Harmonices Mundi by Johannes Kepler Research Paper

Introduction

The work of Johannes Kepler marks a fundamental advance over the Aristotelian physics and Ptolemaic astronomy inherited from antiquity. His immediate predecessor Copernicus, who wrote the first systematic exposition of heliocentric astronomy (De Revolutionibus, 1543), had maintained the earlier postulates: Celestial bodies move with uniform circular motions; the visible motions can be modeled by supplying a sufficient number of deferents and epicycles; no motive force is needed, because the planets move of their own nature. Kepler’s deep-seated belief that the sun emits a force that pervades the cosmos and drives all the celestial bodies on their courses, a belief which meshed well with his Christian faith, led him to investigate the causes of celestial motions and to postulate the existence of a force, now called gravity, which links the cosmos into one organism. His willingness to treat the planets as objects having the same nature as terrestrial bodies enabled him to discard the necessity for circular motions and to adopt elliptical orbits. His mathematical curiosity drove him to make correlations between what had been considered unrelated phenomena and from these phenomena to deduce laws.

Unlike the ancient Greeks who cloaked their discoveries in mystery, Kepler leads us through all the tortuous paths of his creative process, the false starts, crooked turns, agonizing failures, as well as the exhilarating successes. No one has been more unguarded, unsparing, or relentless in sharing with the reader their innermost thoughts and feelings.

Harmonices Mundi

In the Harmonice Mundi, Kepler elaborated concepts first outlined in the Mysterium Cosmographicum. The basic principles of the cosmos are based on geometry: Specifically, the regular polygons are archetypal forms in the human soul as well as in the celestial world; when the planets in their orbits form angles corresponding to the angles of these polygons (for example, 90 degrees square, 60 degrees trine), they inspire and excite the soul. Kepler pursued this line of thought in his astrological writings. Musical harmony is also based on these angles: The ratios of the musical scale (octave, fifth, and so forth) can be derived from the polygons by suitable construction. Furthermore, the planets’ orbital velocities create a harmony: (1) There is a simple ratio between the planet’s velocity at aphelion and its velocity at perihelion; (2) the squares of the periods of revolution of any two planets (the time in which they complete one orbit) are proportional to the cubes of their mean distances from the sun. This latter is Kepler’s third, or harmonic, law, and his joy at its discovery was unbounded. In the final chapters of Harmonice Mundi he assigns musical “notes” to the planets at aphelion and perihelion and demonstrates by the variations in the “tunes” thus played by each planet the eccentricity of the orbit of each: Venus, with an almost circular orbit, plays a monotonous tune; the very eccentric Mercury runs up and down the scale.

The Principles of Harmony in Kepler’s View

The principle of harmony, according to which our universe is a harmonious whole, was another idea that had an overwhelming influence on Kepler. The different parts of the universe are related to each other in a harmonious way, pleasing to the heart and mind. God fashioned our universe not only according to the canons of geometry, but also of harmony. Epistemologically, the principle of harmony implies that our understanding of the universe needs not only to consider geometry but also harmony, the complementary facet to the geometrical nature of the universe.

Nothing could bring so much joy and peace to Kepler’s heart as the contemplation of the harmony of the cosmos. As Caspar says, “These were the thoughts to which he clung during the trials of his life and which distributed light to him in the darkness which surrounded him. They formed the place of refuge, where he felt secure, which he recognized as his true home….” ¹“The magic of the word harmony transported him to another, a pure, paradisiacal world.” ² This idea had an incomparably unique place in his thought and works. As he exclaimed: “I feel carried away and possessed by an unutterable rapture over the divine spectacle of the heavenly harmony.”³ His fascination for harmony was so all-conquering that he wrote, in his letter to Heydonus of London in October, 1605: “May God deliver me from astronomy, so that I can devote all my attention to my work on the harmonies of the universe.”⁴ Kepler was not unaware of the limitations of harmony and of the results he obtained. For instance, in Harmonices Mundi he admitted that the heavenly music he discovered as a result of years of strenuous work was inaudible. Indeed, he accepted that no voices existed in heaven. Still more, he was fully aware that he was considering the movements solely as apparent from the sun and therefore he was dealing with motion as it would be observed by someone from the sun, not from the earth.⁵ And even so he exclaimed: “I do not know why but nevertheless this wonderful congruence with human sound “humano cantu” has such a strong effect on me that I am compelled to pursue this part of the comparison also, even without any solid natural cause.”⁶ Indeed, the thoughts and ideas of harmony had made such a captive of him that he was almost helpless. It was not AN, his most scientific book, but Harmonices Mundi that he considered his best and greatest book, his greatest achievement.

