It is well known that there exist wavelet frames \(\{2^{j/2}\psi(2^jx-k)\}_{j,k\in Z}\) for which the canonical dual frame does not have wavelet structure. In this paper the authors find such a frame with the additional property that other duals, which have the wavelet structure, exist; in fact, infinitely many such duals exist. Similar constructions are possible for wavelet frames \(\{2^{j/2}\psi_l(2^jx-k)\}_{j,k\in Z, l=1,\dots,r}\) generated by \(r\) functions. For such a frame, equivalent conditions for the canonical dual frame to be generated by \(2^Jr\) functions (for some \(J=0,1,2,\dots\)) are given.