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# Fractal dimension and classification of music

Authors:
• 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,
France
Accepted 27 July 1999
Abstract
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-
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
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Chaos, Solitons and Fractals 11 (2000) 2179±2192
*
Corresponding author. Tel.: +33-320-622-233; fax: +33-320-535-593.
E-mail address: iost@lille.emsam.fr (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) [10]. 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 u 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 restrictions. 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 fg 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: f:a;b!R; OSCsf;x max ftÿ jxÿtj<s min ft jxÿtj<s :1 By taking the average of OSCsf;xover the interval [a,b] we have VARsf;a;b 1 bÿaZb a OSCsf;xdx2 then the fractal dimension can be written as Df;a;blim s!02 ÿlogVARsf;a;b logs:3 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 2ÿHf;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 Ma sf;x 1 s2Zs t10Zs t20 fx j 2 6 4t1ÿfxÿt2jadt1dt23 7 5 1=a :4 The function Ka sf;xis de®ned by making the average of Ma sf;xover the whole interval a;b Ka sf;a;b 1 bÿaZxb xa 1 s2Zs t10Zs t20 fx j 2 6 4t1ÿfxÿt2jadt1dt23 7 5 1=a dx:5 By making three times the average of the function, the variance of the estimation of Ka sf;a;bdiminishes. If fis uniformly h olderian and anti-h olderian, there are two positive real numbers cand c0such as for a, real, aP1: csHf;a;bPKa sf;a;bPc0sHf;a;b and Df;a;blim s!02 ÿlogKa sf;a;b logs: 6 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 Df;a;b2ÿHf;a;b. 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 Ka 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 Ka sf;a;bk1ÿ2=a nÿ2kX nÿk ik1X k j0X k l0 fx j "jdxÿfxÿldxja#1=a ;7 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 dimension. 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 later). 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: OSCsf;xcsHf;a;bX p iÿ1 ki si;8 where c;k2R2. Likewise, by integrating the function over the interval a;bwe get VARsf;a;bdsHf;a;bX p i1 ei si:9 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 fg . 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 Class STAT DNL3 DNL1 DNL2 DL Class DNL3 DNL1 DNL2 DL Concerto N 16 16 16 16 String quartet 8888 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 Electronical music N 19 19 19 19 Relaxation music 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 Heavy Metal N 21 21 21 21 Rockn'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 Chopin's Nocturnes N 5 5 5 5 Traditionnal music 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 Progressive Music N 15 15 15 15 Trash music 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 a 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 Fr2 Dinter music r2 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. 1 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 1 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. r2 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 ;xi1;yi1 ;...;xik;yik  fg 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 i;j 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 ;x2;y2 ;x3;y3 ;x4;y4  fg 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 i1;...;2j 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 a 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  5 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). Acknowledgements The authors are very indebted to Professor El Naschie for very valuable comments. References [1] Campbell P. The music of digital computer. Nature 1986;324:523±8. [2] Voss RF, Clark J. 1/fNoise in music and speech. Nature 1975;258:317±8. [3] Voss RF, Clarke J. 1/fNoise in music music from 1/fnoise. J Acoust Soc Am 1978;63(1). [4] Tricot C. J Chim Phys 85 1988:379. [5] Voss RF. Fractals in nature: from characterisation to simulation. In: Peitgen HO, Saupe D, editors. The science of fractal image, New York: Springer, 1988. [6] Schroeder P. Is there such a thing as fractal music? Nature 1987;325:765±6. [7] Campbell P. Nature 325 1987:767. [8] West BJ, Shlesinger, The M. noise in natural phenomena. Am Scientist 1978;78:40±5. [9] Thomsen DE. Making music fractally. Sci News 1980;117:187. [10] Scarpelli AT. 1/frandom tones. Personal Computing 1979;3:17±27. [11] Dubuc B, Quiniou JF, Roques-Carnes C, Tricot C, Zucker SW. Evaluating the fractal dimension of pro®les. Phys Rev A 1989;39(3):1500±12. [12] Hs u KJ, Hs u A. Fractal geometry of music. Proc Natl Acad Sci USA 1990;87:938±41. [13] Hs u KJ. Hs u A. Self-similarity of the 1/fNoise called music. Proc Natl Acad Sci USA 88, 1991:3507±09. [14] Hs u KJ. Fractal geometry of music: From bird songs to Bach, 21±38, in: Applications of fractals and chaos, Berlin: Springer, 1993. [15] 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. [16] 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 [17] Bigerelle M, Iost A. Calcul de la dimension fractale d'un pro®l par la m ethode des autocorr elations moyenn ees norm ees (AMN). C.R. Acad. Sci. Paris, t. 323, S erie II b, 1996:669±74. [18] Bigerelle M, Iost A. Fractal analysis of surface roughness. Noise, numerical and statistical artefacts. 