Automatic Music Composition based on HMM andIdentified Wavelets in Musical Instruments

M.S. Sinith, *K.V.V. MurthyDept. of Electronics and Communication Engineering

College of Engineering, Trivandrum, Indiasinith@ieee.org

Dept. of Electrical EngineeringIndian Institute of Technology Gandhinagar, India.

kvvm@iitgn.ac.in

Abstract—Automatic Music Composition plays a crucial role inthe musical research and can become a tool for the incorporationof artificial intelligence in computer musicology. This paper findsan efficient method for identifying the wavelets and filter bankcoefficients in musical instruments using NLMS algortithm andthe usage of these wavelets for Automatic music Compositionusing Hidden Markov Model. In this paper, a technique toidentify the scaling function and the wavelet functions of thewavelets present in musical instruments, violin and flute, ispresented. NLMS algorithm is used to identify the filter bankcoefficients of wavelet-like elements, found repeating in musicalnotes of the instruments. Pre-trained hidden markov models foreach raga of South Indian Music is used for the composition.The HMM selected has twelve states which represent the twelvenotes in South Indian music. Fundamental frequency trackingalgorithm, followed by quantization is done. The resulting se-quence of frequency jumps of different musical clips of samemusical pattern (Raga) is presented to Hidden Markov Model ofa particular Raga for training. The HMM model of that Ragaalong with the filter coefficient is used to regenerate a piece ofmusic in that particular raga. The methodology is tested in thecontext of South Indian Classical Music, using the wavelet ofclassical music intruments, Flute and Violin.

I. INTRODUCTION

The earliest and most well known survey of digital signalprocessing techniques for the production and processing ofmusical sounds was authored by [1] in 1976. Wavelets forman integral part of digital signal processing today. Analysis ofmusical notes of different musical instruments shows that eachinstrument is associated with a unique wavelet [2]. In widelyadopted Fourier representation in signal processing, one canget the features of the signal either in Time Domain or inFrequency Domain, one at a time. Both are extensively used inanalysis, design and various other applications. However thereare many instances in which the localization in time as wellas localization in frequency, both are required simultaneously.Short duration signals need to be localized in time and smallbandwidth signals localized in frequency. In musical signals,small duration signals or small bandwidth musical pieces areplaced at an effective temporal position to give special effects.They need to be captured in time as well as in frequencysimultaneously. Fourier representation is not suited for such re-quirements. Wavelet transform of finite energy can effectivelycapture the above requirement. Wavelet transform replaces

Fourier Transforms sinusoidal waves by a family generatedby translations and dilations of finite energy signals calledwavelet. Wavelets are manipulated in two ways. The first oneis translation where the position of the wavelet is changedalong the time axis. The second one is scaling. Scaling meanschanging the frequency of the signal. In wavelet analysis, twospecial functions: the (1) wavelet function and (2) the scalingfunction are used. They have unique expressions:

ψ(t) = Σg(n)φ(2t− n) (1)φ(t) = Σh(n)φ(2t− n) (2)

where, g(n) = (−1)1−nh(1− n) (3)

II. FILTER BANK THEORY

As studies reveal wavelets can be generated from a set oftransfer functions H(z) and G(z). A wavelet can be recon-structed from its approximation and detail coefficients [3]. Thewavelet acts as a bank of band pass filters [4]. Most of thewavelet applications are dealt with the coefficients h(n) andg(n) from (1) and (2). These filters form Quadrature MirrorFilters(QMF) whose spectra are mirror images of each other.The filter formed by the mother wavelet acts as constant-Qfilters. Qfactor is given by:

Qfactor = centrefrequency/bandwidth (4)

QMF filters can be used for the filter bank implementation ofDiscrete Wavelet Transform(DWT).The filters form the highpass and low pass filter segment of the QMF filter that givesthe approximation and detailed coefficients of the signal. Theapproximation coefficients obtained at the output of low passfilter is down sampled and passed through another pair ofQMF filters and the process of decomposition continues. Wecan reconstruct the wavelet by using the same QMF filtersby giving the approximation and detail coefficients as filterinputs. In this paper, a new algorithm is proposed to find outthe h(n) and g(n) in an iterative manner. Once h(n) is obtained,g(n) can be obtained using (3). The Scaling function of thestandard wavelets are fed into the Normalized Least MeanSquare(NLMS) algorithm. The role of an adaptive filter is tofind the filter coefficients that give minimum error. The blockdiagram of the adaptive filter is given in fig1. Our algorithm

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Fig. 1. NLMS filter

suggests that by up-sampling the estimated signal obtained atoutput end of the adaptive filter and giving this new signal asinput and the original signal as desired signal, a new set offilter weights can be obtained that would give minimum error.This result however is accomplished by repeating the abovesteps iteratively for many times. This set of filter weights helpto reconstruct the h‘(n) with minimum error. This processwas carried out with all the standard wavelets. For all thesewavelets, with this set of filter weights, their respective h‘(n)and g‘(n) were successfully reconstructed. Thus the waveletand scaling functions of each wavelet for flute and violin isobtained using this algorithm.

III. NORMALIZED LEAST MEAN SQUARE ALGORITHM

In designing an FIR adaptive filter [5], [6] the goal is tofind the weights of filter, wn, at time n that minimizes thequadratic equation, i.e. the prediction error:

ζ(n) = E[e2(n)] (5)

The vector that minimizes ζ(n) may be found by setting thederivates of ζ(n) with respect to w*(k) equal to zero. Theweight-vector update equation used here is

wn+1 = wn + µE[e(n)x∗(n)] (6)

An LMS adaptive filter having p+1 coefficients requirep+1 multiplication and p+1 addition to update the filtercoefficients. In addition, it is necessary to compute the errorSince wn is a vector of random variables, the convergence ofthe LMS algorithm [7] must be considered within a statisticalframework. The main disadvantage of the LMS algorithm isthat it is very sensitive to the changes in the input signal . Asa result, it is very difficult to find an optimum step size orconvergence parameter , which guarantees convergence of thealgorithm, as well as minimizes the time of computation. Asuitable alternative is to use the Normalized LMS algorithm[8], which normalizes the LMS step size with the power ofthe input. When the input becomes too small, the NLMSalgorithm can be modified by adding a small positive value ε

to the power of the input signal.

wn+1 = wn + βe(n)x∗(n)ε+ x(n2)

(7)

In the new algorithm developed [2], h′(n) was tried to bereconstructed using LMS algorithm. Here the desired signalwas down sampled and then up sampled before being givento the filter. But it encountered the problem of sensitivity ofstep size as mentioned above. The obtained h′(n) varied muchfrom the actual value as the step size is varied . This led usto modify our algorithm by incorporating NLMS algorithm.

IV. HIDDEN MARKOV MODEL

Hidden Markov models (HMMs) are mathematical modelsof stochastic processes, i.e. processes which generate randomsequences of outcomes according to certain probabilities. Asimple example of such a process is a sequence of coin tosses.More concretely, an HMM is a finite set of states, each ofwhich is associated with a (generally multidimensional) prob-ability distribution. Transitions among the states are governedby a set of probabilities called transition probabilities. In aparticular state, an outcome or observation can be generated,according to the associated probability distribution. It is onlythe outcome not the state that is visible to an external observer.So states are hidden and hence the name hidden Markovmodel.

In order to define an HMM completely, the followingelements are neededThe number of states of the model, NThe number of observation symbols in the alphabet, M.A set of state transition probabilities

A = {aij}

aij = P{qt+1 = j/qt = i}, 1 ≤ i, j ≤ N

where qt denotes the current state.A probability distribution in each of the states,

B = {bjk}

bjk = P{αt = vk/qt = j}, 1 ≤ j ≤ N, j ≤ N, 1 ≤ k ≤M

where vk denotes the kth observation symbol in the alphabetand t the current parameter vector.The initial state distribution,

π = {πi}

where πi = p(q1 = i), 1 ≤ i ≤ NThus, an HMM can be compactly represented as

λ = {A,B, π}

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Hidden Markov models and their derivatives have been widelyapplied to speech recognition and other pattern recognitionproblems [9]. Most of these applications have been inspired bythe strength of HMMs, ie the possibility to derive understand-able rules, with highly accurate predictive power for detectinginstances of the system studied, from the generated models.This also makes HMMs the ideal method for solving Ragaidentification problems.

A. Selection of Hidden Markov Model

The HMM used in our solution is significantly differentfrom that used in, say word recognition. In the HMM, used inour context [9], each note in each octave represents one state inλ . Thus, the number of states actually required is N=12x3=36(Here, we are considering the three octaves of Indian classicalmusic, namely the Mandra, Madhya and Tar Saptak, each ofwhich consist of 12 notes) as used in [6]. Instead we havefirst mapped the frequencies in the Mandra and Tar Saptakinto the corresponding frequencies in the Madhya. Hence thetotal number of states for the HMM is N=12. HMM in [10] isbased on the frequency difference, where the total number ofstates needed for the HMM is very high compared to the HMMwhose states are taken on the basis of absolute frequencies.Sowe have adopted absolute frequency as the basis for HMM.