The idea of harmony is ancient. For instance, heavenly harmony, i.e., that the distances of the planets ought to be arranged in a pleasingly harmonic pattern, in simple ratios or “proportions,” goes back at least to the ancient Greeks. The Pythagoreans believed in the harmony of the spheres, according to which each heavenly body emitted a sound and these sounds together made a harmonic music.

They knew that the sound’s pitch depended on the speed of the vibrations of a moving body (string, air column, celestial sphere, etc.): the faster the motion, the higher the pitch.

Plato also talked about the harmony of the universe, as is evident from the Timaeus, where the Demiurge created the universe by harmoniously arranging the different elements.⁷ Aristotle, too, spoke of cosmic harmony.⁸ For our purpose the most significant ancient authority to discuss heavenly harmony in detail was Ptolemy. In fact, Kepler himself wrote about it in Harmonices Mundi. After long attempts to get hold of a copy of the Harmonices of Ptolemy, Herwart sent Kepler one to read. And his reaction? He exclaimed: “There, beyond all my expectations and with the greatest wonder, I found almost the whole third book given over to the same contemplation of celestial harmony, fifteen hundred years ago.”⁹ Of course, he clarified that the similarity between his ideas and Ptolemy’s was evident only in the questions asked, not in their solutions, because in Kepler’s view Ptolemy’s work “seemed to have recited a pleasant Pythagorean dream rather than to have aided philosophy.”¹⁰ Nevertheless, Kepler was pleasantly surprised at the sameness of concern between him and the great ancient astronomer and took this as an assurance that “the finger of God” was guiding him.

The music of the spheres was a very popular concept in the Middle Ages and remained so in Kepler’s time. According to this concept, God had arranged the planets in a pleasingly harmonious pattern so that the heavens could sing his praises. Obviously, this belief was based on, or at least was strengthened by, the words of the Psalmist: “The heavens proclaim the glory of God.”

Different views have emerged as to what harmony meant to Kepler. Westman distinguishes two kinds of harmonies: microcosmistic and architectonic. Microcosmistic harmony “refers to the principles of analogy or similarity of structures between the invisible and visible realms.” On the other hand, architectonic harmony “denotes the principles of mathematical beauty and simplicity completeness, equality, regularity, symmetry, the concordance of whole and parts -which are themselves reflected in the visible world.”¹¹ The first kind, therefore, consists in an agreement between an original and a copy, whereas the second kind comprises a cluster of related ideas. The second kind seems to defy any definite and precise description, a difficulty that also attends Wilson’s attempt to specify what harmony is by giving only a possible list of connotations of the term: “Simplicity and symmetry of geometrical arrangement, simplicity of causal explanation, and the architectonic beauty of system created by an artist God.”¹² This partial list is very similar to Westman’s description of architectonic harmony, which also treats of harmony in terms of simplicity, symmetrical arrangement, right fit, etc. Holton, quoting H. Zaiser, attempts to elucidate the matter as follows: “Harmony resides no longer in numbers which can be gained from arithmetic without observation. Harmony is no longer the property of the circle in higher measure than the ellipse. Harmony is present when a multitude of phenomena is regulated by the unity of a mathematical law which expresses a cosmic idea.”¹³ This passage seems to identify harmony with mathematical reducibility, in the sense that harmony exists where various phenomena can be expressed by a mathematical formula. Only a partial description of harmony, mathematical reducibility focuses on only one of harmony’s aspects. This point becomes clear when we study what Kepler himself said about his idea of harmony.

In Harmonices Mundi Kepler gave a detailed, though not so clear, discussion of harmony.¹⁴ He talked of two kinds of harmony: sensible harmony (harmonia sensilis) and insensible harmony (harmonia insensilis). The two are intimately connected because the latter is but the archetype or paradigmatio of the former.¹⁵ According to him, sensible harmony involves four elements: (1) two sensible items of the matching kind that can be compared on the basis of quantity; (2) the soul that can perform the comparison; (3) the inner reception of the sensible; (4) the appropriate proportion or ratio that is understood as harmony. Harmony, therefore, has objective and subjective facets. Agreeable or appropriate, quantifiable relation or proportion constitutes the objective aspect. The mind and the inner reception of the appropriate proportion by the mind make up the subjective aspect. The inner perception of the appropriate mathematical proportion gives rise to sensible harmony. Sensible harmony for Kepler is not a purely intellectual idea but appeals to our aesthetic feelings as well. According to him, our souls have the inborn ability to know what the agreeable proportion is and what is not. The soul is born with archetypal harmonic laws.

