The Music of Pythagoras

by Kitty Ferguson

  [Parmenides] was also associated (as Sotion said) with Ameinias, son of Diochaites, the Pythagorean, a poor man but of good character. It was rather Ameinias that he followed: when Ameinias died he set up a shrine for him (Parmenides came from a famous, wealthy family); and he was led to calm by Ameinias and not by Zeophanes.

  It would seem that if Parmenides “followed” Ameinias, was “led to calm” by him, and thought so highly of him as to set up a shrine, then Parmenides’ own thinking would show traces of Pythagorean ideas. Lured by this clue, scholars have repeatedly attempted to find elements of Pythagoreanism in Parmenides’ writing, with no success.

  In a paradoxical twist, history celebrates Parmenides for insights that he did not claim were correct; for example, that the light of the Moon “always gazing at the rays of the Sun” is reflected light.4 He laid out such ideas in Part 2 of a beautiful, enigmatic poem, after he had warned in Part 1—a guide to the Way of Truth—that what he was going to present in Part 2 was “deceitful.” He was not claiming to present “facts” or even opinions, only what human opinion on these matters might plausibly be at best.

  He argued that those setting out on a voyage of “inquiry” probably mistakenly believed they had a choice between two subjects, things that existed and things that did not exist. But nonexistent things were unthinkable and unsayable, and inquiry into them was “a trail of utter ignorance.” As for what existed, certain things had to be true about it: It had always to have existed, and it had to be indestructible. Otherwise there would be a chance it might at some time not exist, which was unthinkable and unsayable. It had also to be continuous in space and time (no gaps), unchanging and unmoving, and finite. Human senses told one otherwise, admitted Parmenides, but they could not be trusted. So much for any possibility of learning about the world by observing and experiencing it!

  Melissus, another “Eleatic” philosopher, was an admiral from Samos, though his and Pythagoras’ lives there did not overlap. As Aristotle told the story, in 441 B.C. Athens declared war on Samos. The Samians defeated Pericles himself in a sea battle, but Pericles survived and hostilities continued. When a stalemate dragged on, Pericles, bored and underestimating the Samians, led some of his ships away on an expedition. Melissus, commanding Samos’ fleet, took this opportunity to attack, “despising the small number of their ships and the inexperience of their commanders.”5 This time the Athenian fleet suffered a devastating defeat. Samos destroyed many enemy ships, captured war supplies, and gained control of the eastern Mediterranean.

  Melissus also found the time to write a prose version of Parmenides’ Way of Truth, introducing new arguments to support Parmenides but disagreeing with him on key points. Melissus argued that whatever existed had to be infinitely extended in all directions, not be finite as Parmenides thought. For that reason, no more than one thing could be in existence. Melissus believed even more strongly than Parmenides that sense perception was an illusion, that reality was completely different from the way it appeared.

  Zeno, like Parmenides, was from Elea. He came up with forty different arguments to support Melissus’ assertion that only one thing could exist, produced four arguments to show that motion was impossible, and carried similar issues—including the concept of infinity—to even greater extremes, some have said to the point of intellectual nihilism. Zeno is believed to have been the author of a book called Against the Philosophers, which almost certainly meant “Against the Pythagoreans.” His criticisms may have influenced a change in their way of thinking that showed up in the work of Philolaus, about whether a point in geometry has any dimensions.

  The Eleatics’ penchant for a strictly abstract, logical approach and their distrust of sense perceptions was, in turn, a reaction against thinkers like Thales, Anaximander, and Anaximenes, all of whom may have taught Pythagoras. Observational evidence had seemed no illusion to them, and they suggested that substances such as water (Thales) and air (Anaximenes) were fundamental reality. Anaximander had been more abstract, but he implied a “many” that did not gel with the concept that only one thing exists. These ideas set the stage for Philolaus. He chose not to take on the Eleatics directly, but continued to value the possibility of studying nature using the tools of the five senses.

