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Fractal dimension and classification of music

  • Ecole Nationale Supérieure d'Arts et Métiers LILLE

Abstract and Figures

The fractal aspect of different kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great number of different musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal dimension helps discriminate different categories of music. Then, we used an original statistical technique based on the Bootstrap assumption to find 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 different 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 different kinds of music very well: musics could be classified by their fractal dimensions.
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Fractal dimension and classi®cation of music
M. Bigerelle
, A. Iost
Equipe mat
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 dierent kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great
number of dierent musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal
dimension helps discriminate dierent 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 dierent 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 dierent kinds of music very well: musics could be classi®ed by their fractal dimen-
sions. Ó2000 Elsevier Science Ltd. All rights reserved.
1. Introduction
Fractal analysis lead to study of dierent physical phenomena met in dierent sciences as Materials
science, Fluids mechanics, Wear, Chemistry, Botany, etc. This paper aims to analyze the fractal aspect of
dierent musics (rock music, traditional music, classical music and so on...). Using dierent statistical
methods, we will analyze how fractal dimension helps discriminate dierent 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 [1] analyzed the music of digital computer and reviewed dierent 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. Dierent works have already been carried out using spectral analysis in the
lower frequency. Voss [2] 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 dierent
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. [3] remarked that the spectra exponent was constant and could be related to the
fractal aspect of the music. According to some fractal theories [4] which applied on self-ane 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: (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 Pfand 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 dierent musics (classics, blues, medieval, the Beatles, etc.), Voss [5] con®rmed the
1=fspectra. The 1=fspectra of music were also con®rmed by Schroeder [6] and Campbell [7]. 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 [11] 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 [12] 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 [13] the sum Liof
all note intervals iin a composition and found the following fractal relation Lic=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 [14] 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
dierent compositions by considering acoustic frequencies. We have analyzed Hs
u's results [14] 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 [15]. Secondly,
acoustic analysis are dicult 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 dierent diapasons: when a musician writes a score, he analysis
his music with taking considerations into account (Chopin would have composed his Nocturne dierently if
they had ®rst been played on an organ rather than a piano). The second one is the interactions between
dierent 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 dicult when dierent instruments are playing together while
interactions between dierent 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 eect to analyze the spectra (by the Fast Fourier Transform) aects signal analysis and it
becomes dicult to analyze music by the spectra. Secondly, computing fractal dimension by Spectra im-
poses that audio signal is self-ane 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 dicult 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-
anity (or self-similar) hypothesis. The signal will be seen as a function yft(where fis the amplitude
signal in decibels) and we resorted two dierent fractal analysis: The Variation method [16] and the ANAM
method [17]. 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 [1] 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 dierent musics is shown at dierent 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 dierent methods used to calculate the fractal dimension of music. The
Variation [16] and the ANAM [17] 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
this graph.
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 ftÿ
min ft
By taking the average of OSCsf;xover the interval [a,b] we have
VARsf;a;b 1
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 dierent svalues. The slope corre-
sponds to the h
olderian coecient 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 [17].
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
sf;x 1
The function Ka
sf;xis de®ned by making the average of Ma
sf;xover the whole interval a;b
sf;a;b 1
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,
real, aP1:
The fractal dimension is approached by linear regression of logKa
sf;a;bvs. logsfor dierent svalues.
The slope estimates the H
olderian coecient 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;bit is impossible to make the calculation over a;asand 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. Eciency of the two methods
With these methods, the errors due to discretization on the ordinate disappear. It is important to see that
scale eect on yhas no eect 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 dierent 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 ecient 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;xcsHf;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 [18] as follows:
where c;k2R2. Likewise, by integrating the function over the interval a;bwe 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 coecient eonly if
its value is signi®cantly dierent 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 dierent. 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 dierent 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.
