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Zipf's Law, Music Classification, and Aesthetics
Manaris, Bill.
Romero, Juan.
Machado, Penousal.
Computer Music Journal, Volume 29, Number 1, Spring 2005,
pp. 55-69 (Article)
Published by The MIT Press
For additional information about this article
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http://muse.jhu.edu/journals/cmj/summary/v029/29.1manaris.html
Manaris et al.
Zipf’s Law, Music
Classification, and
Aesthetics
55
The connection between aesthetics and numbers
dates back to pre-Socratic times. Pythagoras, Plato,
and Aristotle worked on quantitative expressions of
proportion and beauty such as the golden ratio.
Pythagoreans, for instance, quantified “harmo-
nious” musical intervals in terms of proportions (ra-
tios) of the first few whole numbers: a unison is 1:1,
octave is 2:1, perfect fifth is 3:2, perfect fourth is
4:3, and so on (Miranda 2001, p. 6). The Pythagorean
scale was refined over centuries to produce well-
tempered and equal-tempered scales (Livio 2002,
pp. 29, 186).
Galen, summarizing Polyclitus, wrote, “Beauty
does not consist in the elements, but in the harmo-
nious proportion of the parts.” Vitruvius stated,
“Proportion consists in taking a fixed nodule, in
each case, both for the parts of a building and for the
whole.” He then defined proportion as “the appro-
priate harmony arising out of the details of the work
itself; the correspondence of each given detail
among the separate details to the form of the design
as a whole.” This school of thought crystallized
into a universal theory of aesthetics based on “unity
in variety” (Eco 1986, p. 29).
Some musicologists dissect the aesthetic experi-
ence in terms of separable, discrete sounds. Others
attempt to group stimuli into patterns and study
their hierarchical organization and proportions
(May 1996; Nettheim 1997). Leonard Meyer states
that emotional states in music (sad, angry, happy,
etc.) are delineated by statistical parameters such as
dynamic level, register, speed, and continuity (2001,
p. 342).
Building on earlier work by Vilfredo Pareto, Al-
fred Lotka, and Frank Benford (among others),
George Kingsley Zipf refined a statistical technique
known as Zipf’s Law for capturing the scaling prop-
erties of human and natural phenomena (Zipf 1949;
Mandelbrot 1977, pp. 344–345).
We present results from a study applying Zipf’s
Law to music. We have created a large set of metrics
based on Zipf’s Law that measure the proportion or
distribution of various parameters in music, such as
pitch, duration, melodic intervals, and harmonic
consonance. We applied these metrics to a large cor-
pus of MIDI-encoded pieces. We used the generated
data to perform statistical analyses and train artifi-
cial neural networks (ANNs) to perform various
classification tasks. These tasks include author at-
tribution, style identification, and “pleasantness”
prediction. Results from the author attribution and
Computer Music Journal, 29:1, pp. 55–69, Spring 2005
© 2005 Massachusetts Institute of Technology.
Bill Manaris,
*
Juan Romero,
†
Penousal
Machado,
‡
Dwight Krehbiel,
§
Timothy Hirzel,
*
Walter Pharr,
*
and Robert B. Davis
¶
*Computer Science Department, College of
Charleston
66 George Street, Charleston, SC 29424 USA
{manaris, hirzel, pharr}@cs.cofc.edu
†
Creative Computer Group, RNASA Lab
Faculty of Computer Science, University of A
Coruña, Spain
jj@udc.es
‡
Centre for Informatics and Systems Department of
Informatic Engineering Polo II-University of Coimbra
3030 Coimbra, Portugal
machado@dei.uc.pt
§
Psychology Department, Bethel College
North Newton, KS 67117 USA
krehbiel@bethelks.edu
¶
Department of Mathematics and Statistics, Miami
University
Hamilton, OH 45011, USA
davisrb@muohio.edu
style identification ANN experiments have ap-
peared in Machado et al. (2003, 2004) and Manaris
et al. (2003), and these results are summarized in
this article. Results from the “pleasantness” predic-
tion ANN experiment are new and therefore dis-
cussed in detail. Collectively, these results suggest
that metrics based on Zipf’s Law may capture essen-
tial aspects of proportion in music as it relates to
music aesthetics.
Zipf’s Law
Zipf’s Law reflects the scaling properties of many
phenomena in human ecology, including natural
language and music (Zipf 1949; Voss and Clarke
1975). Informally, it describes phenomena where
small events are quite frequent and large events are
rare. Once a phenomenon has been selected for
study, we can examine the contribution of each
event to the whole and rank it according to its “im-
portance” or “prevalence” (see linkage.rockefeller
.edu/wli/zipf). For example, we may rank unique
words in a book by their frequency of occurrence,
visits to a Web site by how many of them originated
from the same Internet address, and so on.
In its most succinct form, Zipf’s Law is expressed
in terms of the frequency of occurrence (i.e., count
or quantity) of events, as follows:
F~ r
–a
(1)
where Fis the frequency of occurrence of an event
within a phenomenon, ris its statistical rank (posi-
tion in an ordered list), and ais close to 1. In the
book example above, the most frequent word would
be rank 1, the second most frequent word would be
rank 2, and so on. This means that the frequency of
occurrence of a word is inversely proportional to its
rank. For example, if the first ranked word appears
6,000 times, the second ranked word would appear
approximately 3,000 times (1/2), the third ranked
word approximately 2,000 times (1/3), and so on.
Another formulation of Zipf’s Law is
P(f) ~ 1/f
n
(2)
where P(f) denotes the probability of an event of
rank f, and nis close to 1. In physics, Zipf’s Law is a
special case of a power law. When nis 1 (Zipf’s ideal),
the phenomenon is called 1/fnoise or pink noise.
Zipf distributions (e.g., 1/fnoise) have been dis-
covered in a wide range of human and naturally oc-
curring phenomena including city sizes, incomes,
subroutine calls, earthquake magnitudes, thickness
of sediment depositions, extinctions of species, traf-
fic jams, and visits to Web sites (Schroeder 1991;
Bak 1996; Adamic and Huberman et al. 2000; see
also linkage.rockefeller.edu/wli/zipf).
In the case of music, we can study the “impor-
tance” or “prevalence” of pitch events, duration
events, melodic interval events, and so on. For in-
stance, consider Chopin’s Revolutionary Etude. To
determine if its melodic intervals follow Zipf’s Law,
we count the different melodic intervals in the
piece, e.g., 89 half steps up, 88 half steps down, 80
unisons, 61 whole steps up, and so on. Then we plot
these counts against their statistical rank on a log-
log scale. This plot is known as the rank-frequency
distribution.
