Would people living on different ringworlds around the same star measure time differently?

Okay so I have a sci-fi world masquerading as a fantasy world where numerous different genetically-engineered humanoids live on nine different ringworlds orbiting an artificial Earth-mass black hole (which they see as a "black sun"). The black hole's accretion disk provides light to the ringworlds much like a small star would, but if we take it as a given that these rings all revolve around the black hole at various rotational speeds (like the planets in our solar system do relative to the sun), how would the cultures living on these ringworlds measure time differently? Assuming of course that one ring passing into the shadow created by the ring above or below it (depending on which side of the ring you're living) is analogous to a single night.

EDIT: Hmmm... actually you're probably right. I want this whole structure to be Earth-sized so traveling across all the different rings doesn't take much longer than an IRL cross-continental journey apiece, but I also want the black hole to be small enough that the rings don't fall in. Anyone have any suggestions as to what would be an appropriate size that'd balance luminosity and gravity for a planet-sized structure?

As for the shape and arrangement of the structure it's similar in concept to the Game of Thrones opening:

http://i.stack.imgur.com/pqf90.jpg

posted about 8 years ago

Z.Schroeder‭

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