  We define the half-volume spectrum $\{\tilde \omega_p\}_{p\in \mathbb N}$ of
a closed manifold $(M^{n+1},g)$. This is analogous to the usual volume spectrum
of $M$, except that we restrict to $p$-sweepouts whose slices each enclose half
the volume of $M$. We prove that the Weyl law continues to hold for the
half-volume spectrum. We define an analogous half-volume spectrum $\tilde c(p)$
in the phase transition setting. Moreover, for $3 \le n+1 \le 7$, we use the
Allen-Cahn min-max theory to show that each $\tilde c(p)$ is achieved by a
constant mean curvature surface enclosing half the volume of $M$ plus a
(possibly empty) collection of minimal surfaces with even multiplicities.
