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ournal of Statistical Mechanics:An IOP and SISSA journalJ Theory and Experiment

Long-range correlation andmultifractality in Bach’s Inventionspitches

G R Jafari1,2, P Pedram1,2 and L Hedayatifar3

1 Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran2 Department of Nano-science, IPM, PO Box 19395-5531, Tehran, Iran3 Department of Physics, Yazd University, Yazd, IranE-mail: Gjafari@gmail.com, pouria.pedram@gmail.com andleilahedayati@yahoo.com

Received 5 December 2006Accepted 4 April 2007Published 25 April 2007

Online at stacks.iop.org/JSTAT/2007/P04012doi:10.1088/1742-5468/2007/04/P04012

Abstract. We show that some of Bach’s pitch series can be considered asa stochastic process with scaling behaviour. Using the multifractal detrendedfluctuation analysis (MF-DFA) method, frequency series of Bach pitches havebeen analysed. In this view we find the same second-moment exponents (afterdouble profiling) in ranges (1.7–1.8) in his works. Comparing MF-DFA resultsof original series to those for shuffled and surrogate series, we can distinguishmultifractality due to long-range correlations and a broad probability densityfunction. Finally, we determine the scaling exponents and singularity spectrum.We conclude that the fat tail has more effect on the multifractality nature thanlong-range correlations.

Keywords: new applications of statistical mechanics

c©2007 IOP Publishing Ltd and SISSA 1742-5468/07/P04012+15$30.00

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Contents

1. Introduction 2

2. Multifractal detrended fluctuation analysis 52.1. Description of MF-DFA method . . . . . . . . . . . . . . . . . . . . . . . . 52.2. Relation to standard multifractal analysis . . . . . . . . . . . . . . . . . . 7

3. Analysis of music frequency series 8

4. Conclusion 13

Acknowledgments 14

References 14

Pythagoras knew it, but Bach demonstrated it: without mathematics there is nomusic [1].

1. Introduction

Many people think that mathematics and music have some vague sort of affinity, but mostoften the supposed relationship between the two fields turns out to be in details that arenot central to either. The mathematical proportions in musical works are hidden from thelistener from the old days. Thus we are forced to make use of interpretative techniquesin order to search for them, which is problematic from a methodological point of view.Some music historians have very little time for numerology. In addition to mathematicsbeing seen as numerical symbolism, music is closely linked to absolute physical entities,such as frequency and relation between intervals (an interval is the space between twonotes). Already in antiquity this was seen as a natural or cosmic premise on which musicrelied. It is an illustrated fact that not just musical notation, but also the relationshipbetween music and time, has something to do with mathematics and with one of themost significant transformations in music history. Complexity of music can especiallyattract scientific interest. Among a great variety of complex and disordered systems mostmusic parameters such as frequency and pitch (pitch is the sound frequency of any givennote) [2]–[6], amplitude or dynamics (dynamics are the changes in volume during a musicalpiece) [7, 8], intervals and rhythm (rhythm is the structure of the placement of notes inmusical time) can be considered as stochastic processes. Also, some authors try to clusterthe music [9, 10].

In all technicalities, music can be composed of notes. A note is a sign used in music torepresent the relative duration and pitch of sound. In traditional music theory pitch classesare represented by the first seven letters of the Latin alphabet (A, B, C, D, E, F and G)or Do–Re–Mi–Fa–Sol–La–Ti, in Italian notation. Each note is assigned a specific verticalposition on a staff. Since the physical causes of music are vibrations of mechanical systems,their frequencies are often measured in hertz (Hz). These frequencies are mathematically

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Figure 1. Typical sheet music, Invention no 1 by Bach.

related to each other, and are defined around the central note. The current ‘standardpitch’ for this note is 440 Hz. Any note is exactly an integer number of half-steps awayfrom the central note. Let this distance be denoted n. Then, the desired frequency isgiven by

ν = 440 × 2n/12 Hz. (1)