The transition probability A = {aij} represents the proba-bility of note j appearing after note i in a note sequence ofthe Raga represented by λ.

The initial state probability π = {πi} represents the proba-bility of note i being the first note in a note sequence of theRaga represented by λ .

The outcome probability B = {bij} is set according to thefollowing formulaBij = 0,∀i 6= j,Bij = 1,∀i = j.

The last condition takes the hidden character away from, butit can be argued that this setup suffices for the representationof Ragas, as our solution distinguishes between performancesof distinct raga on the basis of the exact order of notes sungin them and not on the basis of the embellishments used.Thus,at each state α in λ, the only possible outcome is note α.

V. RESULTS

The filter bank coefficient of the standard wavelets db2was found iteratively using the NLMS algorithm as in TableI. With this algorithm the wavelet and scaling functionswere reconstructed successfully. New scaling functions andwavelet functions for two music instruments-violin andflute are found. The results are shown in Table II. Therepeating elements were extracted from the classical musicsignals played. Average of these signals were found out withrespect to each instrument. With this average being fed tothe algorithm defined, scaling and wavelet functions of eachinstrument were obtained. The scaling and wavelet functionsare shown in figures [2-5]. The Composition of music isautomatically done using the probability values of the HMM

Fig. 2. Scaling function for violin

Fig. 3. Wavelet function for violin

obtained after training the model with several musical clipsof same Raga. Each required note of the 12 notes in SouthIndian Classical music is obtained by Upsampling and thendownsampling to get the desired pitch and these waves areconcatenated to get the Composition. The HMM of a typicalRaga Kalyani obtained is shown in table III. If the waveletof flute is replaced by Violin, beautiful Composition of Violinis also obtained.

Fig. 4. Scaling function for Flute

.

Acknowledgments

First Author Sinith thanks all his friends who have givenvaluable information relating to the work. we also thank

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TABLE IOBTAINED FILTER COEFFICIENT USING NLMS

Actual Filter Coefficients h(n) Step-Size Obtained filter coefficients[0.683, 1.183, 0.317,−0.183] 0.3 [0.682, 1.666, 0.3172,−0.1801][0.683, 1.183, 0.317,−0.183] 0.7 [0.683, 1.183, 0.317,−0.183][0.683, 1.183, 0.317,−0.183] 1.2 [0.68291.1830.317− 0.183][0.683, 1.183, 0.317,−0.183] 1.9 [0.726, 1.994, 0.329,−0.1802]

TABLE IIFILTER COEFFICIENTS FOR FLUTE AND VIOLIN

Instrument F ilterCoefficientViolin [0.0388, 0.0640, 0.0777, 0.0839, 0.0865, 0.0871, 0.0878, 0.0902, 0.0955, 0.1017, 0.1088, 0.1139, 0.1175, 0.1191, 0.1191, 0.1201, 0.1233, 0.1283, 0.1344, 0.1402]Flute [0.0833, 0.1249, 0.1458, 0.1562, 0.1614, 0.1640, 0.1654, 0.1660, 0.1663, 0.1665, 0.1666, 0.1666, 0.1666, 0.1667, 0.1667, 0.1667, 0.1667, 0.1667, 0.1667, 0.1667]

TABLE IIIHMM FOR A TYPICAL Raga

states 1 2 3 4 5 6 7 8 9 10 11 121 0.3548 0 0 0.2258 0.0323 0.0645 0.0323 0.129 0.1613 0 0 02 0.6667 0.1667 0 0 0 0.1667 0 0 0 0 0 03 0 0 0 0 0 0 1 0 0 0 0 04 0.0833 0.3333 0 0.1667 0.0833 0.1667 0.0833 0 0.0833 0 0 05 0.3333 0.3333 0 0 0.3333 0 0 0 0 0 0 06 0 0 0.1667 0.5 0.1667 0.1667 0 0 0 0 0 07 0 0 0 0 0 0.5 0.5 0 0 0 0 08 0.1 0 0 0 0 0 0 0.6 0.3 0 0 09 0.4444 0 0.1111 0 0 0 0.1111 0.2222 0.1111 0 0 0

10 0 0 0 0 0 0 0 0 0 1 1 111 0 0 0 0 0 0 0 0 0 1 1 112 0 0 0 0 0 0 0 0 0 1 1 1

Fig. 5. Wavelet function for Flute

Mr.Thor, University of Sussex, England for the valuable in-sight given by him during the work.

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