  We introduce a family of domains --- which we call the $\mu_{1,n}$-quotients
--- associated with an aspect of $\mu$-synthesis. We show that the natural
association that the symmetrized polydisc has with the corresponding spectral
unit ball is also exhibited by the $\mu_{1,n}$-quotient and its associated unit
"$\mu_E$-ball". Here, $\mu_E$ is the structured singular value for the case $E
= \{[w]\oplus(z I_{n-1})\in \mathbb{C}^{n\times n}: z,w\in \mathbb{C}\}$, n =
2, 3, 4,... Specifically: we show that, for such an $E$, the Nevanlinna-Pick
interpolation problem with matricial data in a unit "$\mu_E$-ball", and in
general position in a precise sense, is equivalent to a Nevanlinna-Pick
interpolation problem for the associated $\mu_{1,n}$-quotient. Along the way,
we present some characterizations for the $\mu_{1,n}$-quotients.
