How did rhythm originate in humans, and other species? One cross-cultural universal, frequently found in human music, is isochrony: when note onsets repeat regularly like the ticking of a clock. Another universal consists in synchrony (e.g. when individuals coordinate their notes so that they are sung at the same time). An approach to biomusicology focuses on similarities and differences across species, trying to build phylogenies of musical traits. Here we test for the presence of, and a link between, isochrony and synchrony in a non-human animal. We focus on the songs of one of the few singing primates, the lar gibbon (Hylobates lar), extracting temporal features from their solo songs and duets. We show that another ape exhibits one rhythmic feature at the core of human musicality: isochrony. We show that an enhanced call rate overall boosts isochrony, suggesting that respiratory physiological constraints play a role in determining the song's rhythmic structure. However, call rate alone cannot explain the flexible isochrony we witness. Isochrony is plastic and modulated depending on the context of emission: gibbons are more isochronous when duetting than singing solo. We present evidence for rhythmic interaction: we find statistical causality between one individual's note onsets and the co-singer's onsets, and a higher than chance degree of synchrony in the duets. Finally, we find a sex-specific trade-off between individual isochrony and synchrony. Gibbon's plasticity for isochrony and rhythmic overlap may suggest a potential shared selective pressure for interactive vocal displays in singing primates. This pressure may have convergently shaped human and gibbon musicality while acting on a common neural primate substrate. Beyond humans, singing primates are promising models to understand how music and, specifically, a sense of rhythm originated in the primate phylogeny.