1619 first edition
Harmonices Mundi (Latin: The Harmony of the World, 1619) is a book by Johannes Kepler. In the work, written entirely in Latin, Kepler discusses harmony and congruence in geometrical forms and physical phenomena. The final section of the work relates his discovery of the so-called "third law of planetary motion".
It is estimated that Kepler had begun working on Harmonices Mundi sometime near 1599, which was the year Kepler sent a letter to Michael Maestlin detailing the mathematical data and proofs that he intended to use for his upcoming text, which he originally planned to name De harmonia mundi. Kepler was aware that the content of Harmonices Mundi closely resembled that of the subject matter for Ptolemy's Harmonica, but was not concerned, because the new astronomy Kepler would use, most notably the adoption of elliptic orbits in the Copernican system, allowed him to explore new theorems. Another important development that allowed Kepler to establish his celestial-harmonic relationships, was the abandonment of the Pythagorean tuning as the basis for musical consonance and the adoption of geometrically supported musical ratios; this would eventually be what allowed Kepler to relate musical consonance and the angular velocities of the planets. Thus Kepler, could reason that his relationships gave evidence for God acting as a grand geometer, rather than a Pythagorean numerologist.
The concept of musical harmonies intrinsically existing within the spacing of the planets existed in medieval philosophy prior to Kepler. Musica universalis was a traditional philosophical metaphor that was taught in the quadrivium, and was often called the "music of the spheres". Kepler was intrigued by this idea while he sought explanation for a rational arrangement of the heavenly bodies. It should be noted that when Kepler uses the term "harmony" it is not strictly referring to the musical definition, but rather, a broader definition encompassing congruence in Nature and the workings of both the celestial and terrestrial bodies. He notes musical harmony as being a product of man, derived from angles, in contrast to a harmony that he refers to as being a phenomenon that interacts with the human soul. In turn, this allowed Kepler to claim the Earth has a soul because it is subjected to astrological harmony.
Kepler divides The Harmony of the World into five long chapters: the first is on regular polygons; the second is on the congruence of figures; the third is on the origin of harmonic proportions in music; the fourth is on harmonic configurations in astrology; the fifth is on the harmony of the motions of the planets.
Chapters 1 and 2 of The Harmony of the World contain most of Kepler's contributions concerning polyhedra. He is primarily interested with how polygons, which he defines as either regular or semiregular, can come to be fixed together around a central point on a plane to form congruence. His primary objective was to be able to rank polygons based on a measure of sociability, or rather, their ability to form partial congruence when combined with other polyhedra. He returns to this concept later in Harmonices Mundi with relation to astronomical explanations. In the second chapter is the earliest mathematical understanding of two types of regular star polyhedra, the small and great stellated dodecahedron; they would later be called Kepler's solids or Kepler Polyhedra and, together with two regular polyhedra discovered by Louis Poinsot, as the Kepler-Poinsot polyhedra. He describes polyhedra in terms of their faces, which is similar to the model used in Plato's Timaeus to describe the formation of Platonic solids in terms of basic triangles. The book features illustrations of solids and tiling patterns, some of which are related to the golden ratio.
While medieval philosophers spoke metaphorically of the "music of the spheres", Kepler discovered physical harmonies in planetary motion. He found that the difference between the maximum and minimum angular speeds of a planet in its orbit approximates a harmonic proportion. For instance, the maximum angular speed of the Earth as measured from the Sun varies by a semitone (a ratio of 16:15), from mi to fa, between aphelion and perihelion. Venus only varies by a tiny 25:24 interval (called a diesis in musical terms). Kepler explains the reason for the Earth's small harmonic range:
The celestial choir Kepler formed was made up of a tenor (Mars), two bass (Saturn and Jupiter), a soprano (Mercury), and two altos (Venus and Earth). Mercury, with its large elliptical orbit, was determined to be able to produce the greatest number of notes, while Venus was found to be capable of only a single note because its orbit is nearly a circle.
At very rare intervals all of the planets would sing together in "perfect concord": Kepler proposed that this may have happened only once in history, perhaps at the time of creation. Kepler reminds us that harmonic order is only mimicked by man, but has origin in the alignment of the heavenly bodies:
Kepler discovers that all but one of the ratios of the maximum and minimum speeds of planets on neighboring orbits approximate musical harmonies within a margin of error of less than a diesis (a 25:24 interval). The orbits of Mars and Jupiter produce the one exception to this rule, creating the inharmonic ratio of 18:19. The cause of this dissonance might be explained by the fact that the asteroid belt separates those two planetary orbits, as discovered in 1801, 150 years after Kepler's death.
Kepler's previous book, Astronomia nova, related the discovery of the first two of the principles that we know today as Kepler's laws. The third law, which shows a constant proportionality between the cube of the semi-major axis of a planet's orbit and the square of the time of its orbital period, is set out in Chapter 5 of this book, immediately after a long digression on astrology.
A small number of recent compositions either make reference to or are based on the concepts of Harmonices Mundi or Harmony of the Spheres. The most notable of these are: