Harmonic and Timbre Analysis of Tabla Strokes

Indian twin drums mainly bayan and dayan (tabla) are the most important percussion instruments in India popularly used for keeping rhythm. It is a twin percussion/drum instrument of which the right hand drum is called dayan and the left hand drum is called bayan. Tabla strokes are commonly called as `bol', constitutes a series of syllables. In this study we have studied the timbre characteristics of nine strokes from each of five different tablas. Timbre parameters were calculated from LTAS of each stroke signals. Study of timbre characteristics is one of the most important deterministic approach for analyzing tabla and its stroke characteristics. Statistical correlations among timbre parameters were measured and also through factor analysis we get to know about the parameters of timbre analysis which are closely related. Tabla strokes have unique harmonic and timbral characteristics at mid frequency range and have no uniqueness at low frequency ranges. Introduction Among the percussion instruments, ‘tabla’ is one of the most important musical instruments in India. Tabla plays an important role in accompanying vocalists, instrumentalists and dancers in every style of music from classical to light in India, mainly used for keeping rhythm. The 'right hand' drum, called the dayan (also called the dahina, or the tabla) is a conical (almost cylindrical) drum shell carved out of a solid piece of hard wood. The 'open' end is covered by a composite membrane. The 'left hand' drum, called the bayan (also called the duggi) is a hemispherical bowl shaped drum made of polished copper, brass, bronze, or clay. Both of them have an 'open' end, covered by a composite membrane. The drum head (puri), is unique, and is made of goatskin, the lao. There is a weight in the middle, the syahi or gub. The syahi is perfect circle, in the middle of the puri, it is a semi-permanent paste made of coal dust, iron fillings, and rice paste. Around the outside of the puri, is a ring of thicker skin, this is called the chanti, this is not attached to the lao. The puri is laced by buffalo skin straps, baddhi, and tensioned by round wooden ‘chocks’, called gittak. As an accompanying instrument, Tabla serves the purpose of keeping rhythm by repeating a theka (beat-pattern) and adorns the vocal/instrumental music that it is accompanying. In this music, the choice of strokes is precise, each one functioning like a note in a melody; the timbral and rhythmic structures are equally important and carefully integrated into a singing line. Tabla strokes are typically inharmonic in nature but strongly pitched resonant strokes (Raman 1934). The sounds of most drums are characterized by strongly inharmonic spectra; however tablas, especially the dayan are an exception. This was pointed out as early as 1920 by C. V. Raman and S. Kumar. Raman further refined the study in a later paper (Ghosh R N, 1922, Raman C V, 1934, Rao K N, 1938). Thereafter several theoretical and experimental studies were held on the dynamics of the instrument (Ramakrishna B S, 1957, Sarojini T et. al, 1958, Banerjee B M et. al, 1991, Courtney D, 1999). The classical model put forth by Raman represents the sound of tabla-dayan, as having a spectrum consisting of five harmonics; these are the fundamental with its four overtones (Courtney D, 1999). Here we studied the timbre characteristics of tabla strokes. Strokes chosen for analysis Tabla playing has a very well developed formal structure and an underlying "language" for representing its sounds. A tabla `bol's constitutes a series of syllables which correlate to the various strokes of the tabla. Here we have considered nine tabla strokes. Stroke ‘Ta/Na’ executes by lightly pressing the ring finger down in order to mute the sound while index finger strikes the edge. Stroke ‘Ti’ executes by striking the dayan on the 2 nd circle with the index finger and by keeping the finger on that position causes more damping but after striking if the index finger release quickly to give an open tone it produces ‘Teen’. Stroke ‘Ghe’ executes by striking the bayan with middle and index finger keeping the wrist on the membrane but after striking if released quickly it produces ‘Ge’. Stroke ‘Thun’ executes by striking on the centre circle of dayan with index, middle, ring and little fingers together and by quickly releasing. Stroke ‘Tu’ executes by striking at the corner of centre circle of dayan with index finger only and immediately after striking finger will lift. Stroke ‘Te’ executes by striking the dayan with middle and ring finger at the centre of the circle. Stroke ‘Re’ executes by striking the dayan with index finger at the centre of the circle and by keeping the finger on that position causes more damping. Experimental procedure All the strokes were played by eminent tabla players and the sound was recorded in a noise free acoustic room. Membrane of tabla 1, 2 and 3 have diameter 5”, tabla 4 has a diameter 5.5” and tabla 5 has a diameter 6”. We have 5x9 = 45 stroke signals. Each of these sound signals was digitized with sample rate of 44.1 kHz, 16 bit resolution and in a mono channel. All the sound samples are of same length. We used Long Term Average Spectrum (LTAS) for timbre analysis. A rigorous statistical analysis was done based on Principal Component Analysis and Varimax with Kaiser Normalization. Timbre analysis Timbre is defined in ASA (1960) as that quality which distinguishes two sounds with the same pitch, loudness and duration. This definition defines what timbre is not, not what timbre is. Timbre is generally assumed to be multidimensional, where some of the dimensions have to do with the spectral envelope, the time envelope, etc. Many timbre parameters have been proposed to encompass the timbre dimensions. Among all timbre parameters important parameters are irregularity, tristimulus1 (T1), tristimulus 2 (T2), tristimulus 3 (T3), odd and even parameters, spectral centroid and brightness (Park T H 2004, Grey J M 1977, Patranabis A 2011). Beside these we also measured pitch, attack time, difference in frequency and amplitude between two highest peaks of LTAS and average RMS power of each signals (Sengupta R. et. a., 2004). From the LTAS of each signal above mentioned timbre features were measured of which some are harmonic and some perceptual features. From figure 1 to 7 it is observed that stroke 'ta' for all tablas have low brightness hence this stroke possess lower energy for all tablas. Since this stroke executes by striking index finger at the edge and such process of stroke cause weak resonance in the cavity of tabla. Brightness of all other strokes is different for five tablas. So timbre variations are confirmed in five tablas. Brightness and hence energy of all the nine strokes are high for the 3 rd tabla. So it may be assumed that the resonance takes place in 3 rd tabla is the highest among other tablas. But brightness and hence energy of all the nine strokes are low for the 4 th and 5 th tablas. So it may be assumed that the resonance takes place in 4 th and 5 th tablas are the lowest among other tablas. Strokes 'thun', ‘ti’ and ‘re’ for all tablas have high centroid hence this stroke is of high pitched for all the tablas. Since these strokes executes by striking at the vicinity of the circle and such process of stroke cause strong resonance in the cavity of tabla. Centroid of all other strokes is different for five tablas. Tristimulus 1 for stroke ‘ta’ is high for 4 th and 5 th tablas, while tristimulus 2 for ‘tu’, ‘teen’ and ‘ghe’ are high for 4 th and 5 th tablas. Both the tablas show lower tristimulus 3. Besides these strokes all other strokes have lower fundamental and also less energized lower partials, energy pumps up at higher partials. Comparing all strokes it is observed that irregularity among partials are higher for tabla 3 and 4 for the strokes ‘ghe’, ‘tu’ and ‘teen’. No significant difference is observed in odd and even parameters. Stroke 'Ge' of tabla 1 is different from others viz. brightness and centroid both are low and stronger 2nd, 3rd and 4th harmonics and low irregularity. Other tablas show uniformity in timbre for the stroke 'Ge'. Strokes 'Ghe', 'tu','teen' and 'ta' of tabla 3 and 4 are different from others viz. brightness and centroid are too low and stronger 2nd, 3rd and 4th harmonics and higher irregularity, while other three tablas show uniformity and all these four strokes are free stroke. For stroke 'thun', tabla 3 and 4 differ from other tablas only in brightness i.e. its CG of amplitude. Stroke 'ti' of tabla 3 is different from others viz. brightness and centroid both are low, stronger 2nd, 3rd and 4th harmonics and higher irregularity. For strokes 're' and 'te', tabla 3 and 4 differs from other tablas only in irregularity, in which both the strokes are made at the centre circle and both are damped strokes. Damped strokes have higher brightness and spectral centroid than the free strokes. So this concludes the fact that style (nature and intensity) of strokes of player