The Music of Pythagoras

by Kitty Ferguson

  Early in his university career, Kepler realized that theology, mathematics, and astronomy would all be essential in his personal search for truth. He never ceased to be a devoutly religious man, but, as he later wrote, he believed that “God also wants to be known through the Book of Nature.” Perhaps it was in that interest (Kepler would have thought so) that God had placed a superb professor of mathematics and astronomy at the University of Tübingen: Michael Mästlin.

  When Kepler first arrived there in 1589, forty-six years had passed since the death of Copernicus and the publication of his De revolutionibus in 1543. Many scholars were finding Copernicus’ grasp of celestial mechanics and his mathematics invaluable, while choosing to ignore his rearrangement of the cosmos. The University of Tübingen still officially taught Ptolemaic astronomy, and Michael Mästlin made sure his pupils had a good grounding in that, for which Kepler would later be grateful when he sought to overturn it. But Mästlin believed that Copernicus’ system had to be taken literally and that the planets and the Earth do, indeed, orbit the Sun. Kepler also read Nicholas of Cusa and was soon writing: “I have by degrees—partly out of Mästlin’s lectures, partly out of myself—collected all the mathematical advantages which Copernicus has over Ptolemy.” In a letter he wrote later to Mästlin, Kepler called Pythagoras the “grandfather of all Copernicans.”1

  Johannes Kepler

  During his university years, Kepler rapidly became well-read in the classics and also encountered neo-Platonic/Pythagorean thinkers of his own era. He gave all of it a religious and Pythagorean spin of his own: A universe created by God must surely be the perfect expression of a profound hidden order, harmony, simplicity, and symmetry, no matter how complicated and confusing it might appear to people who, like himself, were only beginning to understand it. This was the conviction that set fire to his spiritual and scientific imagination, and that flame would last him a lifetime. He was about to pin this idea to the wall using more precise observations of the heavens and his innate genius for rigorous mathematics: a potent combination.

  While still a student at Tübingen, Kepler openly defended Copernican astronomy in two formal debates, arguing that the planets’ periods and their distances from the Sun made far better sense in the Copernican system; and that if the Sun was indeed (like the Creator) the source of all change and motion, then it might follow that the closer a planet was to the Sun, the faster it would travel. He worked busily and happily on astronomical questions and wrote a piece about how the movements of the heavens would appear to someone on the Moon. Despite all that, it seems not to have occurred to him that he might pursue any career other than as a clergyman.

  Near the end of his fifth university year, Kepler learned that his time at Tübingen was to end immediately, and not in the way he had planned. A Protestant school in southern Austria appealed to the university for a teacher, mainly for mathematics but also with knowledge of history and Greek. Tübingen had decided to send Kepler. Sorely discouraged and frustrated, he made the move to Graz. It was there that, about a year after his arrival, while drawing a diagram on the board for his pupils, he made the startling discovery that a triangle seemed somehow to be dictating the distance between the orbits of Jupiter and Saturn. The triangle was the Pythagorean tetractus.

  The date was January 19, 1595, and Kepler was lecturing about the Great Conjunctions that occur when Jupiter and Saturn, as viewed from the Earth, appear to pass each other. This does not happen often in anyone’s lifetime, for Jupiter overtakes Saturn only approximately every twenty years. Imagine the two planets moving on a great circular belt around the Earth. During the twenty-year interval between two Great Conjunctions, Saturn moves about two thirds of the way around the belt, while Jupiter makes one complete revolution and two thirds of another. The locations of the Great Conjunctions leap forward on the belt by two thirds of the circle every twenty years.

