Kepler built Harmonices Mundi as a sustained argument moving from pure mathematics to cosmology across five books. The first two books lay out the geometry of regular figures, which shapes can be constructed and which combinations of angles produce a genuine harmonic proportion, treating geometry itself as the source of harmony rather than as a separate subject applied to music afterward. The third book turns explicitly to musical theory, deriving the consonant intervals, the octave, the fifth, the third, from the same proportions established geometrically in the first two books, so that music becomes a special case of a more general geometric harmony rather than an independent art.
The fourth book applies that harmonic framework to astrology, an application Kepler treated seriously despite his reputation as a founder of modern astronomy, arguing that harmonic aspects between planets produce real effects on earthly life and weather. It is the fifth book that carries the work's most lasting scientific content: Kepler calculates the angular velocity of each planet as seen from the sun at its closest and farthest points, expresses the ratios between those velocities as musical intervals, assigning each planet its own characteristic "melody," and in the course of that calculation states what is now called his Third Law, that the square of a planet's orbital period is proportional to the cube of its distance from the sun. For Kepler, that law was not a stray mathematical footnote; it was the proof, arrived at after years of searching, that the harmony he had set out to demonstrate in Book I was actually built into the structure of the solar system.