Consider the following definition of a free semicircular family. Let $(A,\tau)$ be a C* probability space (so $\tau$ is a faithful state on $A$, which is a unital C* algebra). Let $NC_2(p)$ be the set of noncrossing partitions (pairings) of $\{1,\dots,p\}$. Take self-adjoint $s_1,\dots,s_n\in A$. Then, the $s_i$ are a free semicircular family if for any even natural $p$ and $k_1,\dots,k_p$ all integers in $1$ through $n$, $$\tau(s_{k_1}\cdots s_{k_p})=\sum\limits_{\pi\in NC_2(p)}\prod\limits_{\{i,j\}\in\pi}\delta_{k_i,k_j}.$$
My question is how I should "picture" the elements $s_i$. Is there a way to write down specific $s_i$s for small $n$, say $n=3$ or $n=4$? Will this be possible on "reasonable" choices of $A$, say some finite-dimensional matrices?