  The torus group $(S^1)^{\ell+1}$ has a canonical action on the odd
dimensional sphere $S_q^{2\ell+1}$. We take the natural Hilbert space
representation where this action is implemented and characterize all odd
spectral triples acting on that space and equivariant with respect to that
action. This characterization gives a construction of an optimum family of
equivariant spectral triples having nontrivial $K$-homology class thus
generalizing our earlier results for $SU_q(2)$. We also relate the triple we
construct with the $C^*$-extension \[
  0\longrightarrow \clk\otimes C(S^1)\longrightarrow C(S_q^{2\ell+3})
  \longrightarrow C(S_q^{2\ell+1}) \longrightarrow 0. \]