In this paper we would like to characterize the complex behaviour of thefrequency of the note signals of Bach’s Inventions and sinfonias through computationof signal parameters scaling exponents, which quantifies the correlation exponents andmultifractality of the signal. The inventions and sinfonias are a collection of short pieceswhich Bach wrote for the musical education of his young pupils. These are among thefinest examples of artistic gems ever written for this purpose, and probably because of thisthey have been very popular among Bach’s pupils and others ever since they were written.The inventions and sinfonias contain two and three music voices respectively. The numberof data in the frequency series is dependent on the piece; one voice contains 1000 notes onaverage. Because of the non-stationary nature of frequency of a music series, and due tothe finiteness of the available data sample, we should apply methods which are insensitiveto non-stationarities, like trends. In order to separate trends from correlations we needto eliminate trends from our frequency data. Several methods are used effectively for thispurpose: detrended fluctuation analysis (DFA) [11], rescaled range analysis (R/S) [12]and wavelet techniques (WTs) [13].

We use the MF-DFA method for analysis and eliminating trends from the data set.This method is a modified version of the DFA method to detect multifractal properties oftime series. The DFA method, introduced by Peng et al [11], has became a widely usedtechnique for the determination of (mono-) fractal scaling properties and the detection oflong-range correlations in noisy, non-stationary time series [14]–[17]. It has successfullybeen applied to diverse fields such as DNA sequences [11, 18], heart rate dynamics [19]–[21], neuron spiking [22], human gait [23], wind speed [24], long-time weather records [25],cloud structure [26], geology [27], ethnology [28], economical time series [29], and solidstate physics [30]. One reason to employ the DFA method is to avoid spurious detectionof correlations that are artefacts of non-stationarity in time series.

The focus of present paper is on the intriguing statistical properties and multifractalnature of frequency series. In particular, figures 1 and 2 show the score of Invention

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n

ν

100 200

800

1000

1200

1400

1600

1800

2000

n

ν

100 200

600

800

1000

1200

1400

1600

1800

2000

Figure 2. Up to down: typical frequency series of Invention no 1 and Sinfoniano 1 by Bach.

no 1 and the frequency fluctuation of Invention no 1 and Sinfonia no 1, respectively. Ingeneral, two different types of multifractality in frequency series can be distinguished.(i) Multifractality due to a fatness of the probability density function (PDF) of thetime series. In this case the multifractality cannot be removed by shuffling the series.(ii) Multifractality due to different long-range correlations in small- and large-scalefluctuations. In this case data may have a PDF with finite moments, e.g. a Gaussiandistribution. Thus corresponding shuffled time series will exhibit mono-fractal scaling,since all long-range correlations are destroyed by the shuffling procedure. If both kindsof multifractality are present, the shuffled series will show weaker multifractality than theoriginal series.

The paper is organized as follows. In section 2 we describe MF-DFA methods indetail and show that scaling exponents determined by the MF-DFA method are identicalto those obtained by standard multifractal formalism based on partition functions. Insection 3, in analysis of frequency series of Bach’s Inventions we also examine the sourceof multifractality in frequency data by comparison of MF-DFA results for the remainingdata set to those obtained by MF-DFA for shuffled and surrogate series. Section 4 closeswith a conclusion.

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2. Multifractal detrended fluctuation analysis

The simplest type of multifractal analysis is based upon standard partition functionmultifractal formalism, which has been developed for multifractal characterization ofnormalized, stationary measurements [31]–[34]. Unfortunately, this standard formalismdoes not give us correct results for non-stationary time series that are affected by trends orthose which cannot be normalized. In the early 1990s an improved multifractal formalismwavelet transform modulus maximum (WTMM) method [13] was developed. This methodis based on wavelet analysis and involves tracing the maximum lines in continuous wavelettransform over all scales. Other methods like multifractal detrended fluctuation analysis(MF-DFA) are based on identification of scaling of the qth-order moment depending onsignal length, and this is a generalization of the standard DFA method in which q = 2.In contrast to the WTMM method, MF-DFA does not require the modulus maximumprocedure, and hence does not require more effort in programming and computing timethan conventional DFA. On the other hand, one should find correct scaling behaviourof fluctuations from experimental data, which are often affected by non-stationary-liketrends. These have to be well distinguished from intrinsic fluctuations of system. Inaddition, often in collected data we do not know the reasons, or even worse the scales, forunderlying trends, and also usually available record data is small. So for reliable detectionof correlations, it is essential to distinguish trends for intrinsic fluctuations from collecteddata. Hurst rescaled-range analysis [12] and other non-detrending methods work wellwhen records are long and do not involve trends, otherwise they might give the wrongresults. DFA is a well established method for determining scaling behaviour of noisy datawhere the data include trends and we do not know their origin and shape [11, 20, 35, 36].

2.1. Description of MF-DFA method

The modified multifractal DFA (MF-DFA) procedure consists of five steps. The first threesteps are essentially identical to the conventional DFA procedure (see e.g. [11], [14]–[17]).Suppose that xk is a series of length N , and it is of compact support, i.e. xk = 0 for aninsignificant fraction of the values only.

• Step 1. Determine the ‘profile’

Y (i) ≡i∑

k=1

[xk − 〈x〉] , i = 1, . . . , N. (2)

Subtraction of the mean 〈x〉 from xk is not compulsory, since it would be eliminatedby later detrending in the third step.

• Step 2. Divide profile Y (i) into Ns ≡ int(N/s) non overlapping segments of equallengths s. Since the length N of series is often not a multiple of considered timescales, a short part at the end of the profile may remain. In order not to disregard thispart of the series, the same procedure should be repeated starting from the oppositeend. Thereby, 2Ns segments are obtained altogether.

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• Step 3. Calculate the local trend for each of 2Ns segments by a least-square fit of theseries. Then determine the variance

F 2(s, ν) ≡ 1

s

s∑

i=1

{Y [(ν − 1)s + i] − yν(i)}2 , (3)

for each segment ν = 1, . . . , Ns and

F 2(s, ν) ≡ 1

s

s∑

i=1

{Y [N − (ν − Ns)s + i] − yν(i)}2 , (4)

for ν = Ns + 1, . . . , 2Ns, where yν(i) is a fitted polynomial in segment ν. Linear,quadratic, cubic or higher order polynomials can be used in the fitting procedure(conventionally called DFA1, DFA2, DFA3, . . ., DFAm) [11, 21]. Since detrending oftime series is done by subtraction of the fitted polynomial from the profile, differentorder DFAs differ in their capability of eliminating trends in series. In (MF-)DFAmtrends of order m in profile (and equivalently order m − 1 in original series) areeliminated. Thus a comparison of results for different orders of DFA allows one toestimate the type of the polynomial trend in time series [15, 16].

• Step 4. Average over all segments to obtain the qth order fluctuation function, definedby

Fq(s) ≡{

1

2Ns

2Ns∑

ν=1

[F 2(s, ν)

]q/2

}1/q

, (5)

where, in general, variable q can take any real value except zero. For q = 2, thestandard DFA procedure is retrieved. Generally we are interested to know howgeneralized q dependent fluctuation functions Fq(s) depend on timescale s for differentvalues of q. Hence, we must repeat steps 2, 3 and 4 for several timescales s. It isapparent that Fq(s) will increase with increasing s. Of course, Fq(s) depends on theDFA order m. By construction, Fq(s) is only defined for s ≥ m + 2.

• Step 5. Determine the scaling behaviour of fluctuation functions by analysing log–log plots of Fq(s) versus s for each value of q. If series xi are long-range power lawcorrelated, then Fq(s), for large values of s, increases as a power-law, i.e.

