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Multifractal analyses of music sequences

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Multifractal analysis is applied to study the fractal property of music. In this paper, a method is proposed to transform both the melody and rhythm of a music piece into individual sets of distributed points along a one-dimensional line. The structure of the musical composition is thus manifested and characterized by the local clustering pattern of these sequences of points. Specifically, the local Hölder exponent and the multifractal spectrum are calculated for the transformed music sequences according to the multifractal formalism. The observed fluctuations of the Hölder exponent along the music sequences confirm the non-uniformity feature in the structures of melodic and rhythmic motions of music. Our present result suggests that the shape and opening width of the multifractal spectrum plot can be used to distinguish different styles of music. In addition, a characteristic curve is constructed by mapping the point sequences converted from the melody and rhythm of a musical work into a two-dimensional graph. Each different pieces of music has its own unique characteristic curve. This characteristic curve, which also exhibits a fractal trait, unveils the intrinsic structure of music.
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Physica D 221 (2006) 188–194
www.elsevier.com/locate/physd
Multifractal analyses of music sequences
Zhi-Yuan Sua, Tzuyin Wub,
aDepartment of Information Management, Chia Nan University of Pharmacy and Science, Tainan 717, Taiwan
bDepartment of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan
Received 9 April 2006; received in revised form 10 July 2006; accepted 9 August 2006
Available online 1 September 2006
Communicated by S. Kai
Abstract
Multifractal analysis is applied to study the fractal property of music. In this paper, a method is proposed to transform both the melody and
rhythm of a music piece into individual sets of distributed points along a one-dimensional line. The structure of the musical composition is
thus manifested and characterized by the local clustering pattern of these sequences of points. Specifically, the local H¨
older exponent and the
multifractal spectrum are calculated for the transformed music sequences according to the multifractal formalism. The observed fluctuations of
the H¨
older exponent along the music sequences confirm the non-uniformity feature in the structures of melodic and rhythmic motions of music.
Our present result suggests that the shape and opening width of the multifractal spectrum plot can be used to distinguish different styles of music.
In addition, a characteristic curve is constructed by mapping the point sequences converted from the melody and rhythm of a musical work into
a two-dimensional graph. Each different pieces of music has its own unique characteristic curve. This characteristic curve, which also exhibits a
fractal trait, unveils the intrinsic structure of music.
c
2006 Elsevier B.V. All rights reserved.
Keywords: Music; Fractal; Multifractal analysis; Multifractal spectrum; H ¨
older exponent
1. Introduction
Nature is full of irregular patterns and complicated phenom-
ena. Despite their complicated appearances, ‘self similarity’,
that is, the similarity between the whole and a small portion
of a system, can be observed in many configurations and phe-
nomena upon closer investigation. Geometry with such scale-
invariant features has now been categorized and designated as
‘fractal’ in literature [1]. Many geometries existing in nature are
fractal, e.g., a mountain’s profile and the shape of snowflakes.
Music, whose origin may be attributed to imitating the har-
mony of nature’s sound, also demonstrates a fractal property
like many other naturally occurring fluctuations do.
Music can be used to express human feelings and emotions
toward nature. A few musical notes can be aligned by a
composer’s will into a beautiful and pleasant song; whereas
the same notes can be arranged into an annoying or discordant
noise if randomly aligned. So what is the mystique of music?
Corresponding author. Tel.: +886 2 33662708; fax: +886 2 23631755.
E-mail address: tywu@ntu.edu.tw (T. Wu).
This is an issue that has been investigated for hundreds of years,
but has not been concluded so far. Fractal theory [1], developed
in the 1970s, provides an innovative tool for the analysis of a
sequence of symbols. By applying fractal tools in the study of
music, researchers, including Voss and Hsu, were surprised to
discover that the self-similarity property, which is ubiquitous
in nature, also exists in music. Such an observation may be
regarded as the first step toward a further understanding of what
music is and explaining how music simulates the harmony of
nature.