In general, the slope of the distribution may range
from 0 to –∞, with –1 denoting Zipf’s ideal. In other
words, this slope corresponds to the exponent nin
Equation 2. The R
2
value can range from 0 to 1, with
1 denoting a straight line. This value gives the pro-
portion of y-variability of data points with respect
to the trend line.
Figure 1a shows the rank-frequency distribution
of melodic intervals for Chopin’s Revolutionary
Etude. Melodic intervals in this piece approximate a
Zipfian distribution with slope of –1.1829 and R
2
of
0.9156. Figure 1b shows the rank-frequency distri-
bution of chromatic-tone distance for Bach’s Air on
the G String. The chromatic-tone distance is the
time interval between consecutive repetitions of
chromatic tones. In this piece, the chromatic-tone
distance approximates a Zipfian distribution with
slope of –1.1469 and R
2
of 0.9319. It should be noted
that the less pronounced fit at the tails (high and
low ranking events) is quite common in Zipf plots
of naturally occurring phenomena.
In many cases, the statistical rank of an event is in-
versely related to the event’s size. Informally, smaller
events tend to occur more frequently, whereas larger
events tend to occur less frequently. For instance,
the statistical rank of chromatic-tone distance in
56 Computer Music Journal
distribution of chromatic-
tone distance for Bach’s
Orchestral Suite No. 3 in D,
movement no. 2, Air on
the G String, BWV 1068.
Figure 1. (a) Rank-
frequency distribution of
melodic intervals for
Chopin’s Revolutionary
Etude, Op. 10 No. 12 in C
minor; (b) rank-frequency
Manaris et al.
Mozart’s Bassoon Concerto in B-flat Major is in-
versely related to the length of these time intervals
(Zipf 1949, p. 337). In other words, plotting the counts
of various distances against the actual distances
(from smaller to larger) produces a near-Zipfian line.
Using size instead of rank on the x-axis generates
a size-frequency distribution. This is an alternative
formulation of Zipf’s Law that has found applica-
tion in architecture and urban studies (Salingaros
and West 1999). This formulation is also used in the
box-counting technique for calculating the fractal
dimension of phenomena (Schroeder 1991, p. 214).
Zipf’s Law has been criticized on the grounds that
1/fnoise can be generated from random statistical
processes (Li 1992, 1998; Wolfram 2002, p. 1014).
However, when studied in depth, one realizes that
Zipf’s Law captures the scaling properties of a phe-
nomenon (Mandelbrot 1977, p. 345; Ferrer Cancho
and Solé 2003). In particular, Benoit Mandelbrot, an
early critic, was inspired by Zipf’s Law and went on
to develop the field of fractals. He states:
Natural scientists recognize in “Zipf’s Laws” the
counterparts of the scaling laws which physics
and astronomy accept with no extraordinary
emotion—when evidence points out their valid-
ity. Therefore physicists would find it hard to
imagine the fierceness of the opposition when
Zipf—and Pareto before him—followed the
same procedure, with the same outcome, in the
social sciences. (Mandelbrot 1977, pp. 403–404)
Zipf-Mandelbrot Law
Mandelbrot generalized Zipf’s Law as follows:
P(f) ~ 1/(1 + b)f
(1 + c)
(3)
where band care arbitrary real constants. This is
known as the Zipf-Mandelbrot Law. It accounts for
natural phenomena whose scaling properties are not
necessarily Zipfian.
Zipf’s Law in Music
Zipf himself reported several examples of 1/fdistri-
butions in music. His examples were processed
manually, because computers were not yet avail-
able. Zipf’s corpus consisted of Mozart’s Bassoon
Concerto in B-flat; Chopin’s Etude in F minor, Op.
25, No. 2; Irving Berlin’s Doing What Comes Natu-
rally; and Jerome Kern’s Who. This study focused
on melodic intervals and the distance between repe-
titions of notes (Zipf 1949, pp. 336–337).
Richard Voss and John Clarke (1975, 1978) con-
ducted a large-scale study of music from classical,
jazz, blues, and rock radio stations recorded continu-
ously over 24 hours. They measured several param-
eters, including output voltage of an audio amplifier,
loudness fluctuations of music, and pitch fluctua-
tions of music. They discovered that pitch and loud-
ness fluctuations in music follow Zipf’s distribution.
Additionally, Voss and Clarke developed a computer
program to generate music using three different ran-
57
(a)
(b)
dom number generators: a white-noise (1/f
0
) source,
a pink-noise (1/f) source, and a brown-noise (1/f
2
)
source. They used independent random-number gen-
erators to control the duration (half, quarter, eighth)
and pitch (various standard scales) of successive
notes. Remarkably, the music obtained through the
pink-noise generators was much more pleasing to
most listeners. In particular, the white-noise genera-
tors produced music that was “too random,” whereas
the brown-noise generators produced music that was
“too correlated.” They noted, “Indeed the sophisti-
cation of this ‘1/fmusic’ (which was ‘just right’) ex-
tends far beyond what one might expect from such a
simple algorithm, suggesting that a ‘1/fnoise’ (per-
haps that in nerve membranes?) may have an essen-
tial role in the creative process” (1975, p. 318) .
John Elliot and Eric Atwell (2000) failed to find
Zipf distributions in notes extracted from audio sig-
nals. However, they used a small corpus of music
pieces and were looking only for ideal Zipf distribu-
tions. On the other hand, Kenneth Hsu and Andrew
Hsu (1991) found 1/fdistributions in frequency in-
tervals of Bach and Mozart compositions. Finally,
Damián Zanette found Zipf distributions in notes
extracted from MIDI-encoded music. Moreover, he
used these distributions to demonstrate that as mu-
sic progresses, it creates a meaningful context similar
to the one found in human languages (see http://
xxx.arxiv.org/abs/cs.CL/0406015).
Zipf Metrics for Music
Currently, we have a set of 40 metrics based on
Zipf’s Law. They are separated into two categories:
simple metrics and fractal metrics.
Simple Metrics
Simple metrics measure the proportion of a particu-
lar parameter, such as pitch, globally. Table 1 shows
the complete set of simple metrics we currently
employ (Manaris et al. 2002). Obviously, there are
many other possibilities, including size of move-
ments, volume, timbre, tempo, and dynamics.
