Since Allen et al. demonstrated 30 years ago that beams with helical wavefronts carry orbital angular momentum (OAM), the OAM of beams has attracted extensive attention and stimulated lots of applications in both classical and quantum physics. Akin to an optical frequency comb, a beam can carry multiple various OAM components simultaneously. A series of discrete, equally spaced, and equally weighted OAM modes comprise an OAM comb. Inspired by the spatially extended laser lattice, we demonstrate both theoretically and experimentally an approach to producing OAM combs through azimuthal binary phases. Our study shows that transition points in the azimuth determine the OAM distributions of diffracted beams. Multiple azimuthal transition points lead to a wide OAM spectrum. Moreover, an OAM comb with any mode spacing is achievable through reasonably setting the position and number of azimuthal transition points. The experimental results fit well with theory. This work presents a simple approach that opens new prospects for OAM spectrum manipulation and paves the way for many applications including OAM-based high-security encryption and optical data transmission, and other advanced applications.