A SU(N)_{L}×SU(N)_{R} gauge theory for a scalar multiplet Y transforming in the bifundamental representation (N,N[over ¯]) preserves, for N>4, an accidental U(1) symmetry first broken at operator dimension N. A vacuum expectation value for Y can break the symmetry to H_{s}=SU(N)_{L+R} or to H_{h}=SU(N-1)_{L}×SU(N-1)_{R}×U(1)_{L+R}. In the first case the accidental U(1) gets also broken, yielding a pseudo-Nambu-Goldstone boson with mass suppression controlled by N. In the second case a global U(1) remains unbroken. The strong CP problem is solved by coupling Y to new fermions carrying color. The first case allows for a Peccei-Quinn solution with U(1)_{PQ} protected by the gauge symmetry up to order N. In the second case U(1) can get broken by condensates of the new strong dynamics, resulting in a composite axion. By coupling Y to fermions carrying only weak isospin, models for axionlike particles can be constructed.