the wobbling pulley. They
disappeared when the string under tension was
pressed against the frame of the pulley.
Since the prong is not at the node we simply
ignored the first "loop" (near the prong) and
measured the loop length on the basis of the
remaining part of the string. Then the theoretical
loop length was calculated. It was always in
reasonable agreement with what was measured.
I still do not know the first node is sometimes
very close to the prong (for example 2 cm) and
sometimes far away, for example, 18 cm.
Ludwik Kowalski