Fq(s) ∼ sh(q). (6)

In general, exponent h(q) may depend on q. For stationary time series such asfractional Gaussian noise (fGn), Y (i) in equation (2) will have a fractional Brownianmotion (fBm) signal, so 0 < h(q = 2) < 1.0. The exponent h(2) is identical to thewell known Hurst exponent H [11, 14, 31]. Also for non-stationary signal, such as fBmnoise, Y (i) in equation (2) will be a sum of the fBm signals, so the correspondingscaling exponent of Fq(s) is identified by h(q = 2) > 1.0 [11, 37]. For a monofractaltime series, h(q) is independent of q, since the scaling behaviour of variance F 2(s, ν) isidentical for all segments ν, and the averaging procedure in equation (5) will just givethis identical scaling behaviour for all values of q. If we consider positive values of q,the segments ν with large variance F 2(s, ν) (i.e. large deviation from the correspondingfit) will dominate the average Fq(s). Thus, for positive values of q, h(q) describes

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the scaling behaviour of segments with large fluctuations. For negative values of q,segments ν with small variance F 2(s, ν) will dominate the average Fq(s). Hence,for negative values of q, h(q) describes the scaling behaviour of segments with smallfluctuations.

2.2. Relation to standard multifractal analysis

For a stationary, normalized series, multifractal scaling exponent h(q) defined inequation (6) is directly related to scaling exponent τ(q) defined by the standard partitionfunction based on multifractal formalism as shown below. Suppose that series xk of lengthN is a stationary, normalized sequence, then the detrending procedure in step 3 of theMF-DFA method is not required, since no trend has to be eliminated. Thus, DFA canbe replaced by standard fluctuation analysis (FA), which is identical to DFA except thedefinition of variance, which is simplified for each segment ν = 1, . . . , Ns. In step 3equation (3) now becomes

F 2FA(s, ν) ≡ [Y (νs) − Y ((ν − 1)s)]2. (7)

Inserting this simplified definition into equation (5) and using equation (6), we obtain

{1

2Ns

2Ns∑

ν=1

|Y (νs) − Y ((ν − 1)s)|q}1/q

∼ sh(q). (8)

For simplicity we can assume that the length N of a series is an integer multiple of scales, obtaining Ns = N/s and therefore

N/s∑

ν=1

|Y (νs) − Y ((ν − 1)s)|q ∼ sqh(q)−1. (9)

This corresponds to the multifractal formalism used e.g. in [32, 34]. In fact, a hierarchyof exponents Hq similar to our h(q) has been introduced based on equation (9) in [32].In order to relate this to standard textbook box counting formalism [31, 33], we employthe definition of a profile in equation (2). It is evident that term Y (νs) − Y ((ν − 1)s) inequation (9) is identical to the sum of numbers xk within each segment ν of size s. Thissum is known as box probability ps(ν) in standard multifractal formalism for normalizedseries xk,

ps(ν) ≡νs∑

k=(ν−1)s+1

xk = Y (νs) − Y ((ν − 1)s). (10)

The scaling exponent τ(q) is usually defined via the partition function Zq(s),

Zq(s) ≡N/s∑

ν=1

|ps(ν)|q ∼ sτ(q), (11)

where q is a real as in the MF-DFA method, discussed above. Using equation (10) we seethat equation (11) is identical to equation (9), and one can obtain the analytical relation

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between two sets of multifractal scaling exponents,

τ(q) = qh(q) − 1. (12)

Thus, we see that h(q) defined in equation (6) for MF-DFA is directly related to classicalmultifractal scaling exponents τ(q). Note that h(q) is different from the generalizedmultifractal dimension

D(q) ≡ τ(q)

q − 1=

qh(q) − 1

q − 1, (13)

that is used instead of τ(q) in some papers. In this case, while h(q) is independentof q for a monofractal time series, D(q) depends on q. Another way to characterize amultifractal series is by looking at the singularity spectrum f(α), which is related to τ(q)via a Legendre transform [31, 33],

α = τ ′(q) and f(α) = qα − τ(q). (14)

Here, α is the singularity strength or Holder exponent, while f(α) denotes the dimensionof a subset of the series that is characterized by α. Using equation (12), we can directlyrelate α and f(α) to h(q),

α = h(q) + qh′(q) and f(α) = q[α − h(q)] + 1. (15)

The Holder exponent denotes monofractality, while in the multifractal case, different partsof the structure are characterized by different values of α, leading to the existence of thespectrum f(α).

