Unusual subextensive configurational entropies that vary with particle size and tend to zero per atom in macroscopic samples are predicted for AMO(3-z)N(z) oxynitrides with perovskite type crystal structures. These materials are crystallographically disordered on the atomic scale, but local anion order produces chains of M-N-M bonds that undergo a 90 degrees turn at each M cation, giving rise to subextensive entropies in materials such as SrTaO2N, LaNbON2, and EuWO1.5N1.5. A general Pauling ice-rules formula is used to calculate the extensive molar entropies for other cases such as SrMoO2.5N0.5 and BaTaO2N. The subextensive oxynitrides are usefully classified as showing an "open order", related to the correlated order of displacements in ferroelectric perovskites such as BaTiO3. This raises the possibility that further open-ordered oxynitride or molecular structures may be accessible, and other states such as spins and charges may also show novel open orders.