For instance, the harmonic consonance metric
operates on a histogram of harmonic intervals
58 Computer Music Journal
Table 1. Our Current Set of 20 Simple Metrics Based On Zipf’s Law
Metric Description
Pitch Rank-frequency distribution of the 128 MIDI pitches
Chromatic tone Rank-frequency distribution of the 12 chromatic tones
Duration Rank-frequency distribution of note durations (absolute duration in seconds)
Pitch duration Rank-frequency distribution of pitch durations
Chromatic-tone duration Rank-frequency distribution of chromatic tone durations
Pitch distance Rank-frequency distribution of length of time intervals between note (pitch)
repetitions
Chromatic-tone distance Rank-frequency distribution of length of time intervals between note (chromatic
tone) repetitions
Harmonic interval Rank-frequency distribution of harmonic intervals within chord
Harmonic consonance Rank-frequency distribution of harmonic intervals within chord based on music-
theoretic consonance
Melodic interval Rank-frequency distribution of melodic intervals within voice
Harmonic-melodic interval Rank-frequency distribution of harmonic and melodic intervals
Harmonic bigrams Rank-frequency distribution of adjacent harmonic interval pairs
Melodic bigrams Rank-frequency distribution of adjacent melodic interval pairs
Melodic trigrams Rank-frequency distribution of adjacent melodic interval triplets
Higher-order intervals Rank-frequency distribution of higher orders of melodic intervals; first-order met-
ric captures change between melodic intervals; second-order metric captures
change between first-order intervals, and so on up to sixth order
Manaris et al.
within each chord in a piece. It counts the number
of occurrences of each interval modulo multiples of
the octave, and it plots them against their conso-
nance ranking. In essence, this metric measures the
proportion of harmonic consonance, or statistical
balance between consonance and dissonance in a
piece. We use a traditional music-theoretic ranking
of harmonic consonance: unison is rank 1, P5 is rank
2, P4 is rank 3, M3 is rank 4, M6 is rank 5, m3 is
rank 6, m6 is rank 7, M2 is rank 8, m7 is rank 9, M7
is rank 10, m2 is rank 11, and the tritone is rank 12.
Simple Zipf metrics are useful feature extractors.
However, they have an important limitation. They
examine a music piece as a whole, ignoring poten-
tially significant contextual details. For instance, the
pitch distribution of Bach’s Air on the G String has a
slope of –1.078. Sorting this piece’s notes in increas-
ing order of pitch would produce an unpleasant mu-
sical artifact. This artifact exhibits the same pitch
distribution as the original piece. Thus, simple met-
rics could be easily fooled in the context of, say
computer-aided music composition, where such met-
rics could be used for fitness evaluation. However, in
the context of analyzing culturally sanctioned music,
this limitation is not significant. This is because cul-
turally sanctioned music tends to be well-balanced
at different levels of granularity. That is, the balance
exhibited at the global level is usually similar to the
balance exhibited at the local level, down to a small
level of granularity, as will be explained shortly.
Fractal Metrics
Fractal metrics handle the potential limitation of
simple metrics in the context of music composi-
tion. Each simple metric has a corresponding fractal
metric (Manaris et al. 2003). Whereas a simple met-
ric calculates the Zipf distribution of a particular at-
tribute at a global level, the corresponding fractal
metric calculates the self-similarity of this distribu-
tion. That is, the fractal metric captures how many
subdivisions of the piece exhibit this distribution at
many levels of granularity.
For instance, to calculate the fractal dimension of
pitch distribution, we recursively apply the simple
pitch metric to the piece’s half subdivisions, quarter
subdivisions, etc., down to the level of single mea-
sures. At each level of granularity, we count how
many of the subdivisions approximate the global
distribution. We then plot these counts against the
length of the subdivision, producing a size-frequency
distribution. The slope of the trend line is the frac-
tal dimension, D, of pitch distribution for this piece.
This allows us to identify anomalous pieces that, al-
though balanced at the global level, may be quite
unbalanced at a local level. This method is similar
to the box-counting technique for calculating the
fractal dimension of images (Schroeder 1991, p. 214).
Taylor et al. (1999) used the box-counting tech-
nique to authenticate and date paintings by Jackson
Pollock. Using a size-frequency plot, they calcu-
lated the fractal dimension, D, of Pollock’s paint-
ings. In particular, they discovered two different
slopes: one attributed to Pollock’s dripping process,
and the other attributed to his motions around the
canvas. Also, they were able to track how Pollock
refined his dripping technique: the slope decreased
through the years, from approximately –1 in 1943 to
–1.72 in 1952.
Experimental Studies: Zipf-Mandelbrot
Distributions in MIDI-Encoded Music
Inspired by the work of Zipf (1949) and Voss and
Clarke (1975, 1978), we conducted two studies to
explore Zipf-Mandelbrot distributions in MIDI-
encoded music. The first study used a 28-piece cor-
pus from Bach, Beethoven, Chopin, Debussy, Handel,
Mendelssohn, Schönberg, and Rodgers and Hart. It
also included seven pieces from a white-noise gen-
erator as a control group (Manaris et al. 2002). The
second study used a 196-piece corpus from various
genres, including Baroque, Classical, Romantic,
Modern, Jazz, Rock, Pop, and Punk Rock. It also in-
cluded 24 control pieces from DNA strings, white
noise, and pink noise (Manaris et al. 2003).
Methodology
Zipf (1949, pp. 336–337) worked with composition
data, i.e., printed scores, whereas Voss and Clarke
(1975, 1978) studied performance data, i.e., audio
59
recorded from radio stations. Our corpus consisted
mostly of MIDI-encoded performances from the
Classical Music Archives (available online at
www.classicalarchives.com).
We identified a large number of parameters of
music that could possibly exhibit Zipf-Mandelbrot
distributions. These attributes included pitch, dura-
tion, melodic intervals, and harmonic intervals,
among others. Table 1 shows a representative sub-
set of these metrics.
Results
Most pieces in our corpora exhibited near-Zipfian
distributions across a wide variety of metrics. In the
first study, classical and jazz pieces averaged near-
Zipfian distributions and strong linear relations
across all metrics, whereas random pieces did not.
Specifically, the across-metrics average slope for
music pieces was –1.2653. The corresponding R
2
value, 0.8088, indicated a strong average linear rela-
tion. The corresponding results for control pieces
were –0.4763 and 0.6345, respectively.
Table 2 shows average results from the second
study. In particular, the 196 music pieces exhibited
an overall average slope of –1.2023 with standard
deviation of 0.2521. The average R
2
is 0.8233 with a
standard deviation of 0.0673. The 24 pieces in the
control group exhibited an average slope of –0.6757
with standard deviation 0.2590. The average R
2
is
0.7240 with a standard deviation of 0.1218. This
suggests that some music styles could possibly be
distinguished from other styles and from non-
musical data through a collection of Zipf metrics.
Music as a Hierarchical Dynamic System
Mandelbrot observed that Zipf-Mandelbrot distri-
butions in economic systems are “stable” in that,
even when such systems are perturbed, their slopes
tend to remain between 0 and –2. Systems with
slopes less than –2, when perturbed, exhibit chaotic
behavior (1977, p. 344). The same stability has also
been observed in simulations of sand piles and vari-
ous other natural phenomena. Phenomena exhibit-
ing this tendency are called self-organized
criticalities (Bak et al. 1987; Maslov et al. 1999).
