The Fifth Hammer: Pythagoras and the Disharmony of the WorldAn ancient tradition holds that Pythagoras invented harmony. It is said that one day, he wandered by a forge and, hearing a wondrous sound come from within, ventured in to investigate. He found five men hammering with five hammers. To his astonishment, he discovered that four of the five hammers stood in a marvelous set of proportions, which, when combined, allowed him to reconstruct the laws of music. But there was also a fifth hammer. Pythagoras saw

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