  We show that for each natural $n>1$ it is consistent that there is a compact
Hausdorff space $K_{2n}$ such that in $C(K_{2n})$ there is no uncountable
(semi)biorthogonal sequence $(f_\xi,\mu_\xi)_{\xi\in \omega_1}$ where
$\mu_\xi$'s are atomic measures with supports consisting of at most $2n-1$
points of $K_{2n}$, but there are biorthogonal systems $(f_\xi,\mu_\xi)_{\xi\in
\omega_1}$ where $\mu_\xi$'s are atomic measures with supports consisting of
$2n$ points. This complements a result of Todorcevic that it is consistent that
each nonseparable Banach space $C(K)$ has an uncountable biorthogonal system
where the functionals are measures of the form $\delta_{x_\xi}-\delta_{y_\xi}$
for $\xi<\omega_1$ and $x_\xi,y_\xi\in K$. It also follows that it is
consistent that the irredundance of the Boolean algebra $Clop(K)$ or the Banach
algebra $C(K)$ for $K$ totally disconnected can be strictly smaller than the
sizes of biorthogonal systems in $C(K)$. The compact spaces exhibit an
interesting behaviour with respect to known cardinal functions: the hereditary
density of the powers $K_{2n}^k$ is countable up to $k=n$ and it is uncountable
(even the spread is uncountable) for $k>n$.
