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By analyzing resonances in the local density of states (LDOS) for a fivefold 2-D quasi-periodic photonic crystal (QPC), we reveal the presence of a well-defined number of defect modes in the spectral region of a low LDOS, which are intrinsic to this class of photonic structures. We may distinguish between surface modes, depending on the truncation of the finite-sized structure, and bulk modes. All defects can be traced to a configuration of closely spaced cylinders. Due to mutual coupling, their collective eigenresonance is strongly shifted into the spectral region of a low LDOS, and they appear as defect modes, which are comparable to acceptor modes in periodic photonic crystals. Control over the geometrical parameters of such resonators allows for a simple adjustment of the defect mode frequencies in QPCs.