We study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals which includes 3n-gaskets. We consider such examples as the Sierpinski gasket, a non-p.c.f. analog of the Sierpinski gasket, the level-3 Sierpinski gasket, a fractal 3-tree, the hexagasket, and one dimensional fractals. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations.

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