  We show how to compute the spectral flow of the odd signature operator $\pm
*d_{a_t}-d_{a_t}*$ along an analytic path of flat connections $a_t$ on a bundle
over a closed odd-dimensional manifold in terms of Massey products in the DGLA
of bundle-valued differential forms. To obtain this information, we set up a
sequence of cochain complexes $\{\calg^*_n,\delta_n\}$, for $n=0,1,2,\ldots$
and Hermitian forms $$Q_n:\calg_n\times\calg_n\ra \bbbC$$ whose signatures
determine the spectral flow through $t=0$. The complexes and Hermitian forms
are constructed using Massey products.