  Kepler had drawn a circle on the chalkboard to represent the great circle of the zodiac belt, and then marked the points in the zodiac where the successive Great Conjunctions occurred, viewed from Earth. If one plotted only three Great Conjunctions, those points were very near to being the corners of an equilateral triangle, but not quite. Beginning another triangle where the first ended (plotting the next conjunctions), the new triangle did not precisely retrace the first one. For example, the fourth conjunction in Kepler’s drawing (the conjunction that occurred in the year 1643) happened at almost the same point as the first (in 1583), and the fifth at almost the same point as the second. Draw lines connecting them and you almost have an equilateral triangle . . . but, again, not quite, and you have not retraced the first triangle. So the triangle “rotates,” as Kepler’s diagram shows. The result is two circles, outer and inner, with the distance between them set by the rotating triangle. Thus, Kepler’s triangle seemed to be mysteriously dictating the distance between the orbits of the first two planets. Interestingly, the radius of the inner circle looked as though it were half that of the outer circle, and observations of the heavens showed that the radius of Jupiter’s orbit was approximately half the radius of Saturn’s.

  Drawing from Kepler’s Mysterium cosmographicum depicting the pattern of Jupiter-Saturn conjunctions and where they happened in the zodiac. The conjunction in 1583 (right) occurred when the two planets were in Aries/Pisces. The conjunction in 1603 (lower left) was in Sagittarius, in 1623 in Leo, in 1643 in Aries, in 1663 in Sagittarius, and so on. If the conjunctions occurred repeatedly in the same positions in the zodiac, Kepler’s drawing would have looked like the insert (upper right). Instead they “progress,” as represented in the central figure.

  An amazed Kepler decided immediately to try the next regular polygon—the square (the triangle has three sides, the square four)—to see whether it would serve similarly for the separation between the orbits of Jupiter and Mars.* If it did, he planned to try a pentagon (five sides) for the separation between the orbits of Mars and Earth, a hexagon for Earth and Venus, and so forth. He hoped the arrangement of the cosmos would resemble this diagram, with the triangle, then the square, then the pentagon, then the hexagon, and so forth, all nested between the separate planetary orbits. The idea failed on the first try, when the square would not work for the known separation between the orbits of Jupiter and Mars.

  Kepler experimented with other regular polygons, searching for a fit, but he realized that given the infinite number of polygons available, success was assured. To the early Pythagoreans, this might have seemed adequate. Not to Kepler, for the question remained, why—among all the possibilities—these particular polygons worked and not others. Why had God chosen to construct the universe in this way and not in some other?

  Though many of his contemporaries considered questions like these naive, they bothered Kepler, who had already been focusing his thinking along two lines of investigation: what reasoning God was using when he made things the way they are; and the physical reasons why the universe operates as it does. Clearly, for Kepler, shuffling through all the polygons and finding five that fit neatly between the six planetary orbits was not satisfactory. Since there were regular polygons to fit any planetary distances one might find, he felt there had to be a scheme that would limit the actual, possible ratios (Saturn to Jupiter, Jupiter to Mars, Mars to Earth, Earth to Venus, Venus to Mercury), accounting for why some ratios, not others, existed in the heavens and there were only six planets.

  It occurred to Kepler that he was making a mistake in trying to apply two-dimensional, flat figures (polygons) to a three-dimensional universe, and he decided to experiment instead with solid figures, the regular polyhedra.* That thought was a Pythagorean knockout. There were, after all, only five regular polyhedra (the Pythagorean or Platonic solids) not an infinite number of possibilities. To Kepler’s immense satisfaction, he found he could fit the five polyhedra into a nested arrangement that quite nicely coincided with the known separations between the six “spheres” in which the planets orbit.†

  Any of the fiv
e regular polyhedra—cube, tetrahedron (pyramid), octahedron, icosahedron, and dodecahedron—can be set inside a sphere so that each of its points touches the sphere; and a smaller sphere can be set inside any polyhedron so that it touches the center of each side of the polyhedron. This was almost certainly what was meant by the cryptic words of the Philolaus fragment: “The bodies in the sphere are five.” So Kepler pictured the solids nesting among the planetary spheres, giving the separation between them just as the triangle had seemed to give the separation between the orbits of Jupiter and Saturn in his drawing on the chalkboard. This “polyhedral theory” appears to have been completely original with Kepler.