11th International conference on surface modi®cation technologies, Paris, 1997. [19] Bigerelle M. Pertinence des param etres physiques une approche probabiliste. ADER mod elisation 1997;1:49±50. [20] Schroeder M. Fractals, chaos, power laws, W.H. Freeman and Company, New York, 1990. [21] Kay M. Did Mozart use the Golden mean? American Scientist, March/April 1993. [22] Haylock D. The Golden section in BeethovenÕs ®fth. Math Teaching 1978;84:56±7. [23] Lendvai E. Duality and synthesis in the music of Bela Bartok, in Module, Proportion, Symmetry, Rythm, G.Kepes, 1966. [24] Lendvai E. Bela Bartok: an analysis of his music, Kahn 1 Averill, 1971. [25] Lendvai E. Some striking proportions in the music of Bela Bartok. Fibonacci Quarterly 1971;9(5):527±8. [26] Howat R. Debussy in proportion: a musical analysis, Cambridge: Cambridge University Press, 1983. [27] Coutney S. Erik Satie, Golden section analysis, music and letters, Oxford University Press, 1996:77(2):242±52. [28] GarlandÕs TH. Fascinating Fibonacci, Dale Seymours Publications, 1987. [29] El Naschie MS. Superstring knots and noncommutative geometry in E1space. Int J Theor Phys 1998;37(12):2935±51. [30] El Naschie MS. Time symmetry duality and cantorian space-time. Chaos Solitons & Fractals 1996;7(4):499±518. [31] 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 ... However, other than those ideal fractals, a lot of physical processes and signals demonstrate similar properties. Natural schematic patterns [150], music signals [17,180] as well as biomedical signals [39] show indeed a complex structure across timescales. The complexity of such signals is typically measured through the fractal dimension, which is higher than their topological dimension. ... ... EEG segment is partitioned into its main bands through bandpass filtering with a 10th order Butterworth filter. We include alpha (8-13 Hz), beta (14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29), and gamma (30-45 Hz) rhythms in our analysis, as well as raw signals, since those have been acknowledged as the most emotion-sensitive [178] and have shown the largest multiscale variability. We select 12 left (Fp1, AF3, F7, F3, FC5, FC1, T7, C3, CP5, CP1, P3, P7) and 12 right (Fp2, AF4, F4, F8, FC2, FC6, C4, T8, CP2, CP6, P4, P8) channels that have shown competitive performance, particularly when their asymmetry is examined, and we assess the proposed features on each set separately. ... Thesis Full-text available The analysis of human emotions is a widely researched topic in the scientific fields of Psychology and Neuroscience, trying to investigate the nature and elicitation mechanisms of our feelings. From a computational perspective, however, it remains rather underexplored. While Artificial Intelligence has made overwhelming progress in modeling rational intelligence, there are yet no highly reliable systems to analyze affect, as considerable barriers exist in this process: Emotion expression can be highly subjective and its interpretation varies depending on the context, whereas it poses an inter-subject variability. Yet, most Signal Processing and Machine Learning studies concentrate on behavioral processing of emotions, through modalities like speech, text and facial expressions. To address the challenges of Affective Analysis, in this thesis we choose to process brain signals, and specifically the Electroencephalogram (EEG), as a means to derive emotional information. Recorded physiological and neural signals are capable of being more objective and reliable affective indicators, whereas they can also contribute to develop human-aid systems for applications like the treatment or rehabilitation from brain diseases. Importantly, we consider music as the means to induce emotions for the EEG recordings, since music is known to have a deep emotional impact on humans. Our approach can be divided into two main parts: In the first one, we analyze the complex structure of the EEG and examine novel feature extraction schemes that are based on two multifractal algorithms, namely Multiscale Fractal Dimension and Multifractal Detrended Fluctuation Analysis. In this way we attempt to quantify the variability of the observed signals' complexity across multiple timescales. Our proposed EEG features surpass widely used baselines on Emotion Recognition, whereas they show competitive results in challenging subject-independent experiments and recognition of arousal, indicating that it is highly correlated with the EEG's fragmented structure. In the second part, we utilize a two-branch neural network as a bimodal EEG-music framework, which learns common latent representations between the EEG signals and their music stimuli in order to examine their correspondence. Through this model, we perform supervised emotion recognition experiments and retrieval of music rankings to EEG input queries. By applying this system to independent subject data, we also extract interesting patterns regarding the latent similarity of brain and music signals, the temporal variation of the music-induced emotions and the activated brain regions in each case. As a whole, this study deals with core problems regarding the