In this work, the elastic wave propagation and dispersion characteristics of a curved tapered frame structure are investigated analytically. Separately, wave propagation through uniform curved and straight tapered beam was reported in the existing literature; however, no literature reports the influence of simultaneous bent and taper on the wave propagation. In particular, the band characteristics for the curved and tapered beam with two types of cross sections, i.e., rectangular and circular, are presented. The paper elucidates that introducing a small periodic bent angle cross section produces a complete, viz., axial and flexural bandgap in the low-frequency region, and conicity enhances the width of the band. It is also evidenced that a curved tapered frame with a solid circular cross section induces a wider bandgap than the rectangular section. A complete first normalized bandwidth of 159% is achievable for the circular cross section and 123% in the case of the rectangular section. The complete result is presented in a non-dimensional framework for wider applicability. An analysis of a finite tapered curved frame structure also demonstrates the attenuating characteristics obtained from the band structure of the infinite structure. The partial wave mode conversion, i.e., generation of coupled axial and flexural mode from a purely axial or flexural mode in an uncoupled medium, is observed. This wave conversion is perceived in reflected and transmitted waves while this curved tapered frame is inserted between the two uniform cross-sectional straight frames.