  The nullity of a minimal submanifold $M\subset S^{n}$ is the dimension of the
nullspace of the second variation of the area functional. That space contains
as a subspace the effect of the group of rigid motions $SO(n+1)$ of the ambient
space, modulo those motions which preserve $M$, whose dimension is the Killing
nullity $kn(M)$ of $M$. In the case of 2-dimensional tori $M$ in $S^{3}$, there
is an additional naturally-defined 2-dimensional subspace; the dimension of the
sum of the action of the rigid motions and this space is the natural nullity
$nnt(M)$. In this paper we will study minimal tori in $S^{3}$ with natural
nullity less than 8. We construct minimal immersions of the plane $R^{2}$ in
$S^{3}$ that contain all possible examples of tori with $nnt(M)<8$. We prove
that the examples of Lawson and Hsiang with $kn(M)=5$ also have $nnt(M)=5$, and
we prove that if the $nnt(M)\le6$ then the group of isometries of $M$ is not
trivial.
