  The $q$-state clock model with the cosine potential has a single phase
transition for $q\leq4$ and two transitions for $q\geq5$. It is shown by Monte
Carlo simulations that the helicity modulus for the five-state clock model
($q=5$) does not vanish at the high-temperature transition. This is in contrast
to the clock models with $q\geq6$ for which the helicity modulus vanishes. This
means that the transition for the five-state clock model differs from the
Kosterlitz-Thouless (KT) transition. It is also shown that this change in the
transition is caused by an interplay between the number of angular directions
and the interaction potential: by slightly modifying the interaction potential,
the KT transition for $q=6$ turns into the same non-KT transition. Likewise,
the KT transition is recovered for $q=5$ when the Villain potential is used.
Comparisons with other clock-model results are made and discussed.
