Fractal dimension and classi®cation of music
, A. Iost
eriaux ENSAM Lille, Laboratoire de M
etallurgie Physique, CNRS UMR 8517, 8 Boulevard Louis XIV, 59046 Lille cedex,
Accepted 27 July 1999
The fractal aspect of dierent kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great
number of dierent musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal
dimension helps discriminate dierent categories of music. Then, we used an original statistical technique based on the Bootstrap
assumption to ®nd a time window in which fractal dimension reaches a high power of music discrimination. The best discrimination is
obtained between 1/44100 and 16/44100 Hertz. We admit that to distinguish some dierent aspects of music well, the high information
quantity is obtained in the high frequency domain. By calculating fractal dimension with the ANAM method, it was statistically
proven that fractal dimension could distinguish dierent kinds of music very well: musics could be classi®ed by their fractal dimen-
sions. Ó2000 Elsevier Science Ltd. All rights reserved.
Fractal analysis lead to study of dierent physical phenomena met in dierent sciences as Materials
science, Fluids mechanics, Wear, Chemistry, Botany, etc. This paper aims to analyze the fractal aspect of
dierent musics (rock music, traditional music, classical music and so on...). Using dierent statistical
methods, we will analyze how fractal dimension helps discriminate dierent kinds of music. As the fractal
dimension is independent of the power of the music, this no-dimensional number could be used to correlate
acoustic aspects of sounds with physical parameters.
Campbell  analyzed the music of digital computer and reviewed dierent techniques used to model
aspects of musical perception. In analogy with the well-known Mandelbrot set, he asked if musical cog-
nition depends on discrete structures such as scales or rhythms and he introduced the notions of a macro-
and a micro-structure of music. Dierent works have already been carried out using spectral analysis in the
lower frequency. Voss  analyzed the power spectrum Sfand found that ¯uctuation in music and pitch
exhibits a 1=fpower spectra. Before a critical value of a frequency, Voss found that Sfvaries in keeping
with 1=f2. According to Voss Sfis not 1=ffor higher frequency (100 Hz±2 kHz), meaning that spectrum
contains much information. A bandpass ®lter ranging between 100 Hz and 10 kHz was used to obtain a 1=f
structure. This frequency domain led the author to analyze aspects of music alongside with the amplitude
(loudness) of the audio signals and did not analyze the acoustic frequency domain. Analyzing three dierent
radio stations over 12 h (Classical, jazz±blues and rock), they found the 1=fspectra. By analyzing music
spectra in 1=f, Voss et al.  remarked that the spectra exponent was constant and could be related to the
fractal aspect of the music. According to some fractal theories  which applied on self-ane functions, a
Chaos, Solitons and Fractals 11 (2000) 2179±2192
Corresponding author. Tel.: +33-320-622-233; fax: +33-320-535-593.
E-mail address: email@example.com (M. Bigerelle).
0960-0779/00/$ - see front matter Ó2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 137-X
relation between the spectra Pfand the fractal dimension Dleads to the following relation Pf/f2Dÿ5.
As a consequence, fractal dimension is unique and equal to 2. By this theory, we have to admit that an-
alyzing fractal aspect in the lower frequency could teach that fractal dimension is constant for all musics.
More recently, analyzing dierent musics (classics, blues, medieval, the Beatles, etc.), Voss  con®rmed the
1=fspectra. The 1=fspectra of music were also con®rmed by Schroeder  and Campbell . As a con-
sequence, fractal music can be created using this fractal spectra: 1=fspectrum is generated and the audio
signal is obtained using the Inverse Fast Fourier Transform [8,9]. This music becomes more pleasant than a
white noise sound and lies between white (independent of the frequency) and brown music ( 1=f2spectra)
However, the precision of the power spectrum method is low  and is particularly due to an important
variance of the power spectrum.
As Voss and Clark failed to ®nd a fractal dimension of acoustic music, Hs
u and Hs
u  analyzed Bach's
composition and found that the change of acoustic sound has a fractal geometry. As a music note could be
de®ned by f=f02i=n, they applied fractal concept considering that it would be more appropriate to de®ne
melody as a succession of note intervals and not a succession of music notes and found the relation
Fc=jD, where Fis the percentage incidence frequency of the note interval between successive notes, Dis
the fractal dimension and jis the note interval. To transform the audio signal into a visual one, Hs
digitized notes of a score, their frequencies were plotted against the successive number of notes in the
composition and a fractal structure was found. With the same technique, Hs
u analyzed  the sum Liof
all note intervals iin a composition and found the following fractal relation Lic=eD, where eis a `music
yardstick' that measures the `length' of the total interval iof a music score, ca constant and Dthe fractal
dimension. An algorithm for music reduction has been devised from these relations. Finally, by analyzing
scores for bird songs, Hs
u  found that most bird songs are not characterized by a fractal relation.
