The Music of Pythagoras

by Kitty Ferguson

  Disappointingly, Aristotle did not really answer the question whether Plato’s view of the relationship between the ideal and the material world was derivative of the Pythagoreans, or original, or somewhere in between. What Aristotle concluded has hung in the air for centuries, with the answer depending on what he meant by one ambiguous Greek sentence. Burkert cut to the heart of the matter:

  Again and again it becomes clear that the Pythagorean doctrine cannot be expressed in Aristotle’s terminology. Their numbers are “mathematical” and yet, in view of their spatial, concrete nature, they are not. They “seem” to be conceived as matter and yet they are something like Form. They are, in themselves, Being, and yet are not quite so.5

  Guthrie put it more simply: “By the use of his own terminology, Aristotle imports an unnecessary confusion into the thought of the early Pythagoreans. It is no use his putting the question whether they employ numbers as the ‘material’ or the ‘formal’ causes of things, since they were innocent of the distinction.”6

  Aristotle gave what is probably the most reliable description of the pre-Platonic Pythagorean concept of “music of the spheres,” grumbling that “it does not contain the truth,” though he admitted it was “ingeniously and brilliantly formulated.” He explained that the Pythagoreans realized that all harmonious-sounding musical intervals were the result of certain numerical ratios in the tuning of an instrument, so “number” was “harmony.” The same numerical ratios determined the arrangement of the cosmic bodies, resulting in a “harmony of the spheres.” Here, said Aristotle, was “what puzzled the Pythagoreans and made them postulate a musical harmony for the moving bodies”:

  It seems that bodies so great must inevitably produce a sound by their movement. Even bodies on Earth do that, although they are not so great in bulk or moving at so high a speed, or so many in number and enormous in size, all moving at a tremendous speed. It is unthinkable that they should fail to produce a noise of surpassing loudness. Taking this as their hypothesis, and also that the speeds of the stars, judging from their distances, are in the ratios of the musical consonances, they affirm that the sound of the stars as they revolve is concordant.

  Some heavenly bodies appear to move faster than others. Aristotle wrote that the Pythagoreans had arrived at the idea that the faster the motion, the higher the pitch it produced, and they had taken this into consideration when allowing the ratios of the relative distances between the bodies to correspond to musical intervals. With the full complement of heavenly bodies, the result was a complete octave of the diatonic scale.*

  What surprises is that Aristotle or anyone could think the eight notes of the scale heard simultaneously would be harmonious. The sound would not be beautiful. There would be cacophony in the heavens. Humans should be glad they cannot hear it. Pity Pythagoras, who, legend says, could! The explanation cannot be that harmonia did not imply audible sound, for Aristotle thought the Pythagoreans believed planetary movement produced actual tones. He never explained how it could be beautiful, but he did give what he thought was the Pythagorean explanation—different from Archytas’—for why ordinary humans do not hear it:

  To solve the difficulty that no one is aware of this sound, they account for it by saying that the sound is with us right from birth and has thus no contrasting silence to show it up; for voice and silence are perceived by contrast with each other, and so all mankind is undergoing an experience like that of a coppersmith, who becomes by long habit indifferent to the din around him.

  AT THE TIME of Aristotle and in later antiquity, it was generally assumed that if one mentioned “Pythagorean mathematics,” an educated person would know what that meant, but in fact the meaning was vague, apparently referring to a tradition that thought it inspiring to discover hidden, true relationships of the sort that, once found, seemed inevitable. Since the evidence about what sixth- and fifth-century Pythagorean mathematics were like is so sparse, we are at a loss to know how authentically Pythagorean this so-called Pythagorean mathematics was. To modern eyes, its vestiges seem feeble by comparison with Euclid’s Elements, which appeared around 300 B.C. Did it really reflect a naive mathematics of Pythagoras himself, and his associates? Or was it “a dilute, popularized selection from what had been originally a rigorous mathematical system”?7 Perhaps it was a hodgepodge of what survived from early, primitive mathematical thought from several sources, mistakenly lumped under the heading “Pythagorean”? Maybe a much more authentically Pythagorean, lively mathematici heritage had moved through Archytas to influence Euclid, while this older, calcified, fading mathematics limped alongside, still bearing the name “Pythagorean.”

