4 M. R. SCHROEDER

discovered de Broglie matter waves: In essence, integer-valued integrals

meant that path differences are divisible by the electron's wavelength without

leaving a remainder.

2. Music and numbers

Ever since Pythagoras, small integers and their ratios have played a

fundamental role in the construction of musical scales. There are good

reasons for this preponderance of small integers both in the production and

perception of music. String instruments, as abundant in antiquity as today,

produce simple frequency ratios when their strings are subdivided into equal

lengths: shortening the string by one half produces the frequency ratio 2:1, the

octave; and making it a third shorter produces the frequency ratio 3:2, the

perfect fifth.

In perception, ratios of small integers avoid unpleasant beats between

harmonics. Apart from the frequency ratio 1:1 ("unison"), the octave is the

most easily perceived interval. Next in importance comes the perfect fifth.

Unfortunately, as a consequence of the fundamental theorem of arithmetic,

musical scales exactly congruent modulo the octave cannot be constructed

from the fifth alone because there are no positive integers k and m such that

r ^ m

3

(1)

However, there are good approximation to (1). Writing

2n

or

log23 =

m

we see that we need a rational approximations to log2 3. The proper way of

doing this is to expand log23 into a continued fraction

log23 = [1, 1, 1,2,2,-.- ]