Harmonices Mundi (Latin: The Harmony of the World, 1619) is a book by Johannes Kepler. In the work 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.
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; and the fifth on the harmony of the motions of the planets.
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:
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 also 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 unharmonic ratio of 18:19. In fact, the cause of Kepler's 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.
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