  There are also differences of opinion about whether there is valid reason to call the five regular solids that Plato featured “Pythagorean” solids.8* The issue is not a simple one, for “knowing about” the solids, or “discovering” them, or “trying to figure them out,” are not the same as “giving them a full mathematical description” or being able to prove that they are the only possible perfect solids. It is uncertain which achievement deserves to be rewarded with having one’s name attached to it.

  Arguing in favor of early Pythagorean knowledge of the solids is the fact that these shapes were familiar in nature and construction. Cubes (and pyramids, for anyone who had been to Egypt) were familiar building shapes, though pyramids often were five-sided including the base, not four-sided tetrahedra. A dodecahedron dating from at least as early as Pythagoras, apparently Etruscan, has been discovered near Padua. Pyrite crystals appear as cubes and also, in southern Italy and on the isle of Elba, in the form of dodecahedra.9 A fluorite crystal is an octahedron; quartz crystals are pyramids and double pyramids; garnet crystals, dodecahedra. Pythagoras would have known about gems and crystals if his father really was a gem engraver, and someone with a Pythagorean cast of mind would surely have been curious about regular, beautiful shapes that appear without any human intervention. It would have been in keeping for someone obsessed with numbers to try to understand them by means of numbers.

  Also favoring an early Pythagorean knowledge of them is that, if the fragment is genuine, fifty to a hundred years after Pythagoras’ death Philolaus knew about the five regular solids but was almost certainly not, himself, their discoverer. In the absence of evidence to show who did or did not discover them, it is not far-fetched to think the five regular solids might legitimately be called Pythagorean.

  Plato associated four of the five solids with the four elements in his Timaeus, as had Philolaus in the fragment that read, “The bodies in the sphere are five: fire, water, earth, and air, and fifthly the hull of the sphere”† But had anyone made that association earlier? The scholar Aëtius, of the second century A.D., thought Pythagoras had:

  There being five solid figures, called the mathematical solids, Pythagoras says earth is made from the cube, fire from the pyramid, air from the octahedron, and water from the icosahedron, and from the dodecahedron is made the “sphere of the whole.”10

  Since “Pythagoras says” was used for what Pythagoras’ followers said, the attribution should probably be read as “the Pythagoreans said.” Aëtius got his information from Theophrastus, a pupil of Aristotle who may have been contradicting his teacher, for Aristotle scoffed that the Pythagoreans had “nothing new to add” to knowledge about the elements. However, Aristotle had so little respect for the idea of associating elements with solids that even if irrefutable evidence had existed that the association originated with the Pythagoreans he would still have dismissed it as “nothing new to add.” Little survives of Theophrastus’ history of philosophy or of the books he wrote about individual philosophers, but more would have been available when Aëtius was doing his research. However, though Philolaus’ fragment associated the elements with the solids, and the solids might have been known to earlier Pythagoreans, the identification of the four elements as fire, water, earth, and air did not originate with them. Philolaus was evidently familiar with the idea from his older contemporary,
the Sicilian poet-philosopher Empedocles, born ten years after Pythagoras’ death.*

  The question whether the Pythagoreans thought of a point as having magnitude seems trivial, but it is related to the question of who first knew about the solids. Zeno, one of the Eleatics, reputedly scorned the Pythagoreans for naively thinking that a point had dimensions like a pebble and that two points (pebbles) touching one another made a line, but that way of thinking made the pyramid easy to “discover” by building a little pebble structure. The Pythagorean preoccupation with the numbers 1, 2, 3, and 4 makes it difficult to believe they did not extend their progression past making a triangle with three pebbles to building a little pyramid with four, or better yet a larger one with 10, the perfect number.

