This paper calculates the density of states on fractals, taking into account the scaling properties of both the volume and connectivity. The authors use a Green's function method that relates the diffusion problem to the density of states. They find that proper mode counting requires a reciprocal space with a new fractional dimensionality, given by $\overline{d} = 2\overline{d}/(2 + \overline{\delta})$, where $\overline{d}$ is the effective dimensionality and $\overline{\delta}$ is the exponent characterizing the diffusion constant's dependence on distance. For percolation clusters, they find $\overline{d} = 4/3$, independent of the Euclidean dimensionality. They discuss the crossover to normal behavior at low frequencies for finite fractals and percolation above the percolation threshold $p_c$. The relevance of their predictions to experimental results on proteins is also discussed. The authors show that the usual relationship between the density of states and the Euclidean dimensionality does not apply. The density of states cannot be described in terms of an anomalous dimensionality alone but requires an additional index describing the internal structure. They determine a fracton dimensionality $\overline{d}$ of the relevant reciprocal space that ensures proper mode counting for a fractal in terms of the index governing diffusion $\overline{\delta}$ and the anomalous or fractal dimension $\overline{d}$. The density of states for free particles and lattice vibrations is determined. These results are applied to polymer chains, the triangular Sierpinski gasket, and percolation networks. The authors were motivated by recent work on the ESR spin-lattice relaxation time of iron in proteins, which suggested an anomalous vibrational density of states due to a fractal structure. However, they included only the anomalous dimensionality $\overline{d}$ in their analysis. The structure of the diffusion equation allows it to be mapped onto a master equation, which has the same form as the free particle Schrödinger equation and the equation of motion for mechanical vibrations. This enables the mapping of the eigenvalue density of states for the quantum vibrational problem onto the eigenvalue density of states of the diffusion problem. The latter can be obtained from the single site Green's function for the diffusion problem. The authors find that the fracton dimensionality $\overline{d}$ is an intrinsic property of the fractal geometry and differs from the mass scaling exponent, the fractal dimensionality, and the diffusion constant scaling exponent. The results are applied to various examples, including a one-dimensional chain, a triangular Sierpinski gasket, and a critical percolation network. The paper also discusses crossover and finite size effects, and the relevance of the results to experimental measurements on proteins.This paper calculates the density of states on fractals, taking into account the scaling properties of both the volume and connectivity. The authors use a Green's function method that relates the diffusion problem to the density of states. They find that proper mode counting requires a reciprocal space with a new fractional dimensionality, given by $\overline{d} = 2\overline{d}/(2 + \overline{\delta})$, where $\overline{d}$ is the effective dimensionality and $\overline{\delta}$ is the exponent characterizing the diffusion constant's dependence on distance. For percolation clusters, they find $\overline{d} = 4/3$, independent of the Euclidean dimensionality. They discuss the crossover to normal behavior at low frequencies for finite fractals and percolation above the percolation threshold $p_c$. The relevance of their predictions to experimental results on proteins is also discussed. The authors show that the usual relationship between the density of states and the Euclidean dimensionality does not apply. The density of states cannot be described in terms of an anomalous dimensionality alone but requires an additional index describing the internal structure. They determine a fracton dimensionality $\overline{d}$ of the relevant reciprocal space that ensures proper mode counting for a fractal in terms of the index governing diffusion $\overline{\delta}$ and the anomalous or fractal dimension $\overline{d}$. The density of states for free particles and lattice vibrations is determined. These results are applied to polymer chains, the triangular Sierpinski gasket, and percolation networks. The authors were motivated by recent work on the ESR spin-lattice relaxation time of iron in proteins, which suggested an anomalous vibrational density of states due to a fractal structure. However, they included only the anomalous dimensionality $\overline{d}$ in their analysis. The structure of the diffusion equation allows it to be mapped onto a master equation, which has the same form as the free particle Schrödinger equation and the equation of motion for mechanical vibrations. This enables the mapping of the eigenvalue density of states for the quantum vibrational problem onto the eigenvalue density of states of the diffusion problem. The latter can be obtained from the single site Green's function for the diffusion problem. The authors find that the fracton dimensionality $\overline{d}$ is an intrinsic property of the fractal geometry and differs from the mass scaling exponent, the fractal dimensionality, and the diffusion constant scaling exponent. The results are applied to various examples, including a one-dimensional chain, a triangular Sierpinski gasket, and a critical percolation network. The paper also discusses crossover and finite size effects, and the relevance of the results to experimental measurements on proteins.