Melody harmonization, which involves generating a chord progression that complements a user-provided melody, continues to pose a significant challenge. A chord progression must not only be in harmony with the melody, but also interdependent on its rhythmic pattern. While previous neural network-based systems have been successful in producing chord progressions for given melodies, they have not adequately addressed controllable melody harmonization, nor have they focused on generating harmonic rhythms with flexibility in the rates or patterns of chord changes. This paper presents AutoHarmonizer, a novel system for harmonic density-controllable melody harmonization with such a flexible harmonic rhythm. AutoHarmonizer is equipped with an extensive vocabulary of 1,462 chord types and can generate chord progressions that vary in harmonic density for a given melody. Experimental results indicate that the AutoHarmonizer-generated chord progressions exhibit a diverse range of harmonic rhythms and that the system's controllable harmonic density is effective.