Canonical curves on surfaces of very low degree

Let $C$ be a non--hyperelliptic curve of genus $g$. We recall some facts about curves endowed with a base--point--free $g^1_4$. Then we prove that if the minimal degree of a surface containing the canonical model of $C$ in $\mathbb P^{g-1}_k$ is $g$, then $7\le g\le 12$ and $C$ carries exactly one $g^1_4$. As a by--product, we deduce that if the canonical model of $C$ in $\mathbb P^{g-1}_k$ is contained in a surface of degree at most $g$, then C is either trigonal or tetragonal or isomorphic to a plane sextic.