3. Analysis of music frequency series

As mentioned in section 2, spurious correlations may be detected if the time series isnon-stationarity, so direct calculation of correlation behaviour, spectral density exponent,fractal dimensions etc does not give us reliable results. It can be checked out thatthe frequency series is non-stationary. One can verify the non-stationarity propertyexperimentally by measuring the stability of the average and variance in a movingwindow for example using scale s. According to the MF-DFA1 method, generalized Hurstexponents h(q) in equation (6) can be found by analysing log–log plots of Fq(s) versus s foreach q (see figure 3 for q = 2). The Hurst exponent is in the range 0 < H < 1. However,the MF-DFA method can only determine positive generalized Hurst exponents, in orderto refine the analysis near the fGn/fBm boundary or strongly anti-correlated signals whenthis exponent is close to zero. The simplest way to analyse such data is to integrate thetime series before the MF-DFA procedure. Hence, we replace the single summation inequation (1), which is describing the profile from the original data, by a double summationby using classification by the signal summation conversion method (SSC). After using theSSC method, fGn switch to fBm and fBm switch to sum-fBm. In this case the relationbetween the new exponent, h∗(q = 2), and h(q = 2) is h(q = 2) = h∗(q = 2) − 1 [37]–[39](recently Movahed et al have proven the relation between the derived exponent from thedouble profile of a series in the DFA method and the h(q = 2) exponent in the appendixof [38]). We find h∗(q = 2) = 1.74 ± 0.03 of Invention no 1 by using the SSC method.Note that the exponents of the new series, after double profiling (SSC), are different frombefore SSC. Therefore we name these new exponents h∗(q), τ ∗(q) and f ∗(α∗).

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s

F(s

)

50 100

103

104

105

h*(2)=

1.74

Figure 3. The log–log plot F (s) versus s for q = 2.0 after double profiling forInvention no 1.

Table 1. Values of h∗(q = 2) and τ∗(q = 2) exponents and width of the singularityspectrum f∗(α∗), Δα, for q = 2.0 of Invention no 1 obtained by MF-DFA1.

h∗(2) τ∗(2) Δα

Original 1.74 ± 0.03 2.48 ± 0.03 1.70Surrogate 1.71 ± 0.03 2.44 ± 0.03 0.80Shuffled 0.50 ± 0.03 0.02 ± 0.03 0.50

Results using the MF-DFA1 method for the frequency signal are shown in figure 4,which shows that the frequency series is a multifractal process as indicated by the strongq dependence of the exponents h∗(q = 2) and τ ∗(q) [39]. The q dependence of themultifractal scaling exponent τ ∗(q) has linear behaviours for q > 0 and q < 0 and theslopes of τ ∗(q) are 1.53± 0.01 and 2.61± 0.01, respectively. The values which are derivedfor the quantities of the MF-DFA1 method for Invention no 1 are given in table 1. We havecalculated the exponent h∗(q) for other inventions and sinfonias as well, and all of themare in the 1.7–1.8 range (figure 5). Figure 4(c) shows that the width of the singularityspectrum, f ∗(α∗), i.e. Δα = α∗(qmax)−α∗(qmin), for the series is approximately, 1.70. Thisvalue shows that the power of the multifractality of interevents is very strong [40]. Thevalues of derived quantities from the MF-DFA1 method are given in table 1. There aresome specific lengths in music that have important effects in music’s statistical parameters.(a) Rhythm. (b) The scale is a set of tones each having a definite pitch (perceivedfundamental frequency of a sound) and each having a specific frequency ratio compared tothe others, that is, each having specific interval relative to all the other pitches, arrangedin a sequence from low to high, or alternatively from high to low. Composers oftentransform musical patterns by moving every note in the pattern by a constant number ofscale intervals. Since the intervals of a scale can have various sizes, this process introduces

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q

h* (q)

-20 -10 0 10 20

1.5

2

2.5

3

(a)

q

τ∗

-20 -10 0 10 20-60

-50

-40

-30

-20

-10

0

10

20

30

(q)

(b)

α∗

f* ()

2 2.5

0.6

0.7

0.8

0.9

1

α∗

(c)

Figure 4. The q dependences of the exponents h∗(q) and τ∗(q) and singularityspectrum f∗(α∗), after double profiling, are shown in the upper to lower panelsrespectively for Invention no 1.

