The number of possible melodies is unfathomably large, yet despite this virtually unlimited potential for melodic variation, melodies from different societies can be surprisingly similar. The motor constraint hypothesis accounts for certain similarities, such as scalar motion and contour shape, but not for other major common features, such as repetition, song length, and scale size. Here we investigate the role of information constraints in shaping these hallmarks of melodies. We measure determinants of information rate in 62 corpora of Folk melodies spanning several continents, finding multiple trade-offs that all act to constrain the information rate across societies. By contrast, 39 corpora of Art music from Europe (including Turkey) show longer, more complex melodies, and increased complexity over time, suggesting different cultural-evolutionary selection pressures in Art and Folk music, possibly due to the use of written versus oral transmission. Our parameter-free model predicts the empirical scale degree distribution using information constraints on scalar motion, melody length, and, most importantly, information rate. These results provide strong evidence that information constraints during cultural transmission of music limit the number of notes in a scale, and suggests that a tendency for intermediate melodic complexity reflects a fundamental constraint on the cultural evolution of melody.