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The Music of Pythagoras CHAPTER 5 “All things known have number” Sixth Century B.C. How an Ancient Brotherhood Cracked THE PYTHAGOREAN DISCOVERY that “all things known have number the Code of the Universe and Lit the —for without this, nothing could be thought of or known”—was made in music. It is well established, as so few things are about Path Pythagoras, that the first natural law ever formulated mathematically was the relationship between musical pitch and the from Antiquity to Outer Space. length of a vibrating harp string, and that it was formulated by the earliest Pythagoreans. Ancient scholars such as Plato’s pupil Xenocrates thought that Pythagoras himself, not his followers or associates, made the discovery. Musicians had been tuning stringed instruments for centuries KITTY FERGUSON by the time of Pythagoras. Nearly everyone was aware that sometimes a lyre or harp made pleasing sounds, and sometimes it 2008 did not. Those with skill knew how to manufacture and tune an instrument so that the result would be pleasing. As with many other discoveries, everyday use and familiarity long preceded any deeper understanding. What did “pleasing” mean? When the ancient Greeks thought of “harmony,” were they thinking of it in the way later musicians and music lovers would? Lyres, as far as anyone is able to know at this distance in time, were not strummed like a modern guitar or bowed like a violin. Whether notes were sung together at the same time is more difficult to say, but music historians think not. It was the combinations of intervals in melodies and scales—how notes sounded when they followed one another—that was either pleasing or unpleasant. However, anyone who has played an instrument on which strings are plucked or struck knows that unless a string is stopped to silence it, it keeps sounding. Though lyre strings may not have been strummed together in a chord, more than one pitch and often several pitches were heard at the same time, the more so if there was an echo. Even when notes are played in succession and “stopped,” human ears and brains have a pitch memory that Ferguson- The Music of Pythagoras p60 from either the square or the rectangular figure. CHAPTER 6 “The famous figure In the line of pebbles that then forms the diagonal or of Pythagoras” hypotenuse of the triangle, the pebbles are not the same distances Sixth Century B.C. from one another as they are in the other two sides, nor are they touching one another. Having all the pebbles in all three sides of a IN THE FIRST OR EARLY SECOND century A.D., Plutarch, the author of triangle at equal distances from their immediate neighbors, or all the famous Parallel Lives, and his team of researchers tried to find touching one another, requires a new figure: Set down one pebble, the earliest reference connecting Pythagoras with the “Pythagorean then two, then three, then four, with all the pebbles touching their theorem.”* They came upon a story in the writing of a man named neighbors. The result is a triangle in which all three sides have the Apollodorus, who probably lived in the century of Plato and same length, an equilateral triangle. Notice that the four numbers Aristotle, that told of Pythagoras sacrificing an ox to celebrate the in this triangle are the same as the numbers in the basic musical discovery of “the famous figure of Pythagoras.”† Plutarch ratios, 1, 2, 3, and 4, and the ratios themselves are all here: concluded that this “famous figure” must have been the Beginning at a corner, 2:1 (second line as compared with first), Pythagorean triangle. Unfortunately Apollodorus was no more then 3:2, then 4:3. The numbers in these ratios add up to 10. The specific than those words “the famous figure of Pythagoras”— Pythagoreans decided 10 was the perfect number. They also which probably indicates that it was so famous he had no need to concluded that there was something extraordinary about this be. equilateral triangle, which they called the tetractus, meaning A modern author could also write “the famous figure of “fourness.” The tetractus was, in a nutshell, the musicalnumerical Pythagoras” and be as certain as Apollodorus apparently was that order of the cosmos, so significant that when a Pythagorean took no reader would think of anything but the “Pythagorean triangle.” an oath, he or she swore “by him who gave to our soul the Even nonmathematicians can often recall the “Pythagorean tetractus.” theorem” from memory: the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. For millennia, anyone who had reason to know anything about this theorem thought Pythagoras had discovered it. Most scholars think it was after Pythagoras’ death that the Pythagoreans found they could construct a tetrahedron (or pyramid)—a four-sided solid—out of four equilateral triangles, and they probably knew this by the time Philolaus wrote the first Pythagorean book in the second half of the fifth century.