This tendency characterizes a complex system that
has come to rest. Because the system has lost en-
ergy, it is bound to stay in this restful state, hence
the “stability” of these states.
Mandelbrot states that, because these stable dis-
tributions are very widespread, they are noticed and
published, whereas chaotic distributions tend not to
be noticed (1977, p. 344). Accordingly, in physics, all
60 Computer Music Journal
Table 2. Average Results Across Metrics for Various Genres from a Corpus of 220 Pieces
Genre Slope R
2
Slope Std. Dev. R
2
Std. Dev.
Baroque –1.1784 0.8114 0.2688 0.0679
Classical –1.2639 0.8357 0.1915 0.0526
Early Romantic –1.3299 0.8215 0.2006 0.0551
Romantic –1.2107 0.8168 0.2951 0.0609
Late Romantic –1.1892 0.8443 0.2613 0.0667
Post Romantic –1.2387 0.8295 0.1577 0.0550
Modern Romantic –1.3528 0.8594 0.0818 0.0294
Twelve-Tone –0.8193 0.7887 0.2461 0.0964
Jazz –1.0510 0.7864 0.2119 0.0796
Rock –1.2780 0.8168 0.2967 0.0844
Pop –1.2689 0.8194 0.2441 0.0645
Punk Rock –1.5288 0.8356 0.5719 0.0954
DNA –0.7126 0.7158 0.2657 0.1617
Random (Pink) –0.8714 0.8264 0.3077 0.0852
Random (White) –0.4430 0.6297 0.2036 0.1184
“natural” distribution
from the interpretation of
this piece as performed by
harpsichordist John
Sankey.
Figure 2. (a) Rank-
frequency distribution of
note durations from the
score of Bach’s Two-Part
Invention No. 13 in A mi-
nor, BWV 784; (b) the more
Manaris et al.
distributions with slope less than –2 are collectively
called black noise, as opposed to brown noise (slope
of –2), pink noise (slope of –1, i.e., Zipf’s ideal), and
white noise (slope of 0). (See Schroeder 1991, p. 122.)
The tendency of music to exhibit rank-frequency
distribution slopes between 0 and –2, as observed in
our experiments with hundreds of MIDI-encoded
music pieces, suggests that perhaps composing mu-
sic could be viewed as a process of stabilizing a hier-
archical system of pitches, durations, intervals,
measures, movements, etc. In this view, a com-
pleted piece of music resembles a dynamic system
that has come to rest.
For a piece of music to resemble black noise, it
must be rather monotonous. In the extreme case,
this corresponds to a slope of negative infinity (–∞),
i.e., a vertical line. Other than the obvious “mini-
malist” exceptions, such as John Cage’s 4'33", most
performed music tends to have some variability
across different parameters such as pitch, duration,
melodic intervals, etc. Figure 2a shows an example
of black noise in music. It depicts the rank-
frequency distribution of note durations from the
MIDI-encoded score of Bach’s Two-Part Invention
No. 13 in A minor. This MIDI rendering has an un-
natural, monotonous tempo. The Zipf-Mandelbrot
slope of –3.9992 reflects this monotony. Figure 2b
depicts the rank-frequency distribution of note du-
rations for the same piece, as interpreted by harpsi-
chordist John Sankey. The Zipf-Mandelbrot slope of
–1.4727 reflects the more “natural” variability of
note durations found in the human performance.
Music Classification and Zipf’s Law
There are numerous studies on music classification,
such as Aucouturier and Pachet (2003), Pampalk et
al. (2004), and Tzanetakis et al. (2001). However,
we have found no references to Zipf’s Law in this
context.
Zipf’s Law has been used successfully for classifi-
cation in other domains. For instance, as mentioned
earlier, it has been used to authenticate and date
paintings by Jackson Pollock (Taylor et al. 1999). It
has also been used to differentiate among immune
systems of normal, irradiated chimeric, and athymic
mice (Burgos and Moreno-Tovar 1996). Zipf’s Law
has been used to distinguish healthy from non-
healthy heartbeats in humans (see arxiv.org/abs/
physics/0110075). Finally, it has been used to distin-
guish cancerous human tissue from normal tissue
using microarray gene data (Li and Yang 2002).
Experimental Studies
We performed several studies to explore the applica-
bility of Zipf’s Law to music classification. The
studies reported in this section focused on author
attribution and style identification.
Author Attribution
In terms of author attribution, we conducted five
experiments: Bach vs. Beethoven, Chopin vs. De-
61
(a)
(b)
bussy, Bach vs. four other composers, and Scarlatti
vs. Purcell vs. Bach vs. Chopin vs. Debussy
(Machado et al. 2003, 2004).
We compiled several corpora whose size ranged
across experiments from 132 to 758 music pieces.
Our data consisted of MIDI-encoded performances,
the majority of which came from the online Classi-
cal Music Archives. We applied Zipf metrics to ex-
tract various features for each piece. The number of
features per piece varied across experiments, ranging
from 30 to 81. This collection of feature vectors was
used to train an artificial neural network. Our train-
ing methodology is similar to one used by Miranda
et al. (2003); in particular, we separated feature vec-
tors into two data sets. The first set was used for
training, and the second set was used to test the
ANN’s ability to classify new data. We experimented
with various architectures and training regimens
using the Stuttgart Neural Network Simulator (see
www-ra.informatik.uni-tuebingen.de/SNNS).
Table 3 summarizes the ANN architectures used
and results obtained from the Scarlatti vs. Purcell
vs. Bach vs. Chopin vs. Debussy experiment. The
success rate across the five-author attribution ex-
periments ranged from 93.6 to 95 percent. This
suggests that Zipf metrics are useful for author at-
tribution (Machado et al. 2003, 2004).
The analysis of the errors made by the ANN indi-
cates that Bach was the most recognizable com-
poser. The most challenging composer to recognize
was Debussy. His works were often misclassified as
scores of Chopin.
Style Identification
We have also performed statistical analyses of the
data summarized in Table 2 to explore the potential
for style identification (Manaris et al. 2003). We
have discovered several interesting patterns.
For instance, our corpus included 15 pieces by
Schönberg, Berg, and Webern written in the twelve-
tone style. They exhibit an average chromatic-tone
slope of –0.3168 with a standard deviation of 0.1801.
The corresponding average for classical pieces was
–1.0576 with a standard deviation of 0.5009, whereas
for white-noise pieces it was 0.0949 and 0.0161, re-
spectively. Clearly, by definition, the chromatic-tone
metric alone is sufficient for identifying twelve-tone
music. Also, DNA and white noise were easily iden-
tifiable through pitch distribution alone. Finally, all
genres commonly referred to as classical music ex-
hibited significant overlap in all of the metrics; this
included Baroque, Classical, and Romantic pieces.
This is consistent with average human competence
in discriminating between these musical styles.
