  A proper total colouring of a graph $G$ is called harmonious if it has the
further property that when replacing each unordered pair of incident vertices
and edges with their colours, then no pair of colours appears twice. The
smallest number of colours for it to exist is called the harmonious total
chromatic number of $G$, denoted by $h_t(G)$. Here, we give a general upper
bound for $h_t(G)$ in terms of the order $n$ of $G$. Our two main results are
obvious consequences of the computation of the harmonious total chromatic
number of the complete graph $K_n$ and of the complete multigraph $\lambda
K_n$, where $\lambda$ is the number of edges joining each pair of vertices of
$K_n$. In particular, Araujo-Pardo et al. have recently shown that
$\frac{3}{2}n\leq h_t(K_n) \leq \frac{5}{3}n +\theta(1)$. In this paper, we
prove that $h_t(K_{n})=\left\lceil \frac{3}{2}n \right\rceil$ except for
$h_t(K_{1})=1$ and $h_t(K_{4})=7$; therefore, $h_t(G) \le \left\lceil
\frac{3}{2}n \right\rceil$, for every graph $G$ on $n>4$ vertices. Finally, we
extend such a result to the harmonious total chromatic number of the complete
multigraph $\lambda K_n$ and as a consequence show that $h_t(\mathcal{G})\leq
(\lambda-1)(2\left\lceil\frac{n}{2}\right\rceil-1)+\left\lceil\frac{3n}{2}\right\rceil$
for $n>4$, where $\mathcal{G}$ is a multigraph such that $\lambda$ is the
maximum number of edges between any two vertices.
