A method is proposed to determine the transfer matrix parameters of a discontinuity in a waveguide with the finite element method (FEM). This is used to characterize open and closed woodwind instrument toneholes and develop expressions for the shunt and series equivalent lengths. Two types of toneholes are characterized: Unflanged toneholes made of thin material, such as found on saxophones and concert flutes, and toneholes drilled through a thick material, such as found on most instruments made of wood. The results are compared with previous tonehole models from the literature. In general, the proposed expressions provide a better fit across a wide range of frequencies and tonehole sizes than previous results. For tall toneholes, the results are in general agreement with previous models. For shorter tonehole heights, some discrepancies from previous results are found that are most important for larger diameter toneholes. Finally, the impact of a main bore taper (conicity) on the characterization of toneholes was investigated and found to be negligible for taper angles common in musical instruments.