  Despite Kepler’s conviction that there were deep, harmonious connections in nature, and his hope that he had found a stunning example, there was a side to his intellectual makeup that set him apart from the Pythagoreans who had decided there must be ten bodies in the cosmos. He did not merely assume that the universe must surely fit his beautiful geometrical scheme without testing it against Copernican theory and the available observational records “to see whether this idea would agree with the Copernican orbits, or if my happiness would be carried away by the wind.” Though the need for such a testing procedure seems obvious today, it did not to those who studied nature in the sixteenth century. Kepler was, in fact, feeling his way into the process that would later be dubbed the scientific method.

  Kepler’s polyhedral theory drawing, from his Mysterium cosmographicum, showing his nesting arrangement of the six planetary spheres and the five Pythagorean/Platonic solids

  Given that there are eight or nine planets orbiting the Sun,* not only the six known in Kepler’s time, and that his polyhedral theory has turned out to be quite off the wall, it is astonishing to read Kepler’s exclamation that “within a few days everything worked, and I watched as one body after another fit precisely into its place among the planets.” He knew, however, that there were better observations than those he was using. Those undertaken by the older Danish astronomer Tycho Brahe were far more precise. Tycho was a reputedly arrogant, supremely talented man whose nose, it was widely known, had been partially hacked off in a youthful duel and reconstructed out of gold and silver. Unfortunately for Kepler, Tycho was behaving like a new Pythagoras or Copernicus: He was keeping his findings to himself and refusing to publish them.

  Kepler finished writing Mysterium cosmographicum, his book about the polyhedral theory, in the winter of 1595–96, and it came off the press in 1597. When he was an old man, Kepler would reminisce that this small volume with the long title (the complete title required about six lines of print) was the point of departure for the path his life would take from that time forward. He could, with some justice, have said the same with regard to its watershed significance for the whole of science. As the eminent historian of science Owen Gingerich has commented, “Seldom in history has so wrong a book been so seminal in directing the future course of science.”

  The game was now afoot in earnest. The polyhedral theory was not a dead end, and the reason was that Kepler—Platonist and Pythagorean when it came to his faith in harmony and symmetry—was a thoroughgoing Aristotelian when it came to his respect for down-to-earth, or at least visible-from-Earth, observational data. He first tried to approach Tycho soon after Mysterium’s publication, but it was a truly labyrinthine four-year trail of events that finally brought him to the moment when Tycho’s logbooks lay open before him. Meanwhile Kepler set off in another even more Pythagorean direction.

  KEPLER MENTIONED MUSIC in Mysterium only once, noting that just as there were five regular solids in geometry, so there were five harmonic intervals in music. He was counting more than the octave, fourth, and fifth of the Pythagoreans. Kepler had begun to give himself a thorough grounding in music theory. In his own musical calculations, he decided to use what is known as “just” tuning, rather than “Pythagorean” tuning. Pythagorean ideas about harmony, based on the ratios among the numbers 1, 2, 3, and 4, considered only the intervals of the octave, fifth, and fourth as being consonant. In “just” tuning, more commonly used in Kepler’s day, as it still is, major and minor thirds and sixths were also recognized as pleasing to the ear.* Kepler thought the addition of these intervals was a great improvement over the music of the ancients, and it was a rare musician or listener of his era (or later) who would not have agreed.

  His comment was no musical theory of astronomy, but in 1599, two years after the publication of Mysterium, Kepler mentioned some ideas about the harmony of the spheres in letters he wrote, first to an Englishman in Padua who he hoped would pass his idea along to Galileo, and then to his patron Herwart von Hohenburg and his old mentor Michael Mästlin in Tübingen. Kepler’s proposal was not exactly the same in all three letters, for his thoughts were developing rapidly regarding a question he had raised in Mysterium: Why does each planet take the time it does to orbit the Sun? Sure that planets farther from the Sun actually move more slowly and are not merely handicapped by being assigned an outer lane in the race, he was pondering what logic might lie behind the planets’ different distances from the Sun and their different velocities.