In fact, by analyzing Hs
u's theory, we have to assume that fractal dimension may not be constant for
dierent compositions by considering acoustic frequencies. We have analyzed Hs
u's results  from a
statistic point of view and we concluded that fractal dimension of frequency of note intervals in music is
statistically constant and was between 1.16 and 1.22. Why should fractal dimension be experimentally
constant in the audio spectra? With some reserves, let us try and give explanations. Firstly, Hs
u used less
information to calculate the fractal dimension (for example, 391 intervals for Bach Invention N°10). Two
more points have to be chosen to estimate the fractal dimension as minutely as possible . Secondly,
acoustic analysis are dicult when frequency of the audio signal is non-stationary as in music. In fact, it is
not that obvious to decorelate to separate the frequency (or pitches) of a note, its duration and the time
interval between two notes: Music is a combination of these frequencies and it seems impossible to impose a
discriminating function. Thirdly, as Hs
u's analysis was carried out by digitizing the note from a score, two
important pieces of information were lost. The ®rst one is the timbre. In any instrument, a note frequency
(fundamental) is not unique and its harmonics gives the instrument's speci®city. For example, on a piano,
ut3pitch (fundamental) is 261 Hz and has two audible harmonics ut4(552 Hz) and sol4(783 Hz). In fact,
music cannot be de®ned as a succession of dierent diapasons: when a musician writes a score, he analysis
his music with taking considerations into account (Chopin would have composed his Nocturne dierently if
they had ®rst been played on an organ rather than a piano). The second one is the interactions between
dierent pitches that could not be analyzed in a score and are an important physiological aspect of the
music (for example, the score of the right-hand of a piano composition is not independent of the score on
the left one). Finally, score analysis would be dicult when dierent instruments are playing together while
interactions between dierent instruments are also aspects of the music.
These remarks lead us to consider that music has to be analyzed as a whole and not in parts. No
mathematical artifact has to be introduced: ®ltering audio message should be prohibited because any cut-o
is suggestive and could introduce into the signal some informations that the composer did not even suggest.
We do not use the power spectra analysis for three reasons. Firstly, as frequencies vary along with time,
window time eect to analyze the spectra (by the Fast Fourier Transform) aects signal analysis and it
becomes dicult to analyze music by the spectra. Secondly, computing fractal dimension by Spectra im-
poses that audio signal is self-ane and this assertion is not proved yet. Finally, by Fourier's analysis, a
lower frequency gets greater variances than a higher frequency and this frequency heteroscedasticity leads
to dicult statistical interpretations of the signal.
2180 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192
As a consequence, to analyze music, we chose to digitize the signal without ®ltering it, with no self-
anity (or self-similar) hypothesis. The signal will be seen as a function yft(where fis the amplitude
signal in decibels) and we resorted two dierent fractal analysis: The Variation method  and the ANAM
method . These two methods provide the advantage of being independent of the power signal (in Db)
and give a good calculated fractal dimension for this kind of stochastic function with poor mathematical
2. The usual method to calculate the fractal dimension
2.1. Music recording
Musics are digitized at the sample rate of 44100 Hz for exactly 2 min, and are recorded from an audio
disk. If the time exceeds 2 min, a 2 min interval is randomly chosen over the time music graph. If records are
stereophonic, then the left and right signals are overimposed to obtain a monophonic signal. We chose the
same time music length to analyze fractal aspect of the music along the same scale and to avoid introducing
any lower frequency statistical bias. We analyzed nearly 5 million data. Signal amplitude is encoded in 16
bit words. This coding and sampling rate were speci®ed in an article by Campbell  who remarks that a
calculation involving ¯oating point operation does not introduce noise into the signal. That is why, we did
not use the integer operation. The signal of two dierent musics is shown at dierent scales in Figs. 1 and 2.
The random choice of music will be explained later.
Fig. 1. Fractal aspect of the music. Plot of the time series music versus the amplitude. Each point is taken at the sample rate of
1/44 100 s during a period of 2 min. Fig. 1(a) is the integral time series of the 5th symphony of Ludwig Van Beethoven, Fig. 1(b) a zoom
of 1 s and Fig. 1(c) a zoom of 0.1 s.
M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192 2181
2.2. Calculation of fractal dimension
We are going to present the two dierent methods used to calculate the fractal dimension of music. The
Variation  and the ANAM  methods. These two methods can be used when the graph is de®ned by
the set of points Ex;fx;a6x6b
where Graph of f:a;b!R. Time music plot is represented by
2.2.1. The variation method
This method was proposed by Dubuc et al. [6,16] and applied to roughness measurements. The s-
oscillation of the function fin xis de®ned as:
OSCsf;x max ftÿ
By taking the average of OSCsf;xover the interval [a,b] we have
then the fractal dimension can be written as
Fig. 2. Fractal aspect of the music. Plot of the time series music versus the amplitude. Each point is taken at the sample rate of
1/44 100 s during a period of 2 min. Fig. 1(a) is the integral time series `Evidence of abominations' by Massacra, a punk music band,
Fig. 1(b) a zoom of 1 s and Fig. 1(c) a zoom of 0.1 s.
