This paper studies the asymptotic spectra for Toeplitz or quasi-Toeplitz matrices of large order and small bandwidth. The analysis is split into one for the boundary condition independent and the dependent spectra, with algorithms for both. For such matrices of large order \((n\geq 100)\) that are nonnormal, the standard QR based eigenvalue algorithms will generally become useless unless a certain diagonal scaling is introduced that limits the growth of the associated eigenvectors. Various applications of finite difference approximations to partial differential equations are included.