  Building on work of Brandt and Terao in their study of $k$-formality, we
introduce a co-chain complex associated to a multi-arrangement and prove that
its cohomologies determine freeness of the associated module of
multi-derivations. This provides a new homological method for determining
freeness of arrangements and multi-arrangements. We work out many applications
of this homological method. For instance, we prove that if a multi-arrangement
is free then the underlying arrangement is $k$-formal for all $k\ge 2$. We also
use this method to completely characterize freeness of certain families of
multi-arrangements in moduli, showcasing how the geometry of multi-arrangements
with the same intersection lattice may have considerable impact on freeness.
New counter-examples to Orlik's conjecture also arise in connection to this
latter analysis.
