The Music of Pythagoras (19 page)

In Corinth, Dionysius met Aristotle’s pupil Aristoxenus, who was collecting information about Pythagoras and the Pythagoreans. Aristoxenus would turn out to be one of the earliest and most valuable sources, for Tarentum was his birthplace and he said his father knew Archytas. From Dionysius, who had been rather useless at nearly everything else, Aristoxenus was able to glean firsthand knowledge about Pythagoreans in the fourth century in Syracuse, not far from the area where the society had originated.

As Dionysius told the story to Aristoxenus, some of his courtiers in Syracuse had spoken disparagingly of the local Pythagoreans as arrogant, pious fakes whose rumored moral strength and superiority would evaporate in a crisis. Other courtiers disagreed, and the two sides contrived a way to settle the dispute. Would one Pythagorean be willing to stake his life on the dependability and faithfulness of another? Would the other’s faithfulness and dependability—to the death—prove deserving of such trust?

The courtiers accused a man named Phintias, a member of the local Pythagorean community, of plotting against Dionysius. When Dionysius sentenced him to death, Phintias asked for a stay of execution for the remainder of the day, long enough to settle his affairs. It was a Pythagorean custom, established by Pythagoras himself, to keep no private property but own all things in common. Phintias was the oldest among the local brotherhood and chiefly in charge of the management of finances. Dionysius and his court, following their plan, allowed him to send for another Pythagorean, Damon, to remain as hostage until his return. To the astonishment of the court, Damon willingly came to stand as personal surety for Phintias. Phintias departed, and the courtiers—sure they had seen the last of him—sneered at Damon for being such a trusting fool. But the faithful Phintias returned at sunset to face his death rather than leave his friend to be executed in his stead. “All present were astonished and subdued,” reported Dionysius, who was so impressed that he embraced the two men and asked to be allowed to join their bond of friendship. Not surprisingly, “they would by no means consent to anything of the kind.” What happened to them then is not known. Plato, so often at court in Syracuse, also likely heard about this incident, but he never wrote about it.

Plato’s activities in Megale Hellas went beyond learning about Pythagoras and Pythagorean teachings, experiencing day-to-day reality in a tyrant’s court, and abortive attempts to tutor Dionysius. He helped Archytas strengthen his position in Tarentum as a minor philosopher king. Archytas went on to play a prominent role in political affairs among the cities of Megale Hellas and Sicily, in accordance with the Pythagorean tradition of wise and able involvement in public service.

I
N A SEARCH
for the real Pythagoras and the Pythagoreans and what they believed and taught, the information about Archytas, Plato, and Dionysius the Younger provides valuable clues. Most significantly, it reveals a link between Plato and a Pythagorean community that still existed in the fourth century
B.C
. in the region where Pythagoras and his followers had had their golden age in the late sixth century. Plato knew, and knew of, other fourth-century Pythagoreans, but after his visits to Tarentum, when he thought “Pythagorean” he was probably mostly thinking of Archytas and his associates. When he thought “Pythagorean mathematics and learning” he was thinking of the mathematics and learning of Archytas.

What was he like, this man who was, for Plato, the best available evidence of what it meant to be Pythagorean and what “Pythagorean knowledge” was? What could Plato have learned from him about Pythagoras and what Pythagorean teaching had been more than a century earlier?

Archytas was known to be a mild-mannered man who ruled in Tarentum through a democratic set of laws—information deduced from the news that these were not always obeyed: Though the “laws” said a man should serve no more than one year, the city “elected” Archytas seven times to the office of
strategos
, or ruling general.
²
Aristoxenus wrote that Archytas was never defeated as a general except once, when his political opponents forced him to resign and the enemy immediately captured his men. Archytas, said Aristoxenus—whose father, he claimed, had known the man in person—was “admired for excellence of every sort.”

There can be no doubt that as a scholar Archytas lived by the great insight that set the Pythagoreans apart from other ancient thinkers: that numbers and number relationships were the key to vast knowledge about the universe. Archytas was a rigorous mathematician who solved an infamous problem in Greek mathematics known as the Delian problem, or doubling a cube, that is, creating a new cube twice the
volume of the first. Archytas’ solution was sophisticated, requiring new geometry using three dimensions—“solid” geometry—and involving the idea of movement.
³
*

Diagram showing how Archytas solved the Delian problem, evidence of how advanced Pythagorean mathematics and geometry had become in little more than one century.
^†

Viewing the world through the eyes of his Pythagorean forebears, Archytas could not avoid pondering the possible hidden, underlying numbers and geometry. “Why are the parts of plants and animals (except for the organs) all round?” he asked, “of plants, the stems and branches; of animals, the legs, thighs, arms, thorax? Neither the whole animal nor any part is triangular or polygonal.” He suspected there was a “proportion of equality in natural motion, since all things move proportionately, and this is the only motion that returns back to itself, so that when it occurs it produces circles and rounded curves.”

Later scholars, among them Euclid and Ptolemy, agreed that Archytas’ precise work in the mathematics of music was fundamentally linked with the earliest Pythagorean mathematics and music theory.
⁴
Archytas extended the study of numerical ratios between notes of the scale and showed that if you defined a whole tone as the interval separating the fourth and fifth notes of the scale (such as F and G in a scale beginning with C), as Greek music theorists were doing, then a whole tone could not be divided into two equal halves.
*
This had dramatic implications, for it was an example of something obviously present in the real world that could not be measured precisely. A different example, discovered in the right triangle, had famously caused the first Pythagoreans to have a devastating crisis of faith in the rationality of the universe, but incommensurability seemed no longer to disturb Pythagoreans like Archytas in the fourth century
B.C
.

