  For two orthonormal bases of a $d$-dimensional complex Hilbert space, the
notion of complete incompatibility was introduced recently by De Bi\`{e}vre
[Phys. Rev. Lett. 127, 190404 (2021)]. In this work, we introduce the notion of
$s$-order incompatibility with positive integer $s$ satisfying $2\leq s\leq
d+1.$ In particular, $(d+1)$-order incompatibility just coincides with the
complete incompatibility. We establish some relations between $s$-order
incompatibility, minimal support uncertainty and rank deficiency of the
transition matrix. As an example, we determine the incompatibility order of the
discrete Fourier transform with any finite dimension.