Table 1
Results of fractal dimensions using dierent regressions and ®tting methodsa
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 dierent 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. Discussion
4.1. Method used to calculate fractal dimension
Firstly, we compare dierent 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 dierent 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 dierent kinds of musics give dierent 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
Dinter music
Dintra music 0:0092
0:000387 23:72:
The value of Fis highly signi®cant and we can arm with a probability error inferior to 1=100 000 that
fractal dimension can discriminate the categories of music.
We will now calculate the statistics (eects) in each class. To estimate fractal eect, 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 dier
signi®cantly and none of its subsets are tested. The Ducan test uses the studentized range statistic and we
get: bp1ÿ1ÿa
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 dierent.
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 dierences 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 dierences
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 dierents 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 dierent 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 dierent 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
with the
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;kthat discriminates with the best
statistical characteristic among the dierent 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;kvalue that gives the greatest variance between dierent classes of music [19,22]. A probability function,
that represents the probability to arm that fractal dimension could be dierent 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
Table 3
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 dierent
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
Table 2
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=44 100s and cor-
respond to the frequency domains 3;44kHz. 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 [20].
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)
(punk music).
Fig. 4. Results of the bootstrap analysis of variance. The abscisse represents the class i;jwhere fractal dimension Dlc
i;jare calculated.
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
probability values.
2188 M. Bigerelle, A. Iost / Chaos, Solitons and Fractals 11 (2000) 2179±2192
Table 4
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 arm
that Di;jdiscriminates fractal dimension better the others Di;j; Di®nf: Probability to arm 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 dierent 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.
Table 5
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.
6. Conclusion
That study proved that dierent 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 e1 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 Qgcould be encoded as a Cantor space. A universal power law was found between this
quantity and the factor scale Qgagÿu. 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.
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Cell studies play an important role in the basis of studies on cancer diagnosis and treatment. Reliable viability assays on cancer cell studies are essential for the development of effective drugs. Lens-less digital in-line holographic microscopy (DIHM) has become a powerful tool in the characterization and viability analysis of microparticles such as cancer cells due to its advantages such as high efficiency, low cost, and flexibility to integrate with other components. This study is designed to perform viability tests using fractal dimensions of alive and dead cancer cells based on digital holographic microscopy and machine learning. In the in-line holography configuration, a microscopy assembly consisting of inexpensive components was built using an LED source, and the images were reconstructed using computational methods. The standard U.S. air force target was used to evaluate the capability of our imaging setup then holograms of stained cancer cells were recorded. To characterize individual cells, 19 different rotational invariant fractal dimension values were extracted from the images as features. An artificial neural network technique was employed for the classification of fractal features extracted from cells. The artificial neural network was compared with four other machine learning techniques through five different classification performance measures. The empirical results indicated that artificial neural networks performed better than compared classification techniques with accuracies of 99.65%. The method proposed in this paper provides a new method for the study of cell viability which has the advantages of high accuracy and potential for laboratory application.
It is known that bird vocalizations and music share acoustic similarities. Unsurprisingly, bird sounds inspired a number of music composers. In music, complexity plays an important role in auditory attractiveness. Would sound complexity also be important to explain the attractiveness of bird vocalizations to humans? In our study, we experimentally assessed the preference for vocalizations according to their level of complexity, indicated by objective measurements. Further, given that men and women enjoy music similarly, we verified whether the taste for the sound of birds differs between the sexes. The study was conducted on 114 adults living in a rural district in the northeast of Brazil. The results showed a significant and linear preference for sounds, with the most complex ones being preferred. Moreover, both men and women were attracted to the songs of these animals. For the first time, the importance of complexity in humans’ appreciation for bird vocalizations has been objectively demonstrated. Our results stress the relationship between bird vocalizations and music for people. In addition to its theoretical nature, this study might be useful to predict, based on the sound complexity, which birds would be subject to a higher risk of capture, information that would help in reducing the loss of biodiversity. Moreover, giving the apparently universal aspect of bird song attraction to humans, it would be advisable in terms of conservation efforts to elect singing birds as flagship species. We hope that our research will serve as a motivation for further efforts in this area, as it clearly brings important insights into ethnozoology and other interdisciplinary fields.
