The Music of Pythagoras

by Kitty Ferguson

  When von Hohenburg wrote expressing concern that the numbers were not really a fit, and that the theory was perhaps based on suspicion and not really demonstrated, Kepler replied:

  First, I think that aside from a few propositions, I have proposed not an ironclad demonstration but one which nevertheless stands, in the absence of contrary argument. Second, the suspicion is not entirely false. For man is the image of the Creator, and it may be that in certain matters pertaining to the adornment of the world the same things appear to man as to God.

  His own ideas of harmony, he believed, were in synchrony with the Creator’s; the remaining difficulties in his theories would not be difficult to overcome, and he would soon have them solved. He had little idea of the arduous intellectual journey that lay before him, or that twenty years would pass before he arrived at the great “harmonic theory” that would continue to be regarded as correct in the twenty-first century.

  In August 1599, von Hohenburg mentioned in a letter to Kepler an opinion of Ptolemy on the number of consonant intervals there were in music. Kepler eagerly wrote back to say that if Ptolemy’s book would not overburden the messenger, he very much hoped von Hohenburg would send it. In two more letters he continued to ask, and in July 1600, von Hohenberg finally complied. The book was a poor Latin translation of Ptolemy’s Harmonics, and Kepler later complained that he could hardly make sense of it. Nevertheless, it amazed him that Ptolemy’s speculations were not far different from his own, though “to be sure, much was still lacking in the astronomy of that age; and Ptolemy, having begun badly, could plead desperation. Like the Scipio of Cicero, he seems to have recited a kind of Pythagorean dream rather than advancing philosophy.”3

  The late summer and autumn of 1600, when Kepler was first reading the Harmonics, was not a convenient moment to consider harmonic theories more deeply. The previous winter, Kepler had joined Tycho Brahe at Benatky Castle near Prague, where the imperious astronomer was then in residence under the patronage of the Holy Roman Emperor Rudolph II. Kepler had arrived anticipating a fruitful collaboration and thinking that his hopes of being able to consult Tycho’s phenomenal astronomical data were about to be realized. Instead he had found himself having to cope with a difficult, paranoid, secretive old man who treated him more like an untrustworthy and unpaid servant than a collaborator and would allow him only tantalizing, inadequate glimpses of the precious data. Tycho’s longing to gain immortality with his own Tychonic system made him highly suspicious of young Kepler, who openly preferred the Copernican system.* Kepler’s hope of improving his financial situation had sent him on a fruitless journey back to Graz, seeking a continuance of his salary as District Mathematician (in absentia) there. But in the summer Kepler found himself not better off but worse. His health was failing, and a drastic turn in the Counter-Reformation in Catholic Graz suddenly made him a penniless Protestant refugee. Reluctantly, when all other possibilities failed (including an appeal to his old mentor Michael Mästlin for a job at Tübingen) he settled with his frightened family in Prague, even more at the mercy of Tycho Brahe than before. Kepler set aside Ptolemy’s book and his own ideas about harmony, but only temporarily. He had meanwhile not by any means abandoned his polyhedral theory. For him, that theory, his studies of the harmony of the spheres, his great revision, later, of all of astronomy in the light of Tycho’s observations—and much else—were not isolated, disconnected efforts. They were all part of a unity of thought and work.

  By January 1607, the wheel of fortune had turned again for Kepler. Tycho had died in the autumn of 1601, and Kepler had been the heir to Tycho’s position and duties in Rudolph’s court. With Tycho’s observational logs finally lying open before him (albeit with Tycho’s relatives all too ready to snatch them away and intermittently succeeding) Kepler had spent more than half a decade beating his brains over this data and his own calculations, using all the mathematical skill he could muster and inventing new mathematics to work out the true orbit of Mars. At one desperate point he had been almost ready to relinquish his Pythagorean faith and admit that the orbit of Mars simply did not make mathematical sense at all. He had even taken issue with God about it, in words he might have used to express disappointment about a human colleague: “Heretofore we have not found such an ungeometrical conception in his other works!”