  Philolaus dealt with old questions: How did everything (the cosmos, or “world order”) begin? What basics—“first principles” (archai)—had to be in place, or had to be true, in order for anything else to happen? He answered in the first sentence of his book: “Nature in the cosmos was fitted together harmoniously from unlimited things and limiting things, both the cosmos as a whole and all things within it.” The key words were “harmoniously,” “unlimited,” and “limiting.”* The ideas of the “unlimited” and the “limiting” were older than the Eleatics or the Pythagoreans, the first known mention having come from Anaximander. “Harmoniously” was uniquely Pythagorean.

  If Pythagoras did study with Anaximander, he learned that for him the first principle was something more abstract than Thales’ water. It was the “limitless” (or the “unlimited”)—characterless, indefinite, unbounded by time or space. The primordial description in Genesis of the earth “without form and void” is close to the same idea. So is the late-twentieth-century scientific concept that describes the universe (or “pre-universe”) as a state of wobbling quantum nothingness from which anything (or nothing) could have emerged. The “limitless” was a situation with no differentiation, no choices made, no orders given, no laws laid down that would allow or compel some things to happen but not others. It was the “limiting” that was responsible for differentiation. Anaximander did not, however, think of the limitless only as a situation that preexisted the world, that came first chronologically and ended when the heavens and the world emerged. The limitless was a fundamental background to eternally continuing cycles of destruction and generation. He associated the limitless with time.

  If Philolaus can be taken as an example, Pythagoreans also believed that the fundamental principles limitless and limiting were both needed in order for anything else to exist, which raised a problem. The two principles were discordant and opposed to one another; how could they work together to produce anything? There had to be another first principle. The limitless and the limiting “must necessarily be locked together by a harmony if they are to be held together in a world.” Harmony had also to be a “first principle,” one of the archai, maybe the most fundamental of all.

  The word harmonia was not coined by Pythagoras or Philolaus. As early as Homer’s Odyssey, it meant joining or fitting together. In carpentry it meant a wooden nail or peg. In music it referred to the stringing of a lyre with strings of different tension. The Pythagoreans gave it new importance. In the ratios of music, they felt they had found an actual link between harmonia on the everyday level and the harmonia that helped create the universe and that bound it together. They had come to think of the ratios of musical harmony as exemplifying the primordial organizing principle of the universe.

  The musical interval of the octave was the “first consonance,” which Philolaus identified by the name harmonia. The “second consonance” was the interval of a fifth; the next was the interval of a fourth. Add the four numbers in these ratios (1, 2, 3, 4) and the result is 10, the perfect number.*

  The numbers 1, 2, 3, and 4 had additional significance for Philolaus. They underlay the progression from point to line to surface to solid:

  A point (on the left) is 1; a line is 2 (defined by two points, one at each end); a surface is 3 (defined by three points, one at each corner); a solid is 4 (defined by four points, one at each corner).

  Later, Speusippus, Plato’s nephew and successor as head of his Academy, explained what he understood Philolaus to have meant: “The point is the first principle leading to magnitude, the line the second, the surface the third, the solid the fourth.” This sounded more complicated, but it allowed the progression to apply to other shapes and solids besides the triangle and pyramid; for example,
a square, with four corners, led to a cube, with eight.

  According to Philolaus, the Pythagoreans took the number 10 and ran with it. Aristotle later commented that they “construct the whole heavens out of numbers.” Ten being the perfect number, there had to be ten major heavenly bodies, though no one could see ten in the sky. Also, said Philolaus, there had to be fire both at the center of the universe and at the highest point, surrounding everything, at the outermost circumference or “uppermost level.” This was partly observable, for the stars were fires out on the periphery, but what about the center? Here, according to Philolaus, the Pythagoreans made a leap that set them far ahead of their contemporaries. The Earth could not be the center of the cosmos, nor, for that matter, could the Sun. The center had to be a “central fire,” a fiery “hearth of the universe” around which the Earth, the Moon, the Sun, the five planets, and the stars revolve. As the scholar Aëtius—probably of the second century A.D.—described it, “Unlike other philosophers, who say that the Earth is at rest, Philolaus the Pythagorean said that it revolves about the fire in an inclined circle like the sun and moon.” In the centuries when no one—with the singular, brilliant exception of Aristarchus of Samos, who proposed a sun-centered cosmos in the third century B.C.—was willing to consider a moving Earth that was not the center of the cosmos, scholar after scholar tried to show, or simply assumed, that the Pythagoreans could not really have meant this.

