Periodic structures are known to exhibit alternating propagating and attenuating band characteristics over the frequency range. In this work periodic bar, beam , ladder frame, and square grid structures are suitably synthesized to achieve a maximal attenuating frequency band. A common geometric framework that of an Eigenvalue map is developed. The concept is shown to be applicable to a wide class of periodic structures. It is shown that the finite structure designed based on periodic structure theory has superior isolation characteristics in comparison to a homogeneous or an arbitrarily chosen periodic structure. Based on the periodic structure theory, practically useful design guidelines are proposed for superior attenuation characteristics.