  Yet eventually the universe and the Creator had turned out, in an unexpected way, to have lived up to Pythagorean standards after all. The planetary orbits, Kepler discovered, were elliptical, not circular, and with that hard-won realization everything fell into place. Kepler had been able to engineer an entire revision of astronomy in the light of Copernican theory and his own. He had found precisely how a planet’s velocity changes as it moves closer to and farther away from the Sun in its orbit. He had painstakingly chronicled his “war with Mars” and stated his first two laws of planetary motion in his book Astronomia nova, and had sent to press the manuscript for this great work that was the fruit of his, and Tycho’s, labors.* The book would win Kepler immortality. The campaign that had begun in Denmark at Uraniborg when Tycho first decided to train his fabulous instruments on Mars was over, and Kepler had awarded the victory to Copernicus, not to Tycho or Ptolemy. Even so, Kepler still clung to his polyhedral theory as possibly being the underlying logic of the solar system, though he now knew that it could not account for all the proportions.

  That January, 1607, a letter arrived from von Hohenburg. He was trying to find a copy of Ptolemy’s Harmonics, evidently having forgotten that he had sent a copy of that very book to Kepler six and a half years earlier. Kepler reminded him of that previous gift but asked whether he would now please find him a copy in the original Greek.

  In March, the book arrived. It included commentaries by Porphyry and the fourteenth-century monk Barlaam of Seminara (the man who had tried unsuccessfully to teach Petrarch Greek). Barlaam argued that the text appearing as Ptolemy’s last three chapters was not authentic. This was particularly disappointing to Kepler, who was sure that it was in those very chapters that Ptolemy must have showed how to use harmonic principles to derive the parameters of his planetary models. Kepler made plans to publish an edition of Harmonics in Greek with his own commentary, in which he would explain and then refute Ptolemy’s theories, then compare them with his own. He also planned to undertake a new Latin translation. In the table of contents for the final volume of his Harmonice mundi, he listed the commentary and also an appendix of about thirty pages, translating what he felt were the most relevant parts of Ptolemy’s book and re-creating Ptolemy’s text in the suspect chapters. But when Harmonice mundi appeared in 1619, the actual appendix included, instead, only an apology that the promised material, begun ten years earlier but interrupted by a move from Prague to Linz “combined with many other troubles,” was not there.

  “Many other troubles” was, sadly, an understatement. Soon after New Year’s, 1611, Kepler’s three children had contracted smallpox. His six-year-old son, Friedrich, who had been a particular delight to Kepler, died. Troops led by a cousin of Kepler’s patron, Emperor Rudolph II, overran Prague and rioted with vigilantes in the streets surrounding Kepler’s house. Rudolph, always an exceedingly eccentric, reclusive ruler, and by then somewhat over the brink of madness, abdicated the throne. Kepler’s wife died of a fever that July, and Rudolph himself expired the next winter. Even before the emperor’s death, Kepler had foreseen the end of his usefulness in Prague and had accepted a position as teacher and district mathematician in Linz—a job on about the level of the one in which he had begun his career seventeen years before in Graz. Though he was widely known and respected, famous for his Astronomia nova, nothing better was available because of an earlier statement still on record that he had made and would not recant, that he believed a Calvinist also was a “brother in Christ.” That opinion disqualified him from any position at a Lutheran university. His salary as imperial mathematician continued, theoretically, but in reality he was still trying to collect years of back pay fr
om the undependable imperial treasury. If all of that had not been enough to distract him, he found it imperative, with two motherless children, to look for a new wife, and he remarried about two years after his first wife’s death. In December 1615, disaster struck again. His mother was accused of witchcraft. In the next three years, while he defended her and struggled to keep his own reputation from being destroyed in the process, he and his new wife lost two infant daughters and also his much loved stepdaughter, the daughter of his first wife by a previous marriage.

