  For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$,
Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of
$\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$,
with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$
recovering the prismatic and syntomic cohomology of $R$, respectively. We give
an alternate construction of these filtrations that applies also when $R$ is a
well-behaved commutative ring spectrum; for example, we can take $R$ to be
$\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$.
We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$
and observe that, when $p \ge 3$, the motivic spectral sequence for
$V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.