Thus an essential element of harmony is the appropriate, agreeable, pleasing, quantifiable relation or proportion between the items under consideration. What constitutes the appropriate or the agreeable is not clearly defined in Kepler. It may come from simplicity, from symmetric arrangement, from regularity, from unifiability, from reducibility, etc. The element of appropriateness is open ended. This open endedness may explain the lack of definiteness in the description of harmony by scholars like Wilson and Westman (in his architectonic harmony). From this discussion it follows that Zaiser’s view of harmony as reducibility is just one instance of harmony, not the only one.

Although Kepler’s description of harmony is vague, his general position is quite clear. God is not just a dry mathematician, not just a dry logician, but is also a lover of aesthetic beauty. God is a musician. In creating the universe, God used not only the laws of geometry but also those of harmony, particularly the laws of musical harmony. The different things in the world have been arranged, not at random, but according to definite proportions that are pleasing to mind and heart. The different parts of the universe are related to each other by definite harmonic ratios. Hence any attempt to unravel the laws of nature has to take into account the harmonic relations. In the Harmonices Mundi Kepler applied the considerations of musical harmony to the study of the structure and laws of nature. His investigations revealed that the ratios of the angular velocities of the planets at the extremities of their orbits were the basic musical intervals: 4/5 (major third) for Saturn, 5/6 (minor third) for Jupiter, 2/3 (perfect fifth) for Mars, 15/16 (major semitone) for Earth, 24/25 (minor semitone) for Venus, and 5/12 (octave plus a minor third) for Mercury. These ratios give the music the cosmic creator sang and still sings. Kepler believed that this discovery was an achievement of utmost importance. For him these results were no mere speculation. Indeed, for him the Harmonices Mundi was a valuable book of cosmology wherein he had been able to uncover how the universe existed in its innermost bosom (“qualis existat penitissimo sinu”). It provided him with the first genuine model of the universe (“prima universitatis exempla genuina.”)¹⁶ He believed that essential ideas with regard to the origin and nature of the universe, ideas most valuable for a correct understanding of the universe, were unlocked as a result of these studies. He was convinced that this assisted him greatly in his goal to read the mind and plan of God.

However, there were several problems. His mode of thinking and developing ideas in the Harmonices Mundi is quite strange and very different from what we are used to. In the case of musical instruments, the length of the vibrating string and the frequency or number of vibrations is the relevant factors for harmony. Corresponding to them in astronomy we should consider distance of planets and their velocity. But in Kepler’s work the harmony discovered is neither between the distances nor between the orbital velocities of the planets. Rather, it is only between their angular velocities with respect to the sun. Also the velocities considered are only at the aphelion and perihelion of the planetary orbits. How could Kepler have jumped to a momentous conclusion concerning cosmic harmony under such restricted and limited conditions? How could he have exulted so much at this discovery? The findings seem incomplete, to say the least. However, he could justify himself on the basis of his belief that the universe is a harmonious, unified whole. For if the universe is closely interconnected, then what is true for a part of it must be true for other parts as well.

There were also problems with observational agreement. For instance, Kepler found that the agreement between theory and observational results in the case of Jupiter and Mars was poor.¹⁷ However, here he had a ready explanation. He reminded the readers that in the polyhedral theory, where the position was considered, it was Jupiter and Mars that had the best fit. It should come as no surprise, then, that we get a poor fit for the same planets when we consider motion. In his view for a particular case if we get the best result with respect to the position of the planet, then we should expect to get the worst result for the same case with respect to motion. In other words, we cannot expect the best accuracy in both position and motion (velocity); they are complementary to each other. Undoubtedly, here one is reminded of Heisenberg’s uncertainty principle, although, despite his familiarity with Kepler’s writings, he does not seem to have noticed this idea about complementarity’s in Kepler. Surprisingly strange and ad hoc though Kepler’s justification may seem, his explanation is not so weak. If he is correct, the poorest result in the polyhedral theory must be the most accurate in the harmonic theory, and, sure enough, such is the case. Saturn-Jupiter gave the poorest agreement with observation in the geometrical theory, whereas they showed the best agreement in the harmonic theory.¹⁸ Clearly, Kepler was neither a loose nor a careless thinker. And his idea also argues for the complementary nature of geometry and harmony.