  Earth, Moon, Sun, five planets, and the “outer fire” (stars) added up to nine things to “dance around the center.” Since there had to be ten, the Pythagoreans decided there was a “counter-earth,” closer to the central fire than the Earth. The central fire and counter-earth were never visible from the Earth, because in their revolutions Earth and counter-earth were always “opposite” one another. Aristotle commented, not too approvingly:

  Any agreements that they found between number and harmony on the one hand, and on the other the changes and divisions of the universe and the whole order of nature, these they collected and applied; and if something was missing, they insisted on making their system coherent. For instance they regarded the decad as something perfect, and as embracing the whole nature of number, whence they assert that the moving heavenly bodies are also ten; and since there are only nine to be seen, they invent the counter-earth as a tenth.6

  The central fire and counter-earth were certainly consistent with the Eleatic view that human sense perceptions were not trustworthy for finding out what was true about the universe, for neither could be perceived with any of the five senses.

  The Pythagoreans could have picked up the idea that the Moon shone by reflected light from Parmenides, or perhaps from Anaxagoras, but according to Philolaus the light it reflected was not the Sun. Instead, both Moon and Sun caught the light and heat of the central fire and the outer fire. The Sun, like glass, filtered these through to the Earth. Living beings inhabited the Moon and probably the counter-earth, though because of their positioning the inhabitants of Earth and counter-earth never saw one another. The Moon was home to “living creatures and plants that are bigger and fairer than ours. Indeed the animals on it are fifteen times as powerful and do not excrete, and the day is correspondingly long.” This must have been calculated from the fact that the lunar “day” lasts fifteen Earth days. It would be consistent with Earth and counter-earth orbiting the central fire for Philolaus to have thought the Earth was a sphere. Though this does not appear in any of the fragments, Aristotle and another later author, Alexander Polyhistor, wrote that Pythagoreans in the late fifth century (that would have included Philolaus) and early fourth century B.C. believed that the Earth was spherical.

  Night and day on Earth, wrote Philolaus, were produced by the Earth’s and Sun’s positions relative to one another, and the apparent rotation of the planets and Sun were in part the result of movement of the Earth. The Pythagoreans were not only the first to realize that what we see, from Earth, as the heavenly motions is a combination of movement in opposite directions; they were also far ahead of their contemporaries in recognizing that the movement of the Earth itself contributes to the picture.

  Philolaus linked all of this to the origin of the cosmos, when harmonia reconciled the limitless and the limiting. The discovery of the ratios of musical harmony had provided a brilliant metaphor for the interaction of the limited and the unlimited. The whole range of musical pitches, stretching infinitely in opposite directions, higher and lower, and including an infinite number of possible pitches “between” the tones usually heard in music, represented the unlimited. When this infinite continuum of possible pitches was sorted out (limited) according to one series of ratios and not another, the result was order and beauty. The infinite possibilities still existed, higher, lower, and between the notes, but the “unlimited” was thus disciplined and brought into harmony within an order, a kosmos.7

  The “first thing to be harmonized,” wrote Philolaus, was the central fire. The central fire was the number 1; the outer fire, the number 2. The ratio 2:1 represented the musical octave, so an octave separated the two extremes of the cosmos. Some Pythagoreans went so far as to suggest that the periodic motions of the nine orbiting bodies around the central fire were related to the musical ratios, and their revolutions produced the “music of the spheres,” but that idea did not appear in Philolaus’ book, at least in the fragments that survived.

  Philolaus’ cosmic arrangement was odd and imaginative. In spite of Aristotle’s disparagement, it must be admitted that the Pythagoreans were clearly capable of independent, outside-the-box thinking. This was not storytelling or myth-making, but drawing conclusions by deciding “This must be so, on the basis of what we already know about the cosmos and the numerical rules by which things work”—a giant step in the direction of what has become the time-honored way of developing scientific theories.

  Philolaus made clear that Pythagoras believed in and taught reincarnation (transmigration of the soul) and that the soul was immortal, tied up with a divine, universal soul to which it might someday return. The way Philolaus applied the idea of harmonia to the soul showed up in Plato’s dialogues and in Aristotle, who wrote, “There seems to be in us a sort of affinity to musical modes and rhythms, which makes some philosophers say that the soul is a harmonia, others that it possesses harmonia.”8

  Philolaus evidently played an important role in forming Plato’s impression of Pythagoras and Pythagorean teaching, though judging from the dialogues they never met in person. Plato knew Philolaus’ work through members of surviving Pythagorean communities, and from Socrates. In Plato’s dialogue Phaedo, his characters are supposed to have learned of the soul’s harmonia from Philolaus. Simmias, who has listened to him in Thebes, says:

  And in point of fact I fancy that you yourself are well aware, Socrates, that we mostly hold a view of this sort about the soul: we regard the body as held together in a state of tension by the hot, the cold, the dry and the moist, and so forth, and the soul as a blending or harmonia of these in the right and due proportion.9

  Simmias must have been listening to others besides Philolaus, because his interpretation sounds more like that of the medical scholar Alcmaeon of Croton. Alcmaeon’s lifetime may have overlapped Pythagoras’, and he was probably a Pythagorean himself. If not, he was close to them and clearly reflected Pythagorean thinking about opposites when he wrote, “What preserves health is an equilibrium of the powers . . . health is a balanced mixture of opposites.”10

  Because a body could go so out of synchrony in sickness and death as to lose any suggestion of harmony, Plato’s Simmias worried that his soul could not be immortal. Echecrates, another Pythagorean character in the dialogue, also comments uneasily,

  This teaching that the soul is a kind of harmonia has had, and still has, a strong hold on me, and when you mentioned it I was reminded that I too had believed it. Now, it is as if I were starting at the beginning again. I terribly need another argument that can persuade me that the souls of the dead do n
ot die with them.11

  The argument Echecrates and Simmias needed to hear—the best Pythagorean shoring up of their faltering faith—had appeared earlier in the same dialogue when Socrates expressed surprise that Simmias and his friend Cebes, both described as students of Philolaus, were ignorant of Philolaus’ teaching about suicide. Socrates admits that this is part of a “secret doctrine”: We are put into the world by the gods, who take care of us, and we must not leave the world until the moment they have chosen, even though death, when finally permitted, is like getting out of prison. A Philolaus fragment in the writing of the early Christian scholar Clement of Alexandria echoes the idea: “This Pythagorean Philolaus says: ‘The ancient theological writers and prophets also bear witness that the soul is yoked to the body as a punishment, and buried in it as in a tomb.’ ”12 Unfortunately, souls had a tendency to become too fond of bodily existence. Plato attempted to put this issue in its proper perspective in one of the most Pythagorean statements of his Phaedo:

  [The soul that is not completely purified] has always associated with the body and tended it, filled with its lusts and so seduced by its passions and pleasures as to think that nothing is real except what is bodily, what can be touched and seen and eaten and made to serve sexual enjoyment.13

  Though Plato’s characters Simmias, Echecrates, and Cebes had misgivings, Philolaus, Plato, Socrates, the Pythagoreans before them—including Alcmaeon—and Pythagoras himself all clearly believed that souls were immortal. Bodily health was harmony; sickness and death a breakup of that harmony, but this physical harmony was not the ultimate harmony. There was a universal harmony to which every Pythagorean aspired to escape from the tedious round of earthly reincarnations. The soul was set in the body by means of numbers and an immortal harmonia, and its quest for the divine level was dependent on number. Plato’s thinking took off from there.