A new type of high-n discrete Alfvén eigenmodes is shown to exist in the large-α (α ≡ −q2Rdβ/dr) second ballooning stable toroidal plasmas. Here, n is the toroidal mode number, q is the safety factor, β is the ratio between plasma and magnetic pressures, and R and r are, respectively, the major and minor radii. These magnetohydrodynamic eigenmodes are bounded by the α-induced potential wells along the magnetic field line and, thus, exist even in absence of the toroidal Alfvén frequency gap. Due to negligible continuum damping via wave energy tunneling, these large-α toroidal Alfvén eigenmodes are quasimarginally stable and, thus, could be readily destabilized by energetic particles. © 2004 American Institute of Physics.