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 (γ/ω) 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 (k95). We find that the scaling of γ/ω versus k95 for medium-n Toroidal AEs, with n = 3 and n = 7, follows the same trend previously measured and explained theoretically for the n = 1 and n = 2 modes. Theoretical analysis of these measurements has been performed using the LEMan code, with the results in good agreement for the magnetic configurations where there is only a very minor up/down asymmetry in the poloidal cross-section of plasma. These experimental results further confirm 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.