A parallelism of the fractal geometry of natural landscape and that of music suggests that music can be investigated through a visual representation of acoustic signals. The parallelism inspires us to make musical abstracts by scaling the original down to a half quarter or eighth of its original length. An algorithm for music reduction has been devised. The self-similarity of Bach’s music has been demonstrated by this analysis. Bird songs, nursery rhymes and classical music are distinguished by their diatonic scale. Bird songs and nursery rhymes are not well-structured successions of tones, dominated by unison or seconds (i = 0,1,2). A proper combination of selected songs can, however, include enough variety to achieve a fractal geometry. The progress to baroque and classical composers is manifested by the approximation to fractal geometry in Bach’s and Mozart’s music, simulating the harmony of nature. This harmony is absent in modern music.
Music critics have compared Bach's music to the precision of mathematics. What "mathematics" and what "precision" are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot's fractal geometry and the precision is the deviation from a log-log linear plot.
Within a general theory, a probabilisticjustification for a compactification which reduces aninfinite-dimensional spacetime $E^{{\text{(}}\infty {\text{)}}} (n = \infty )$ to afour-dimensional one (DT = n = 4) isproposed. The effective Hausdorff dimension of this spaceis is given by $\langle \dim _{\text{H}} E^{{\text{(}}\infty {\text{)}}} \rangle = d_{\text{H}} = 4 + \Phi ^3 ,{\text{ where }}\Phi ^3 = 1/[4 + \Phi ^3 ]$ is a PV number and φ = (√5– 1)/2 is the golden mean. The derivation makes use of various results from knot theory,four-manifolds, noncommutative geometry, quasiperiodictiling, and Fredholm operators. In addition somerelevant analogies between $E^{{\text{(}}\infty {\text{)}}} $ , statistical mechanics, and Jones polynomials are drawn.This allows a better insight into the nature of theproposed compactification, the associated $E^{{\text{(}}\infty {\text{)}}} $ space, and thePisot–Vijayvaraghavan number 1/φ3= 4.236067977 representing its dimension. This dimensionis in turn shown to be capable of a naturalinterpretation in terms of the Jones knot invariant andthe signature of four-manifolds. This brings the work near to the context of Witten andDonaldson topological quantum field theory.
It is shown that the subtle mean, which is the third power of the Golden number, has some quite interesting properties. These properties connecting diverse fields such as knot theory, subfactors, noncummutative geometry, Cantorian spacetime and quasi crystals are discussed and illustrated. It is conjectured that the subtle mean is the mean dimension of actual spacetime at the resolution of quantum physics.
Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex forms found in nature. Shapes such as coastlines, mountains and clouds are not easily described by traditional Euclidean geometry. Nevertheless, they often possess a remarkable simplifying invariance under changes of magnification. This statistical self-similarity is the essential quality of fractals in nature. It may be quantified by a fractal dimension, a number that agrees with our intuitive notion of dimension but need not be an integer. In Section 1.1 computer generated images are used to build visual intuition for fractal (as opposed to Euclidean) shapes by emphasizing the importance of self-similarity and introducing the concept of fractal dimension. These fractal forgeries also suggest the strong connection of fractals to natural shapes. Section 1.2 provides a brief summary of the usage of fractals in the natural sciences. Section 1.3 presents a more formal mathematical characterization with fractional Brownian motion as a prototype. The distinction between self-similarity and self-affinity will be reviewed. Finally, Section 1.4 will discuss independent cuts, Fourier filtering, midpoint displacement, successive random additions, and the Weierstrass-Mandelbrot random function as specific generating algorithms for random fractals. Many of the mathematical details and a discussion of the various methods and difficulties of estimating fractal dimensions are left to the concluding Section 1.6.