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The dating of a possible European business cycle has been inconclusive. At this stage, there is no consensus on the existence of such a cycle, or of its periodicity and amplitude, or of the relationship of individual member countries to that cycle. Yet cyclical convergence is the key consideration for countries that wish to be members of the currency union. The confusion over whether and to what degree the UK is converging on the cycles of its European partners, or whether its cycle is more in line with the US, is one example of this lack of consensus. Moreover, countries will vary in the components and characteristics that make up their output cycles at any moment, as well as in the state of their cycle. In this paper we show how to decompose a business cycle into a time-frequency framework. This allows us to decompose movements in output, both at the European level and in member countries, into their component cycles and allows those component cycles (and the coherence between them) to vary in their importance and cyclical characteristics. That then allows us to determine if the inconclusive convergence results obtained so far have appeared because member countries have some cycles in common, but diverge at other frequencies.