Musical scales have both general and culture-specific properties. While most common scales use octave equivalence and discrete pitch relationships, there seem to be no other universal properties. This paper presents an additional property across the world's musical scales that may qualify for universality. When the intervals of 998 (just intonation) scales from the Scala Archive are represented on an Euler lattice, 96.7% of them form starconvex structures. For the subset of traditional scales this percentage is even 100%. We present an attempted explanation for the star-convexity feature, suggesting that the mathematical search for universal musical properties has not yet reached its limits.