  We introduce discrepancy values, quantities inspired by the notion of the
spectral spread of Hermitian matrices. We define them as the discrepancy
between two consecutive Ky-Fan-like seminorms. As a result, discrepancy values
share many properties with singular values and eigenvalues, yet are
substantially different to merit their own study. We describe key properties of
discrepancy values, and establish several tools such as representation
theorems, majorization inequalities, convex formulations, etc., for working
with them. As an important application, we illustrate the role of discrepancy
values in deriving tight bounds on the norms of commutators.