  Speusippus, Plato’s pupil and nephew, attributed the point–line–surface–solid progression to Pythagoreans before Archytas. It was a more primitive way of arriving at a solid than Archytas’ use of “movement.” Even the use of movement may have come before Archytas, and leads easily to a square and cube. An example appears in a reference from the Skeptic philosopher Sextus Empiricus, who flourished at the turn of the second to the third century A.D. He called this a “scheme of the Pythagoreans”: “Some say that body is formed from one point. This point by flowing produces a line, the line by flowing makes a surface, and this when moved into depth generates a body in three dimensions.”11

  The sophistication of geometry in a Pythagorean community a little more than a century after Pythagoras’ death—as witness Archytas’ solution for doubling the cube—makes it ludicrous to insist that earlier Pythagoreans could not have discovered the five regular solids. Nevertheless, the man who first arrived at a complete mathematical understanding of them was not a Pythagorean. He was Theaetetus, a friend of Plato who was killed in 369 B.C. Whatever was known about the regular solids earlier, Theaetetus, with his description of the octahedron and the icosahedron, finished the job.

  In the end, in spite of differing viewpoints about the solids and the “Pythagorean mathematics” of late antiquity, there is widespread consensus that the first Pythagoreans opened up a new way of thinking about, appreciating, and using numbers, representing a watershed and having very long lasting impact. Their profound musical/mathematical discovery was as modern as tomorrow’s science news, as timeless as any discovery ever made, but most of the true mathematical connections and relationships in nature were hidden too deep for them to find. Even Kepler, in the sixteenth century A.D., with a Pythagorean certainty that such relationships existed, spent a good part of his lifetime searching for them on too superficial a level and was surprised when he had to admit that nature followed her own far cleverer mathematics, not his. In spite of the Pythagorean faith in the power of numbers, they had no inkling of how far numbers would lead humankind. Working out the implications of their discovery would take centuries.

  ALONG WITH ARISTOTLE, three other authors who lived during the latter part of the fourth century B.C. were the earliest and most reliable sources used by Porphyry and Iamblichus. They were Heracleides Ponticus, of Plato’s Academy, and Aristoxenus of Tarentum and Dicaearchus of Messina, both Aristotle’s pupils. Heracleides Ponticus, like Plato, wrote dialogues. He used the character “Pythagoras” as a spokesman, telling stories about his former lives and calling himself philosophos, lover of wisdom. Other Pythagoreans in the dialogues, Hicetas and Ecphantus, taught that the Earth rotates.* Heracleides believed that the Earth rotates, and that this makes it appear to humans as though the stars are moving.†

  Dicaearchus was Porphyry’s and Iamblichus’ source about Pythagoras’ arrival in Croton and his success among the young men, the city rulers, and the women. Dicaearchus claimed that in his own time the memory of the revolts that ended Pythagorean rule was still vivid in Magna Graecia. He revered Pythagoras as a moral teacher and social reformer, but he believed in no sort of immortality and scorned the idea that anyone could remember former lives, joking that Pythagoras had been a beautiful courtesan in one reincarnation. A man of extensive learning and a scientist with an independent turn of mind, an admirer of Pythagoras and yet not an unqualified admirer, Dicaearchus had his ear to the ground at a time when the oral record could be extremely trustworthy, in the region where Pythagoras had lived and flourished—all of which increases the likelihood that what he reported was genuine.

  Aristoxenus, like Dicaearchus, did not toe the Pythagorean line precisely. He dismissed the idea of the soul being more than a harmony of the body’s various components, and his music theory took a different direction from Archytas’. The information Porphyry and Iamblichus attributed to Aristoxenus probably came from his biography of Pythagoras—thought to have been the first written—but neither Porphyry nor Iamblichus ever actually saw Aristoxenus’ and Dicaearchus’ books.* The information they took from them came indirectly, through other writers who lived during the centuries in between.

