We propose a classification scheme for the commensurate phases in the family of pseudo one-dimensional structures formed from the mixed hcp stacking of A3O9 and oxygen-deficient A3A'O6 layers. We envisage these compounds as composites of two substructures having common a and b unit cell parameters but different parameters c1 and c2. Use of the ratio c1/c2 facilitates the prediction of new commensurate structures while allowing for the commonly incommensurate nature of materials in this family. The structures of the new commensurate phases Ba5CuIr3O12, Ba14Cu3Ir8O33, Ba16Cu3Ir10O39, and Ba9Cu2Ir5O21 are predicted and subsequently verified by powder X-ray diffraction and HRTEM. Ba5CuIr3O12 has a 10 layer structure with space group P321, a = 10.143 82(8) Å, c = 21.6553(2) Å; Ba14Cu3Ir8O33 has a 14 layer structure with space group P321, a = 10.145 85(8) Å, c = 29.9574(3) Å; Ba16Cu3Ir10O39 has a 16 layer structure with space group P321, a = 10.136 43(7) Å; c = 35.0616(3) Å; Ba9Cu2Ir5O21 has an 18 layer structure with space group R32, a = 10.144 64(11) Å, c = 38.2455(6) Å. Sequences of trigonal prismatic sites and octahedral sites run in chains parallel to z, with Ba cations located between the chains; the distribution of iridium and copper cations in the octahedral and trigonal prismatic sites is disordered in each case. Electron diffraction patterns and lattice images show evidence for modulation in the structures of Ba5CuIr3O12, Ba14Cu3Ir8O33, and Ba16Cu3Ir10O39 but not in that of Ba9Cu2Ir5O21. The magnetic susceptibilities of all four phases obey a modified Curie-Weiss law above 150 K, with no long-range magnetic order observed between 5 and 300 K. They are all electrical insulators.