2182 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192
Df;a;bis associated with the graph of the function fde®ned over the interval a;b. The fractal dimension
is obtained by linear regression of log VARsf;a;bversus log sfor dierent svalues. The slope corre-
sponds to the h
olderian coecient Hf;a;band the fractal dimension is given by Df;a;b
2.2.2. The ANAM method
This method was recently proposed by Bigerelle and Iost .
Given two positive real numbers a<b,fis a C0class function such as f:aÿs;bs!R;aP1, a real
number. The function Ma
sf;xis then de®ned as
The function Ka
sf;xis de®ned by making the average of Ma
sf;xover the whole interval a;b
By making three times the average of the function, the variance of the estimation of Ka
If fis uniformly h
olderian and anti-h
olderian, there are two positive real numbers cand c0such as for a,
The fractal dimension is approached by linear regression of logKa
sf;a;bvs. logsfor dierent svalues.
The slope estimates the H
olderian coecient Hf;a;band the fractal dimension is given by
In order to discretize Eq. (5) by a numerical method, we use a ®rst-order numerical integration which
gives a good evaluation of Ka
sf;a;bwith less consuming calculation time. The problem of the boarders ±
as in other methods ± must now to be considered. In point of fact, if the function fis de®ned over the
interval a;bit is impossible to make the calculation over a;asand bÿs;b. The integration of
sf;a;bis made over the interval as;bÿs(Window size) so as to avoid introducing a bias in the
calculation (that is to say that Ka
sf;a;bwould depend on the number of discretized points). Let us de®ne
by x1;f1;x2;f2;...;xn;fnthe discretizing points of the graph f;xiÿxiÿ1dx;i22;n, the sample
rate), we obtain
nbeing the number of discretized points from the graph of the ffunction and skdx. The graph of the
logkdxversus log Ka
sf;a;bis a straight line, where the slope converges to the H
older exponent. A
nonlinear regression and a correction methods could also be used and will be used in our analysis. As it was
proved than the fractal dimension does not depend on the avalues. To minimize the numerical complexity
of Eq. (7) and to avoid the evaluation of the a-power evaluation, we retain the value a1.
M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192 2183
2.2.3. Eciency of the two methods
With these methods, the errors due to discretization on the ordinate disappear. It is important to see that
scale eect on yhas no eect on the determination of the fractal dimension (this assertion is not veri®ed for
the well-known Box counting method). In this study, we want to search discriminating power of fractal
dimension of music and we do not want this result to be correlated with the power amplitude of the time
series. That is why we used the Variation and the ANAM methods.
By applying the two methods on dierent function graphs, we proved that the ANAM method is more
precise than the others (around 30%) when Windows size is small and that the number of discretized points
is not too important.
Otherwise, the ANAM method cannot be used when the windows size is too large: the three sums lead to
an important computational time when kin Eq. (7) increases. This fact is well-known to numerical
methods: increasing precision increases computational time. When the fractal dimension of the graph is
uniform (fractal dimension does not depend on the analysis scale) and no high frequency noises are in-
troduced, then the ANAM method can be used with a small window and estimates precisely the fractal
The main problem is how to quantify these phenomena on a physical pro®le: we have to analyze the
variation of the fractal function on all the time domains. The oscillation methods can be used to study these
domains by an ecient algorithm based on the properties of the Max and Min functions. But some
problems due to the information lost about the Max and Min functions lead to calculate a large window to
well-appreciate fractal dimensions.
We think that the Variation method has to be used for all the windows in a ®rst approach. Then, if
we want a more precise analysis, the ANAM method has to be applied (see physical justi®cations
2.2.4. Corrected methods
If fis h
olderian and anti-h
olderian, then OSCsf;xcsHf;a;b.Ifsdecreases, the error made on the
oscillation rises logarithmically (because the number of points decreases linearly with s). The relation can be
stated  as follows:
where c;k2R2. Likewise, by integrating the function over the interval a;bwe get
We must ®nd Hf;a;b,d,e, which minimize this function and the problem can be resolved by nonlinear
regression. In fact, there are no mathematical reasons, except for the discretization phenomenon, that
VARsf;a;bis a function with a term 1=s. Thanks to a statistical analysis, we take the coecient eonly if
its value is signi®cantly dierent from 0 (Student's test). Moreover, it can be proved during the nonlinear
regression that dsHf;a;band e=sare orthogonal by regression. By adding the term e=sin the regression
model, the expectation of Hf;a;bis unchanged. We also use another term f=s2if this term is statistically
signi®cant and so on. The method presented to estimate the errors due to discretization is justi®ed for the
Variation method and can be applied to the ANAM method. This method improves results of all methods
speci®cally when fractal dimension increases.
