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This paper reports the first quantitative analysis of the measurements of the damping rate (gamma/omega) for stable Alfvén Eigenmodes (AEs) with toroidal mode number (n) in the range |n|=3-15 as function of the edge plasma elongation (kappa95). We find that the damping rate gamma/omega vs. kappa95 for medium-n Toroidal AEs, with n=3 and n=7, increases for increasing elongation, i.e. its scaling vs. kappa95 follows the same trend previously measured and explained theoretically for the n=1 and n=2 TAE modes. Theoretical analysis of the measurements for the n=3 TAEs has been performed using the LEMan code. The results are in good agreement (within a factor 2) for all the magnetic configurations where there is only a very minor up/down asymmetry in the poloidal cross-section of the 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.