We investigate the scalar-wave resonances of systems composed of identical Neumann-type inclusions arranged periodically around a circular ring. Drawing on natural similarities with the undamped Rayleigh-Bloch waves supported by infinite linear arrays, we deduce asymptotically the exponentially small radiative damping in the limit where the ring radius is large relative to the periodicity. In our asymptotic approach, locally linear Rayleigh-Bloch waves that attenuate exponentially away from the ring are matched to a ring-scale WKB-type wave field. The latter provides a descriptive physical picture of how the mode energy is transferred via tunnelling to a circular evanescent-to-propagating transition region a finite distance away from the ring, from where radiative grazing rays emanate to the far field. Excluding the zeroth-order standing-wave modes, the position of the transition circle bifurcates with respect to clockwise and anti-clockwise contributions, resulting in striking spiral wavefronts.