  We prove that Buchholz's system of fundamental sequences for the $\vartheta$
function enjoys various regularity conditions, including the Bachmann property.
We partially extend these results to variants of the $\vartheta$ function,
including a version without addition for countable ordinals. We conclude that
the Hardy functions based on these notation systems enjoy natural monotonicity
properties and majorize all functions defined by primitive recursion along
$\vartheta(\varepsilon_{\Omega+1})$.