  After Aristotle there were no attempts in antiquity to draw a distinction between pre-Platonic Pythagorean doctrine and Plato. Beginning with Plato’s pupils Speusippus and Xenocrates, no one for centuries would make a distinction between Platonism and Pythagoreanism at all. Almost without exception, everyone would accept what Plato taught in his Timaeus and his “oral doctrine” (reported by Aristotle) as the teaching of the early Pythagoreans. In the eyes of the educated world, Plato was a Pythagorean.

  BY THE TURN of the centurY in 300 B.C., the world of classical Greece, of Plato and Aristotle, and of strong and often warring city-states like Athens, Sparta, and Thebes had ended.12 The rise of a power from the north—the kingdom of Philip the Great of Macedonia—was heralding a new era. Less than forty years after Philip had become king of Macedonia in 359, his son (traditionally Aristotle’s pupil) Alexander the Great had conquered not only Greece but also Egypt and the entire Persian empire to the east, as far as present-day Afghanistan, Pakistan, and the Indus River. The culture and learning of Greece and its colonies and of the conquered peoples mixed and, to an impressive extent, enriched one another.

  After Alexander died in 323 B.C., though the city-states had not vanished entirely and change was slow in remoter regions such as Magna Graecia, his short-lived and sprawling empire became three “successor states” under his former generals and associates. Mainland Greece became part of Macedon. The Seleucid dynasty controlled Syria. Egypt was ruled by the Ptolemies, the dynasty that would later include Cleopatra. At the time of Alexander’s death (and Aristotle’s, for he died a year later, in 322), Athens was still the hub of the intellectual world, but Alexandria, with the wealth of the Ptolemys lavished on literature, the arts, mathematics, science, and a library and museum would soon rival and eventually eclipse her.

  Around 300 B.C., Euclid, who lived in Alexandria, gave mathematics and geometry a new form of life, surpassing all others in antiquity for putting the power of numbers to use in a truly significant and comprehensive manner. Euclid personified the Pythagorean intellectual and philosophical conviction that mathematics was a precious guide to truth, and he was even known to use a Pythagorean aphorism, but he did not consider himself a Pythagorean nor did he belong to a Pythagorean community.*

  Euclid in a sixteenth-century engraving

  Euclid’s Elements is one of the premier intellectual achievements of all time, foundational for later mathematicians and geometers. It was both a comprehensive summary and treatment of what had been discovered before him, and wondrously original, and Euclid did not clearly distinguish between what was new and what was old. He knew the Pythagorean theorem and included it in Book I as “Proposition 47,” never referring to it as “Pythagorean” but also never claiming it was his own discovery or mentioning another origin. His knowledge of early Pythagorean mathematics and astronomy appears to have come mostly through Archytas, though modern experts who have analyzed the Elements believe that many of the results which appear in it13 predated Archytas, and that some of the material was extremely old.14 Archytas had previously built on some of this
earlier work, and his discoveries, particularly his number theory, were incorporated by Euclid in the Elements Book VIII.

  By Euclid’s standards, a feeling of inevitability and a few examples did not constitute a “proof.” The so-called Pythagorean mathematics of his own contemporaries did not fall in happily with his higher abstraction.15 That tradition nevertheless wheezed along and proved tenacious beyond all expectation. Iamblichus preferred it:

  Pythagorean mathematics is not like the mathematics pursued by the many. For the latter is largely technical and does not have a single goal, or aim at the beautiful and the good, but Pythagorean mathematics is preeminently theoretical; it leads its theorems toward one end, adapting all its assertions to the beautiful and the good, and using them to conduce to being.16

  Though Euclid was translated into Latin and not unknown in the Middle Ages, the premier mathematical textbook of those later centuries would be in the “Pythagorean” mathematical tradition, not his.* However, and in spite of Iamblichus’ opinion, Euclid’s Elements resonates with joy and appreciation for the beauty of the subject he was exploring as no one had before. Though modern mathematicians still carry forward the ancient Pythagorean/Platonic belief in the beautiful rationality of numbers, and even tend to be suspicious of anything claiming to be mathematical truth that is not beautiful, it is Euclidean technical rigor that guards the gate of beauty.

  CHAPTER 11

  The Roman Pythagoras