In the early 1930's the great linguist George Kingsley Zipf made a
momentous discovery while poring over some statistics on the English
language: he had discovered a hidden ``music'' in the structure of
our language. In fact, this music appears in all other human languages,
and seems to be a deep order underlying many other processes, such as
the distribution of populations of cities. 

The discovery that Zipf had made is as follows. Take a very large novel
written in English (or French, or German, ...) and write down the most common
word, the second most common word, third most common, and so on. The first
few words you should have written down are `the', `of', `to', and 'a'.
Next to these words, write down their frequency; that is, write down the
percentage of words in the novel that are the particular target word.
Now make a plot, where the x-axis is the rank of the word
(`the' has rank 1, `of' has rank 2, etc.) and the y-axis is the frequency.
Upon plotting the points with coordinates (rank, frequency), you should
notice something quite strange: the points appear to fit a simple,
very smooth curve!

The reason this is surprising is that if one knew little about the frequency
of English words, one would probably guess that the points should be scattered
about in a somewhat random manner (well, they would still be ``monotone
decreasing'', so at least would have *some* structure).
However, it turns out that, at least for the first couple
hundred words, they more-or-less fit the following smooth function:

f = 0.07/r = (7 percent)/r, f = frequency and r = rank.

As an example, take the word `of'. This has rank 2, so the formula predicts
that its frequency should be f = 0.07/2 = 0.035, or 3.5 percent.

The formula cannot be a good fit for words too far down the list, because
the sum of the frequencies must equal 1, and yet the sum

1 + 1/2 + 1/3 + 1/4 + ...,

known as the ``harmonic series'', gets bigger and bigger the more terms
one adds (in the parlance of math, it ``diverges to infinity'').

The ``harmonic series'' is where the idea for the title of this essay comes
from, because these ratios 1, 1/2, 1/3, and so on, are directly related
to the fundamental frequencies at which a string can vibrate: take a
taut piece of string, and strum or pluck it as you would a guitar. The string should
settle down to a ``standing wave'', whose ``wavelength'' is some fraction
1/n of the length of the string. This 1/n is called a ``harmonic number''
for obvious reasons.

Not content with just describing his eponymous law, Zipf sought to
elucidate the fundamental mathematical forces behind it.
With time, he formulated a theory which posits that its main cause is
the many millions of interactions between speakers and listeners,
over many, many years.

Since the time of Zipf other explanations have been proposed for his
law, one of which is that slightly restricted classes of random processes,
such as those producing human languages, are behind it. Others have
argued that certain of these random models are not realistic, though
they do produce the ``power law''-type behavior appearing in Zipf's law.

Perhaps with time, better mathematical frameworks will surface, which will
reveal Zipf's law to be a consequence of a vastly more general theory,
from which deep insights about our world will emerge.