interpretation of complex EEG signals and illustrates multiple ways that music stimulates the brain activity. ... In our analysis we employ two different methods to evaluate A(M), the VAR method [33][34][35] and the ANAM method [36]. We follow reference [37] for their implementation. ... Article Full-text available We present a new method, based on fractal analysis, to characterize the output of a physical detector that is in the form of a set of real-valued, discrete physical measurements. We apply the method to gravitational-wave data from the latest observing run of the Laser Interferometer Gravitational-wave Observatory. We show that a measure of the fractal dimension of the main detector output (strain channel) can be used to determine the instrument status, test data stationarity, and identify non-astrophysical excess noise in low latency. When applied to instrument control and environmental data (auxiliary channels) the fractal dimension can be used to identify the origins of noise transients, non-linear couplings in the various detector subsystems, and provide a means to flag stretches of low-quality data. ... In both the sub-problems, the proposed ODE-Net has outperformed the remaining three models. (ii) The application of nonlinear methodologies for source modeling indicates the importance of non-deterministic/ chaotic approaches in understanding the underlying intricacies of speech/music signals [57][58][59][60][61][62][63]. In this context fractal analysis of the signal which reveals the geometry embedded in acoustic signals assumes significance. ... Article Full-text available Music is often considered as the language of emotions. The way it stimulates the emotional appraisal across people from different communities, culture and demographics has long been known and hence categorizing on the basis of emotions is indeed an intriguing basic research area. Indian Classical Music (ICM) is famous for its ambiguous nature, i.e. its ability to evoke a number of mixed emotions through only a single musical narration, and hence classifying evoked emotions from ICM becomes a more challenging task. With the rapid advancements in the field of Deep Learning, this Music Emotion Recognition (MER) task is becoming more and more relevant and robust, hence can be applied to one of the most challenging test case i.e. classifying emotions elicited from ICM. In this paper we present a new dataset called JUMusEmoDB which presently has 1600 audio clips (approximately 30 s each) where 400 clips each correspond to happy, sad, calm and anxiety emotional scales. The initial annotations and emotional classification of the database was done based on an emotional rating test (5-point Likert scale) performed by 100 participants. The clips have been taken from different conventional ‘raga’ renditions played in two Indian stringed instruments – sitar and sarod by eminent maestros of ICM and digitized in 44.1 kHz frequency. The ragas, which are unique to ICM, are described as musical structures capable of inducing different moods or emotions. For supervised classification purposes, we have used Convolutional Neural Network (CNN) based architectures (resnet50, mobilenet v2.0, squeezenet v1.0 and a proposed ODE-Net) on corresponding music spectrograms of the 6400 sub-clips (where every clip was segmented into 4 sub-clips) which contain both time as well as frequency domain information. Along with emotion classification, instrument classification based response was also attempted on the same dataset using the CNN based architectures. In this context, a nonlinear technique, Multifractal Detrended Fluctuation Analysis (MFDFA) was also applied on the musical clips to classify them on the basis of complexity values extracted from the method. The initial classification accuracy obtained from the applied methods are quite inspiring and have been corroborated with ANOVA results to determine the statistical significance. This type of CNN based classification algorithm using a rich corpus of Indian Classical Music is unique even in the global perspective and can be replicated in other modalities of music also. The link to this newly developed dataset has been provided in the dataset description section of the paper. This dataset is still under development and we plan to include more data containing other emotional as well as instrumental entities into consideration. ... In our analysis we employ two different methods to evaluate A(M), the variation (VAR) method [33][34][35] and the ANAM method [36]. We follow Ref. [37] for their implementation. ... Preprint We present a new method, based on fractal analysis, to characterize the output of a physical detector that is in the form of a set of real-valued, discrete physical measurements. We apply the method to gravitational-wave data from the latest observing run of the Laser Interferometer Gravitational-wave Observatory. We show that a measure of the fractal dimension of the main detector output (strain channel) can be used to determine the instrument status, test data stationarity, and identify non-astrophysical excess noise in low latency. When applied to instrument control and environmental data (auxiliary channels) the fractal dimension can be used to identify the origins of noise