  In this article we determine feasible parameter sets for (what could
potentially be) commutative association schemes with noncyclotomic eigenvalues
that are of smallest possible rank and order. A feasible parameter set for a
commutative association scheme corresponds to a standard integral table algebra
with integral multiplicities that satisfies all of the parameter restrictions
known to hold for association schemes. For each rank and involution type, we
generate an algebraic variety for which any suitable integral solution
corresponds to a standard integral table algebra with integral multiplicities,
and then try to find the smallest suitable solution. Our main results show the
eigenvalues of commutative association schemes of rank 4 and nonsymmetric
commutative association schemes of rank 5 will always be cyclotomic. In the
rank 5 cases these our conclusions rely on calculations done by computer for
Gr\"obner bases or for bases of rational vector spaces spanned by polynomials.
We give several examples of feasible parameter sets for small symmetric
association schemes of rank 5 that have noncyclotomic eigenvalues.
