  In this paper we study the large $N$ solution to matrix models describing the
partition functions of 3d supersymmetric gauge theories on $S^3$. The model we
focus on has a single $U(N)$ gauge group and fundamental fields, whose number
scales with $N$, also known as the Veneziano limit. The novel point in our
model is our choice of masses for the fundamental fields. Instead of vanishing
or fixed masses, we consider a linear distribution. We show that the model can
still be solved by standard large $N$ techniques and explore the different
phases of the model. We also commend about other natural mass distributions.