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n

h* (2)

10 20 301

1.2

1.4

1.6

1.8

2

Figure 5. The second-moment exponents, h∗(q = 2), after double profiling, ofvarious inventions and sinfonias in their number ordering.

subtle melodic and harmonic variation into the music. This variation is what gives scalarmusic much of its complexity. (c) The bar or measure is a segment of time defined as agiven number of beats (the basic time unit of a piece) of a given duration. Moreover, inPersian music (Eastern music) there are other characteristic lengths such as Gushe, whichis based on the change of the tonic or stop notes. The existence of these characteristiclengths can affect music complexity. The music styles and more importantly composerscan use various characteristic lengths in the music. We have checked a few pieces of othergenres of music like jazz and Persian traditional music (Eastern music) and obtained h∗(2)in the ranges (1.75–1.9) and (1.3–1.7), respectively. We find more characteristic lengthsin the Persian traditional music, which leads to a decrease of the exponents.

Usually in the MF-DFA method, deviation from a straight line in the log–log plotof equation (6) occurs for small scales s. This deviation limits the capability of DFAto determine the correct correlation behaviour for very short scales and in the regime ofsmall s. The modified MF-DFA is defined as follows [15]:

Fmodq (s) = Fq(s)

〈[F shufq (s′)]2〉1/2 s1/2

〈[F shufq (s)]2〉1/2 s′1/2

(for s′ � 1), (16)

where 〈[F shufq (s)]2〉1/2 denotes the usual MF-DFA fluctuation function, defined in

equation (5), averaged over several configurations of shuffled data taken from originaltime series, and s′ ≈ N/40. The values of the h∗(q) exponent obtained by modified MF-DFA1 methods for frequency time series is 1.70± 0.03. The relative deviation of the newexponent which is obtained by the modified MF-DFA1 in comparison to the MF-DFA1for original data is less than 5%.

Now, we are interested to determine the source of the multifractality. In general, twodifferent types of multifractality in time series can be distinguished. (i) Multifractality

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due to a fatness of the probability density function (PDF) of the time series. In this casethe multifractality cannot be removed by shuffling the series. (ii) Multifractality due todifferent correlations in small- and large-scale fluctuations. In this case, data may have aPDF with finite moments, e.g. a Gaussian distribution. Thus corresponding shuffled timeseries will exhibit mono-fractal scaling, since all long-range correlations are destroyed bythe shuffling procedure. If both kinds of multifractality are present, the shuffled series willshow weaker multifractality than the original series. The easiest way to clarify the type ofmultifractality is by analysing corresponding shuffled and surrogate time series. Shufflingof the time series destroys long-range correlation, therefore if the multifractality belongsonly to long-range correlation we should find h∗

shuf(q) = 1.5. The multifractality naturedue to the fatness of the PDF signals is not affected by the shuffling procedure. On theother hand, to determine multifractality due to the broadness of the PDF, the phase ofdiscrete Fourier transform (DFT) coefficients of the frequency time series are replacedwith a set of pseudo-independent distributed uniform (−π, π) quantities in surrogatemethod. The correlations in surrogate series do not change, but the probability functionchanges to a Gaussian distribution. If multifractality in the time series is due to a broadPDF, h∗(q) obtained by the surrogate method will be independent of q. If both kinds ofmultifractality are present in frequency time series, shuffled and surrogate series will showweaker multifractality than the original one.