1.1. Frequency ratio between music tones
When comparing two tones, a frequency ratio of small
number integers (e.g. 1:2 (an octave), 2:3 (a fifth), etc.,
under the circumstance of ‘just intonation’) indicates a more
harmonious sound than a ratio of larger number integers
(e.g. 5:6 (a minor third), 15:16 (a minor second), etc.). Just
intonation is a system of tuning in which all of the intervals
can be represented by ratios of whole numbers, with a strongly-
implied preference for the smallest numbers compatible with
a given musical purpose. Unfortunately this definition, while
0167-2789/$ - see front matter c
2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.physd.2006.08.001
Z.-Y. Su, T. Wu / Physica D 221 (2006) 188–194 189
accurate, does not convey much to those who are not already
familiar with the art and science of tuning. The piano and
almost all modern keyboard instruments follow the twelve-
tone scale; i.e. an octave with a frequency ratio of 1:2 is
divided geometrically by even intervals into 12 semitones, each
corresponding to one of the seven white or five black keys on
the piano, and their frequency and fundamental frequency f0
satisfy the exponential function of fj/f0=2j/12 . The twelve-
tone scale differs from just intonation in frequency ratio; e.g. a
perfect fourth consists of 5 semitones with a frequency ratio of
25/12 =1.3348, which is close to 4/3; a perfect fifth consists
of 7 semitones with a frequency ratio of 27/12 =1.4983,
which is nearly 3/2. Both are ratios of smaller integers. A
diminished fifth, however, has 6 semitones with a frequency
ratio of 26/12 =1.4142, which is almost 1000/707. This is not
a ratio of small integers. Therefore, such an interval has been
traditionally considered dissonant and is rarely used in classical
pieces.
1.2. Music as 1/f noise
Before discussing the relationship between music and fractal
theory, let us focus on a particular type of noise — 1/fnoise
first. Mandelbrot proposed that there is a kind of sound in
which the quality is unaffected by changes in play speed, and
called this sound ‘scaling noise’ [1]. The plainest example
of scaling noise is ‘white noise’. Suppose a time series is
produced in accordance with temporal variations of white noise,
a calculation of its power spectral density S(f)reveals that the
relationship between S(f)and fcan be stated as S(f)fβ,
where scaling exponent β=0, indicating its monotonousness
at whatever play speed. In other words, white noise is a mixture
of frequency components from a wide range that are randomly
and completely combined; its features are utmost randomness
and totally unrelated points. Brownian noise is another type of
scaling noise with scaling exponent β=2. It depicts Brownian
movement or random walk, with the strongest correlation
among points within a characteristic time scale.
On the other hand, after conducting a spectral analysis
on various types of music, including classical music (Bach,
Mozart, Beethoven . . . ) and modern jazz, Voss and Clarke [2,
3] discovered that musical works of various melodies and
styles share a similar tendency toward a 1/fspectrum. In fact,
music featuring a 1/fspectrum happens to be a 1/fnoise
intermediary between the flat spectrum of white noise and
the steep 1/f2spectrum of Brownian noise. It is a kind of
scaling noise, too. However, neither white noise nor Brownian
noise can be called music; the former is so random and
unassociated that it becomes uninteresting, while the latter has
over-emphasized connections and lacks charm. Only 1/fnoise
can merge the randomness and orderliness into a naturally
pleasant and attractive whole [4,5].
1.3. Fractal geometry in music
Observation of time series of 1/fnoise with various time
scales reveals statistical self-similarity. That is to say, any
enlargement or reduction of the timeline would not affect
the tendency of fluctuation. Mandelbrot called such behavior
scale invariance. Furthermore, 1/fnoise features a long-
range correlation, or retaining memory over a rather long
period of time. Coincidently, nature is saturated with the 1/f
phenomenon, as seen in a mountain contour and the fluctuation
of a river’s water level, whose variations also have the traits
of scale invariance and long-range correlation. The spectral
analysis in the study by Voss and Clarke substantiated the
assumption that music imitates characteristics of temporal
variations demonstrated by nature and the universe, and that
music features fractal geometry.
As mentioned above, Voss and Clarke, from their analysis
on the power spectrum S(f)of musical signals of various
styles, observed fractal distribution approximating to 1/fin
power spectra of both loudness and frequency fluctuation
(waves of melody). However, they also pointed out that such a
phenomenon is not found in all ranges of frequency; instead,
it is only so between 100 Hz and 10 kHz. In cases of high
frequency (100 Hz–2 kHz), S(f)is not molded as 1/f.
