  We construct a canonical family of even periodic $\mathbb{E}_{\infty}$-ring
spectra, with exactly one member of the family for every prime $p$ and
chromatic height $n$. At height $1$ our construction is due to Snaith, who
built complex $K$-theory from $\mathbb{CP}^{\infty}$. At height $2$ we replace
$\mathbb{CP}^{\infty}$ with a $p$-local retract of $\mathrm{BU} \langle 6
\rangle$, producing a new theory that orients elliptic, but not generic, height
$2$ Morava $E$-theories.
  In general our construction exhibits a kind of redshift, whereby
$\mathrm{BP}\langle n-1 \rangle$ is used to produce a height $n$ theory. A
familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and
Yagita, relates the $K(n)$-localization of our height $n$ ring to work of
Peterson and Westerland building $E_n^{hS\mathbb{G}^{\pm}}$ from
$\mathrm{K}(\mathbb{Z},n+1)$.
