  Roseman moves are seven types of local modification for surface-link diagrams
in $3$-space which generate ambient isotopies of surface-links in $4$-space. In
this paper, we focus on Roseman moves involving triple points, one of which is
the famous tetrahedral move, and discuss their independence. For each diagram
of any surface-link, we construct a new diagram of the same surface-link such
that any sequence of Roseman moves between them must contain moves involving
triple points (and the numbers of triple points of the two diagrams are the
same). Moreover, we can find a pair of two diagrams of an $S^2$-knot such that
any sequence of Roseman moves between them must involve at least one
tetrahedral move.