Hence Voss and Clarke suggested that, within a certain range,
signal fluctuations of most musical works feature long range
correlation, and the exponents of the power spectrum may also
be associated with fractal content of music.
In the 1990s, Hsu and Hsu [6] discovered from analysis
of music scores by Bach and Mozart that, in general, the
difference in pitch jbetween two successive notes (i.e. the
melody) and the frequency of their appearance Fhave an
exponential relation, which can be stated as FjD, where
Dis dimension. Values of the exponent Din various musical
scores range between 1 and 3, but they are not integers. As
the dimension is not a whole number, the frequency of pitch
variation in music can be categorized as fractal geometry. In
order to visualize music, Hsu and Hsu [7] used the jvalue to
represent each musical note in a score, marked them in order of
appearance on coordinate axes (x,y), forming a curve, and then
diminished the sequence length by labeling points at intervals
of 2, 4, and 8 . . . points. The reduced curve looked much the
same as the original one, and the style remained unaffected.
Therefore, musical scores share the feature of self-similarity
with fractal geometry [8].
In addition, in a recent study Shi [9] employed the
calculation method of the Hurst exponent to examine the pitch
sequence fashioned in folk songs and piano pieces. Their
results indicated that music sequences have the property of long
range correlation and the fundamental principle of music is the
balance between repetition and contrast. Further, Bigerelle and
Iost [10] applied the ‘Variance Method’ to study the fractal
dimensions in 180 musical works of various styles. Based
on statistical results, they proposed that various music pieces
could be categorized by fractal dimension. Madison [11] used
a similar approach to study different musical scores with Hurst
exponents, which were found thereafter to play an important
role in the emotional expression of musical performance. The
study by Manaris et al. [12] of a 220-piece corpus (baroque,
classical, romantic, 12-tone, jazz, rock, DNA strings, and
random music) revealed that esthetically pleasing music might
be describable under the Zipf–Mandelbrot law. Gunduz and
Gunduz [13] studied the mathematical structures of six songs
190 Z.-Y. Su, T. Wu / Physica D 221 (2006) 188–194
by treating them as complex systems. They also calculated the
fractal dimension of a scattering diagram constructed from the
six songs’ melody.
From the above literature review, it is noticed that the Fourier
power spectrum, as well as the analysis methods used by many
previous investigators to compute the Hurst exponent and the
value of fractal dimension only refer to the ‘mean’ properties of
the overall sequence. However, it is well-established experience
that naturally evolving geometries and phenomena are rarely
characterized by a single scaling ratio; different parts of a
system may be scaling differently. That is, the clustering pattern
is not uniform over the whole system. Such a system is better
characterized as ‘multifractal’ [1]. A multifractal can be loosely
thought of as an interwoven set constructed from sub-sets
with different local fractal dimensions. Real world systems are
mostly multifractal in nature. Music too, as will be shown later
in this paper, has non-uniform property in its movement. It is
therefore necessary to re-investigate the musical structure from
the viewpoint of the multifractal theory.
2. Multifractal analysis
2.1. Multifractal formalism
There are two common approaches for multifractal
formalism:
(i) Generalized dimensions [14]
Suppose points with a total number of Nare distributed in
the space. Weighing local mass density pi(r)of points with
different exponents q(moment) would lead to the definition of
generalized dimensions:
Dq=1
q1lim
r0
log P
i
pi(r)q
log r(1)
where pi(r)=Ni(r)/Nis the portion of points that fall
within the ith sub-cover with size r,qis the given weight,
and Dqis the generalized dimension. If local densities of point
distributions in a fractal set are scattered unevenly, its Dqvalue
varies with the given weight q. When q<1, Dqreflects
the fractal dimension of low-density point distributions in the
set (or dispersive areas); while for q>1, Dqreflects the
fractal dimension of high-density point distributions in the set
(or dense areas). By definition, D0is just the conventional box-
counting dimension, D1is the information dimension and D2
is the correlation dimension.
