In this paper, a model for the interaction of the strings with the frets in a guitar or other fretted string instruments is introduced. In the two-polarization representation of the string oscillations it is observed that the string interacts with the fret in different ways. While the vertical oscillation is governed by perfect or imperfect clamping of the string on the fret, the horizontal oscillation is subject to friction of the string over the surface of the fret. The proposed model allows, in particular, for the accurate evaluation of the elongation of the string in the two modes, which gives rise to audible dynamic detuning. The realization of this model into a structurally passive system for use in digital waveguide synthesis is detailed. By changing the friction parameters, the model can be employed in fretless instruments too, where the string directly interacts with the neck surface.