  Topological Spherical T-duality was introduced by Bouwknegt, Evslin and
Mathai in [BEM15] as an extension of topological T-duality from $S^1$ bundles
to $\mathrm{SU}(2)$-bundles endowed with closed 7-forms. This notion was
further extended to sphere bundles by Lind, Sati and Westerland [LSW16] as a
duality between $S^{2n-1}$-bundles endowed with closed $(4n-1)$-forms. We
generalise this relation one step further and define T-duality for
$S^{2n-1}$-bundles endowed with closed odd forms of arbitrary degree. The
degree of the form determines the dimension of the fibers of the dual spaces.
We show that $T$-duals exist and, as in the previous cases, $T$-dual spaces
have isomorphic twisted cohomology.
