  We define a new large $N$ limit for general $\text{O}(N)^{R}$ or
$\text{U}(N)^{R}$ invariant tensor models, based on an enhanced large $N$
scaling of the coupling constants. The resulting large $N$ expansion is
organized in terms of a half-integer associated with Feynman graphs that we
call the index. This index has a natural interpretation in terms of the many
matrix models embedded in the tensor model. Our new scaling can be shown to be
optimal for a wide class of non-melonic interactions, which includes all the
maximally single-trace terms. Our construction allows to define a new large $D$
expansion of the sum over diagrams of fixed genus in matrix models with an
additional $\text{O}(D)^{r}$ global symmetry. When the interaction is the
complete vertex of order $R+1$, we identify in detail the leading order graphs
for $R$ a prime number. This slightly surprising condition is equivalent to the
complete interaction being maximally single-trace.