(ii) Multifractal spectrum [15,16]
The other multifractal formalism is to calculate the local
scaling exponent of point distribution, also called the H¨
older
exponent α:
α=lim
r0
log pi(r)
log r.(2)
The physical significance of αis that α=1 indicates uniform
distribution of points, while α < 1 and α > 1 represent
‘dense inside and dispersive outside’ and ‘dispersive inside and
dense outside’ types of point distribution, respectively. Now let
n(α)dαdenote the number of sub-covers with the local scaling
exponent ranging between αand α+dα. If the original point
set features a multifractal distribution, then n(α) and the size of
sub-cover ragain has a power-law relation:
n(α) rf(α) .(3)
In this equation, the power f(α) can be viewed as the fractal
dimension of the set formed by sub-sets with a local scaling
exponent of α. The correlation diagram of f(α) and αis called
the multifractal spectrum of the point distribution.
By an analogy to well-known relationships in thermodynam-
ics [15,16], it is induced that f) and αare related to the gen-
eralized dimension Dqand qvia a Legendre transformation:
α=d
dq[(q1)Dq](4)
f(q)=qd
dq[(q1)Dq] − (q1)Dq.(5)
Common approaches first calculate the generalized dimension
Dqand then use Eqs. (4) and (5) to find αand f(α). The
prerequisite of this procedure, however, is that Dqmust be a
smooth function of q. For signals adopted from nature, such
postulation is not appropriate. Hence other researchers [17,18]
proposed another method to obtain the multifractal spectrum
f(α) directly from the weighted pi(r); i.e. set
µi(r,q)=pi(r)q,X
i
pi(r)q(6)
then
α(q)=lim
r0P
i
µi(r,q)log pi(r)
log r(7)
f(q)=lim
r0P
i
µi(r,q)log µi(r,q)
log r.(8)
In this paper, the direct formulations (7) and (8) are used
to determine the multifractal spectrum of point distribution in
music.
2.2. Conversion of musical melody and rhythm into sequences
of points
Melody and rhythm are the two important elements of music.
Conventionally, melody is defined as successive changes in
pitch (tone) in an ordered arrangement of sounds, and rhythm is
defined as successive changes in tone duration of the arranged
sounds [6–8]. Before applying the multifractal analysis, the
melody and rhythm of a music piece must be converted into
sequences that are amenable to the multifractal formalism. The
method we propose in this study is stated as follows.
Point distribution of musical melody is constructed by
dividing an octave evenly into 12 pitches, as per defined in the
twelve-tone scale. The first note of the music piece is chosen as
the base point and a black point is placed at the first position
of an imaginary line. If the absolute value of pitch difference
between the second note and the first note (i.e. melody) is m,
Z.-Y. Su, T. Wu / Physica D 221 (2006) 188–194 191
Fig. 1. Point distributions converted from the melody and rhythm of the music
score Gavotte.
another black point is placed at the (m+1)th position of the
line. This procedure is then repeated until the point distribution
diagram of musical melody is completed.
As for the construction of point sequence for rhythm, a
shortest measure of time is first selected, e.g., the sixteenth
note, and a black point is marked on the first position of the
line. If the first note of the music piece is a sixteenth note,
a black point is placed at the 2nd position of the line. If the
second note is an eighth note, another black point is placed
at the 4th position of the line, indicating that the temporal
interval between this note and the following note (i.e. rhythm) is
42=2 beats. Continuous repetition of this process completes
the point distribution diagram of rhythm. As an example, Fig. 1
shows the one-dimensional point distributions of melody and
rhythm transformed from the music score Gavotte by Gossec.
Just by observation, we can easily discern the non-uniform
structures in both the melodic and rhythmic motions of the
music piece.
2.3. Calculations of αand f )
Calculation of the H¨
older exponent αis performed as
follows. Choosing any position i(whether a black point or
blank) on the line as the center of an interval, after selecting
various interval sizes r, the portion of black points of total
number Nthat reside in the interval of radius r, denoted by
pi(r), is calculated. The results are plotted on the diagram of
log rlog pi(r), and the slope of the curve represents the local
H¨
older exponent at this center position. Similarly, the procedure
can be repeated at every position ion the line, and the variation
of the H¨
older exponent along the line is obtained. The variation
curve of the H¨
older exponent against position reveals how the
local point distribution changes in density.
