  Periodic Geometry studies isometry invariants of periodic point sets that are
also continuous under perturbations. The motivations come from periodic
crystals whose structures are determined in a rigid form but any minimal cells
can discontinuously change due to small noise in measurements. For any integer
k>=0, the density function of a periodic set S was previously defined as the
fractional volume of all k-fold intersections (within a minimal cell) of balls
that have a variable radius t and centers at all points of S. This paper
introduces the density functions for periodic sets of points with different
initial radii motivated by atomic radii of chemical elements and by continuous
events occupying disjoint intervals in time series. The contributions are
explicit descriptions of the densities for periodic sequences of intervals. The
new densities are strictly stronger and distinguish periodic sequences that
have identical densities in the case of zero radii.
