I agree with Larry Lobel's finding in the Player Piano Co. catalog.

Years ago I was given this friendly hint when rebuilding the main and
feeder bellows on an old reed organ.  In that organ, some thin wooden
"slats"  (as I would call them)  were glued to the inside of the bellows
cloth to create a natural fold, almost like the binding of a book. In the
restoration, I used some heavy cardboard and achieved excellent results.
Being lighter than the wooden slats, it made the pumping a bit easier.

I'm sure that the pneumatics of a player piano are much finer and require
more precision, but it would be great to hear from the experts in our
audience about the similarities and contrasts.  Humbly submitted,

Stephen Matta¶
stevematta@mindspring.com