The multifractal spectrum of the point distribution is
determined according to Eqs. (7) and (8). First, a point
distribution resembling the one shown in Fig. 1 is covered by
boxes with size r. If the probability of a black point falling
into the ith box is pi(r), then µican be derived from Eq. (6)
with specified weighting exponent q, and values of Pµilog pi
and Pµilog µican be calculated based on the given box
size rand weighting q. Next, the value of ris changed, and
the corresponding values of Pµilog piand Pµilog µiare
re-calculated. Continuing this way, the results are plotted on
diagrams of log rPµilog piand log rPµilog µi. Proper
scaling regions are identified in these diagrams, and the slopes
of the curves within the scaling ranges are calculated by the
least-square fitting method. These slopes are the values of α(q)
and f(q), respectively. The whole process is then repeated for
various values of weighting exponent qchosen between −∞
and +∞. The curve traced by αf(α) is the multifractal
spectrum of the point distribution.
Fig. 2. Scaling of pi(r)with box size r.
Fig. 3. Variation of H¨
older exponents along the melody sequences converted
from three different music scores. Data values of Le Cygne and Gavotte have
been shifted upward by 2 and 4 units respectively for clarity.
3. Results and discussion
Three music scores were analyzed in this study: (I) Gossec’s
Gavotte, (II) Saint-Sa¨
ens’ Le Cygne, and (III) Ave Maria
by Bach and Gounod. After the melody and rhythm of the
scores were converted into point distributions, the scaling
exponent of local point density (i.e. the H¨
older exponent) and
the multifractal spectrum of each sequences of points were
calculated by using the methods described in the previous
section.
Fig. 2 shows a typical log–log plot of point density pi(r)
vs. radius rof the interval centered at position i=300 of the
melody sequence transformed from the score Ave Maria. The
smallest radius of the interval has a width of 2, while the largest
radius can extend to a length of dozens of positions. To avoid
the boundary effect (since the sequence is finite in length), the
largest radius of the box is limited to 1/10 of the total length of
the sequence, in this case, about 74. The local H¨
older exponent
αis then obtained from a linear fitting of the data points within
the scaling range.
Detailed temporal organizations of the melodic and rhythmic
motions of music can be analyzed by inspecting the local values
of the H¨
older exponent α.Figs. 3 and 4show the variations
of the H¨
older exponent αalong the sequences converted from
the melodies and rhythms of the three different musical scores
mentioned above. Irregular fluctuations of the curves around
the value of α=1 are apparent in these figures. The H¨
older
exponent (also called the local crowding index) defined in Eq.
(2) reflects the invariant scaling nature of the population density
of point distribution in a small region centered at position iwith
those in the vicinity of increasing sizes. Variation in αvalue
with position isignifies changes in the local clustering pattern
192 Z.-Y. Su, T. Wu / Physica D 221 (2006) 188–194
Fig. 4. Variation of H¨
older exponents along the rhythm sequences converted
from three different music scores. Data values of Le Cygne and Gavotte have
been shifted upward by 2 and 4 units respectively for clarity.
Fig. 5. Typical point distributions with different αvalues.
of point distribution along the line. The geometric interpretation
of the H¨
older exponent αis most instructively illustrated by
Fig. 5 where point distributions corresponding to the special
cases α=1, α=1/2 and α=2 are compared. It is
seen that an αvalue less than one denotes a densely occupied
region surrounded by a sparse vicinity, while an αvalue greater
than one represents a less-populated region surrounded by a
dense vicinity. The fluctuating αcurves shown in Figs. 3 and
4clearly suggest that arrangements of the melody and rhythm
of music are highly non-uniform in structure. In obtaining the α
curve, once again, only the middle 1/10 to 9/10 portion of the
sequence was analyzed, and the maximal box size rwas limited
to 1/10 of the total length of the sequence so as to avoid edge
effect.
