It's interesting to see all the different approaches to cutting the segments. I didn't want to mess with thin slices, so I made a sequence of glue/part operations.
First of all, I turned both blanks (maple and blackwood) round between centers. Then I parted the maple and trimmed one end on each square. I also trimmed one end of the blackwood blank square.
Next, I glued the blackwood to the first maple blank. I chucked it up and parted the blackwood off (leaving a thin slice attached). I repeated the process with the second maple piece, and then with the blackwood again (only leaving a thicker slice).
Then I drilled, tubed, and turned the upper (segmented) and lower blanks. Although my segments weren't particuarly thin, I could easily have made them so.
I hope that helps,
Eric