2.2.5. Regression analysis
In the non-corrected methods, fractal dimension can be calculated by the linear regression or by the non-
least-square. When in the linear case or in the nonlinear case fractal dimension can be dierent. The best
one is the nonlinear model.
2184 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192
3. Experimental result
We will now draw the experimental design. 12 categories of music were chosen in Concerto, String
quartet, Electronic music, Heavy metal, Jazz, Chopin's Nocturne, Progressive music, Relaxation music,
Rock'n'roll, Symphonies, Traditional music, Trash music (like punk music). All the titles were randomly
chosen in a music data bank thanks to keywords given to the dierent kinds. We chose to take a 2 min
random sample that represents a sample size of 5 000 000 data. For all the titles of music we take exactly the
same number of points to avoid biasing the fractal dimension estimation and we divide this interval in
dyadic interval of 2kpoints with k21::18
. So the fractal aspect of music is studied between 4=44 100 and
24 s. We compute fractal dimension by the variation method without error correction with a linear ®tting
(log±log plot) and the nonlinear ®tting. Then we analyze the data with error correction by nonlinear re-
gression. All the results we obtained are shown in Table 1.
Results of fractal dimensions using dierent regressions and ®tting methodsa
Class STAT DNL3 DNL1 DNL2 DL Class DNL3 DNL1 DNL2 DL
Concerto N 16 16 16 16 String
MIN 1.684 1.691 1.689 1.709 1.744 1.744 1.745 1.728
MAX 1.880 1.857 1.871 1.799 1.820 1.811 1.818 1.761
MEAN 1.823 1.814 1.821 1.776 1.782 1.777 1.781 1.746
STD 0.052 0.045 0.049 0.022 0.022 0.020 0.021 0.011
N 19 19 19 19 Relaxation
10 10 10 10
MIN 1.743 1.743 1.744 1.734 1.805 1.799 1.805 1.763
MAX 1.873 1.857 1.867 1.800 1.886 1.867 1.880 1.817
MEAN 1.835 1.823 1.831 1.768 1.863 1.847 1.858 1.787
STD 0.031 0.027 0.030 0.020 0.023 0.019 0.021 0.018
N 21 21 21 21 Rock`n'Roll 19 19 19 19
MIN 1.819 1.808 1.815 1.744 1.775 1.771 1.774 1.728
MAX 1.883 1.865 1.876 1.809 1.917 1.891 1.907 1.833
MEAN 1.864 1.849 1.858 1.790 1.871 1.853 1.864 1.793
STD 0.018 0.015 0.016 0.015 0.027 0.023 0.026 0.026
Jazz N 15 15 15 15 Symphony 17 17 17 17
MIN 1.7538 1.751 1.753 1.690 1.690 1.693 1.693 1.728
MAX 1.8784 1.858 1.870 1.787 1.850 1.836 1.845 1.773
MEAN 1.8359 1.822 1.830 1.749 1.792 1.785 1.790 1.752
STD 0.0329 0.028 0.031 0.030 0.046 0.041 0.044 0.015
N 5 5 5 5 Traditionnal
18 18 18 18
MIN 1.727 1.728 1.728 1.719 1.724 1.724 1.725 1.715
MAX 1.819 1.809 1.816 1.743 1.849 1.836 1.845 1.782
MEAN 1.779 1.774 1.778 1.731 1.797 1.790 1.795 1.745
STD 0.034 0.030 0.032 0.011 0.031 0.028 0.030 0.022
N 15 15 15 15 Trash
26 26 26 26
MIN 1.781 1.778 1.781 1.745 1.864 1.850 1.859 1.786
MAX 1.880 1.859 1.872 1.797 1.911 1.888 1.902 1.839
MEAN 1.836 1.825 1.832 1.771 1.894 1.873 1.886 1.816
STD 0.030 0.025 0.028 0.014 0.011 0.009 0.010 0.011
N: Number of dierent musics in each class used to calculate fractal dimension; MIN: Minimal value of the fractal dimension in the
given class; MAX: Maximal value of the fractal dimension in the given class; MEAN: Mean of the fractal dimension in the given class;
STD: Standard deviation of the fractal dimension in the given class; DNL3: Fractal dimension calculated using second-order cor-
rection models; DNL2: Fractal dimension calculated using the ®rst-order correction models; DNL1: Fractal dimension calculated
using the nonlinear least-square method without error correction of error; DL: Fractal dimension calculated using the linear least-
square methods (without correction of error).