* The word tetractus, however, was in use during Pythagoras’ lifetime. It hints that there was more “fourness” to the idea than the fact that 4 Ferguson- The Music of Pythagoras p67 Ferguson- The Music of Pythagoras p68 skilled draftsman can produce right triangles that no human eye can see are not absolutely precise—this without knowledge of the Pythagorean theorem. Just as tuning a harp is an “ear thing”—and was, long before anyone understood the ratios of musical harmony —the use of right triangles in design was an “eye thing.” Such judgments of harmony, figures, and lines are intuitive for human beings, and the mathematical relationships that lie hidden in nature and the structure of the universe often manifest themselves in the everyday world in useful ways long before anyone thinks of looking for explanations or deep relationships. Yet at certain times and places in history and prehistory—for reasons about which it is only possible to speculate— For many who learned the formula in school and always circumstances have been right to call forth a longing to look thought of it only in terms of squaring numbers, rather than beyond the surface. Among the Pythagoreans there was a strong involving actual square shapes, it came as an almost chilling and unusual motivation. Investigation like this was the road by revelation when Jacob Bronowski in his television series The which one could escape the tedious round of reincarnations and Ascent of Man attached a square to each side of a right triangle and rejoin the divine level of existence. One cannot summarily dismiss showed what the equation really means. The space enclosed in the the tradition that they discovered the theorem, though, contrary to square “on the hypotenuse” is exactly the same amount of space as popular belief for centuries, they were definitely not the first to do is enclosed in the other two squares combined. The whole matter so. suddenly took on a decidedly Pythagorean aura. Clearly this was No one knows how or when the “Pythagorean theorem” was something that might indeed have been discovered and is true in a first discovered, but it happened long before Pythagoras. way that does not require a trained mathematician or even a Archaeologists have found the theorem on tablets in Mesopotamia mathematical mind to recognize. In fact, using numbers is only one dating from the first half of the second millennium B.C., a thousand of several ways of discovering it and proving it is true. years before his lifetime. It was already so well known then that it Bronowski pointed out that right angles are part of the most was being taught in scribal schools. In other regions, evidence of primitive, primordial experience of the world: early knowledge of the theorem is less conclusive but still interesting. Egyptian builders knew how to create square corners There are two experiences on which our visual with an astounding and mysterious degree of precision, perhaps by world is based: that gravity is vertical, and that the using a technique that earned them the nickname “rope pullers” horizon stands at right angles to it. And it is that among their Greek contemporaries. There is a hint about what that conjunction, those cross-wires in the visual field, meant, perhaps, from circa 1400 B.C. in a wall painting in a tomb at which fixes the nature of the right angle.1 Thebes, where Porphyry’s story had Pythagoras spending most of his time while in Egypt. The painting shows men measuring a field Bronowski did not mean that experiencing the world in this with what looks like a rope with knots or marks at regular way necessarily leads immediately, or ever, to the discovery of the intervals.2 Possibly they were using the rope to create right angles, Pythagorean theorem. Indeed, all over the ancient world, long taking a length of rope 12 yards long, making it into a loop, and before Pythagoras, right angles were used in building and marking it off with three notches or knots so as to divide it into surveying, and right triangles appeared decoratively.* Without lengths 3, 4, and 5 yards long. Three, four, and five are a “triple” drawing tools, a draftsman can produce right triangles, and a of whole-number unit measurements that create a right triangle, Ferguson- The Music of Pythagoras p69 Ferguson- The Music of Pythagoras p70 and holding the loop at the three marks and pulling it tight would speculate that it did. The Cynic philosopher Onesicritus traveled have given them one.

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