  We introduce a matrix product state (MPS) with an incommensurate periodicity
by applying the spin-rotation operator of each site to a uniform MPS in the
thermodynamic limit. The spin rotations decrease the variational energy with
accompanying translational symmetry breaking and the rotational symmetry
breaking in the spin space even if the Hamiltonian has the both symmetries. The
optimized pitch of rotational operator reflects the commensurate/incommensurate
properties of spin-spin correlation functions in the $S=1/2$ Heisenberg chain
and the $S=1/2$ ferro-antiferro zigzag chain.
