Flooring Tile - tile over both cement board and top of floor joist

I would like to lower the cement board base for a bathroom tile installation by removing the existing plywood subfloor, installing 14 1/2" strips of 3/4'' plywood between the floor joists(16 "O.C.) then installing 14 1/2 "strips of cement board over the plywood subfloor so that the finished base will level. Will the exposed tops of the floor joists cause expansion/contraction problems that will cause the tile to work loose? If this will not work, is there another alternative to lower the base, other than to plane down the tops of the floor joists?

thanks in advance,
eldodog

I don't think the cement board will provide enough rigidity to compensate for the plywood having been cut into pieces. The cut plywood won't supply any rigidity or continuity at all.

The alternative would be a "mud job," which will work, but it takes a bit more expertise. If you are interested, we can help.