The frequencies of vibration of a tuning fork can be affected by the force used to strike it. Depending upon the magnitude, direction, and the number and location of the points of application, vibrational modes other than the fundamental mode could be excited. Each of these vibrational modes have a different frequency. The mode frequencies are characteristic, but not all necessarily need to be excited every time. For a tuning fork, the fundamental mode, the clang mode, the asymmetric in-plane mode, the out of plane bending mode, and the asymmetric out of plane modes are shown here as animations.
The frequency engraved on the tuning fork is for the fundamental mode. As a device designed to act as pitch reference, the other modes (of a high-quality fork) should not be easy to excite strongly, and should damp out quickly. But you should be able affect how much the auxiliary modes are excited by how you strike the fork.