This paper reports the results of recent experiments performed on the JET tokamak on Alfvén Eigenmodes (AEs) with toroidal mode number (n) in the range n = 3-15. The stability properties and the use of these medium-n AEs for diagnostic purposes is investigated experimentally using a new set of compact in-vessel antennas, providing a direct and real-time measurement of the frequency, damping rate and amplitude for each individual toroidal mode number. First, we report on the development of a new algorithm for mode detection and discrimination using the Sparse Signal Representation theory. The speed and accuracy of this algorithm has made it possible to use it in our plant control software, allowing real-time tracking of individual modes during the evolution of the plasma background on a 1ms time scale. Second, we report the first quantitative analysis of the measurements of the damping rate for stable n = 3 and n = 7 Toroidal AEs as function of the plasma elongation. The damping rate for these modes increases for increasing elongation, as previously found in JET for n = 0-2 AEs. A theoretical analysis of these JET data has been performed with the LEMan, CASTOR and TAEFL codes. The LEMan and TAEFL results are in good agreement with the measurements for all magnetic configurations where there is only a minor up/down asymmetry in the plasma poloidal cross-section. The CASTOR results indicate that continuum damping is not the only mechanism affecting the stability of these medium-n AEs. The diagnostic potential of these modes has being confirmed during the recent gas change-over experiment, where independent measurements of the effective plasma isotope ratio AEFF have been provided in addition to the more routinely employed spectroscopic and gas-balance ones. These data shows a slight difference in the measurement of AEFF when using n<5 and n>7 modes, suggesting a radial dependence in the effective plasma isotope ratio.