  The recently elucidated structure of the delafossite YCuO$_{2.5}$ reveals a
Cu-O network with nearly independent $\Delta$ chains having different
interactions between the $s=1/2$ spins. Motivated by this result, we study the
$\Delta$ chain for various ratios $J_{\rm bb}/J_{\rm bv}$ of the base-base and
base-vertex interactions. By exact diagonalization and extrapolation, we show
that the elementary excitation spectrum, which (within numerical error) is the
same for total spins $S_{\rm tot}=0$ and 1, has a gap only in the interval
$0.4874(1) \leq J_{\rm bb}/J_{\rm bv} \leq 1.53(1)$. The gap is dispersionless
for $J_{\rm bb}/J_{\rm bv}=1$, but has increasing $k$-dependence as $J_{\rm
bb}/J_{\rm bv}$ moves away from unity, related to the instability of dimers in
the ground state.
