We report the results of Raman measurements performed on polycrystalline samples of superconducting Tl2Can-1Ba2CunO4+2n (n=1,2,3) with well-characterized transport, magnetic, and structural properties. The room-temperature first-order Raman spectra of polycrystalline samples of predominantly Tl2Ba2CuO6, Tl2CaBa2Cu2 O8, and Tl2Ca2Ba2 Cu3O10 phases, with respective superconducting transition temperatures of 10, 110, and 125 K display a systematic variation with the number of CuO2 layers and great similarities with the Raman spectrum of YBa2Cu3O7. Comparison with Raman spectra of the impurity phase candidated BaCO3, BaCuO2, CaCuO2, Tl2O3, and TlCa5Cu6Ox allows identification of intrinsic Raman modes of the superconducting phases. Factor group analysis of the three body-centered tetragonal structures, space group 14mmm, yields the mode classifications and the atoms involved in the modes of A1g and B1g symmetry corresponding to c-axis motion that produce the strongest modulation of the polarizability. The systematic variation between the spectra of the three phases and comparison with the well-characterized YBa2Ca3Ox Raman spectrum suggests mode assignments for the high-frequency modes of A1g and B1g symmetry involving oxygen motion. © 1988 The American Physical Society.