As a result of toroidal rotation, sequences of new global modes are predicted to arise in magnetically confined plasmas. The frequencies of these Alfvén modes lie inside gaps of the continuous magnetohydrodynamic (MHD) spectrum that are created or enlarged by toroidal flow. The numerically obtained results are compared with an analytical investigation, yielding a useful criterion for mode existence. Because of their low frequencies, these modes may be easily destabilized by energetic particles. Because of their sensitivity to the Mach number however, these modes can provide a valuable extension to MHD spectroscopy by giving information on the rotational velocity.