  Kepler thought, as the Pythagoreans and others had before him, that the planets, moving through something like air, must produce a sound, just as the strings of a musical instrument would if hung in a breeze, and he believed the sound was harmonious. Only two people had anticipated Kepler’s precise linking of music and planetary movement: John Scotus Eriugena, in the ninth century, and Giorgio Anselmi of Parma, in the early fifteenth.2 Eriugena had recognized that if you associated a musical pitch with a planet, and if pitch depended on the planet’s distance from the Earth, then you had to include in your theory the way the pitch would change as a planet moved and changed its distance—which planets clearly did, especially in Eriugena’s arrangement with some of the planets orbiting the Sun. Anselmi had not imagined each planet as having an individual tone but rather as singing its own melody in counterpoint with the others. In devising his eight-octave planetary scale, he had taken into consideration the planets’ orbital periods. The result was a great cosmic symphony.

  In 1599, Kepler was considering the possibility that the velocities (in his word, the “vigor”) of the six planets might be related to one another in the same relationships that would produce a harmonious chord if translated into lengths of strings on a musical instrument. For example, a relationship of 3:4 between the velocities of Saturn and Jupiter, used as the relationship between two string lengths, would produce the interval of the fourth. So one could think of the “interval” between Saturn and Jupiter as a musical fourth. Kepler calculated the proportions of the velocities of the planets as 3:4 for Saturn to Jupiter, 4:8 (1:2) for Jupiter to Mars, 8:10 (4:5) for Mars to Earth, 10:12 (5:6) for Earth to Venus, and 12:16 (3:4) (Venus and Mercury). Translating those ratios into musical intervals, he worked out a chord composed of (starting from the lowest note) intervals of a fourth, an octave, a major third, a minor third, and another fourth. In modern notation, an example of this chord would be:

  Kepler’s 1599 planetary chord

  Kepler had chosen the velocities with the aim of having a harmonious chord, and now he found that by doing so he had produced musical intervals that were close to the spatial intervals between the planets in his polyhedral theory. The planetary orbital periods had been well known since antiquity, so he was able to proceed to calculate how large the different orbits had to be in relation to one another if the planets, with these known periods, were traveling at the velocities his musical intervals predicted. He compared the results with the orbital sizes calculated from Copernican theory and found that his harmonic theory was in somewhat better agreement than his polyhedral theory.

  Kepler’s own summation of what he had learned from each theory was that with the harmonic theory he could calculate the planets’ distances from the Sun, relative to one another; and with the polyhedral theory he could calculate the thickness of the empty spaces between the spheres in wh
ich the planets orbited.

  Kepler wrote to Mästlin that he had found a clever way to connect his polyhedral theory to three of the five intervals in his chord. The cube was the polyhedron that separated the orbits of Saturn and Jupiter. Three flat squares meet at each corner of a cube, and the corner of each of the three squares is a 90-degree angle. Add those three 90-degree angles together and you get 270 degrees. The ratio between 270 and 360 (the number of degrees in a complete circle) is 3:4. It seemed appropriate that the musical interval (the fourth) that required the ratio of string lengths 3:4 was the one that defined the space interval between Saturn and Jupiter. Kepler found similar relationships working for the intervals between Jupiter and Mars, and Earth and Venus.

  Kepler thought that he had made good progress with his harmonic theory, and that the harmony he was discovering reflected the mind of the Creator and was surely carried out in the cosmos. He confided to Mästlin and von Hohenburg, late in the summer of 1599, that he felt as though he had “a bird under a bucket.” He was soon writing to von Hohenburg that he was planning a work titled Harmonice mundi.