We discuss how the modal response of violin plates changes as their shape varies. Starting with an accurate 3D scan of the top plate of a historic violin, we develop a parametric model that controls a smooth shaping of the interior of the plate, while guaranteeing that the boundary is the same as the original violin. This allows us to generate a family of violin tops whose shape can be smoothly controlled through various parameters that are meaningful to a violin maker: from the thickness in different areas of the top to the location, angle, and dimensions of the bass bar. We show that the interplay between the different parameters affects the eigenmodes of the plate frequencies in a nonlinear fashion. We also show that, depending on the parameters, the ratio between the fifth and the second eigenfrequencies can be set to match that used by celebrated violin makers of the Cremonese school. As the parameterisation that we define can be readily understood by violin makers, we believe that our findings can have a relevant impact on the violin making community, as they show how to steer geometric modifications of the violin to balance the eigenfrequencies of the free plates.