  This paper characterizes singularities with Mather minimal log discrepancies
in the highest unit interval, i.e., the interval between $d-1$ and $d$, where
$d$ is the dimension of the scheme. The class of these singularities coincides
with one of the classes of (1) compound Du Val singularities, (2) normal
crossing double singularities, (3) pinch points, and (4) pairs of non-singular
varieties and boundaries with multiplicities less than or equal to 1 at the
point. As a corollary, we also obtain one implication of an equivalence
conjectured by Shokurov for the usual minimal log discrepancies.