Subsequent analyses of variance (ANOVA) re-
vealed significant differences among some genres.
For instance, twelve-tone music and DNA were
identifiable through harmonic-interval distribution
alone. Similarly to author attribution, we expect
62 Computer Music Journal
Table 3. Author Attribution Experiment with Five Composers from Various Genres
SCARLATTI VS. PURCELL VS. BACH VS. CHOPIN VS. DEBUSSY
Test Set MSE
Train Patterns (%) Test Patterns (%) Architecture Cycles Errors Success Rate (%) Train Test
652 (86%) 106 (14%) 81-6-5 10000 6 94.4 0.00005 0.07000
4000 6 94.4 0.00325 0.10905
81-12-5 10000 6 94.4 0.00313 0.11006
4000 5 95.3 0.00321 0.10201
541 (71%) 217 (29%) 81-6-5 10000 11 95 0.00386 0.09076
4000 11 95 0.00199 0.10651
81-12-5 10000 14 93.6 0.00194 0.14195
4000 11 95 0.00388 0.09459
MSE = Mean-Square Error.
Manaris et al.
that a combination of metrics will be sufficient for
style identification. To validate this hypothesis, we
are currently conducting a large ANN-based style-
identification study.
Aesthetics and Zipf’s Law
Arnheim (1971) proposes that art is our answer to
entropy and the Second Law of Thermodynamics.
As entropy increases, so do disorganization, ran-
domness, and chaos. In Arnheim’s view, artists sub-
consciously tend to produce art that creates a
balance between chaos and monotony. According to
Schroeder, this agrees with George Birkhoff’s The-
ory of Aesthetic Value:
[F]or a work of art to be pleasing and interesting,
it should neither be too regular and predictable
nor pack too many surprises. Translated to math-
ematical functions, this might be interpreted as
meaning that the power spectrum of the func-
tion should behave neither like a boring ‘brown’
noise, with a frequency dependence 1/f
2
, nor like
an unpredictable white noise, with a frequency
distribution of 1/f
0
. (Schroeder 1991, p. 109)
As mentioned earlier, in the case of music, Voss
and Clarke (1975, 1978) have shown that classical,
rock, jazz, and blues music exhibits 1/fpower spec-
tra. Also, in our study, 196 pieces from various genres
exhibited an average Zipf-Mandelbrot distribution
of approximately 1/f
1.2
across various music attri-
butes (Manaris et al. 2003).
In the visual domain, Spehar et al. (2003) have
shown that humans show an aesthetic preference
for images exhibiting a Zipf-Mandelbrot distribu-
tion between 1/f
1.3
and 1/f
1.5
. Finally, Mario Livio
(2002, pp. 219–220) has demonstrated a connection
between a Zipf-Mandelbrot distribution of 1/f
1.4
and
the golden ratio (0.61803 . . . ).
Zipf’s Law and Human Physiology
Boethius believed that musical consonance “pleases
the listener because the body is subject to the same
laws that govern music, and these same proportions
are to be found in the cosmos itself. Microcosm and
macrocosm are tied by the same knot, simultaneously
mathematical and aesthetic” (Eco 1986, p. 31).
One connection between near-1/fdistributions in
music and human perception is the physiology of
the human ear. The basilar membrane in the inner
ear analyzes acoustic frequencies and, through the
acoustic nerve, reports sounds to the brain. Interest-
ingly, 1/fsounds stimulate this membrane in just
the right way to produce a constant-density stimu-
lation of the acoustic nerve endings (Schroeder
1991, p. 122). This corroborates Voss and Clarke’s
finding that 1/fmusic sounds “just right” to human
subjects, as opposed to 1/f
0
music, which sounds
“too random,” and 1/f
2
music, which sounds “too
monotonous” (Voss and Clarke 1978).
Functional magnetic resonance imaging (fMRI)
and other measurements are providing additional
evidence of 1/factivity in the human brain (Zhang
and Sejnowski 2000; see also arxiv.org/PS_cache/
cond-mat/pdf/0208/0208415.pdf). According to Carl
Anderson, to perceive the world and generate adap-
tive behaviors, the brain self-organizes via sponta-
neous 1/fclusters or bursts of activity at various
levels. These levels include protein chain fluctua-
tions, ion channel currents, synaptic processes, and
behaviors of neural ensembles. In particular,
“[e]mpirical fMRI observations further support the
association of fractal fluctuations in the temporal
lobes, brainstem, and cerebellum during the expres-
sion of emotional memory, spontaneous fluctua-
tions of thought and meditative practice” (Anderson
2000, p. 193).
This supports Zipf’s proposition that composers
may subconsciously incorporate 1/fdistributions
into their compositions because they sound right to
them and because their audiences like them (1949,
p. 337). If this is the case, then in certain styles such
as twelve-tone and aleatoric music, composers may
subconsciously avoid such distributions for artistic
reasons.
Experimental Study: “Pleasantness” Prediction
We conducted an ANN experiment to explore the
possible connection between aesthetics and Zipf-
63
Mandelbrot distributions at the level of MIDI-
encoded music. In this study, we trained an ANN
using Zipf-Mandelbrot distributions extracted from
a set of pieces, together with human emotional re-
sponses to these pieces. Our hypothesis was that
the ANN would discover correlations between Zipf-
Mandelbrot distributions and human emotional
responses and thus be able to predict the “pleasant-
ness” of music on which it had not been trained.
Methodology
We used a corpus of twelve excerpts of music. These
were MIDI-encoded performances selected by a
member of our team with an extensive music the-
ory background. Our goal was to identify six pieces
that an average person might find pleasant and six
pieces that an average person might find unpleas-
ant. All excerpts were less than two minutes long to
minimize fatigue for the human subjects. Table 4
shows the composer, name, and duration of each
excerpt.
We collected emotional responses from 21 sub-
jects for each of the twelve excerpts. These subjects
were college students with varied musical back-
grounds. The experiment was double-blind in that
neither the subjects nor the people conducting the
experiment knew which of the pieces were presumed
as pleasant or unpleasant.
Subjects were instructed to report their own emo-
tional responses during the music by using the
mouse to position an “X” cursor within a two-
dimensional space on a computer monitor. The hor-
izontal dimension represented “pleasantness,” and
the vertical dimension represented “activation” or
arousal. The system recorded the subject’s cursor
coordinates once per second. Positions were
recorded on scales of 0–100 with the point (50, 50)
representing emotional indifference or neutral reac-
tion. Table 4 shows the average pleasantness rating
and standard deviation.
Much psychological evidence indicates that
“pleasantness” and “activation” are the fundamen-
tal dimensions needed to describe human emotional
responses (Barrett and Russell 1999). Following es-
tablished standards, the emotion labels “excited,”
“happy,” “serene,” “calm,” “lethargic,” “sad,”
“stressed,” and “tense” were placed in a circle
around the space to assist the subjects in the task.
