  We consider four measures of contextuality, chosen for being based on the
fundamental properties of the notion of contextuality, and for being applicable
to arbitrary systems of measurements, both without and with disturbance. We
have previously shown that no two of them are functions of each other: as
systems of measurements change, either of them can change, while the other
remains constant. This means that they measure different aspects of
contextuality, and we proposed that rather than picking just one measure of
contextuality in one specific sense, one could use all of them to characterize
a contextual system by its pattern of contextuality. To study patterns of
contextuality, however, one needs a systematic way of varying systems of
measurements, which requires their convenient parametrization. We have
convenient parametrization within the class of cyclic systems that have played
a dominant role in the foundations of quantum mechanics. However, they cannot
be used to study patterns of contextuality, because within this class the four
measures of contextuality have been shown to be proportional to each other. In
this concept paper, we introduce hypercyclic systems of measurements. They
generalize cyclic systems while preserving convenient parametrization. We show
that within this class of systems, the same as for systems at large, no two of
the measures of contextuality are functions of each other. This means that
hypercyclic systems can be used to study patterns of contextuality.
