Harmonices Mundi (Latin: The Harmony of the World; full title Harmonices mundi libri V, “The Five Books of the Harmony of the World”) is a book by the German astronomer Johannes Kepler, published in 1619 in Linz. The work, written entirely in Latin, treats harmony and congruence in geometrical forms and physical phenomena. It is divided into five chapters: the first on regular polygons, the second on the congruence of geometric figures, the third on the origin of harmonic proportions in music, the fourth on harmonic configurations in astrology, and the fifth on the harmony of the motions of the planets. The book's final section contains Kepler's discovery of what is now called the third law of planetary motion. Interpretations of the work's place in the history of science differ sharply, ranging from a landmark of mathematical astronomy to a late expression of Pythagorean and astrological cosmology; see harmonices-mundi-historiographical-status-viewpoint-debate.
Kepler began working on the ideas behind Harmonices Mundi around 1599. The book followed his 1609 Astronomia Nova, in which he had published the first two laws of planetary motion (elliptical orbits and the equal-area law); Astronomia Nova related the discovery of the first two principles now known as Kepler's laws. For fuller background on the intervening decade and Kepler's broader astronomical program, see Astronomia Nova and harmonices-mundi-history.
Kepler formulated the third law on 15 May 1618 (New Style) and published it the following year in Book V of Harmonices Mundi. The law states that the ratio of the cube of a planet's orbital semi-major axis to the square of its orbital period is the same constant for every planet. In Book V, Kepler also describes the orbital motion of the planets and argues that this motion nearly matches musical harmonies, deriving these from the ratio of each planet's maximum to minimum angular speed at perihelion and aphelion. Kepler assigned musical scales to the six known planets-Saturn, Jupiter, Mars, Earth, Venus, and Mercury-with the highest and lowest notes of each set by the planet's angular velocity relative to the Sun.
The book sits within the much older tradition of the musica universalis or “music of the spheres,” an idea generally credited to Pythagoras and developed within the medieval quadrivium, which held that the proportions among the motions of celestial bodies form a form of music-mathematical rather than audible. Kepler departed from this tradition in one key respect: where medieval philosophers spoke of the music of the spheres only metaphorically, Kepler argued for physical harmonic relationships actually present in the planets' motions. He also abandoned Pythagorean tuning in favor of geometrically derived musical ratios, which he believed showed God acting as a geometer rather than a numerologist. Kepler's framing of the work as theological as much as astronomical, and the relationship between his religious commitments and his mathematics, is treated at harmonices-mundi-religious-motivation-viewpoint; see also Heliocentrism - Protestant Viewpoint for the wider confessional context of early heliocentric astronomy.
One unresolved feature of Kepler's harmonic scheme was the ratio he calculated between Mars and Jupiter, 18:19, which corresponded to no musical interval he recognized and which he could not explain. The 1801 discovery of the asteroid belt in the gap between those two orbits is sometimes noted in retrospective discussions of the book, though Kepler's harmonic framework did not predict it in any rigorous sense; see harmonices-mundi-mars-jupiter-gap-debate.
Soon after publication, Kepler became embroiled in a priority and substance dispute with the English physician and Hermetic philosopher Robert Fludd, who had published his own harmonic theory shortly before. Kepler had attacked Fludd's theory of world harmony in an appendix to Harmonices Mundi, and Fludd responded in turn, with Kepler issuing a further reply. Contemporaries characterized the exchange as firm but collegial scholarly disagreement, in contrast to attacks on Fludd by other critics that sought to damage his reputation through association with magic. Some later musicological assessments hold that Kepler and Marin Mersenne judged Fludd's method false and his Pythagorean approach outmoded; the extent to which this view is dominant in the field is disputed. Details of the dispute are at harmonices-mundi-fludd-controversy.
Harmonices Mundi was, by some accounts, the least-read of Kepler's major works in the years immediately following its publication, with few references to it in subsequent astronomical literature compared to his later Epitome Astronomiae Copernicanae. Its modern reputation rests substantially on Book V's third law rather than on the harmonic project as a whole; this gap between contemporary and later reception is discussed further at harmonices-mundi-reception-history.