These labels, in effect, helped the subjects discern
the semantics of the selection space. Similar meth-
ods for continuous recording of emotional response
to music have been used elsewhere (Schubert 2001).
It is worth emphasizing that the subjects were not
64 Computer Music Journal
Table 4. Twelve Pieces Used for Music Pleasantness Classification Study
Human Rating
Composer Piece Duration Average Std. Dev.
Beethoven Sonata No. 20 in G, Opus 49. No. 2 1'00" 72.84 11.83
Debussy Arabesque No. 1 in E (Deux Arabesques) 1'34" 78.30 17.96
Mozart Clarinet Concerto in A, K.622 (first movement) 1'30" 67.97 12.43
Schubert Fantasia in C minor, Op. 15 1'58" 68.17 13.67
Tchaikovsky Symphony 6 in B minor, Op. 36, second movement 1'23" 68.59 13.52
Vivaldi Double Violin Concerto in A minor, F. 1, No. 177 1'46" 63.12 15.93
Bartók Suite, Op. 14 1'09" 42.46 14.58
Berg Wozzeck (transcribed for piano) 1'38" 35.75 15.79
Messiaen Apparation de l’Eglise Eternelle 1'19" 39.75 17.12
Schönberg Pierrot Lunaire (fifth movement) 1'13" 44.00 15.85
Stravinksy Rite of Spring, second movement (transcribed for piano) 1'09" 43.19 15.58
Webern Five Songs (1. “Dies ist ein Lied”) 1'26" 39.74 13.04
The first six pieces were rated by subjects as “pleasant” overall; the last six pieces were rated as “unpleasant” overall.
(Neutral is 50.)
Manaris et al.
reporting what they liked or even what they judged
as beautiful. We are not aware of any studies relat-
ing how one’s musical preferences or formal train-
ing might affect one’s reporting of pleasantness.
However, there is evidence that pleasantness and
liking are not the same (Schubert 1996). Also, it has
been shown that pleasantness represents a more
useful predictor of emotions than liking when using
the above selection space in the music domain (Ri-
tossa and Rickard 2004).
For the ANN experiment, we divided each music
excerpt into segments. All segments started at 0:00
and extended in increments of four seconds. That is,
the first segment extended from 0:00 to 0:04, the
second segment from 0:00 to 0:08, the third seg-
ment from 0:00 to 0:12, and so on. We applied Zipf
metrics to extract 81 features per music increment.
Each feature vector was associated with a desired
output vector of (1, 0) indicating pleasant and (0, 1)
indicating unpleasant. This generated a total of 210
training vectors.
We conducted a twelve-fold, “leave-one-out,”
cross-validation study. This allowed for twelve pos-
sible combinations of eleven pieces to be learned
and one piece to be tested. We experimented with
various ANN architectures. The best one was a
feed-forward ANN with 81 elements in the input
layer, 18 in the hidden layer, and two in the output
layer. Internally, the ANN was divided into two 81
×9 ×1 “Siamese-twin” pyramids, both sharing the
same input layer. One pyramid was trained to recog-
nize pleasant music, the other unpleasant. Classifi-
cation was based on the average of the two outputs.
Results
Table 5 shows the results from all 12 experiments.
The ANN performed extremely well with an aver-
age success rate of 98.41 percent. All pieces were
classified with 100 percent accuracy, with one ex-
ception: Berg’s piece was classified with only 80.95
percent accuracy. The ANN was considered suc-
cessful if it rated a music excerpt within one stan-
dard deviation of the average human rating; this
covers 68 percent of the human responses.
There are two possibilities for this “failure” of the
ANN. Either our metrics fail to capture essential as-
pects of Berg’s piece, or the other eleven pieces do
not contain sufficient information to enable the in-
terpretation of Berg’s piece.
Figure 3a displays the average human ratings for
Vivaldi’s Double Violin Concerto in A minor, F. 1.,
No. 177. Figure 3b shows the pleasantness ratings
predicted by the ANN for the same piece. The
65
Table 5. Summary of Results from Twelve-Fold, Cross-Validation ANN Experiment, Listed by Composer of
Test Piece
MSE
Composer Success Rate (%) Cycles Train Test
Beethoven 100.00 32200 0.008187 0.003962
Debussy 100.00 151000 0.001807 0.086451
Mozart 100.00 222200 0.004430 0.003752
Schubert 100.00 592400 0.001982 0.004851
Tchaikovsky 100.00 121400 0.004268 0.004511
Vivaldi 100.00 431600 0.003870 0.009643
Bartók 100.00 569200 0.001700 0.008536
Berg 80.95 4600 0.015412 0.100619
Messiaen 100.00 35200 0.008392 0.001315
Schönberg 100.00 8000 0.016806 0.015803
Stravinksy 100.00 311200 0.004099 0.002693
Webern 100.00 468600 0.002638 0.013540
Average 98.41 245633 0.006133 0.021306
Std. Dev. 5.27 212697 0.004939 0.032701
of 50 denotes a neutral re-
sponse. (b) Pleasantness
classification by ANN of
the same piece having
been trained on the other
11 pieces.
Figure 3. (a) Average pleas-
antness (o) and activation
(x) ratings from 21 human
subjects for the first 1 min,
46 sec of Vivaldi’s Double
Violin Concerto in A mi-
nor, F. 1, No. 177. A rating
ANN prediction approximates the average human
response.
Relevance of Metrics
The analysis of ANN weights associated with each
metric gives an indication of its relevance for a par-
ticular task. A large median value suggests that, for
at least half of the ANNs in the experiment, the
metric was useful in performing the particular task.
There were 13 metrics that had median ANN
weights of at least 7. Table 6 lists these metrics in
descending order with respect to the median. It also
lists the mean ANN weights, standard deviations,
and the ratio of standard deviation and mean.
Among the metrics with the highest medians,
two of them stand out: harmonic consonance and
chromatic tone. This is because they have a high
mean and relatively small standard deviation, as in-
dicated by the last column of Table 6. It can be ar-
gued that these metrics were most consistently
relevant for “pleasantness” prediction across all
twelve experiments.
As mentioned earlier, harmonic consonance cap-
tures the statistical proportion of consonance and
dissonance in a piece. “Pleasant” pieces in our cor-
pus exhibited similarities in their proportions of
harmonic consonance: the slope ranged from
–0.8609 (Schubert, 0:08 sec) to –1.8087 (Beethoven,
0:40) with an average of –1.2225 and standard devia-
tion of 0.1802. “Unpleasant” pieces in our corpus
also exhibited similarities in their proportions of
harmonic consonance; in this case, however, the
slope ranged from –0.2284 (Schönberg, 0:24) to
–0.9919 (Berg, 0:20) with an average of –0.5343 and
standard deviation of 0.1519. Owing to the overlap
between the two ranges, the ANN had to rely on ad-
ditional metrics for disambiguation.