M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192 2185
4.1. Method used to calculate fractal dimension
Firstly, we compare dierent regression methods used to calculate fractal dimension. Methods used to
calculate fractal dimension by a linear least-square (DL) and the nonlinear least-square without error
correction (DNL1) give dierent results. In fact, we want to calculate a result on the same equation. In most
of the bibliography about fractal dimension calculation, it can be noticed that the log transformation is
performed being easier to compute (no iterative algorithm). But, fractal dimension is biased by this method
and nonlinear fractal dimension has to be computed by a nonlinear square regression. If we analyse errors
for the 189 time series studied, we notice that error between linear and nonlinear estimations on the fractal
dimension is majored by 0.05. As fractal dimension lies between 1 and 2, one gets a maximal error of 5%.
Secondly, best results in terms of statistics correlations are obtained using model DNL3. For those reasons,
all results discussed later are studied using model DNL3.
4.2. Statistical analysis
Now, we are going to see if dierent kinds of musics give dierent fractal dimensions. We decide to carry
out an analysis of variance for the variable fractal dimension and the classes are de®ned by the kind of
music. This analysis shows that
Dintra music 0:0092
The value of Fis highly signi®cant and we can arm with a probability error inferior to 1=100 000 that
fractal dimension can discriminate the categories of music.
We will now calculate the statistics (eects) in each class. To estimate fractal eect, we ®rst study the
distribution of each fractal dimension in each class. We apply the Kolmogorov Smirnov test to verify the
Gaussian adequation (Table 2). So we cannot reject at the 0.05 signi®cant level that fractal dimension
probability density function obeys the Gaussian law. This assumption proves that fractal dimension is a
good estimator of music signal. With this assumption, a good test to compare means between classes is the
Ducan test if we want to obtain simultaneous con®dence intervals. We use this test because it gets a greater
power using multiple-stage tests. The Ducan test ®rst tests the homogeneity of all the means at a level bk.If
the result is a rejection, then each subset of kÿ1 means is tested at level bkÿ1; otherwise, the procedure
stops. In general, if the hypothesis of homogeneity of a set of pmeans is rejected at the bklevel, then each
subset of pÿ1 means is tested at the bpÿ1level; otherwise, the set of pmeans is considered not to dier
signi®cantly and none of its subsets are tested. The Ducan test uses the studentized range statistic and we
pÿ1where Ducan's method controls the comparison error rate at the alevel. Table 3
gives results of Duncan test for a0:05. For each test, the accuracy of the test is given. Means are classi®ed
from the highest to the lowest. Means with the same letter are not signi®cantly dierent.
4.3. Music analyses
The statistical analyses of the fractal dimension scatters the music into four groups. The highest fractal
dimension is obtained for Trash music. This music is high speed, musicians principally play the drums and
the guitar, voices are loud then we get a high fractal dimension. Then we can group the Rock'n'roll and
Heavy-metal musics. The dierences between these two kinds of music are not signi®cant because they get
The Purpose of Analysis of Variance. In general, the purpose of analysis of variance (ANOVA) is to test for signi®cant dierences
between means. This name is derived from the fact that in order to test for statistical signi®cance between means, we are actually
comparing (i.e., analyzing) variances. At the heart of ANOVA is the fact that variances can be divided up, that is, partitioned.
Dinter music represents the variance of the fractal dimension between the dierents classes of music and r2
Dintra music the variance of the
fractal dimension for a given class of music. The greater Fis, the more discriminate the fractal dimension is.
2186 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192
the same origin and are played with similar instruments (Drums, distortioned guitars and bass). Relaxation
music belongs to the same class. These surprising results have to be explained: relaxation music has no
tempo. In fact, by analyzing the time series, we can remark that there is no peak. So the analysis consider
this music as a pink noise and so that the fractal dimension is great. Then the third group contains rhythmic
music with low tempo as for progressive music, concerto, jazz and electronic music. Finally, the fourth
group is composed of musics with low tempo, few percussions and a much slower rhythm as in traditional
music, symphony, string quartet and Chopin's nocturnes.
