A new set of compact in-vessel antennas has been built and installed in JET to provide for the first time the direct measurement of the damping rate (/) of stable Alfvén Eigenmodes (AEs) with toroidal mode number (n) in the range n=3-15. This paper reports the first quantitative analysis of the measurements of the damping rate for these modes as function of the edge plasma elongation (95). We find that the scaling of / vs. 95 for medium-n Toroidal AEs, with n=3-7, follows the same trend previously measured and explained theoretically for the n=1 and n=2 modes. This confirms the possibility of using the edge shape parameters as a real-time actuator for control of the stability of alpha-particles driven AEs in burning plasma experiments, such as ITER.