  Spectral asymptotics of the Neumann problem for the Sturm-Liouville equation
with generalized derivative of a self-similar generalized Cantor type function
as a weight are considered. The spectrum is shown to have a periodicity
property for a wide class of Cantor type self-similar functions, and also the
weaker "quasi-periodicity" condition is demonstrated under certain mixed
boundary conditions. This allows for a more precise description of the main
term of the asymptotics of the counting function of eigenvalues. Previous
results by A. A. Vladimirov, I. A. Sheipak are generalized.