The need for taking principles of harmony into account in the investigations of the laws of nature was a direct result of Kepler’s emphasis on empirical inquiry. In MC he was quite certain that geometry would unlock the secrets of nature. Thus although he did notice discrepancies between his theory and the observed data available to him, at first he tried to explain them away. However, upon coming to know about Tycho Brahe’s accurate observations, he wanted to check his theory with the Dane’s treasure of observations. This comparison with observational data revealed that his theory based on geometry was not fully accurate, which brought home to him the notion that geometry alone could not discover the laws of nature. He found that when he took into consideration the harmonic laws also, he could get the correct results. Both geometry and harmony had to be brought in if a complete answer was sought.

The incompleteness of geometrical considerations and the consequent need to introduce harmonic principles Kepler explained in the Epitome. According to him, “The archetype of the mobile universe is formed not only from the five regular solids from which the planetary paths and the number of motions would be fixed; but is also formed by the harmonic proportions of the six parts in agreement with which the motions themselves have been tuned to the idea of a certain celestial music, or of a six-part harmonic chord.”¹⁹ He expressed the complementary nature of geometry and harmony in Harmonices Mundi as well: “They [the harmonies] provided, so to speak, nose, eyes, and other members to the statue, whereas the latter [the regular solids] prescribed only the rough external quantity of its mass.”²⁰ He argued that the fact that geometry alone was inadequate and hence the harmonies had to be invoked, far from diminishing the beauty of the universe, enriched it and rendered it more complete.

For this reason, in the same way that it is not usual for the bodies of animated beings, or the volumes (blocks) of stones to be shaped in conformity with the absolute standard of some geometric form, but that they should deviate from the external spherical shape, however elegant it might be (nevertheless, retaining the exact measure of its volume), in order that the body may acquire the organs necessary to life and that the stone may receive the image of the animated being; so, the proportions which should be prescribed for the planetary spheres by the solid figures, were less [in value], and as they affected only the body and its material substance, they ought to yield to the harmonies as much as necessary in order to make them come closer and contribute to the beauty of the motions of the spheres.²¹

According to this passage, geometry cannot take a supreme and unyielding position but must make concessions to harmony, insofar as they are necessary to bring about the best and most beautiful universe.

This principle had its reverberations in his religious thinking as well. Earlier, God was only a geometer, but now God has become a musician also. The initial order God placed at creation was such that the motion of all the planets together gave out a harmonious polyphony. Much of Harmonice Mundi has been termed mystic fantasy, but this fantasy, as always, was founded on carefully observed fact and aimed at explicating the dynamics and structure of the solar system. Kepler’s imaginative mind structured and explained data in terms strange to us, but this exuberant imagination did succeed in discovering his three laws of planetary motion.

Conclusion

The Harmonice mundi of Johannes Kepler (1571-1630) is a milestone in the history of science. In it, what is now known as Kepler’s third law of planetary motion is announced. This law, which relates the speed with which a planet moves to its distance from the sun, provided the key to Newton’s deduction of the law of gravity. More than three hundred years later, Kepler’s third law still appears at the frontier of science it is central to the modern “dark-matter” problem. If we actually turn to the Harmonice mundi itself, however, we find that Kepler’s great discovery is merely one small ingredient in a wild potpourri of number mysticism, music, and astrology, of which Arthur Koestler remarked,

The Harmony of the World is… the climax of his lifelong obsession. What Kepler attempted here is, simply, to bare the ultimate secret of the universe in an all-embracing synthesis of geometry, music, astrology, astronomy and epistemology. It was the first attempt of this kind since Plato, and it is the last to our day.

Kepler was obsessed with an idea that goes back to Pythagoras and the Orphic mystery religions of ancient Greece, and is a recurrent theme in Renaissance poetry and literature. This idea is that the workings of the world are governed by relations of harmony and, in particular, that music is associated with the motion of the planets the music of the spheres.

The notion of “music of the spheres” has historically been a vague, mystical, and elastic one; but we can start to define and appreciate it with the following thoughts. The bases of music are rhythm and harmony. Rhythm is ordered recurrence in time; we say two players are in rhythm if they are hitting notes at the same time or, more generally, if there is an orderly relation between the times when they play notes a steady bass may underlie an intricate melody. As the planets move around the sun, they repeat their orbits periodically; thus there is already a primitive kind of rhythm in their motion. Kepler wished to find relations between the motions of the different planets, to show that in some sense they are all moving to a single cosmic rhythm.

