  A $t$-tone $k$-coloring of $G$ assigns to each vertex of $G$ a set of $t$
colors from $\{1,..., k\}$ so that vertices at distance $d$ share fewer than
$d$ common colors. The {\it $t$-tone chromatic number} of $G$, denoted
$\tau_t(G)$, is the minimum $k$ such that $G$ has a $t$-tone $k$-coloring.
Bickle and Phillips showed that always $\tau_2(G) \le [\Delta(G)]^2 +
\Delta(G)$, but conjectured that in fact $\tau_2(G) \le 2\Delta(G) + 2$; we
confirm this conjecture when $\Delta(G) \le 3$ and also show that always
$\tau_2(G) \le \ceil{(2 + \sqrt{2})\Delta(G)}$. For general $t$ we prove that
$\tau_t(G) \le (t^2+t)\Delta(G)$. Finally, for each $t\ge 2$ we show that there
exist constants $c_1$ and $c_2$ such that for every tree $T$ we have $c_1
\sqrt{\Delta(T)} \le \tau_t(T) \le c_2\sqrt{\Delta(T)}$.
