  In this paper, we introduce a new wavelet tool for studying the degree of
non-periodicity of time series that is based on some recently defined tools,
such as the \textit{windowed scalogram} and the \textit{scale index}. It is
especially appropriate for non-stationary time series whose characteristics
change over time and so, it can be applied to a wide variety of disciplines. In
addition, we revise the concept of the scale index and pose a theoretical
problem: it is known that if the scale index of a function is not zero then it
is non-periodic, but if the scale index of a function is zero, then it is not
proved that it has to be periodic. This problem is solved for the particular
case of the Haar wavelet, thus reinforcing the interpretation and applicability
of the scale index as a useful tool for measuring non-periodicity. Finally, we
discuss the relationship between non-periodicity and unpredictability,
comparing the new wavelet tool with the sample entropy.