  In the winter of 1618, Kepler was too distracted with grief to concentrate on tedious calculations needed for the Rudolfine Tables—astronomical tables based on Tycho’s observations, on which Kepler had been working intermittently for many years. “Since the Tables require peace,” he wrote, “I have abandoned them and turned my mind to developing the Harmony.” In Astronomia nova, he had completely reconstructed astronomy, and this meant not only that the harmony project had taken on much greater proportions but also that Kepler had a much stronger and more comprehensive foundation on which to work. He was now dealing with what he knew to be a real planetary system, whose mathematics and geometry he understood better than anyone else alive.

  “The Harmony” referred to the book he had barely begun in Graz when he had first considered linking the velocities of the planets with musical harmony and had shared his thoughts with von Hohenburg and Mästlin. That period of Kepler’s life had also been a time of mourning, for the death of his first and second child. Now tragedy had again decimated his family, and there must have seemed little evidence of a rational, loving God, but Kepler turned again to the effort to reveal what he believed was God’s marvelous wisdom and rationality to be discovered in nature.

  When Kepler began laying out the table of contents for the book that would be Harmonice mundi, he decided the moment had come to revive his plans to translate Ptolemy’s Harmonics. Then the Thirty Years War broke out and the scarcity of manpower made it impossible to get material printed. Not until 1864, more than two centuries after his death, did an edition of Harmonice mundi appear that included his Latin translation of Ptolemy’s Harmonics. It had survived in manuscript form.

  Nevertheless, in 1618, Kepler was well acquainted with Ptolemy’s Harmonics and had also researched what Aristotle and Pliny had written, centuries before Ptolemy, about the Pythagoreans. He decided that Ptolemy must have been trying to describe and improve on Pythagorean teachings about the harmony of the heavens but had not made it clear what those teachings had been.4 Kepler chose to accept Pliny’s opinion that Pythagoras had assigned a musical pitch to each of the eight heavenly bodies (five planets, stars, Moon, Sun) and linked the distances between them with distances (intervals) between those pitches. Kepler concluded that the Pythagorean heavenly scale must have begun with the Moon, not the Earth, because in an Earth-centered cosmos Earth would not move, and a body at rest, making no sound, has no pitch associated with it. It is something of a mystery why he thought that Pythagoras would have visualized an Earth-centered cosmos. Copernicus had used the Pythagorean concept of a central fire as a precedent for his own rearrangement of the cosmos, and Kepler too liked to point to that precedent. He had read about it in Aristotle’s De Caelo. Perhaps he believed the central fire was an idea from later Pythagoreans, for example Philolaus, and that Pythagoras himself must have treated the Earth as the unmoving center.

  Kepler based his own reconstruction of the scale Pythagoras had used on the Pythagorean reverence for the intervals of the fourth and fifth and on the intervals Pliny had chosen. In Kepler’s reconstruction, the Moon was A, Mercury B flat, Venus B, Sun D, Mars E, Jupiter F, Saturn F sharp, the stars A. The first four notes (A, B flat, B, D) were separated by intervals of a half step, a half step, and a step and a half. The second four (E, F, F sharp, A) were separated by that same sequence of intervals.* The two groups were separated by a whole step (D to E, or Sun to Mars), and that whole step was between the fourth and fifth of the scale, two significant Pythagorean notes because they reflected the ratios 3:4 and 2:3.

  Kepler’s reason for deciding that Pythagoras had associated the lowest note with the nearest heavenly body, the Moon, rather than start his scale on the most distant body, was that he thought Pythagoras, observing the sky, would have seen the higher, more distant planets appear to be moving faster than the lower ones and would not have realized that one component of that movement was the rotation of the Earth itself. The entire sky appears to rotate westward, making one complete rotation every twenty-four hours, while each planet moves in a motion contrary to that. The result of the combined motion is to fool an observer into thinking that the more distant planets (actually the slowest moving) are moving fastest.* Again, Kepler must have been trying to work out what Pythagoras himself would have thought, not later Pythagoreans, who probably did understand that the observed motion of a planet, the Sun, or the Moon, was the combination of two motions. Scholars still regard this understanding as one of the triumphs of early Greek astronomy, probably coming from the Pythagoreans.5 When Kepler got around to constructing his own planetary scales and chords, he had Saturn, not the Moon, sounding the lowest note.

