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proportions of optimal aesthetic value. It is claimed they are embedded in great
works of art, architecture, and music.
Psychologist Gustav Fechner is credited with conducting the first empirical stud-
ies of human aesthetic response in the 1860s. His experiments seemed to show that
golden rectangles had the greatest appeal relative to other aspect ratios. But subse-
quent studies have cast strong doubt on those results. As noted in a special issue
of the journal
Empirical Studies of the Arts
, there were methodological flaws and
cultural bias in previous confirmatory studies (McCormack
2008
, Holger
1997
).
In addition, Livio has credibly debunked supposed Golden Ratio use in works in-
cluding the Great Pyramids, the Parthenon, the Mona Lisa, compositions by Mozart,
and Mondrian's late paintings. However, he notes that use of the Golden Ratio as
an aesthetic guide has become something of a self-fulfilling myth. For example, Le
Corbusier's Modulator, a design aid for proportions, was consciously based on the
Golden Ratio (Livio
2003
).
On a bit firmer ground is a principle credited to linguist George Kingsley Zipf
commonly referred to as
Zipf 's law
. As first applied to natural language, one can
begin with a large body of text and tally every word counting each occurrence. Then
list each word from the most to the least frequent. The observed result is that for the
frequency
P
i
of a given word with a given rank
i
:
1
i
a
P
i
≈
(10.1)
where the exponent
a
is near 1 (Zipf
1949
).
Manaris et al. (
2005
;
2003
) note that this power law relationship has not only
been verified in various bodies of musical composition, but also “colours in images,
city sizes, incomes, music, earthquake magnitudes, thickness of sediment deposi-
tions, extinctions of species, traffic jams, and visits of websites, among others.”
They go on to show how Zipf metrics can be used to classify specific works as
to composer, style, and an aesthetic sense of “pleasantness”. In addition Machado
et al. (
2007
) apply Zipf's law in the creation of artificial art critics. Much earlier
work showed that both frequency and loudness in music and speech conform to a
1
/f
statistical power law. The authors suggest using 1
/f
distributions in generative
music (Voss and Clarke
1975
).
Studies by Taylor have shown that late period “drip” paintings by Jackson Pol-
lock are fractal-like. He has also suggested that the fractal dimension of a given
Pollock painting is correlated with its aesthetic quality. Fractals are mathematical
objects that exhibit self-similarity at all scales. Examples of real world objects that
are fractal-like in form include clouds, mountains, trees, and rivers. In the case of
Pollock's paintings the fractal dimension is a measure of the degree to which the
canvas is filled with finely detailed complex structures. A paint mark with a fractal
dimension of 1 will no more fill the canvas with detailed structures than a typical
straight line. A paint mark with a fractal dimension of 2 will entirely fill the canvas
with fine detail. These correspond well with our everyday topological sense of one
and two dimensional spaces (Peitgen et al.
1992
).
Pollock's paint marks exhibit detail between these two extremes, and have a non-
integer dimension somewhere between 1 and 2. When measured empirically the
