The Music of Pythagoras (11 page)

Aristoxenus told a story having to do with another harmonic ratio experiment that involved Hippasus of Metapontum, and this experiment has particular significance because it is one of the reasons scholars are willing to attribute the discovery of the musical ratios to Pythagoras and his immediate associates. Hippasus, himself a contemporary of Pythagoras, made four bronze disks, all equal in diameter but of different thicknesses. The thickness of one “was 4/3 that of the second, 3/2 that of the
third, and 2/1 that of the fourth.” Hippasus suspended the disks to swing freely. Then he struck them, and the disks produced consonant intervals. This experiment is correct in terms of the physical principles involved, for the vibration frequency of a free-swinging disk is directly proportional to its thickness. Whoever designed and executed this experiment understood the basic harmonic ratios, or learned to understand them from doing the experiment, and the way the story was told suggests that the musical ratios were already known and Hippasus made the four disks to demonstrate them. According to Aristoxenus, the musician Glaucus of Rhegium, one of Croton’s neighboring cities, played on the disks of Hippasus, and the experiment became a musical instrument.

To Walter Burkert, a meticulous twentieth-century scholar, the blacksmith tales make “a certain kind of sense.” In ancient lore, the Idaean Dactyls were wizards and the inventors of music and blacksmithing. According to Porphyry, Pythagoras underwent the initiation set by the priests of Morgos, one of the Idaean Dactyls. A Pythagorean aphorism stated that the sound of bronze when struck was the voice of a daimon—another connection between blacksmithing and music or magical sound. “The claim that Pythagoras discovered the basic law of acoustics in a smithy,” writes Burkert, may have been “a rationalization—physically false—of the tradition that Pythagoras knew the secret of magical music which had been discovered by the mythical blacksmiths.”
¹

W
HEN THE
P
YTHAGOREANS
, with their discovery of the mathematical ratios underlying musical harmony, caught a glimpse of the deep, mysterious patterned structure of nature, the conviction became overwhelming that in numbers lay power, even possibly the power that had created the universe. Numbers were the key to vast knowledge—the sort of knowledge that would raise one’s soul to a higher level of immortality, where it would rejoin the divine.

However revolutionary, one of the most significant insights in the history of knowledge had to be worked out, at the start, in the context of an ancient community, ancient superstitions, ancient religious perceptions, without any of the tools or assumptions of later mathematics, geometry, or science, without any scientific precedent or a “scientific method.” How
would
one begin? The Pythagoreans turned to the world itself and followed up on the suspicion that there was something special about the numbers 1, 2, 3, and 4 that appeared in the musical ratios.
Those numbers were popping up in another line of investigation they were pursuing.

They had at their fingertips a simple but productive way of working with numbers. Maybe at first it was a game, setting out pebbles in pleasing arrangements. Most of the information about “pebble figures” and the connections with the cosmos and music that the Pythagoreans found in them comes from Aristotle. He knew about Pythagorean ideas of “triangular numbers,” the “perfect” number 10, and the
tetractus
.

The dots that still appear on dice and dominoes are a vestige of an ancient way of representing natural numbers, the positive integers with which everyone normally counts. Dots and strokes stood for numbers in Linear B, the script the Mycenaeans used for the economic management of their palaces a thousand years before Pythagoras, and also in cuneiform, an even older script. Pebble figures were a related way of visualizing arithmetic and numbers, but they seem to have been unique to the Pythagoreans.

By tradition, Pythagoras himself first recognized links between the pebble arrangements and the numbers he and his colleagues had discovered in the ratios of musical harmony. Two of the most basic arrangements worked as follows: Begin with one pebble, then place three, then five, then seven, etc.—all odd numbers—in carpenter’s angles or “gnomons,” to form a square arrangement.
^*

Or, begin with two pebbles and then set out four, then six, then eight, etc.—all even numbers—and the result is a rectangle.

That is easier to understand visually than verbally, one reason to use pebbles.

Pythagoras and his associates were alert for hidden connections. The pebble figures of the square and rectangle dictated a division of the world of numbers into two categories, odd and even, and this struck
them as significant. It was a link with what they were thinking of as the two basic principles of the universe, “limiting” and “limitless.” “Odd” they associated with “limiting”; “even” with “limitless.”

Another way of manipulating the pebbles was to cut a triangle from either the square or the rectangular figure.

In the line of pebbles that then forms the diagonal or hypotenuse of the triangle, the pebbles are not the same distances 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 triangle at equal distances from their immediate neighbors, or all touching one another, requires a new figure: Set down one pebble, then two, then three, then four, with all the pebbles touching their neighbors. The result is a triangle in which all three sides have the same length, an equilateral triangle. Notice that the four numbers in this triangle are the same as the numbers in the basic musical ratios, 1, 2, 3, and 4, and the ratios themselves are all here: Beginning at a corner, 2:1 (second line as compared with first), then 3:2, then 4:3. The numbers in these ratios add up to 10. The Pythagoreans decided 10 was the perfect number. They also concluded that there was something extraordinary about this equilateral triangle, which they called the
tetractus
, meaning “fourness.” The
tetractus
was, in a nutshell, the musicalnumerical order of the cosmos, so significant that when a Pythagorean took an oath, he or she swore “by him who gave to our soul the
tetractus
.”

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 was the largest number in the ratios. The tetrahedron or pyramid is a solid in which each face is a
tetractus
, but which also uses the number 4 in other manners—4 faces, 4 points.

When Aristotle, in the fourth century
B.C
., was researching the Pythagoreans, he found a list of connections they made between numbers and abstract concepts. He apparently could not discover what they connected with the numbers 6 and 8.

1 Mind

2 Opinion

3 The number of the whole

4 Justice

5 Marriage

6 ?

7 Right time, due season, or opportunity

8 ?

9 Justice

10 Perfect

It is not difficult to understand how Mind might be 1 and Opinion 2. Justice appears twice because of an association with squareness. The Greeks did not think of 1 as a number. “Number” meant plurality, more than 1. So, for them, the smallest number that is the square of any whole number was 4.
^*
The first number that is the square of an odd number is 9, and that, too, they associated with justice. The idea that
“square” means an evened score—with all need for retaliation at an end—still shows up in the colloquial phrase “That makes us square.” Marriage (5) was the sum of the first odd and even numbers (2 and 3). The link between 7 and “right time” or “due season” reflected wider Greek thought. Life happened in multiples of 7. A child could be born after 7 months in the womb, cut teeth 7 months later, reach puberty at 14, and (if a boy) grow a beard at 21.

The Pythagoreans followed one line of thought that seems particularly odd today, accustomed as most of us are to thinking of squares and cubes of numbers but not of other geometric shapes possibly connected with them in a similar manner. The “square” of 4 was 16, but the “triangle” of 4 was 10, the perfect number. Both ideas were equally picturable with pebbles. Stacking the pebbles so as to discover that the “cube” of 4 was 64, you might just as easily pile them up another way so that the “pyramid” of 4 was 20. Montessori teaching exploits the delight of playing games like this with little objects like pebbles—in the case of Montessori, beads.