5. Multifractal approach
In the previous paragraph, we have analyzed the fractal dimension on all the time series intervals as-
suming that fractal dimension is uniform. A frequency approach (Fig. 3(a) and (b)) shows dierent fractal
scales and a multifractal measure has to be studied. In this work, fractal dimension would not be ap-
proached by frequency analysis. For a multifractal structure, we can postulate that the log±log plot con-
tains information by its linear properties on a given interval. We propose to analyze the dierent scales by
this original following technique. Firstly, we calculated all possible slopes. If nis the number of points in the
log±log plot and xi;yi
one of these points, we propose to calculate `local' fractal dimension by the slope
obtained with the least means square method from points xi;yi
following conditions: i6nÿ1, ik6nand kP1. One obtains the local fractal dimension Di;ikand
®nally for each kind of music a set of local fractal dimension Dlc
i;jis obtained, where lcis the lmusic of the
music class c. As all the music time series get the same number of points, nwill be constant and all Dlc
correspond to the same frequency interval. To ®nd the better pair i;kthat discriminates with the best
statistical characteristic among the dierent classes of music, we used a private method called the para-
metric bootstrap analysis of variance. The aim of this method is to search, by a bootstrap technique, the
i;kvalue that gives the greatest variance between dierent classes of music [19,22]. A probability function,
that represents the probability to arm that fractal dimension could be dierent between classes, is ob-
tained. The lower this probability number, the better the discrimination by the fractal analysis. While
sorting out this probability through ascending sequences (Fig. 4), it is shown that the ®rst 10 windows get a
good discrimination power (Table 4). It could also be shown that the ®rst three fractal dimensions are
statistically the best: This means that the best window is obtained for the x1;y1
Results from the Duncan test. Means of fractal dimension are computed by the Variation method with a large window. Means are
arranged from the highest to the lowest. Means with same letter are not signi®cantly dierent
Duncan grouping Duncan grouping Mean NClass
A 1.89402 26 Trash
B A 1.87119 19 Rock
B 1.86457 21 Heavy
B 1.86389 10 Relax
C 1.83678 15 Progr
C 1.83591 15 Jazz
C 1.83513 19 Elect
C 1.82391 16 Conce
D 1.79730 18 Tradi
D 1.79202 17 Symph
D 1.78208 8 Quartet
D 1.77978 5 Noctu
Results from the critical values of the Kolmogorov Smirmov test about Gaussian assumption. Each number represents the error
probability of rejecting Gaussian adequation. As the usual critical values to not reject Gaussian hypothesis is 0.05, all distributions
could be considered as a Gaussian probability density function
Conce Elect Heavy Jazz Noctu Progr Quart Relax Rock Symph Tradi Trash
0.73 0.88 0.86 0.90 0.94 0.88 0.97 0.79 0.75 0.93 0.96 0.92
M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192 2187
points. It physically leads to two important remarks. First one, as fractal dimension is theoretically cal-
culated for small x(Eqs. (4) and (6)), under fractal music hypothesis, fractal dimension has to be computed
for small x. As a result, we decide to use the ANAM method to calculate the fractal dimension of music. In
fact, we do not use the ANAM method to ®rst study the music. The reason is the high computational time
due to the maximal windows size of one million points (k1 000 000, n5 000 000). Secondly, it was
shown that the ANAM method is more appropriate when the window size is small (10±100 points). When k
is greater than 200, the Variation method gives results nearly as good as the ANAM method. An example
of the windows size in¯uence is given in Fig. 5 for a white noise (D2). As the best window size are equal
to 16, analysis were performed by the corrected ANAM method with a 16 points window size. These re-
marks were con®rmed by the experimental analysis: when xis small, the F-value (discrimination power
function) is divided by a factor 2 ( F54) using the Variation method and divided by a factor 7 (F173)
using the nonlinear ANAM corrected method.
Fractal dimensions calculated with small windows rather than large ones are lower: when windows size
becomes too large, we mentally visualize a noise: this fact can be con®rmed by the autocorrelation functions
(Fig. 6(a) and (b)).
Best values lead to one analyze music fractal aspects in time domain 1=44 100;16=44 100s and cor-
respond to the frequency domains 3;44kHz. These domains are included in the audible frequency with no
lower frequencies. Schroeder had studied the structure of concert hall. He concluded than when listening to
speech or music, the ear unconsciously switches to a short time analysis and high resolution frequency
responses become an important aspect of pleasing sound .
We can remark that classi®cation is more precise using the ANAM method. Now music is divided into
eight groups by the Ducan method (Table 5) against four groups by the ®rst study with Variation method.
