We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions. The applied methodology is flexible, and it is shown in the case of GCD and LCM matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice. These results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices, which we briefly consider as well. We investigate the effectiveness of the obtained bounds for select examples involving number-theoretical lattices.