For a connected graph \(G\) of order \(3\) or more and an edge coloring \(c: E(G) \to \mathbb{Z}_k\) (\(k \geq 2\)) where adjacent edges may be colored the same, the color sum \(s(v)\) of a vertex \(v\) of \(G\) is the sum in \(\mathbb{Z}_k\) of the colors of the edges incident with \(v\). The edge coloring \(c\) is a modular \(k\)-edge coloring of \(G\) if \(s(u) \neq s(v)\) in \(\mathbb{Z}_k\) for all pairs \(u, v\) of adjacent vertices in \(G\). The modular chromatic index \(\chi_m'(G)\) of \(G\) is the minimum \(k\) for which \(G\) has a modular \(k\)-edge coloring. It is shown that \(\chi(G) \leq \chi_m'(G) \leq \chi(G)+1\) for every connected graph \(G\) of order at least 3, where \(\chi(G)\) is the chromatic number of \(G\). Furthermore, it is shown that \(\chi_m'(G) = \chi(G) + 1\) if and only if \(\chi(G) \equiv 2 \pmod{4}\) and every proper \(\chi(G)\)-coloring of \(G\) results in color classes of odd size.