We addressed how rhythm complexity influences auditory–motor synchronization in musically trained individuals who perceived and produced complex rhythms while EEG was recorded. Participants first listened to two-part auditory sequences (Listen condition). Each part featured a single pitch presented at a fixed rate; the integer ratio formed between the two rates varied in rhythmic complexity from low (1:1) to moderate (1:2) to high (3:2). One of the two parts occurred at a constant rate across conditions. Then, participants heard the same rhythms as they synchronized their tapping at a fixed rate (Synchronize condition). Finally, they tapped at the same fixed rate (Motor condition). Auditory feedback from their taps was present in all conditions. Behavioral effects of rhythmic complexity were evidenced in all tasks; detection of missing beats (Listen) worsened in the most complex (3:2) rhythm condition, and tap durations (Synchronize) were most variable and least synchronous with stimulus onsets in the 3:2 condition. EEG power spectral density was lowest at the fixed rate during the 3:2 rhythm and greatest during the 1:1 rhythm (Listen and Synchronize). ERP amplitudes corresponding to an N1 time window were smallest for the 3:2 rhythm and greatest for the 1:1 rhythm (Listen). Finally, synchronization accuracy (Synchronize) decreased as amplitudes in the N1 time window became more positive during the high rhythmic complexity condition (3:2). Thus, measures of neural entrainment corresponded to synchronization accuracy, and rhythmic complexity modulated the behavioral and neural measures similarly.