In astronomy, Archytas puzzled over the question of whether the cosmos is infinitely large, and was notorious for asking: “If I come to the limit of the heavens, can I extend my arm or my staff outside, or not?” He replied that whatever the answer—yea or nay—if he were out there performing this experiment, he could not actually be at the limit of the heavens. If he could not extend his arm or staff farther, something beyond the supposed limit had to be stopping him.
⁵

In archaic-sounding litanies, the first Pythagoreans had asked “What are the isles of the blessed?” and answered “The Sun and the Moon.” Archytas brought this up-to-date in a more sophisticated catechism, asking “What is calm?” and answering as a parent might answer a child, with an example: “What is a man?” “Daddy is a man.” Similarly, Archytas’ reply to “What is calm?” was “Smoothness of the sea.” His catechism, however, implied more than “example answers,” for he liked to connect the specific with the general, reflecting the Pythagorean doctrine of the unity of all being, and he enjoyed thinking about the relationship between the whole and the parts or particulars. His questions and answers about the weather and the sea were particular cases of deeper questions about smoothness and motion. The problem of dividing a whole tone into equal halves was a particular case of a mathematical discovery about
ratios that could not be equally divided. His observations about the roundness in trees, plants, and animals were particular manifestations of a “proportion of equality in natural motion.” Archytas was convinced of a tight connection between understanding the universe, or anything else, as a whole and understanding the details. Plato wrote in his
Republic
that this paragraph from Archytas was “the teaching of the Pythagoreans”:

The students of mathematics [by this, Archytas meant students among the Pythagoreans] seem to me to have attained excellent knowledge, and it is not surprising that they have correctly understood how things stand in each matter. For since they have obtained knowledge of the nature of the universe as a whole, they will have come to have a good view of how each thing stands in particular. Concerning the speed and risings and settings of the heavenly bodies they have handed down to us clear knowledge, concerning geometry and numbers, and not least concerning music. For these studies seem to be sisters.

When Archytas wrote about such matters as smoothness and non-smoothness of the sea he was reflecting another Pythagorean traditional favorite—opposites (smoothness/lack of smoothness; motion/lack of motion)—and for him that line of thought inevitably led back to thinking about infinity. Can something be infinitely calm? Or infinitely uncalm? Or infinitely smooth; infinitely rough?

As a politician and general, Archytas was convinced of what he was sure his Pythagorean forebears had demonstrated: The unity of all things had to include ethics and politics. The value of mathematics extended to the political arena. In the following fragment, “reason” could also be translated as “calculation.” To a Pythagorean like Archytas, the two meanings were probably synonymous.

When reason/calculation is discovered, it puts an end to civil strife and reinforces concord. Where this is present, greed disappears and is replaced by fairness. It is by reason/calculation that we are able to come to terms in dealings with one another. By this means do the poor receive from the affluent and the rich give to the needy, both parties convinced that by this they have what is fair.

Plato, of course, could not have agreed more. The ability to use “reason” or “calculation” would make a philosopher king a superbly able ruler.

For Archytas, the concept of unity meant he should also apply a Pythagorean search for deeper levels of mathematical understanding to optics, physical acoustics, and mechanics. His is the earliest surviving explanation of sound by “impact,” with stronger impacts giving higher pitches, but he nodded to his Pythagorean forebears by insisting this was a theory that had been handed down to him. By “impact,” Archytas meant impact on the air—whipping a stick through the air, playing a high note on a pipe by making the pipe as short as possible (making a stronger pressure on the air, he thought), and the sound of the wind whistling higher pitches as its speed increased, or a “bull-roarer.” That last was an instrument used in the mystery religions, a flat piece of wood on the end of a rope. Whirling it around in the air like a giant slingshot produced a fearsome howling sound; the faster the whirling the higher the pitch.
⁶

One of the most widely known, influential, and enduring Pythagorean ideas passed down through Archytas to Plato was the concept of the “music of the spheres,” the music Archytas and his Pythagorean forebears thought the planets made as they rushed through the heavens. Here is Archytas’ explanation for why humans never hear it:

Many sounds cannot be recognized by our nature, some because of the weakness of the blow (impact), some because of the great distance from us, and some because their magnitude exceeds what can fit into our hearing, as when one pours too much into narrow mouthed vessels and nothing goes in.

According to Pythagorean tradition, only Pythagoras could hear this music.

Archytas was a generous man, kind to slaves and children. He invented toys and gadgets, including a wooden bird (a duck or a dove) that could fly. Aristotle was impressed by “Archytas’ rattle,” “which they give to children so that by using it they may refrain from breaking things about the house; for young things cannot keep still.”
⁷

T
HIS, THEN, WAS
the science, mathematics, music theory, and political philosophy that Plato, from Archytas, learned to think of as
Pythagorean. Through Plato, much of the image of Pythagoras and Pythagorean thought in Western civilization is traceable to Archytas’ window on Pythagoras.

How unclouded was this window? Archytas regarded himself as an authentic Pythagorean, true to the earliest traditions and teachings. In his era, oral accounts could still be accurate, especially in a continuing community that considered it vitally important to keep an ancient memory alive and clear. In many ways, Archytas was probably a good reflection of what it had meant to be Pythagorean when Pythagoras himself walked the paths of Megale Hellas. However, he was one of the
mathematici
, the school of Pythagoreanism that believed following in Pythagoras’ footsteps meant diligently seeking and increasing knowledge. The Pythagorean ideals that underlay Archytas’ thought and work led him to newer discoveries. He was among the great scholars and mathematicians of his era, by reputation the teacher of the mathematician Eudoxus. If Archytas had focused only on the knowledge of the first Pythagoreans, this would have been impossible.