  This article presents, in an illustrative fashion, a first step towards an
extension of the spectral theory of constant length substitutions. Our starting
point is the general observation that the symbolic picture (as defined by the
substitution rule) and its geometric counterpart with natural prototile sizes
(as defined by the induced inflation rule) may differ considerably. On the
geometric side, an aperiodic inflation system possesses a set of exact
renormalisation relations for its pair correlation coefficients. Here, we
derive these relations for some paradigmatic examples and infer various
spectral consequences. In particular, we consider the Fibonacci chain, revisit
the Thue--Morse and the Rudin--Shapiro sytem, and finally analyse a twisted
extension of the silver mean chain with mixed singular spectrum.
