Harmonices Mundi

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"Harmony of the Worlds" redirects here. For the episode of Cosmos: A Personal Voyage, see Harmony of the Worlds (Cosmos: A Personal Voyage).
1619 first edition

Harmonices Mundi[1] (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".[2]

History[edit]

It is estimated that Kepler had begun working on Harmonices Mundi sometime near 1599, which was the year Kepler sent a letter to 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.[3]

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 referred to as the “music of the spheres.” Kepler was intrigued by this idea while he sought explanation for a rational arrangement of the heavenly bodies.[4] 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.[5]

Content[edit]

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.[6]

A small stellated dodecahedron
A great stellated dodecahedron

In Chapter 1 and 2 of The Harmony of the World comes 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 plain 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. Within the second chapter is the earliest mathematical understanding of two types of regular star polyhedra, the small stellated dodecahedron and the great stellated dodecahedron; they would later be referred to as Kepler's solids.[7] 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.[8]

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).[6] Kepler explains the reason for the Earth's small harmonic range:

The Earth sings Mi, Fa, Mi: you may infer even from the syllables that in this our home misery and famine hold sway.[9]

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.[6]

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.[10] Kepler reminds us that harmonic order is only mimicked by man, but has origin in the alignment of the heavenly bodies:

Accordingly you won’t wonder any more that a very excellent order of sounds or pitches in a musical system or scale has been set up by men, since you see that they are doing nothing else in this business except to play the apes of God the Creator and to act out, as it were, a certain drama of the ordination of the celestial movements. (Harmonice Mundi, Book V).[6]

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 unharmonic ratio of 18:19.[6] 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,[11] immediately after a long digression on astrology.

See also[edit]

References[edit]

  1. ^ The full title is Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler's The Harmony of the World).
  2. ^ Johannes Kepler, Harmonices Mundi [The Harmony of the World] (Linz, (Austria): Johann Planck, 1619), p. 189. From the bottom of p. 189: "Sed res est certissima extactissimaque quod proportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis mediarum distantiarum, id est Orbium ipsorum; … " (But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialternate proportion [i.e., the ratio of 3:2] of their mean distances, that is, of the actual spheres, … "
    An English translation of Kepler's Harmonices Mundi is available as: Johannes Kepler with E.J. Aiton, A.M. Duncan, and J.V. Field, trans., The Harmony of the World (Philadelphia, Pennsylvania: American Philosophical Society, 1997); see especially p. 411.
  3. ^ Field, J. V. (1984). A Lutheran astrologer: Johannes Kepler. Archive for History of Exact Sciences, Vol. 31, No. 3, pp. 207-219.
  4. ^ Voelkel, J. R. (1995). The music of the heavens: Kepler's harmonic astronomy. 1994. Physics Today, 48(6), 59-60.
  5. ^ Field, J. V. (1984). A Lutheran astrologer: Johannes Kepler. Archive for History of Exact Sciences, Vol. 31, No. 3, pp. 207-219.
  6. ^ a b c d e Brackenridge, J. (1982). Kepler, Elliptical Orbits, and Celestial Circularity: A Study in the Persistence of metaphysical Commitment Part II. Annals Of Science, 39(3), 265.
  7. ^ Cromwell, P. R. (1995). Kepler's work on polyhedra. Mathematical Intelligencer, 17(3), 23.
  8. ^ Field, J. V. (1984). A Lutheran astrologer: Johannes Kepler. Archive for History of Exact Sciences, Vol. 31, No. 3, pp. 207-219.
  9. ^ Schoot, A. (2001). Kepler's Search for Form and Proportion. Renaissance Studies: Journal Of The Society For Renaissance Studies, 15(1), 65-66
  10. ^ Walker, D. P. (1964). Kepler’s celestial music. Journal of the Warburg and Courtauld Institutes, Vol. 30, pp. 249
  11. ^ Schoot, A. (2001). Kepler's Search for Form and Proportion. Renaissance Studies: Journal Of The Society For Renaissance Studies, 15(1), 65-66

Additional reading[edit]

  • Johannes Kepler, The Harmony of the World. Tr. Charles Glenn Wallis. Chicago: Great Books of the Western World. Pub. by Encyclopædia Britannica, Inc., 1952.
  • "Johannes Kepler," in The New Grove Dictionary of Music and Musicians, Ed. Stanley Sadie. 20 vol. London, Macmillan Publishers Ltd., 1980. ISBN 1-56159-174-2

External links[edit]