Harmony, we now realize, can be considered a special form of rhythm. Sound is vibrations of air; pure musical tones are produced when the vibrations are of a particularly simple and regular form that is, when they are periodic or, to say it another way, when they repeat themselves regularly in time. Two tones harmonize if their intervals of repetition are in rhythm or, in more mathematical language, if their periods are in proportion. Kepler naturally did not have our modern understanding of the nature of sound, but in the third book of Harmonice mundi he attempted to make other, difficult to understand but perhaps somehow related, connections between musical harmony and mathematical proportion.

Endnotes

1. Bruce Brackenridge, “Kepler, Elliptical Orbits, and Celestial Circularity: A Study in the Persistence of Metaphysical Commitment,” part 2, Annals of Science 39 ( 1982), 285.
2. Caspar, Max, and Walther von Dyck, Franz Hammer, and Volker Bialas, eds. Johannes Kepler Gesammelte Werke. 22 vols. Munich: Deutsche Forschungsgemeinshaft and the Bavarian Academy of Sciences, 1937
3. Caspar, Max. Johannes Kepler. Stuttgart: Kohlhammer, 1948. Translated by C. D. Hellman. New York, 1962.
4. Curtis Wilson, “Horrocks, Harmonies, and the Exactitude of Kepler’s Third Law, Science and History”, Studia Copernicana 16 (Ossolineum, 1978), p. 238.
5. Gerald Holton, “Johannes Kepler’s Universe: Its Physics and Metaphysics,” in Thematic Origins of Scientific Thought (Harvard University Press, 1973), p. 349.
6. Koestler, Authur. The Sleepwalkers. New York: Macmillan, 1959.
7. Owen Gingerich, “Kepler, Galilei, and the Harmony of the World,” in The Eye of Heaven: Ptolemy, Copernicus, Kepler (New York, 1992). pp. 18-19
8. Thomas S. Kuhn, The Copernican Revolution ( Cambridge: Harvard University Press, 1957), p. 134.
9. Westman, “Kepler’s Theory of Hypothesis and the ‘Realist Dilemma’,” Studies in History and Philosophy of Science 3 ( 1972), 252.

Footnotes

1. Quoted by Caspar, Johannes Kepler, 288.
2. Ibid., p. 267.
3. Johannes Kepler Gesammelte Werke. 22 vols. p. 480.
4. Johannes Kepler Gesammelte Werke. 22 vols. ll. 91-92.
5. Gingerich, “Kepler, Galilei, and the Harmony of the World,” pp. 18-19.
6. Johannes Kepler Gesammelte Werke. 22 vols. VI, p. 329
7. Plato, Timaeus cited in The Works of Aristotle, tr. W. D. Ross
8. Aristotle, De Caelo, 290B, in The Works of Aristotle, tr. W. D. Ross ( Oxford: Oxford University Press, 1947).
9. Johannes Kepler Gesammelte Werke. 22 vols. VI, p. 289
10. Ibid., p. 289
11. Westman, “Kepler’s Theory of Hypothesis and the ‘Realist Dilemma’,” Studies in History and Philosophy of Science 3 ( 1972), 252.
12. Curtis Wilson, “Horrocks, Harmonies, and the Exactitude of Kepler’s Third Law, Science and History”, Studia Copernicana 16 (Ossolineum, 1978), p. 238.
13. Holton, “Johannes Kepler’s Universe,” p. 349.
14. See Johannes Kepler Gesammelte Werke. 22 vols. VI, pp. 211
15. Kepler believed that the insensible or pure harmonies existed in the mind serving as models for the sensible harmonies.
16. Johannes Kepler Gesammelte Werke. 22 vols. VI, p. 323
17. Bruce Brackenridge, “Kepler, Elliptical Orbits, and Celestial Circularity: A Study in the Persistence of Metaphysical Commitment,” part 2, Annals of Science 39 ( 1982), 285.
18. Ibid., p. 81
19. Gerald Holton, “Johannes Kepler’s Universe: Its Physics and Metaphysics,” in Thematic Origins of Scientific Thought (Harvard University Press, 1973), p. 349
20. Curtis Wilson, “Horrocks, Harmonies, and the Exactitude of Kepler’s Third Law, Science and History”, Studia Copernicana 16 (Ossolineum, 1978), p. 238
21. Thomas S. Kuhn, The Copernican Revolution ( Cambridge: Harvard University Press, 1957), p. 134.