High-order plasma shaping (mainly elongation and shift, as opposed to low-order toroidicity) is shown, under certain conditions, to open gaps in the coupled shear-Alfvén and acoustic continua at frequencies significantly above the values predicted by previous theories. Global eigenmodes in these gaps, which lie between the frequencies of geodesic acoustic modes and toroidicity-induced Alfvén eigenmodes, are found unstable to hot-ion populations typical of tokamak operation, whilst their fundamental resonances with circulating particles are shown to take place at velocities near the geometric mean of the Alfvén and sound speeds. Therefore, such eigenmodes are expected to be observed near the predicted frequencies at operating tokamaks, playing a still unexplored role in magnetohydrodynamic spectroscopy as well as in the stability of next-step fusion experiments.