  Equiangular tight frames (ETFs) and biangular tight frames (BTFs) - sets of
unit vectors with basis-like properties whose pairwise absolute inner products
admit exactly one or two values, respectively - are useful for many
applications. A well-understood class of ETFs are those which manifest as
harmonic frames - vector sets defined in terms of the characters of finite
abelian groups - because they are characterized by combinatorial objects called
difference sets.
  This work is dedicated to the study of the underlying combinatorial
structures of harmonic BTFs. We show that if a harmonic frame is generated by a
divisible difference set, a partial difference set or by a special structure
with certain Gauss summing properties - all three of which are generalizations
of difference sets that fall under the umbrella term "bidifference set" - then
it is either a BTF or an ETF. However, we also show that the relationship
between harmonic BTFs and bidifference sets is not as straightforward as the
correspondence between harmonic ETFs and difference sets, as there are examples
of bidifference sets that do not generate harmonic BTFs. In addition, we study
another class of combinatorial structures, the nested divisible difference
sets, which yields an example of a harmonic BTF that is not generated by a
bidifference set.
