Mammalian circadian clocks consist of complex regulatory loops mediated through--at least--morning, daytime and night-time DNA elements. To prove the transcriptional logic of mammalian clocks, we developed an in cellulo mammalian cell-culture system that allowed us to design and implement artificial transcriptional circuits. Here we show that morning activation and night-time repression can yield the transcriptional output during the daytime, and similarly that daytime activation and morning repression can yield night-time transcriptional output. We also observed that the diverse transcriptional outputs of other phases can be generated through the expression of simple combinations of transcriptional activators and repressors. These results reveal design principles not only for understanding the continuous transcriptional outputs observed in vivo but also for the logical construction of artificial promoters working at novel phases. Logical synthesis of artificial circuits, with an identified structure and observed dynamics, provides an alternative strategy applicable to the investigation of complex biological systems.