144 Chapter 9
only as a result of that special skill, combined with his mathematical acumen, could he
have achieved such a factorization, long before computers or any other mechanical calcula-
tors. The same fascination with the pure manipulation and calculation of numbers also
pervades his musical Tentamen , which contains many tables of numbers that have some
importance in his musical scheme. Study of Euler ’ s early notebooks (around 1726) shows
that, as he prepared his Tentamen , he was not then aware of Leibniz ’ s (1714) view that
“ music charms us, even though its beauty consists only in the harmonies of numbers and
in a calculation, which we do not perceive but which the soul nevertheless carries out, a
calculation concerning the beats or vibrations of sounding bodies, which are encountered
at certain intervals. ”^32 After he met Goldbach, Euler became aware of these views, which
Leibniz had described to Goldbach in a letter of April 1712.^33 Though Euler conceived his
musical theories independently, Leibniz ’ s writings supported them, for Euler had set out
a precise scheme whereby the soul might accomplish its musical counting quite con-
sciously. Given Euler ’ s staggering calculational abilities, including lightning mental com-
putations, one can readily imagine that he himself may have been able to compute what
he was hearing, perhaps even in “ real time. ” At the least, his Tentamen contains his retro-
spective account of musical awareness in terms of explicit arithmetic.
The juxtaposition of the musical and arithmetical concerns in Goldbach ’ s correspon-
dence with Leibniz helps underline the many ways in which these two themes arguably
overlapped and intersected in Euler ’ s mind through his interchanges with Goldbach. Yet
even before then, Euler ’ s absorption in the intricate arithmetic of his music theory provided
the fertile ground on which his ensuing interest in number theory grew. The modern
concept of “ pure mathematics ” should not blind us to the many ways in which, in Euler ’ s
time and before, no hard barrier separated it from the “ applied ” branches of what we now
call physics, engineering, music theory — disciplinary names that he would neither have
known nor separated absolutely. Nothing would have been more natural for Euler than to
follow his intricate musical arithmetic into the further studies of the properties of numbers
that only came to be called “ number theory ” in the aftermath of his own work.
Looking back to the Tentamen , many of Euler ’ s musical arguments directly imply arith-
metical problems that lead straight to the more general questions he later addressed about
the properties of numbers. His definition s – n + 1 for the gradus suavitatis of a musical
interval involves counting the n prime factors of the interval ’ s exponent and their sum s
( box 9.1 ), which are central topics in his ensuing number theoretical work. The Pythago-
reans had begun to investigate perfect numbers (each equal to the sum of its proper divi-
sors, such as 6 = 1 + 2 + 3) and pairs of amicable numbers, for which each is the sum of
the other ’ s proper divisors, such as 220 and 284.^34 Both types of numbers became important
for Euler, but he had already laid the groundwork for their study in his Tentamen. In 1747,
Euler published thirty new pairs of amicable numbers, compared to the four pairs previ-
ously known, listing them in a format that recalls his diagrams ranking musical intervals
in his Tentamen.^35