The characteristics and cragginess of the αcurve can
be further quantitatively analyzed by inspecting the Hurst
exponent of the sequence. Mandelbrot and Van Ness [19]
generalized the expression of the diffusion law of a Brownian
motion x(t)(a random-walk sequence) into the form
1xH(T)= h|xH(t+T)xH(t)|2i1/2TH(9)
where 1xH(T)denotes the mean distance traveled in the time
span Tand His the Hurst exponent, which ranges between 0
and 1. The corresponding motion (time sequence) xH(t)is now
favorably called the ‘fractional Brownian motion’ (fBm). For
H>1/2, the graph of xH(t)is less rugged-looking (smoother)
than that of the Brownian motion (H=1/2), and xH(t)tends
to be increasing in the future if it is increasing in the past
(i.e. persistence in trend). For H<1/2, the graph of xH(t)is
more rugged-looking than that of H=1/2, and xH(t)tends to
be decreasing in the future if it is increasing in the past (i.e. anti-
persistence in trend). The Hurst exponent Hwas calculated for
the above various αcurves, and the results are summarized in
Table 1
The Hurst exponent Hfor αcurves obtained from various music pieces
Gavotte Le Cygne Ave Maria
Melody 0.5748 0.4221 0.6377
Rhythm 0.5600 0.4344 0.7361
Fig. 6. Multifractal spectrum for the melody sequence of Gavotte. Cross marks
represent the error ranges in obtaining the values of αand f(α).
Table 1. Among the three music pieces, Le Cygne has Hurst
exponent Hless than 0.5 both in melody and in rhythm, the
corresponding αcurves in Figs. 3 and 4indeed look more
rugged in profiles than the other two music pieces do. Note that
the values of the Hurst exponent for all curves are not close to
1, indicating the αcurves are self-affine in structure rather than
self-similar.
The multifractal feature of music sequence is characterized
by its spectrum f(α). The graph of multifractal spectrum
(f(α) αcurve) generally shows the shape of a parabola that
is concave downward. The maximum of the curve occurs at
q=0, where f) corresponds to the box-counting dimension
D0of the point set. For q=1, f(α) is equal to the information
dimension D1and the slope of the f(α) curve is equal to 1.
The opening (α(−∞)α(+∞)) of the parabola reflects the
degree of irregularity in the distribution of the point set. A
wide opening parabola indicates that points are not uniformly
distributed along the line; rather, the tendency is to form
clusters of different sizes and densities. In the special case of
a monofractal, the parabola degenerates to a point.
Fig. 6 shows the multifractal spectrum curve for the melody
sequence of Gavotte, in which cross marks denote the error
ranges of αand f(α) values. The wide opening of the
graph again indicates a non-uniform clustering structure of the
sequence. Fig. 7 provides a comparison between the spectra
obtained from the melodies of the three different music pieces
(with error bars removed from the plot for clarity). It is
observed that the opening size of each curve follows the order:
Gavotte >Ave Maria >Le Cygne.Fig. 8 is the multifractal
spectrum curve for the rhythm of Gavotte. Again, it is a
downward opening parabola in shape, except the width of the
opening is smaller than that of the melody sequence. In Fig. 9,
spectra obtained from the rhythms of the three music scores
are compared. It is found that the opening size of the curve
Z.-Y. Su, T. Wu / Physica D 221 (2006) 188–194 193
Table 2
Main fractal data derived from multifractal spectra for different music pieces
Music scores α(−∞)=D−∞ f(0)=D0f(1)=D1α(+∞)=D+∞ α (−∞)α(+∞)
Melody Gavotte 1.561 1.000 0.942 0.740 0.821
Le Cygne 1.384 1.000 0.974 0.798 0.586
Ave Maria 1.526 1.000 0.964 0.762 0.764
Rhythm Gavotte 1.323 1.000 0.977 0.772 0.551
Le Cygne 1.439 0.997 0.963 0.757 0.682
Ave Maria 1.432 1.000 0.977 0.796 0.636
Fig. 7. Comparison of multifractal spectra for melody sequences obtained from
different music scores.
for Gavotte is the smallest, while those of Le Cygne and Ave
Maria are about the same. A larger opening size in the melody
spectrum reflects a broader variation in pitch between notes,
implying that the music may sound more bright and active. A
larger opening size in the spectrum of rhythm, on the other
hand, reveals more conspicuous variation in beats of the music,
suggesting richer emotional traits in the expression of the
music. Such differences in melody and rhythm, which are also
perceivable as we actually listen to these three music works,
are consistent with the results demonstrated in their multifractal
spectra. Thus multifractal spectrum analysis shows a great
potential to become one of the effective tools in discerning
and classifying different musical styles. Various relevant main
fractal data derived from multifractal spectra for different music
pieces are summarized in Table 2 for further reference.