Fig. 3. Power spectra of pitch ¯uctuations. (a) and (b) are, respectively, calculated from the signal Fig. 1(a) (classics) and Fig. 2(a)
Fig. 4. Results of the bootstrap analysis of variance. The abscisse represents the class i;jwhere fractal dimension Dlc
The ordinate represents the critical probability value under no discrimination of the music by the fractal dimension. Prob is the value of
this probability. Mean, median, p5 and p95 are, respectively mean, the median, the 5th quantile and the 95th quantile of the critical
2188 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192
Analysis of discrimation power by the bootstrap analysis of variance versus the intervals where fractal dimension is computed. Di;j: Fractal dimension is calculated with the 2i,2
of the time windowsa
Classe In¯uent fProb Mean Std P95 P5 Di®nf Difsup S1 S2 S3 S4 S5 S6 S7 S8 S9 S10
D1;4O 54.07 1.25E )53 8.52E )49 7.18E )48 6.32E )51 7.68E )66 98.96 0.94 1
D2;3O 54.66 5.84E )54 1.92E )49 1.82E )48 1.80E )51 2.60E )66 99.05 0.95 1 2
D1;3O 53.68 2.07E )53 3.08E )49 2.84E )48 1.30E )50 9.33E )65 99.04 1.14 1 2 3
D2;4O 50.09 2.56E )51 2.90E )46 2.41E )45 5.22E )48 2.25E )64 98.33 1.61 2 3 4
D1;5O 47.84 5.98E )50 4.47E )45 3.92E )44 7.95E )46 2.64E )62 98.05 1.83 2 3 4 5
D1;2O 44.61 6.78E )48 6.05E )44 4.25E )43 1.04E )44 3.14E )56 97.36 2.67 3 4 5 6
D3;4O 40.49 4.02E )45 7.13E )40 5.35E )39 2.60E )41 1.31E )56 96.59 3.43 3 4 5 6 7
D2;5O 40.15 6.99E )45 1.88E )40 1.33E )39 1.34E )40 1.87E )56 96.48 3.55 4 5 6 7
D1;6O 36.68 2.30E )42 6.64E )38 5.01E )37 2.08E )38 1.76E )53 95.49 4.46 4 5 6 7 8
D3;5O 27.43 9.05E )35 9.52E )31 7.61E )30 4.03E )31 8.22E )45 88.75 11.29 5 6 7 8 9
D2;7O 27.28 1.24E )34 1.15E )31 6.73E )31 2.19E )31 1.91E )44 88.40 11.62 5 6 78910
D2;6O 26.88 2.86E )34 3.41E )30 2.81E )29 2.53E )30 3.62E )44 88.18 11.97 6 78910
In¯uent: O means that fractal dimension discriminates the music at 0.05 con®dence level; f: Fisher-Snodecor value; Prob: Probability that fractal dimension are identical for all music
categories; Means: Means of prob; Std.: Standard deviation of Prob; p95: 95% of values of prob are less than p95; p5: 5% of values of prob are less than p5; Difsup: Probability to arm
that Di;jdiscriminates fractal dimension better the others Di;j; Di®nf: Probability to arm that Di;jdoes not discriminate fractal dimension better the others Di;j; S1...Sn: Classi®cation
variables: If jmusics get the same values of Sj then power discrimination is equal at the 0.05% level.
M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192 2189
As a result, Music classi®cation becomes very logical. Firstly, the fractal dimension of relaxation music
becomes less important: as the tempo is not much analyzed, then pink noise aspect disappears. Secondly,
Heavy-metal gets the same fractal dimension that Trash music because this one gets the same origin as
Heavy metal. Rock music lies between Heavy-metal and Progressive music and is well distinguished, this
music is original and is less `powerful' than heavy but more than progressive music. Progressive music has
the same fractal dimension as the electronic one: we could explain this fact by the abundance of synthesizers
used in progressive music. Electronic music gets the same fractal dimension as a concerto and traditional
Fig. 5. Comparaison of the Oscillation and ANAM methods using dierent windows size by computing fractal dimension on a white
noise. OSCLN: Linear Variation method; OSC2: Nonlinear Variation method without error correction; OSC3: Nonlinear Variation
method with error correction; ANAMLN: Linear ANAM method; ANAM2: Nonlinear ANAM method without error correction;
ANAM3: Nonlinear ANAM method with error correction.
Results from the Duncan test with a maximum of 16 points in the windows size computed by the corrected ANAM method
Duncan grouping Duncan grouping Duncan grouping Mean NClass
A 1.52914 21 Heavy
A 1.49108 26 Trash
B 1.43589 19 Rock
C 1.359 15 Progressi
C D 1.334 19 Electronic
D E 1.291 16 Concerto
D E 1.288 18 Traditio
F E 1.261 15 Jazz
F E G 1.246 8 Quartet
F G 1.231 17 Symphonie
G 1.200 10 Relax
H 1.112 5 Nocturne
Fig. 6. Autocorrelation functions of two time series music. Fig. (a) and (b) are, respectively, calculated from the signal Fig. 1(a)
(classics) and Fig. 2(a) (punk music).
2190 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192
music: that could be explained by the fact that electronic music gets the tempo of traditional music with
synthesized instruments and sounds as with those used in a concerto. Then jazz lies between traditional
music and the string quartet: Jazz could be considered as a traditional one with fewer musicians playing
together like string quartet music. Then symphonies lie between string quartet and relaxation. This sur-
prising result could be related to the abundance of symphony instruments playing together on a low fre-
quency tempo: by applying central theorem limit, sounds sum become smoother and decrease fractal
dimension. Then, well distinguished, lie Chopin's nocturnes that get the lower fractal dimension.
