We give subquadratic algorithms that, given two necklaces each with $n$ beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for $p = 1$, $p$ even, and $p = ınfty$. For $p$ even, we reduce the problem to standard convolution, while for $p = ınfty$ and $p = 1$, we reduce the problem to $(min, +)$ convolution and $(median, +)$ convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting $X + Y$ problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the $X + Y$ matrix. All of our algorithms run in $o(n^2)$ time, whereas the obvious algorithms for these problems run in $Θ(n^2)$ time.