Euler: The Mathematics of Musical Sadness 141
mental concepts ( figure 9.3 ). To the more complex of these harmonies, Euler adds a
sort of figured bass notation. When played in order, these complex harmonies become
progressively more audacious, even weird ( ♪ sound example 9.2); indeed, his “ figured
bass ” is more a kind of shorthand notation than anything conforming to the musical usage
of his time.
To simplify calculations in his Tentamen , Euler was one of the first to treat musical
ratios with logarithms, which reduce multiplication to addition and division to subtrac-
tion.^25 This musical application then induces Euler to take a new mathematical step,
because expressing a logarithm ’ s magnitude calls for the use of irrational numbers, in
general.^26 For instance, using logarithms to calculate the ratio of the octave to the fifth,
Euler gets decimals, which he then converts to the expression1 1
1 1
21
2
+
+
+
+
.
He can then obtain approximations by truncating the denominator of this continued frac-
tion at successive points downward.^27 While preparing for the publication of his Tentamen ,
Euler wrote “ On Continued Fractions ” (1737), the first sustained treatment of this new
kind of mathematical object.^28 He realized that continued fractions, as they emerged in his
musical treatment, were ideal arenas for considering irrational numbers, each of which
turns out to correspond to a unique continued fraction, which displays the inner structure
of that number in a different (and often more perspicuous) way than its decimal expansion.
On the other hand, Euler demonstrated that the converse is not true, for it is possible to
express any ordinary (rational) fraction as a continued fraction.^29 Among irrational quanti-
ties, the celebrated “ golden ratio ” φ has the beautifully simple formφ =+
+
+
+1
1
1 1
1 1
1
,
which exposes an inner structure not manifest in its decimal expansion,
15 + =
21 6180339887.. Then too, that most familiar of irrational numbers has the ...beguiling expression21 1
2 1
21
2
=+
+
+
+
,
φ=