To check the nature of the multifractality, we compare the fluctuation function Fq(s)for the original series (after cancellation of the sinusoidal trend) with the result from thecorresponding shuffled, F shuf

q (s), and surrogate series, F surq (s). Differences between these

two fluctuation functions and the original one directly indicate the presence of long-rangecorrelations or broadness of the probability density function in the original series. Thesedifferences can be observed in a ratio plot of Fq(s)/F

shufq (s) and Fq(s)/F

surq (s) with respect

to s [39]. Since anomalous scaling due to a broad probability density affects both Fq(s)and F shuf

q (s) in the same way, only multifractality due to correlations will be observed in

Fq(s)/Fshufq (s). The scaling behaviours of these ratios are

Fq(s)/Fshufq (s) ∼ sh∗(q)−h∗

shuf (q) = sh∗cor(q), (17)

Fq(s)/Fsurq (s) ∼ sh∗(q)−h∗

sur(q) = sh∗PDF(q). (18)

If only the fatness of the PDF is responsible for the multifractality, one should haveh∗(q) = h∗

shuf(q) and h∗cor(q) = 0. On the other hand, deviations from h∗

cor(q) = 0 indicatethe existence of correlations, and the q dependence of h∗

cor(q) indicates that multifractalityis due to long range correlation. If only correlation multifractality is present, one findsh∗

shuf(q) = 1.5. If both distribution and correlation multifractality are present, bothh∗

shuf(q) and h∗sur(q) will depend on q. The q dependence of exponent h∗(q) for original,

surrogate and shuffled time series are shown in figures 6. The q dependence of hcor andh∗

PDF shows that the multifractality nature of the frequency time series is due to both thebroadness of the PDF and long-range correlation. The absolute value of h∗

PDF(q) is greaterthan h∗

cor(q), so the multifractality due to correlation is weaker than the multifractalitydue to fatness.

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q

h(q)

-20 -10 0 10 200

0.5

1

1.5

2

2.5

3Original

Shuffled

Surrogate

*

Figure 6. The new exponent, h∗(q), after double profiling, as a function of q forthe original, surrogate and shuffled data.

Deviation of h∗sur(q) and h∗

shuf(q) from h∗(q) can be determined by using the χ2 testas follows:

χ2� =

N∑

i=1

[h∗(qi) − h∗�(qi)]

2

σ(qi)2 + σ�(qi)2, (19)

symbol ‘’ can be replaced by ‘sur’ and ‘shuf’, to determine the confidence level ofh∗

sur and h∗shuf to the new exponents, h∗(q), of the original series, respectively. The

reduced chi-squares χ2ν� = χ2

�N (N is the number of degrees of freedom) for shuffled and

surrogate time series are 16.07, 21.30, respectively. The widths of the singularity spectrumf ∗(α∗), i.e. Δα = α∗(qmin) − α∗(qmax), for original, surrogate and shuffled time series areapproximately 1.04, 0.65 and 0.92 respectively. These values conclude that multifractalitydue to the fat tail is dominant [40].

Values of the exponents h∗(q = 2.0) and τ ∗(q = 2) and the width of the singularityspectrum f ∗(α∗), Δα, for the original, shuffled and surrogate frequency series obtainedwith the MF-DFA1 method are reported in table 1.

4. Conclusion

The MF-DFA method allows us to determine multifractal characterization of non-stationary and stationary time series. We have shown the MF-DFA1 result of frequencyseries of Bach’s Inventions. Applying the MF-DFA1 method demonstrates that frequencyseries have long-term correlation. We calculated the second-moment exponent after thenew profile for other inventions and sinfonias and they are in the 1.7–1.8 range. The qdependence of h∗(q) and τ ∗(q) shows that the frequency series has multifractal behaviour.

doi:10.1088/1742-5468/2007/04/P04012 13

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Long-range correlation and multifractality in Bach’s Inventions pitches

By comparing the second-moment exponent of the original time series with shuffled andsurrogate ones, we have found that multifractality due to the broadness of the probabilitydensity function has more contribution than correlation in inventions.

Acknowledgments

We would like to thank M Sadegh Movahed for reading the manuscript and usefulcomments. GRJ would like to acknowledge the hospitality extended during his visitsat the IPAM, UCLA, where this work was started.

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