Let \({\mathbf f}=\left( f_1,\dots ,f_n\right)\) be an \(n\)-tuple of elements of a unital commutative ring \(R\) and define \({\mathbf f}^\perp\) to be the set of \(n\)-tuples \(\left( g_1,\dots,g_n\right)\) of elements of \(R\) such that \(g_1f_1+\cdots +g_nf_n=0\). The ring \(R\) is said to be coherent if, for each natural number \(n\) and each \(n\)-tuple \({\mathbf f}\), the set \({\mathbf f}^\perp\) is finitely generated in the sense that there exist elements \({\mathbf g}_1,\dots ,{\mathbf g}_d\) in \({\mathbf f}^\perp\) such that every element \({\mathbf g}\) in \({\mathbf f}^\perp\) has the form \({\mathbf g}= r_1{\mathbf g}_1 +\cdots +r_d{\mathbf g}_d\) for some elements \(r_1,\dots,r_d\) in \(R\). In [J. Funct.Anal.21, 76–87 (1976; Zbl 0317.46049)], W.S.McVoy and L.A.Rubel showed that \(H^\infty\) is coherent while the disk algebra \(A\) is not. More generally, if \(S\) is a subset of the circle \({\mathbf T}\), then the algebra \(A_S\) consisting of those functions that are analytic on the disk \({\mathbf D}\) and continuous and bounded on \({\mathbf D}\cup S\) lies between \(A=A_{\mathbf T}\) and \(H^\infty=A_\emptyset\). In this paper, the author uses similar techniques to show that, if the set \(S\) contains any of its limit points, then the algebra \(A_S\) is not coherent. It remains an open question whether the converse statement is true or not.