Yes, it is possible to have music that consists solely of rhythm without any melodic or harmonic elements. This type of music is often referred to as "pure rhythm" or "rhythmic music." It can involve various percussion instruments or body percussion (like clapping or stomping) that emphasize rhythm, tempo, and dynamics.
In many musical traditions, especially in genres like African drumming, certain forms of dance music, and minimalist compositions, rhythm plays a central role while melodic and harmonic components may be minimal or entirely absent. Additionally, some contemporary music practices, such as certain forms of electronic music or experimental compositions, may prioritize rhythm as the primary element, creating a compelling musical experience based solely on rhythmic patterns.
Rhythm alone can evoke emotions, create tension, and inspire movement, showing that sound isn't the only defining feature of music; rhythm can stand on its own as a significant musical element.