10 Chapter 1
that fit together to make the harmonia of an octave, which he also uses as a verb, to har-
monize, literally meaning to “ octavize ” : “ Nature in the cosmos was harmonized from
unlimiteds and limiters, both the whole cosmos and all things in it. ” Cosmetein means to
put in order and thus beautify (hence “ cosmetics ” ). The cosmos can be an ordered, intel-
ligible whole only because it is harmonized , organized into octaves using alternate mix-
tures of what Philolaus called “ limiters ” and “ unlimiteds, ” which came to be identified
with odd ( “ limiter ” ) and even ( “ unlimited ” ) numbers.^5
Several ancient accounts weave the moment of discovery into a famous, though prob-
lematic, myth. Dictating a simplified account for a noble lady in the second century c.e .,
Nicomachus gave the earliest extant version: Pythagoras was engrossed in trying to find
“ some instrumental aid for the hearing ” that would be comparable to the measuring rod
or compasses used by sight or the balances used for weight and touch. Then,happening by some heaven-sent chance to walk by a blacksmith ’ s workshop, he heard the hammers
beating iron on the anvil and giving out sounds fully concordant in combination with one another,
with the exception of one pairing: and he recognized among them the consonance of the octave and
those of the fifth and the fourth. ... Overjoyed at the way his project had come, with god ’ s help
[ kata theon ], to fulfillment, he ran into the smithy, and through a great variety of experiments [ peirai ]
he discovered that what stood in direct relation to the difference in the sound was the weight of the
hammers, not the force of the strikers or the shapes of the hammer-heads or the alteration of the
iron which was being beaten.^6Writing as the minister of a barbarian king four centuries later, Boethius transmitted this
story with some additions: “ in order to test [ inquiriebat ] this theory more clearly, [Pythago-
ras] commanded the men to exchange hammers among themselves. But the property of
the sounds was not contingent on the muscles of the men, but rather followed the
exchanged hammers. ”^7 Thus, even though Pythagoras came to the smithy “ by the favor
of a god, ” what happened there was not a supernatural or miraculous event worked by
divine power. Instead, we visit an everyday workplace and behold ordinary events revealed
as wondrous. Pythagoras ’ s epiphany led him not to worship or quiet contemplation but
to human actions inquiring into that mysteriously unified sound. Had this been the
scene of a divine revelation, what followed might be understood as inappropriate or even
profane because Pythagoras actually tried to stop or alter the very thing he found most
wonderful.
This test was the archetypal experiment, a trial ( peira ) in which Pythagoras tests the
source of the wonderful sound through an action that attempts to alter it, rather than
through purely verbal or rational means. In Homer, the verb peira ō can mean an assault
that tests the enemy ’ s strength, but also an attempt to gain information, to test someone ’ s
character or fidelity. In Boethius ’ s version, Pythagoras tested the weight of the five
hammers in the shop and found that four of them produced consonant intervals based on
the ratios of their weights: the hammers whose weight was in ratio of 1:2 sounded an
octave; those of 2:3, a perfect fifth; those of 3:4, a perfect fourth. Each consonance yoked