Music and the Origins of Ancient Science 17
fractions, which he does seem to have heard of: “ For you surely know that if someone
undertakes to divide One itself in speech, those who are skilled in these matters laugh and
won ’ t allow it. On the contrary, if you break [the One] up, they multiply it, taking care
that one should never appear not one, but many parts. ”^28 He notes that we implicitly take
any such supposed parts or fractions each to be one in itself, thus negating the premise
that we had “ broken ” the One into many pieces.
For Plato, these counting numbers, with the One as their supreme source, stand as the
touchstone of knowledge as such, the prime example showing the possibility of human
reason, of knowing anything with certainty. We literally count on the difference between
one and two and three as the most certain things we know, which indeed define knowledge
itself. Thus, the Greek insistence on the utter distinction between number and magnitude
is not merely terminological fussiness but the central point on which they grounded their
search for truth. To admit, as moderns do, “ irrational numbers, ” “ imaginary numbers, ”
or “ surreal numbers ” on equal terms with the integers implicitly rejects and ignores
the fundamental insight of logic: countable entities cannot be confused with the endless
divisibility possible (and necessary) for uncountable magnitudes, such as geometric lines.^29
By holding fast to this distinction, Euclid and Plato balanced the realms of the rational
and the irrational, giving coordinate but separate domains to each, respecting both by
never mixing them. This widely held set of assumptions will be of great importance at
several points in this book, as they came to be challenged and replaced with the modern
alternatives.
These fundamental mathematical premises are deeply grounded in musical findings.
The primal Pythagorean ratios place music on the side of arithmetic, not geometry. For
instance, if one tries to “ hear ” the irrational ratio formed by the relation of the diagonal
of a square to its side by approximating that interval on two strings, the result is very close
to the tritone, the interval later notorious as the diabolus in musica ( ♪ sound example 1.4).
Ironically, a perfect geometric division corresponds to a highly dissonant musical interval,
whereas the arithmetic division into simple ratios corresponds to the primal consonances.
On this basis, it seems plausible that the fifth hammer that Pythagoras discarded as “ dis-
sonant with all ” was irrational with respect to them; as Adrastus of Aphrodisias put it, the
irrational is mere “ noise [ psophos ] ” that should not even “ be called notes [ phthongoi ], but
only sounds [ ē choi ]. ” If so, the rejection of the dissonant hammer initiated the ancient
separation between numbers and magnitudes, arithmetic and geometry.^30
Indeed, the same fundamental problem plagues the division of any melodic interval. If,
for instance, one tries to lay out equal tones (9:8) to fill out an octave, it turns out that five
tones undershoot an octave but six tones overshoot it ( ♪ sound example 1.5); to create
modes or scales, it is necessary to introduce some kind of “ half tone ” to fill out the missing
space between five tones and an octave.^31 But, as the Greeks already realized, a perfectly
equal division of a tone (as of an octave) would require the use of an irrational magnitude,
as will become of crucial importance in chapter 4.