Chromatic tone captures the uniform distribu-
tion of pitch, which is characteristic of twelve-tone
and aleatoric music. Such music was rated consis-
tently by our subjects as rather “unpleasant.” The
chromatic tone slope for “unpleasant” pieces ranged
from –0.0578 (Webern, 0:48) to –1.4482 (Stravinsky,
0:32), with an average of –0.6307 and standard devi-
ation of 0.3985. On the other hand, the chromatic
tone slope for “pleasant” pieces ranged from –0.4491
(Debussy, 0:16) to –1.8848 (Mozart, 0:68), with an
average of –1.3844 and standard deviation of 0.3075.
The chromatic tone metric was less relevant for
classification than harmonic consonance owing to
the greater overlap in the ranges of slopes between
“pleasant” and “unpleasant” pieces. Other relevant
metrics include chromatic-tone distance, pitch du-
ration, harmonic interval, harmonic and melodic
interval, harmonic bigrams, and melodic bigrams.
Discussion
These results indicate that, in most cases, the ANN
is identifying patterns that are relevant to human
aesthetic judgments. This supports the hypothesis
66 Computer Music Journal
(a)
(b)
Manaris et al.
that there may be a connection between aesthetics
and Zipf-Mandelbrot distributions at the level of
MIDI-encoded music.
It was interesting to note that harmonic conso-
nance approximated a 1/fdistribution for pieces
that were rated as pleasant and a more chaotic 1/f
0.5
distribution for pieces that were rated as unpleas-
ant. Because the emotional responses used in this
study were actually psychological self-report mea-
sures, this suggests the influence of a higher level of
organization. Also, because an emotional measure
is involved, this likely reflects some higher-level
pattern of intellectual processing that exhibits 1/f
organization. This processing likely draws upon
other, non-auditory information in the brain.
Conclusions
We propose the use of Zipf-based metrics as a basis
for author- and style-identification tasks and for the
assessment of aesthetic properties of music pieces.
The experimental results in author-identification
tasks, where an average success rate of more than
94 percent was attained, show that the used set of
metrics, and accordingly Zipf’s Law, capture mean-
ingful information about the music pieces. Clearly,
the success of this approach does not imply that
other metrics or approaches are irrelevant.
As noticed by several researchers, culturally sanc-
tioned music tends to exhibit near-ideal Zipf distri-
butions across various parameters. This suggests
the possibility that combinations of Zipf-based met-
rics may represent certain necessary but not suffi-
cient conditions for aesthetically pleasing music.
This is supported by our pleasantness study where
an ANN succeeds in predicting human aesthetic
judgments of unknown pieces with more than 98
percent accuracy.
The set of 40 metrics used in these studies repre-
sent only a small subset of possible metrics. The
analysis of ANN weights indicates that harmonic
consonance and chromatic tone were related to hu-
man aesthetic judgments. Based on this analysis
and on additional testing, we are trying to deter-
mine the most useful metrics overall and to develop
additional ones.
It should be emphasized that the metrics pro-
posed in this article offer a particular description of
the musical pieces, where traditional musical struc-
tures such as motives, tonal structures, etc., are not
measured explicitly. Statistical measurements, such
as Zipf’s Law, tend to focus on general trends and
thus can miss significant details. To further explore
the capabilities and limitations of our approach, we
are developing an evolutionary music generation
system in which the proposed classification
methodology will be used for fitness assignment.
67
Table 6. Statistical Analysis of ANN Weights for Metrics Used in the “Pleasantness” Prediction ANN Ex-
periment (Ordered by Median)
Metric Median Mean Std. Dev. Std. Dev./Mean
Harmonic-melodic interval (simple slope) 57.43 64.22 45.09 0.70
Harmonic consonance (simple slope) 44.54 44.48 17.13 0.39
Harmonic bigram (simple slope) 37.76 41.37 31.88 0.77
Pitch duration (simple slope) 32.34 32.53 20.01 0.62
Harmonic interval (simple R
2
)23.54 23.25 15.18 0.65
Chromatic-tone distance (simple slope) 21.82 27.69 16.00 0.58
Chromatic tone (simple slope) 19.93 21.83 7.75 0.36
Melodic bigrams (simple slope) 15.74 20.83 17.82 0.86
Duration (simple R
2
)9.38 10.23 7.60 0.74
Harmonic interval (simple slope) 8.39 8.21 5.90 0.72
Fourth high order (fractal slope) 8.16 8.26 4.86 0.59
Melodic interval (fractal slope) 7.81 8.37 3.42 0.41
Harmonic bigram (simple R
2
)7.46 10.28 11.64 1.13
Once developed, this system will be included in a
hybrid society populated by artificial and human
agents, allowing us to perform further testing in a
dynamic environment.
In closing, our studies show that Zipf’s Law, as
encapsulated in our metrics, can be used effectively
in music classification tasks and aesthetic evalua-
tion. This may have significant implications for
music information retrieval and computer-aided
music analysis and composition, and may provide
insights on the connection among music, nature,
and human physiology. We regard these results as
preliminary; we hope they will encourage further
investigation of Zipf’s Law and its potential applica-
tions to music classification and aesthetics.
Acknowledgments
This project has been partially supported by an in-
ternal grant from the College of Charleston and a
donation from the Classical Music Archives. We
thank Renée McCauley, Ramona Behravan, and
Clay McCauley for their comments. William
Daugherty and Marisa Santos helped conduct the
ANN experiments. Brian Muller, Christopher Wag-
ner, Dallas Vaughan, Tarsem Purewal, Charles Mc-
Cormick, and Valerie Sessions helped formulate and
implement Zipf metrics. Giovanni Garofalo helped
collect human emotional response data for the
ANN pleasantness experiment. William Edwards,
Jr., Jimmy Wilkinson, and Kenneth Knuth provided
early material and inspiration.
References
Adamic, L. A., and B. A. Huberman. 2000. “The Nature of
Markets in the World Wide Web.” Quarterly Journal of
Electronic Commerce 1(1):5–12.
Anderson, C. M. 2000. “From Molecules to Mindfulness:
How Vertically Convergent Fractal Time Fluctuations
Unify Cognition and Emotion.” Consciousness & Emo-
tion 1:2:193–226.
Arnheim, R. 1971. Entropy and Art: An Essay on Disor-
der and Order. Berkeley: University of California
Press.
Aucouturier, J.-J., and F. Pachet. 2003. “Representing Mu-
sical Genre: A State of the Art.” Journal of New Music
Research 32(1):83–93.
