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fundamental. Putting the bridge on the monochord so it touches the string changes the notes that are played. When you place the bridge exactly in the middle of the monochord, touching the center of the string, each half of the string plays the same note: a tone exactly one octave higher than the string’s fundamental. Shifting the bridge slightly might divide the string so that one side has three-fifths of the string and the other has two-fifths; in this case Pythagoras noticed that plucking the string segments creates two notes that form a perfect fifth, which is said to be the most powerful and evocative musical relationship. Different ratios gave different tones that could soothe or disturb. (The discordant tritone, for instance, was dubbed the “devil in music” and was rejected by medieval musicians.) Oddly enough, when Pythagoras put the bridge at a place that did not divide the string into a simple ratio, the plucked notes did not meld well. The sound was usually dissonant and sometimes even worse. Often the tone wobbled like a drunkard up and down the scale.
    To Pythagoras, playing music was a mathematical act. Like squares and triangles, lines were number-shapes, so dividing a string into two parts was the same as taking a ratio of two numbers. The harmony of the monochord was the harmony of mathematics—and the harmony of the universe. Pythagoras concluded that ratios govern not only music but also all other types of beauty. To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature was as simple as understanding the mathematics of proportions.

    Figure 7: The mystical monochord
    This philosophy—the interchangeability of music, math, and nature—led to the earliest Pythagorean model of the planets. Pythagoras argued that the earth sat at the center of the universe, and the sun, moon, planets, and stars revolved around the earth, each pinned inside a sphere (Figure 8). The ratios of the sizes of the spheres were nice and orderly, and as the spheres moved, they made music. The outermost planets, Jupiter and Saturn, moved the fastest and made the highest-pitched notes. The innermost ones, like the moon, made lower notes. Taken all together, the moving planets made a “harmony of the spheres,” and the heavens are a beautiful mathematical orchestra. This is what Pythagoras meant when he insisted, “All is number.”

    Figure 8: The Greek universe
    Because ratios were the keys to understanding nature, the Pythagoreans and later Greek mathematicians spent much of their energy investigating their properties. Eventually, they categorized proportions into 10 different classes, with names like the harmonic mean. One of these means yielded the most “beautiful” number in the world: the golden ratio.
    Achieving this blissful mean is a matter of dividing a line in a special way: divide it in two so that the ratio of the small part to the large part is the same as the ratio of the large part to the whole (see appendix B). In words, it doesn’t seem particularly special, but figures imbued with this golden ratio seem to be the most beautiful objects. Even today, artists and architects intuitively know that objects that have this ratio of length to width are the most aesthetically pleasing, and the ratio governs the proportions of many works of art and architecture. Some historians and mathematicians argue that the Parthenon, the majestic Athenian temple, was built so that the golden ratio is visible in every aspect of its construction. Even nature seems to have the golden ratio in its design plans. Compare the ratios of the size of any two succeeding chambers of the nautilus, or take the ratio of clockwise grooves to counterclockwise grooves in the pineapple, and you will see that these ratios approach the golden ratio (Figure 9).
    The pentagram became the most sacred symbol of the Pythagorean brotherhood