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The vibrational spectrum of a one-dimensional disordered chain of atoms with light isotopic substitutional impurities is calculated numerically for frequencies near a local mode of an isolated impurity atom. It is shown that in the entire frequency region accessible to the calculation, the spectrum does indeed have the characteristic features of a fractal. Series of lines clustered around the lines of an isolated impurity atom or of two and three impurity atoms are examined. It is shown that in these cases a self-similar structure is preserved at arbitrary reductions of the scale both outside and inside the concentration broadening region. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]