Organisms use circadian clocks to generate 24-h rhythms in gene expression. However, the clock can interact with other pathways to generate shorter period oscillations. It remains unclear how these different frequencies are generated. Here, we examine this problem by studying the coupling of the clock to the alternative sigma factor $sigC$ in the cyanobacterium $Synechococcus\; elongatus$. Using single-cell microscopy, we find that $psbAI$, a key photosynthesis gene regulated by both $sigC$ and the clock, is activated with two peaks of gene expression every circadian cycle under constant low light. This two-peak oscillation is dependent on $sigC$, without which $psbAI$ rhythms revert to one oscillatory peak per day. We also observe two circadian peaks of elongation rate, which are dependent on $sigC$, suggesting a role for the frequency doubling in modulating growth. We propose that the two-peak rhythm in $psbAI$ expression is generated by an incoherent feedforward loop between the clock, $sigC$ and $psbAI$. Modelling and experiments suggest that this could be a general network motif to allow frequency doubling of outputs.