We examined by low-energy electron diffraction and scanning tunneling microscopy the surface of thin Cu films on Pt(111). The Cu/Pt lattice mismatch induces a moire modulation for films from 3 to about 10 ML thickness. We used angle-resolved photoemission spectroscopy to examine the effects of this structural modulation on the electronic states of the system. A series of hexagonal- and trigonal-like constant energy contours is found in the proximity of the Cu(111) zone boundaries. These electronic patterns are generated by Cu sp-quantum well state replicas, originating from multiple points of the reciprocal lattice associated with the moire superstructure. Layer-dependent strain relaxation and hybridization with the substrate bands concur to determine the dispersion and energy position of the Cu Shockley surface state.