Finally, if we map the point-sequence version of the melody
and rhythm of a music piece (e.g., the ones given in Fig. 1) into
a two-dimensional graph with abscissa denoting the rhythmic
motion and ordinate the melodic motion, a fractal curve as
shown in Fig. 10 is obtained. The curve is full of many
‘plateaus’ and ‘steps’ of different sizes — a diagram similar
to the ‘Devil’s Staircase’ as exhibited by the behaviors of many
dynamical systems described in their parameters’ planes. The
same method can be applied to diagrammatize the melody
and rhythm of every music score into such a characteristic
curve. Different pieces of music may vary tremendously in
their characteristic curves; some may be rather smooth, others
very steep and rugged. Such a mapping and the resulting
characteristic curve offer yet another way to distinguish the
style of a musical work. By the same token, if the behavior or
Fig. 8. Multifractal spectrum for the rhythm sequence of Gavotte. Cross marks
represent the error ranges in obtaining the values of αand f(α).
Fig. 9. Comparison of multifractal spectra for rhythm sequences obtained from
different music scores.
geometric structure of any dynamical system shows a similar
fractal curve in its parameters’ plane, attempts can also be made
to map out its analogical music score using this approach.
4. Conclusion
In this paper, we have proposed a novel method to convert
the melody and rhythm of a music work into distributed point
sets. The fractal properties of these sequences of points can
then be explored quantitatively by calculating their local H¨
older
exponents αand multifractal spectra f(α)αcurves according
to the multifractal formalism. Three musical pieces, Gavotte,
Le Cygne and Ave Maria were transformed and analyzed in this
study. Erratic fluctuation in the αvalue along the point sequence
194 Z.-Y. Su, T. Wu / Physica D 221 (2006) 188–194
Fig. 10. Characteristic curves (Devil’s Staircase) for different music pieces.
reveals how the local point distribution changes in density and
hence implies a complex and irregular clustering structure in
the melodic and in the rhythmic motion of music.
The multifractal spectra of the melody and rhythm
sequences obtained from the three music pieces all show a
familiar shape of an inverted and downward-opening parabola.
A wide opening of the parabola indicates that points are
not uniformly distributed along the sequence, which once
again confirms the non-uniformity structure of the musical
movement. Our present results show that the opening sizes
of the spectra for the three melodies under study are in the
following order: Gavotte >Ave Maria >Le Cygne; while the
order of the opening sizes of the spectra for the three rhythms
is: Le Cygne >Ave Maria >Gavotte. The physical meaning
as represented by these multifractal spectra is that a relatively
larger opening in the f ) curves of melody and rhythm reflects
the music featuring a more drastic fluctuation in pitch and
richer variation in beat. Thus we are in the position to say that
one piece of music is more melodious or more rhythmic than
the other. The present analysis suggests that the multifractal
spectrum and its relevant fractal data can be used to distinguish
and classify different styles of music.
As a final point, a fractal geometry full of many ‘plateaus’
and ‘steps’ of different sizes is constructed by mapping the
point sequences converted from the melody and rhythm of
a music piece into a two-dimensional curve. The geometry
resembles the ‘Devil’s Staircase’ — a fractal set describing
the dynamics of many interesting dynamical systems in proper
parameters’ planes. This fractal curve characterizes the melodic
as well as the rhythmic motions of music, and is unique to
each different piece of music. We have therefore provided an
innovative means to disclose the intrinsic property of music.
Acknowledgement
The authors would like to express their gratitude for
the financial support from the Tzong Jwo Jang Educational
Foundation for this study.
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