  Having reconstructed what he believed might have been the actual Pythagorean scale, Kepler set it beside Ptolemy’s and considered which he preferred. Both suffered from having an Earth-centered cosmos in mind. But Kepler was fond of Pythagoras’ scale—“altogether more elegant and richer in mysteries” than Ptolemy’s—because it seemed to him to give more importance to the planets’ motions. On the other hand, he gave Ptolemy points for having recognized that there must be a “divine axiom” that determined the number and sizes of the spheres.

  Before Kepler could decide for himself what the harmony of the heavens might be, and what ratios might underlie it, he had to determine which intervals were agreeable to the human ear. He took a great deal of trouble differentiating between types of intervals. There were the usual octaves, fourths, fifths, thirds, and sixths, all of which he called consonantia (they were harmonious when the two tones sounded simultaneously). Then there were several intervals that he called concinna that sounded pleasant following one another in a melody but not when played simultaneously. These included a “major tone” and “minor tone” (roughly equivalent to a whole step and a half step) and two other intervals that were smaller than the interval between adjacent keys on a piano. Finally there were three tiny intervals that Kepler dubbed the “doubtful concinna.” They were not particularly pleasant to hear under any circumstances.

  Kepler combined the intervals into two kinds of musical scales. One had a major third and sixth in it and was the durus scale, close to what we call the major scale. (The major scale beginning on C, for example, includes the intervals C to E and C to A.) The other, with a minor third and sixth, was the mollis scale, close to what we call the minor. (The minor scale beginning on C, for example, includes the intervals C to E flat and C to A flat.) Likewise, chords based on major thirds and sixths were durus; chords based on minor thirds and sixths were mollis.* It requires no musical training to hear the difference between the two scales or chords and experience the emotional effect of this difference: the durus (major) is happy and the mollis (minor) sad. Why these sounds have any influence over human emotions is still a mystery, but the early Pythagoreans, had they known about thirds and sixths, would surely not have been surprised.

  One of Kepler’s goals in the research that lay behind Harmonice mundi was to find out whether two proposals were true: First, that certain ratios between pitches have a special “nobility” and importance and are embodied in the arrangement and movements of the solar system. Second, that the influence of music on the human soul depends on these ratios. As the Pythagoreans had known, musical intervals are the way mathematical ratios show up in sound. One usually encounters written-out music in the form of notes drawn on and between horizontal lines on a page, and seldom does one realize that it would be possible to write out the music more precis
ely (though less practically) as a long string of mathematical ratios. If one wrote out all the continually shifting mathematical proportions among the planets, would the result, played as music, sound harmonious and pleasing to human ears? In Book V, Chapter 9 of Harmonice mundi, Kepler explained why he was convinced—after a prodigious amount of study and calculation—that the details of planetary astronomy, the continually changing speeds and distances of the planets in relationship with one another, were as harmonious and pleasing as could possibly be. He also showed how this best-of-all-possible harmonious arrangements inevitably (even though God had created it) fell a bit short of perfection.

  Most significant for the history of astronomy, Book V began with an ecstatic statement about discovering the relationship between the planets’ orbital radiuses and their orbital periods, even though Kepler had not yet made this discovery when he began writing his book:

  At last I brought it into the light, and beyond what I had ever been able to hope, I laid hold of Truth itself: I found among the motions of the heavens the whole nature of Harmony, as large as that is, with all of its parts. It was not in the same way which I had expected—this is not the smallest part of my rejoicing—but in another way, very different and yet at the same time very excellent and perfect.6

  Kepler felt that this discovery—his third law of planetary motion, or “harmonic law”—was so important that it was essential to go back and insert those sentences to let his readers know what was coming. It was also in Book V that he included a list giving a view of the astonishing mind of Kepler exactly where he was, beginning with a flat statement that sounds completely unremarkable to modern ears but was a block-buster in his time. The list included his three planetary laws, which are still celebrated among the greatest discoveries in astronomy, but also—surprisingly—his old polyhedral theory.