The main role of a violin bridge is to hold the strings and to transmit the vibration of the strings to the violin body. Violin makers have been empirically aware of the fact that a bridge is an important element which influences violin timbre. Thus, a bridge can be regarded as a mechanical filter in the transmission and be used to compensate weak or too strong areas in the resonance of the violin body. The filtering characteristics of a bridge depend on the geometry and material distribution of the bridge. In this paper, the sensitivity of band-averaged frequency response with respect to geometric design variables is derived. Then, topology optimization is applied to obtain optimal violin bridges for desired filtering characteristics. Numerical results show that the proposed optimization process can be a viable tool to design a bridge according to prescribed characteristics for musical performance.