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
However, there are good approximation to (1). Writing
log23 =
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,-.- ]
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