Content uploaded by Tzuyin Wu

Author content

All content in this area was uploaded by Tzuyin Wu on Mar 20, 2020

Content may be subject to copyright.

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)∝f−β,

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 F∝j−D, 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

q−1lim

r→0

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

r→0

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) ∼r−f(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(α) and αare related to the gen-

eralized dimension Dqand qvia a Legendre transformation:

α=d

dq[(q−1)Dq](4)

f(q)=qd

dq[(q−1)Dq] − (q−1)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

r→0P

i

µi(r,q)log pi(r)

log r(7)

f(q)=lim

r→0P

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

4−2=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 (α)

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 r−log 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 r−Pµilog piand log r−Pµilog 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 α−f(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)|2i1/2∝TH(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(α) 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(α ) 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.

References

[1] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York,

1983.

[2] R.F. Voss, J. Clarke, 1/fnoise in music and speech, Nature 258 (1975)

317–318.

[3] R.F. Voss, J. Clarke, 1/fnoise in music: Music from 1/fnoise, J. Acoust.

Soc. Am. 63 (1) (1978) 258–263.

[4] P. Campbell, Replies: Is there such a thing as fractal music? Nature 325

(1987) 766.

[5] R.F. Voss, Fractals in nature: From characterization to simulation, in:

H.-O. Peitgen, D. Saupe (Eds.), The Science of Fractal Images, Springer,

New York, 1988, pp. 21–70.

[6] K.J. Hsu, A.J. Hsu, Fractal geometry of music, Proc. Natl Acad. Sci. USA

87 (1990) 938–941.

[7] K.J. Hsu, A.J. Hsu, Self-similarity of the 1/fnoise called music, Proc.

Natl Acad. Sci. USA 88 (1991) 3507–3509.

[8] K.J. Hsu, Fractal geometry of music: From bird songs to Bach,

in: A.J. Crilly, R.A. Earnshaw, H. Jones (Eds.), Applications of Fractals

and Chaos, Springer-Verlag, New York, 1993, pp. 21–39.

[9] Y. Shi, Correlations of pitches in music, Fractals 4 (1996) 547–553.

[10] M. Bigerelle, A. Iost, Fractal dimension and classiﬁcation of music, Chaos

Solitons Fractals 11 (2000) 2179–2192.

[11] G. Madison, Properties of expressive variability patterns in music

performances, J. New Music Res. 29 (2000) 335–356.

[12] B. Manaris, D. Vaughan, C. Wagner, J. Romero, R.B. Davis, Evolutionary

music and the Zipf–Mandelbrot law: Developing ﬁtness functions for

pleasant music, Lecture Notes in Computer Science 2611 (2003) 522–534.

[13] G. Gunduz, U. Gunduz, The mathematical analysis of the structure of

some songs, Physica A 357 (2005) 565–592.

[14] P. Grassberger, Generalized dimensions of strange attractors, Phys. Lett.

A 97 (1983) 227–230.

[15] U. Frisch, G. Parisi, On the singularity structure of fully developed

turbulence, in: M. Ghil, R. Benzi, G. Parisi (Eds.), Turbulence and

Predictability in Geophysical Fluid Dynamics and Climate Dynamics,

North-Holland, Amsterdam, 1985, pp. 84–88.

[16] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shraiman,

Fractal measures and their singularities: The characterization of strange

sets, Phys. Rev. A 33 (1986) 1141–1151.

[17] A. Chhabra, C. Meneveau, R.V. Jensen, K.R. Sreenivasan, Direct

determination of the f(α) singularity spectrum and its application to fully

developed turbulence, Phys. Rev. A 40 (1989) 5284–5294.

[18] A. Chhabra, R.V. Jensen, Direct determination of the f(α ) singularity

spectrum, Phys. Rev. Lett. 62 (1989) 1327–1330.

[19] B.B. Mandelbrot, J.W. Van Ness, Fractional brownian motions, fractional

noises and applications, SIAM Rev. 10 (4) (1968) 422–437.