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We theoretically analyze the spectrum of phonons of a one-dimensionalquasiperiodic lattice. We simulate the quasicrystal from the classic system ofspring-bound atoms with a force constant modulated by the Aubry-Andr\'e model,so that its value is slightly different in each site of the lattice. From theequations of motion, we obtained the equivalent phonon spectrum of theHofstadter butterfly, characterizing a multifractal. In this spectrum, weobtained the extended, critical and localized regimes, and we observed that themultifractal characteristic is sensitive to the number of atoms and the$\lambda$ parameter of our model. We also verified the presence of borderstates for phonons, where some modes in the system boundaries presentvibrations. Through the measurement of localization of the individualdisplacements in each site, we verify the presence of a phase transitionthrough the Inverse Participation Rate (IPR) for $\lambda= 1.0 $, where thesystem changes from extended to localized.

Phononic Topological States in 1D Quasicrystals