Tight-binding operators are bounded linear operators on $\ell^2(\Gamma)$, where $\Gamma$ is a countable discrete set in $\mathbb R^N$, and they serve as fundamental models in solid-state physics: the points of $\Gamma$ represent atomic positions. In the case $\Gamma=\mathbb Z^n$, one obtains an ideal crystal: the atomic arrangement repeats periodically, and translation symmetry makes the spectrum of the operator accessible through Fourier methods. Quasicrystals are different. They exhibit long-range order without exact periodic repetition, so they are ordered but not repeating like a crystal. A basic mathematical example is given by the vertex set of a Penrose tiling, a nonperiodic tiling of the plane built from a small number of tile shapes that never repeats exactly, yet still displays striking global structure. Once considered a mere mathematical curiosity, quasicrystals were first observed in the laboratory in 1984 and later identified in naturally occurring materials in 2009. In this talk, I will present a new approach for approximating the spectrum of tight-binding operators defined on general discrete sets Gamma. The method uses only information from finite local patches and comes with rigorous upper and lower error bounds. When the underlying set also has finite local complexity -- meaning, roughly, that only finitely many local patterns can occur up to translation -- this yields an explicit algorithm for numerical spectral approximation. In particular, it opens the door to computer-assisted proofs of spectral gaps for models of quasicrystals. My talk is based on joint work with Paul Hege and Massimo Moscolari (Physical Review B 2022 and Mathematics of Computation 2025).