That study proved that dierent kinds of music could be discriminated by their fractal dimension. Our
experimental hypotheses lead us to admit that music gets some fractal aspects in the audible frequencies. By
the fractal method used, fractal criteria become independent of the sound power: fractal dimension be-
comes a number that quanti®es the acoustic space occupation (when fractal dimension increases, the sound
power versus time becomes more chaotic) and would be easy to use. As it was shown, fractal dimension
discriminates musics according to their dynamic aspects. It is well known that the Golden Mean number
pÿ1=2 plays a signi®cant rule for various composers that use Fibonacci Number when they composed
music [21±28]. In the e1 space theory, the fractal dimension of a randomly construct triadic Cantor set is
equal to the Golden Mean [29±31]. Using a dyadic signal decomposition, we have ®nd that the quantity of
information Qgcould be encoded as a Cantor space. A universal power law was found between this
quantity and the factor scale Qgagÿu. The fractal dimension calculated from the ANAM method is
linearly correlated with a. The greater ais, the lower Dis; meaning that the signal entropy increases with D.
However uappears to be quite constant and independent of the sort of music (around 0.3). Some audio
compression or ®ltering algorithms could be deduced from this relation. These results will be published
when some theoretical aspects will be stated. However, this Dcriterion could be used with other signal
parameters to distinguish sounds (physical applications using sound analysis as for example sonar detection
or vocal cognition).
The authors are very indebted to Professor El Naschie for very valuable comments.
 Campbell P. The music of digital computer. Nature 1986;324:523±8.
 Voss RF, Clark J. 1/fNoise in music and speech. Nature 1975;258:317±8.
 Voss RF, Clarke J. 1/fNoise in music music from 1/fnoise. J Acoust Soc Am 1978;63(1).
 Tricot C. J Chim Phys 85 1988:379.
 Voss RF. Fractals in nature: from characterisation to simulation. In: Peitgen HO, Saupe D, editors. The science of fractal image,
New York: Springer, 1988.
 Schroeder P. Is there such a thing as fractal music? Nature 1987;325:765±6.
 Campbell P. Nature 325 1987:767.
 West BJ, Shlesinger, The M. noise in natural phenomena. Am Scientist 1978;78:40±5.
 Thomsen DE. Making music fractally. Sci News 1980;117:187.
 Scarpelli AT. 1/frandom tones. Personal Computing 1979;3:17±27.
 Dubuc B, Quiniou JF, Roques-Carnes C, Tricot C, Zucker SW. Evaluating the fractal dimension of pro®les. Phys Rev A
u KJ, Hs
u A. Fractal geometry of music. Proc Natl Acad Sci USA 1990;87:938±41.
u KJ. Hs
u A. Self-similarity of the 1/fNoise called music. Proc Natl Acad Sci USA 88, 1991:3507±09.
u KJ. Fractal geometry of music: From bird songs to Bach, 21±38, in: Applications of fractals and chaos, Berlin: Springer, 1993.
 Charkaluk E, Bigerelle M, Iost A. Characterization of rough surfaces, in: Proceedings of the Fourth European Conference on
Advanced Materials and Processes EUROMAT 95, Venise, 1995:511±12.
 Dubuc B, Dubuc S. Error bounds on the estimation of fractal dimension. SIAM J Numer Anal 1996;33:602±26.
M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192 2191
 Bigerelle M, Iost A. Calcul de la dimension fractale d'un pro®l par la m
ethode des autocorr
C.R. Acad. Sci. Paris, t. 323, S
erie II b, 1996:669±74.
 Bigerelle M, Iost A. Fractal analysis of surface roughness. Noise, numerical and statistical artefacts. 11th International conference
on surface modi®cation technologies, Paris, 1997.
 Bigerelle M. Pertinence des param
etres physiques une approche probabiliste. ADER mod
 Schroeder M. Fractals, chaos, power laws, W.H. Freeman and Company, New York, 1990.
 Kay M. Did Mozart use the Golden mean? American Scientist, March/April 1993.
 Haylock D. The Golden section in BeethovenÕs ®fth. Math Teaching 1978;84:56±7.
 Lendvai E. Duality and synthesis in the music of Bela Bartok, in Module, Proportion, Symmetry, Rythm, G.Kepes, 1966.
 Lendvai E. Bela Bartok: an analysis of his music, Kahn 1 Averill, 1971.
 Lendvai E. Some striking proportions in the music of Bela Bartok. Fibonacci Quarterly 1971;9(5):527±8.
 Howat R. Debussy in proportion: a musical analysis, Cambridge: Cambridge University Press, 1983.
 Coutney S. Erik Satie, Golden section analysis, music and letters, Oxford University Press, 1996:77(2):242±52.
 GarlandÕs TH. Fascinating Fibonacci, Dale Seymours Publications, 1987.
 El Naschie MS. Superstring knots and noncommutative geometry in E1space. Int J Theor Phys 1998;37(12):2935±51.
 El Naschie MS. Time symmetry duality and cantorian space-time. Chaos Solitons & Fractals 1996;7(4):499±518.
 ElNaschie MS. A notes on the subtle mean in physics. Chaos Solitons & Fractals 1999;10(1):147±53.
2192 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192