Did the ancient Greeks know the natural overtone series?

Music theory, which developed in the course of a single musical-mathematical-astronomical synthesis, does not give a direct answer to the question posed – "did the ancient Greeks know the natural overtone series" as a reasonable system. Nevertheless, the musical theory in the preserved sources-original works, their fragments, compilations – came quite close to understanding of the existening of the phenomenon much later called "natural overtone series". The article attempts to find a correspondence between individual mathematically calculated intervals, the main system-forming generic tetrachordal structures (diaton, chroma, enharmony) and segments of the natural overtone series as evidence of the unity of the scientific and empirical (auditory) approach to achieving the best sound of Music, universal Harmony. The article is a revised and expanded version of the Lecture for students and teachers of conservatories "Musical acoustics in the works of ancient scientists" (Novosibirsk, 2009).