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Harmonices Mundi
See also: Harmonices Mundi

Musica universalis, which had existed since the Greeks, as a metaphysical concept was often taught in quadrivium, and this connection between music and astronomy intrigued Johannes Kepler, and he devoted much of his time after publishing the Mysterium Cosmographicum (Mystery of the Cosmos) looking over tables and trying to fit the data to what he believed to be the true nature of the cosmos. In 1619 Kepler published Harmonices Mundi (literally Harmony of the Worlds), expanding on the concepts he introduced in Mysterium and positing that musical intervals and harmonies describe the motions of the six known planets of the time. He believed that this harmony, while inaudible, could be heard by the soul, and that it gave a “very agreeable feeling of bliss, afforded him by this music in the imitation of God.” In Harmonices, Kepler laid out an argument for a creator who had made an explicit connection between geometry, astronomy, and music, and that the planets were arranged in intelligently.

Harmonices is split into five books, or chapters. The first and second books give a brief discussion on regular polyhedron and their congruences, reiterating the idea he introduced in Mysterium that the five regular solids known about since antiquity define the orbits of the planets and their distances from the sun. Book three focuses on defining musical harmonies, including consonance and dissonance, intervals (including the problems of just tuning), their relations to string length, and what makes music pleasurable to listen to. In the fourth book Kepler presents a metaphysical basis for this system, along with arguments for why the harmony of the worlds appeals to the intellectual soul in the same manner as the harmony of music appeals to the human soul. Here he also uses the naturalness of this harmony as an argument for heliocentrism. In book five, Kepler describes in detail the orbital motion of the planets and how this motion nearly perfectly matches musical harmonies. Finally, after a discussion on astrology in book five, Kepler ends Harmonices by describing his third law, which states that for any planet the cube of the semi-major axis of its elliptical orbit is proportional to the square of its orbital period. In the final book of Harmonices, Kepler explains how the ratio of the maximum and minimum angular speeds of each planet (its speeds at the perihelion and aphelion) is very nearly equivalent to a consonant musical interval. Furthermore, the ratios between these extreme speeds of the planets compared against each other create even more mathematical harmonies. These speeds explain the eccentricity of the orbits of the planets in a natural way that appealed to Kepler’s religious beliefs in a heavenly creator.

While Kepler did believe that the harmony of the worlds was inaudible, he related the motions of the planets to musical concepts in book four of Harmonices. He makes an analogy between comparing the extreme speeds of one planet and the extreme speeds of multiple planets with the difference between monophonic and polyphonic music. Because planets with larger eccentricities have a greater variation in speed they produce more “notes.” Earth’s maximum and minimum speeds, for example, are in a ratio of roughly 16 to 15, or that of a semitone, whereas Venus’ orbit is nearly circular, and therefore only produces a singular note. Mars, which has the largest eccentricity, has the largest interval, a minor tenth, or a ratio of 12 to 5. This range, as well as the relative speeds between the planets, led Kepler to conclude that the solar system was comprised of two basses (Saturn and Jupiter), a tenor (Mars), two altos (Venus and Earth), and a soprano (Mercury), which had sung in “perfect concord,” at the beginning of time, and could potentially arrange themselves to do so again. He was certain of the link between musical harmonies and the harmonies of the heavens and believed that “man, the imitator of the Creator,” had emulated the polyphony of the heavens so as to enjoy “the continuous duration of the time of the world in a fraction of an hour.”

Kepler was so convinced in a creator that he was convinced of the existence of this harmony despite a number of inaccuracies present in Harmonices. Many of the ratios differed by an error greater than simple measurement error from the true value for the interval, and the ratio between Mars’ and Jupiter’s angular velocities does not create a consonant interval, though every other combination of planets does. Kepler brushed aside this problem by making the argument, with the math to support it, that because these elliptical paths had to fit into the regular solids described in Mysterium the values for both the dimensions of the solids and the angular speeds would have to differ from the ideal values to compensate. This change also had the benefit of helping Kepler retroactively explain why the regular solids encompassing each planet were slightly imperfect.