Driven many-body systems have been shown to exhibit discrete time crystal phases characterized by broken discrete time-translational symmetry. This has been achieved generally through a subharmonic response, in which the system undergoes one oscillation every other driving period. Here, we demonstrate that classical time crystals do not need to resonate in a subharmonic fashion but instead can also exhibit a continuously tunable anharmonic response to driving, which we show can emerge through a coresonance between modes in different branches of the dispersion relation in a parametrically driven medium. This response, characterized by a typically incommensurate ratio between the resonant frequencies and the driving frequency, is demonstrated by introducing a time crystal model consisting of an array of coupled pendula with alternating lengths. Importantly, the coresonance mechanism is the result of a bifurcation involving a fixed point and an invariant torus, with no intermediate limit cycles. This bifurcation thus gives rise to a many-body symmetry-breaking phenomenon directly connecting the symmetry-unbroken phase with a previously uncharacterized phase of matter, which we call an anharmonic time crystal phase. The mechanism is shown to generalize to driven media with any number of coupled fields and is expected to give rise to anharmonic responses in a range of weakly damped pattern-forming systems, with potential applications to the study of nonequilibrium phases, frequency conversion, and acoustic cloaking.
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