If a music roll is not going to be transported or otherwise jiggled
around, I know of no reason to put a rubber band around it. Even it
were to be mailed or shipped or moved about, I would use a bit of cloth
tape loosely tied, as John Philipps suggests in the MMD article cited
by Robbie. Besides the rubber disintegration problem, a too-tight
rubber band will leave a permanent indentation in the first few layers
of the roll paper.
Your roll didn't come with a rubber band around it; so don't add one.
Rubber bands and music rolls just don't go together.
Irondequoit, New York