Music and the Origins of Ancient Science 11
an odd with an even number, corresponding to the “ limiter ” and “ unlimited ” Philolaus
described. These ratios are successive pairs taken in the sequence 1:2:3:4, which Boethius
also expresses in the sequence 6:8:9:12: within the octave (6:12 or 1:2), the fourth (6:8 or
3:4) and the fifth (6:9 or 2:3) find their place. This then implied, without adding any new
information (or hammers), that between the interval of the fifth and the fourth emerges
the ratio 8:9, later called a tone or whole step because it is the step between these two
intervals ( ♪ sound example 1.2), which according to Nicomachus “ was in itself discordant,
but was essential to filling out the greater of these intervals. ”
While Nicomachus reconciles the discordant hammer as “ essential ” to the greater inter-
val, Boethius tells that “ the fifth hammer, which was dissonant with all, was rejected. ”
This, too, should be taken as part of the foundational myth: an experiment requires rec-
ognizing and dealing with dissonance , the part of an experience that does not “ sound
together ” with the rest. Clearly, there is peril here: how to decide what is “ dissonance ”
that should be set aside, without throwing out some crucial piece of information? At this
primal scene of Pythagorean science, confronting dissonance represents the price and also
the potential danger of the knowledge achieved through the test and through principled
reconsideration or rejection of some experiences in order that others may stand out more
intelligibly. We shall return to the identity of this fifth hammer.^8
Boethius notes that Pythagoras continued his examination after he left the smithy: he
tested the pitches of strings of different lengths, some stretched by different weights; he
tried pipes, “ using some twice as long as others, as well as fitting in the other proportions ”
and glasses filled with different amounts of water by weight ( figure 1.1 ). “ Thus he made
his belief complete by various experiments, ” for which Boethius now specifically uses the
word experientia , whose literal meaning is “ something lived through as a trial or even
peril ” ( ex-perire ). Yet a ten-pound hammer does not ring differently than a six-pounder
(as you can hear for yourself in ♪ sound example 1.3); one wonders whether the smiths
tried to set Pythagoras straight or whether he even talked to them. Perhaps the word
sphur ō n ( “ hammer ” ) used in some texts was a misreading or corruption of sphaira
( “ sphere ” or “ disc ” ).^9 If so, what Pythagoras may have heard were the pitches sounded by
various-sized metal discs, which could conceivably have behaved in the numerical ratios
recorded, whereas hammers could not. Still, the tale refers to the hammers as the smiths ’
tools, rather than objects forged in the shop.
Nevertheless, strings do behave as Boethius recounts in describing the “ ruler ” or mono-
chord ( kan ō n ) that Pythagoras developed ( figure 1.2 ), which “ is fixed and firm under the
study of anyone. ”^10 A single stretched string mounted against a graduated ruler allows
stopping the string at lengths that will realize various proportions; indeed, the proportions
1:2, 2:3, and 3:4 ring out their respective intervals as the story has them ( ♪ sound example
1.2) if the tension is held constant, but not otherwise. Simple pipes of the same diameter,
material, and construction also will sound these intervals, as will water glasses, according
to the Boethian story. The difficulties with some of these stories were probably known to