Bak, P. 1996. How Nature Works: The Science of Self-
Organized Criticality. New York: Springer-Verlag.
Bak, P., C. Tang, and K. Wiesenfeld. 1987. “Self-Organized
Criticality: An Explanation for 1/f Noise.” Physical Re-
view Letters 59:381–384.
Barrett, L. F., and J. A. Russell. 1999. “The Structure of
Current Affect: Controversies and Emerging Consen-
sus.” Current Directions in Psychological Science
8(1):10–14.
Burgos, J. D., and P. Moreno-Tovar. 1996 “Zipf-Scaling
Behavior in the Immune System.” Biosystems 39(3):
227–232.
Eco, U. 1986. Art and Beauty in the Middle Ages. H.
Bredin, trans. New Haven: Yale University Press.
Elliot, J., and E. Atwell. 2000. “Is Anybody Out There?
The Detection of Intelligent and Generic Language-
Like Features.” Journal of the British Interplanetary
Society 53(1/2):13–22.
Ferrer Cancho, R., and R. V. Solé. 2003. “Least Effort and
the Origins of Scaling in Human Language.” Proceed-
ings of the National Academy of Sciences, U.S.A
100(3):788–791.
Hsu, K. J., and A. Hsu. 1991. “Self-Similarity of the ‘1/f
Noise’ Called Music.” Proceedings of the National
Academy of Sciences, U.S.A. 88(8):3507–3509.
Li, W. 1992. “Random Texts Exhibit Zipf’s-Law-Like
Word Frequency Distribution.” IEEE Transactions on
Information Theory 38(6):1842–1845.
Li, W. 1998. “Letter to the Editor.” Complexity 3(5):9–10.
Li, W., and Y. Yang. 2002. “Zipf’s Law in Importance of
Genes for Cancer Classification using Microarray
Data.” Journal of Theoretical Biology 219:539–551.
Livio, M. 2002. The Golden Ratio. New York: Broadway
Books.
Machado, P., et al. 2003. “Power to the Critics—A Frame-
work for the Development of Artificial Critics.” Pro-
ceedings of 3rd Workshop on Creative Systems, 18th
International Joint Conference on Artificial Intelli-
gence (IJCAI 2003). Coimbra, Portugal: Center for
Informatics and Systems, University of Coimbra,
pp. 55–64.
Machado, P., et al. 2004. “Adaptive Critics for Evolution-
ary Artists.” Proceedings of EvoMUSART2004—2nd
European Workshop on Evolutionary Music and Art.
Berlin: Springer-Verlag, pp. 437–446.
Manaris, B., T. Purewal, and C. McCormick. 2002. “Pro-
gress Towards Recognizing and Classifying Beautiful
Music with Computers: MIDI-Encoded Music and the
68 Computer Music Journal
Manaris et al.
Zipf-Mandelbrot Law.” Proceedings of the IEEE South-
eastCon 2002. New York: Institute of Electrical and
Electronics Engineers, pp. 52–57.
Manaris, B., et al. 2003. “Evolutionary Music and the
Zipf-Mandelbrot Law: Progress towards Developing Fit-
ness Functions for Pleasant Music.” Proceedings of
EvoMUSART2003—1st European Workshop on Evolu-
tionary Music and Art. Berlin: Springer-Verlag,
pp. 522–534.
Mandelbrot, B. B. 1977. The Fractal Geometry of Nature.
New York: W. H. Freeman.
Maslov, S., C. Tang, and Y.-C. Zhang. 1999. “1/f Noise in
Bak-Tang-Wiesenfeld Models on Narrow Stripes.”
Physical Review Letters 83(12):2449–2452.
May, M. 1996. “Did Mozart Use the Golden Section?”
American Scientist 84(2):118.
Meyer, L. B. 2001. “Music and Emotion: Distinctions and
Uncertainties.” In P. N. Juslin and J. A. Sloboda, eds.
Music and Emotion—Theory and Research. Oxford:
Oxford University Press: 341–360.
Miranda, E. R. 2001. Composing Music with Computers.
Oxford: Focal Press.
Miranda, E. R., et al. 2003. “On Harnessing the Electroen-
cephalogram for the Musical Braincap.” Computer Mu-
sic Journal 27(2):80–102.
Nettheim, N. 1997. “A Bibliography of Statistical Appli-
cations in Musicology.” Musicology Australia 20:94–
106.
Pampalk E., S. Dixon, and G. Widmer. 2004. “Exploring
Music Collections by Browsing Different Views.”
Computer Music Journal 28(2):49–62.
Ritossa, D. A., and N. S. Rickard. 2004. “The Relative
Utility of ‘Pleasantness’ and ‘Liking’ Dimensions in
Predicting the Emotions Expressed in Music.” Psychol-
ogy of Music 32(1):5–22.
Salingaros, N. A., and B. J. West. 1999. “A Universal Rule
for the Distribution of Sizes.” Environment and Plan-
ning B: Planning and Design 26:909–923.
Schroeder, M. 1991. Fractals, Chaos, Power Laws: Min-
utes from an Infinite Paradise. New York: W. H. Free-
man.
Schubert, E. 1996. “Enjoyment of Negative Emotions in
Music: An Associative Network Explanation.” Psy-
chology of Music 24(1):18–28.
Schubert, E. 2001. “Continuous Measurement of Self-
Report Emotional Response to Music.” In P. N. Juslin
and J. A. Sloboda, eds. Music and Emotion— Theory
and Research. Oxford: Oxford University Press,
pp. 393–414.
Spehar, B., et al. 2003. “Universal Aesthetic of Fractals.”
Computers and Graphics 27:813–820.
Taylor, R. P., A. P. Micolich, and D. Jonas. 1999. “Fractal
Analysis Of Pollock’s Drip Paintings.” Nature
399:422.
Tzanetakis, G., G. Essl, and P. Cook. 2001. “Automatic
Musical Genre Classification of Audio Signals.” Pro-
ceedings of 2nd Annual International Symposium on
Music Information Retrieval. Bloomington: University
of Indiana Press, pp. 205–210.
Voss, R. F., and J. Clarke. 1975. “1/f Noise in Music and
Speech.” Nature 258:317–318.
Voss, R. F., and J. Clarke. 1978. “1/f Noise in Music: Mu-
sic from 1/f Noise.” Journal of the Acoustical Society
of America 63(1):258–263.
Wolfram, S. 2002. A New Kind of Science. Champaign,
Illinois: Wolfram Media.
Zhang, K., and T. J. Sejnowski. 2000. “A Universal Scaling
Law between Gray Matter and White Matter of Cere-
bral Cortex.” Proceedings of the National Academy of
Sciences, U.S.A 97(10):5621–5626.
Zipf, G. K. 1949. Human Behavior and the Principle of
Least Effort. New York: Addison-Wesley.
69
