that those weights were related in the ratios 2:1, 3:2, and 4:3, presupposing that the vibration and sound of hammers are directly proportional to their weight, which is not the case. Pythagoras then took weights equaling those of the hammers and hung them from strings of equal length. He plucked the taut strings and heard the same intervals—another supposed discovery based on false premises, for the account incorrectly assumes that the frequency of vibration of a string is proportional to the number of units of weight hanging from it. However, it is easy to imagine Pythagoras, or his followers, or both, performing such experiments and considering, with more understanding and skill than those who later ignorantly repeated the tales, what could be learned from the successes and failures. The manner in which these stories came down in history as the way Pythagoras
made
the discovery could be an example of how knowledge is sometimes preserved while the manner of its discovery, and true understanding of it, are lost. Such a loss would be explained if, as some have supposed, the more sophisticated knowledge of Pythagoras was largely forgotten with the breakup of Pythagorean communities after his death.
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 thethird, 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,
made
the discovery could be an example of how knowledge is sometimes preserved while the manner of its discovery, and true understanding of it, are lost. Such a loss would be explained if, as some have supposed, the more sophisticated knowledge of Pythagoras was largely forgotten with the breakup of Pythagorean communities after his death.
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 thethird, 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,
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