A string \alpha\in\Sigma^n is called {\it p-periodic},
if for every i,j \in \{1,\dots,n\}, such that i\equiv j \bmod p,
\alpha_i = \alpha_{j}, where \alpha_i is the i-th place of \alpha.
A string \alpha\in\Sigma^n is said to be period(\leq g),
if there exists p\in \{1,\dots,g\} such that \alpha is {\it p-periodic}.

An \epsilon
property tester for period(\leq g) is a randomized algorithm, that
for an input \alpha distinguishes between the case that \alpha is in
period(\leq g) and the case that
one needs to change at least \epsilon-fraction of the letters of
\alpha, so that it will become period(\leq g). The complexity of
the tester is the number of letter-queries it makes to the input.
We study here the complexity of \epsilon testers for period(\leq g)
when g varies in the range 1,\dots,\frac{n}{2}.
We show that there exists a surprising exponential phase
transition in the query complexity around g=\log n.
That is, for every \delta > 0 and for each g,
such that g\geq (\log{n})^{1+\delta},
the number of queries required and sufficient for testing period(\leq g)
is polynomial in g.
On the other hand, for each g\leq \frac{log{n}}{4}, the number of
queries required and sufficient for testing period(\leq g)
is only poly-logarithmic in g.

We also prove an exact asymptotic
bound for testing general periodicity. Namely,
that 1-sided error, non adaptive \epsilon-testing
of periodicity (period(\leq \frac{n}{2}))
is \Theta(\sqrt{n\log{n}}) queries.