transients, non-linear couplings in the various detector subsystems, and provide a means to flag stretches of low-quality data. ... Fractal scaling seems to be an appropriate methodology for exploring acoustic complexity, the application of different methods depending on the nature of the investigated objects and on the goal of the analysis (Halley et al., 2004). Applications can be found in the field of acoustics and music (Lyamshev and Adreev, 1997;Makabe and Muto, 2014;Bigerelle and Iost, 2000). However, such a fractal approach is more appropriate to explore the complexity of acoustic communities rather than acoustic indices. ... Article Full-text available We investigate the possible presence of ‘long time’ memory in the auto-correlations of biophonic activity of environment sound. The study is based on recordings taken at two sites located in the Parco Nord of Milan (Italy), characterized by a wooded land, rich in biodiversity and exposed to different sources and degrees of anthropogenic disturbances. The audio files correspond to a three-day recording campaign (1-min recording followed by 5-min pause), from (17:00) April 30 to (17:00) May 3, 2019, which have been transformed into ecoacoustic indices time series. The following eight indices have been computed: Acoustic Complexity Index (ACI), Acoustic Diversity Index (ADI), Acoustic Evenness Index (AEI), Bio-acoustic Index (BI), Acoustic Entropy Index (H), Acoustic Richness index (AR), Normalized Difference Soundscape Index (NSDI) and Dynamic Spectral Centroid (DSC). We have grouped the indices carrying similar sound information by performing a principal component analysis (PCA). This allows us to reduce the number of variables from eight to three by retaining a large (≳80%) variance of the original variables. The time series corresponding to the reduced set of new variables have been analyzed, and both seasonal and possible long term trend components have been extracted. We find that no trends are present, i.e. the resulting time series are stationary, and the auto-correlations of the three selected PCA dimensions and associated residuals (obtained after extracting the seasonal components) can be determined. The calculations reveal the presence of a “memory” of few (≲5) hours long in the environment sound, for the two sites considered, which is quantified by the Hurst exponent, H. For Site 1, we find an overall effective Hurst exponent, Hdim≃0.88, for all three dimensions, and Hres≃0.75 for the residuals. For Site 2, the exponents are slightly smaller, amounting to 0.80 and 0.60, respectively. We attempt to correlate the Hurst exponents with a quality index obtained from an aural survey, aimed at determining the sound components, such as biophonies, technophonies and geophonies, at the two sites. We conclude that the higher the Hurst exponents, the higher are the periodic-structured sounds, corresponding to stronger long-term biophonic activity. We find that Site 1 has a more structured environment sound than Site 2, also consistent with the major presence of tall trees surrounding the location of the acoustic sensor at the former. ... Previous knowledge suggests that music signals have a complex behavior: at every instant components (in micro and macro scale: pitch, timbre, accent, duration, phrase, melody etc.) are closely linked to each other [12,13] . These properties are peculiar of systems with chaotic, self organized, and generally, nonlinear behavior. ... Article Full-text available Hindustani classical music is entirely based on the "Raga" structures. In Hindustani music, a "Gharana" or school refers to the adherence of a group of musicians to a particular musical style. Gharanas have their basis in the traditional mode of musical training and education. Every Gharana has its own distinct features; though within a particular Gharana, significant differences in singing styles are observed between generations of performers, which can be ascribed to the individual creativity of that singer. This work aims to study the evolution of singing style among four artists of four consecutive generations from Patiala Gharana. For this, alap and bandish parts of two different Ragas sung by the four artists were analyzed with the help of non linear multifractal analysis (MFDFA) technique. The multifractal spectral width obtained from the MFDFA method gives an estimate of the complexity of the signal. The observations from the variation of spectral width give a cue towards the scientific recognition of Guru-Shisya Parampara (teacher-student tradition) - a hitherto much-heard philosophical term. From a quantitative approach this study succeeds in analyzing the evolution of singing styles within a particular Gharana over generations of artists as well as the effect of globalization in the field of classical music. Article 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. Article 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. Chapter 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. Article 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. Article 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.
Article
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.
Article
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.