Let
R
R
be a Noetherian
N
r
\mathbb {N}^r
-graded ring generated in degrees
d
1
,
…
,
d
r
\textbf {d}_1, \dots , \textbf {d}_r
which are linearly independent vectors over
R
\mathbb {R}
, and let
a
\mathfrak a
be an ideal in
R
0
R_\textbf {0}
. In this paper, we investigate the asymptotic behavior of the grade of the ideal
a
\mathfrak a
on the homogeneous components
M
n
M_\textbf {n}
of a finitely generated
Z
r
\mathbb {Z}^r
-graded
R
R
-module
M
M
and show that the periodicity occurs in a cone.