  We consider the localized nonlinear breathing modes that emerge in
heterogeneous granular configurations of two materials with a periodicity of
three and four beads. We examine as characteristic examples chains with 1 steel
and 2 alumnium beads, as well as ones with 1 steel and three aluminum beads. We
analyze the higher order gaps that emerge in such settings and explore the
intrinsic localized modes that bifurcate from the edge of the upper bands. A
generic surprising feature of such states is that they appear to be more robust
than their counterparts bifurcating from the edges of the lower bands. Direct
numerical simulations, using driving of the system at suitable frequencies
through an actuator or taking advantage of the modulational instabilities of
extended band edge states in the system illustrate the spontaneous formation of
localized modes within the corresponding nearest gaps.