of 1 st tabla is different than others. Also style of strokes is similar for the players of tabla 3 and 4. From table 2, it is observed that tristimulus 2 (T2) and tristimulus 3 (T3) are highly correlated while tristimulus 1 (T1) is weakly correlated with T2 and T3. This concludes the fact that fundamental (corresponds to T1) of different tabla strokes are different. Also fundamental is weak compared to its harmonics, while mid and higher frequency partials behave similarly. Odd and even harmonics are equally proportionate and are highly correlated to each other and so tabla strokes are harmonically good to hear. T2 and T3 both have high correlation with irregularity and spectral centroid. This concludes the fact that high frequency partials have higher order of irregularity among partials. Also brightness (i.e. centre of gravity of amplitude) and spectral centroid (i.e. centre of gravity of frequency) are highly correlated. Fig. 1: Variation of brightness Fig. 2: Variation of centroid Fig. 3: Variation of tristimulus 1 Fig. 4: Variation of tristimulus 2 Fig. 5: Variation of tristimulus 3 Fig. 6: Variation of irregularity Fig. 7: Variation of odd parameter Fig. 8: Variation of even parameter Table 2. Correlation coefficients of various timbre parameters Correlations Between different Timbre parameters Brightn ess Tristim ulus1 Tristim ulus2 Tristi mulus 3 Odd param eter Even Paramet er Spectral Irregulari ty Spectral inharmo nicity Spectral Centroid pitch Attack time Average RMS power Diff in freq of 2 peaks Diff in amp of 2 peaks Brightn ess Pearson Correla tion 1 .472 -.572 .482 -.228 -.318 -.911 -.269 .851 -.169 .439 -.628 .111 .138 Sig. (2tailed) .238 .139 .226 .587 .442 .002 .519 .007 .689 .277 .095 .794 .744 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Tristim ulus1 Pearson Correla tion .472 1 -.793 .386 -.817 -.073 -.285 .034 .527 -.181 .177 -.046 .044 .167 Sig. (2tailed) .238 .019 .346 .013 .864 .495 .937 .179 .667 .674 .913 .918 .692 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Tristim ulus2 Pearson Correla tion -.572 -.793 1 -.868 .492 .328 .229 .144 -.687 -.055 -.430 -.185 .140 .044 Sig. (2tailed) .139 .019 .005 .215 .427 .586 .734 .060 .897 .288 .661 .741 .917 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Tristim ulus3 Pearson Correla tion .482 .386 -.868 1 -.079 -.439 -.115 -.247 .611 .231 .507 .317 -.247 -.203 Sig. (2tailed) .226 .346 .005 .853 .276 .785 .556 .108 .582 .199 .444 .555 .629 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Oddpar ameter Pearson Correla tion -.228 -.817 .492 -.079 1 -.516 .165 -.460 -.210 .498 .293 .145 .039 .057 Sig. (2tailed) .587 .013 .215 .853 .191 .696 .251 .617 .209 .481 .732 .927 .894 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 EvenPa rameter Pearson Correla tion -.318 -.073 .328 -.439 -.516 1 .147 .733 -.435 -.581 -.786 -.168 -.129 -.344 Sig. (2tailed) .442 .864 .427 .276 .191 .728 .039 .281 .131 .021 .691 .760 .403 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Spectra lIrregul arity Pearson Correla tion -.911 -.285 .229 -.115 .165 .147 1 .257 -.677 .260 -.177 .829 -.225 -.286 Sig. (2tailed) .002 .495 .586 .785 .696 .728 .538 .065 .534 .675 .011 .591 .493 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Spectra linharm onicity Pearson Correla tion -.269 .034 .144 -.247 -.460 .733 .257 1 -.234 -.794 -.227 -.013 -.553 -.353 Sig. (2tailed) .519 .937 .734 .556 .251 .039 .538 .577 .019 .588 .975 .155 .390 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Spectra lCentro id Pearson Correla tion .851 .527 -.687 .611 -.210 -.435 -.677 -.234 1 .003 .561 -.317 .093 .367 Sig. (2tailed) .007 .179 .060 .108 .617 .281 .065 .577 .994 .148 .444 .827 .371 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 pitch Pearson Correla tion -.169 -.181 -.055 .231 .498 -.581 .260 -.794 .003 1 .170 .368 .604 .135 Sig. (2tailed) .689 .667 .897 .582 .209 .131 .534 .019 .994 .687 .369 .113 .749 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 attackti me Pearson Correla tion .439 .177 -.430 .507 .293 -.786 -.177 -.227 .561 .170 1 .001 -.127 .028 Sig. (2tailed) .277 .674 .288 .199 .481 .021 .675 .588 .148 .687 .998 .764 .947 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 average RMSpo wer Pearson Correla tion -.628 -.046 -.185 .317 .145 -.168 .829 -.013 -.317 .368 .001 1 -.372 -.053 Sig. (2tailed) .095 .913 .661 .444 .732 .691 .011 .975 .444 .369 .998 .365 .900 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 diff_in _freq_o f_2_pe aks Pearson Correla tion .111 .044 .140 -.247 .039 -.129 -.225 -.553 .093 .604 -.127 -.372 1 .219 Sig. (2tailed) .794 .918 .741 .555 .927 .760 .591 .155 .827 .113 .764 .365 .602 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 diff_in _amp_ of_2_p eaks Pearson Correla tion .138 .167 .044 -.203 .057 -.344 -.286 -.353 .367 .135 .028 -.053 .219 1 Sig. (2tailed) .744 .692 .917 .629 .894 .403 .493 .390 .371 .749 .947 .900 .602 N 8 8 8 8 8 8 8 8 8 8 8 8 8 8 **. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed). Factor analysis for timbre parameters We are working with a huge number of timbre parameters but considering all the parameters simultaneously is not a good option as the parameters might be correlated among themselves. Hence to understand the underlying pattern, factor analysis has been undertaken. The essential purpose of factor analysis is to describe, if possible, the covariance relationships among many variables in terms of a few underlying, but unobservable, random quantities called factors. Suppose variables can be grouped by their correlations i.e., suppose all variables within a particular group are highly correlated among themselves but have relatively small correlations with variables in different group. Then it is conceivable that each group of variables represents a single underlying construct, or factor, that is responsible for the observed correlations and chooses a variable from each group if possible for data reduction. Here the whole data set including 14 parameters of timbre can be classified into 5 underlying factors, which can explain 93.874 % of the total variation in the dataset. For factor analysis, we have used the varimax orthogonal rotation procedure through Principle Component Analysis and have considered five underlying factors. Factors produced in the initial extraction phase are often difficult to interpret. This is because the procedure in this phase ignores the possibility that variables identified to load on or represent factors may already have high loadings (correlations) with previous factors extracted. This may result in significant cross-loadings in which many factors are correlated with many variables. This makes interpretation of each factor difficult, because different factors are represented by the same variables. The rotation phase serves to “sharpen” the factors by identifying those variables that load on one factor and not on another. The ultimate effect of the rotation phase is to achieve a simpler, theoretically more meaningful factor pattern. The size of the factor loadings (correlation coefficients between the variables and the factors they represent) will help in the interpretation. As a general rule, variables with large loadings indicate that they are representative of the factor, while small loadings suggest that they are not. It should be kept in mind that negative factor loading implies negative correlation with the underlying factor and the other loadings. For example, in the rotated component matrix T2 has a negative loading while T3 has a positive loading (.747), which means that T2 and T3 are negatively correlated and also T2 has a negative correlation with the underlying Factor 1. Table 3: Descriptive Statistics Mean Std. Deviation Brightness 12.350850 1.9059274 Tristimulus1 .017563 .0146337 Tristimulus2 .079800 .0272315 Tristimulus3 .902487 .0179607 Odd parameter .479525 .0171888 Even Parameter .502688 .0097811 Spectral Irregularity .134263 .0324775 Spectral inharmonicity -.986013 2.4384462 Spectral Centroid 24.425163 .1749130 pitch 259.041650 54.3488050 Attack time .012350 .0022078 Average RMS power -12.865000 6.8755000 Diff in freq of 2 peaks 303.041675 111.4243042 Diff in amp of 2 peaks 8.029175 4.8769795 This is the table for descriptive statistics for all the timbre parameters. Table 4: table of communalities of timbre parameters Communalities Initial Extraction Brightness 1.000 .987 Tristimulus1 1.000 .921 Tristimulus2 1.000 .996 Tristimulus3 1.000 .919 Odd parameter 1.000 .988 Even Parameter 1.000 .941 Spectral Irregularity 1.000 .959 Spectral inharmonicity 1.000 .893 Spectral Centroid 1.000 .891 pitch 1.000 .986 Attack time 1.000 .784 Average RMS power 1.000 .982 Diff in freq of 2 peaks 1.000 .919 Diff in amp of 2 peaks 1.000 .976 Extraction Method: Principal Component Analysis. This is the table of communalities which shows how much of the variance in the variables has been accounted for by the extracted factors. For instance 99.6% of the variance in Tristimulus2 is accounted for while 78.4% of the variance in Attack time is accounted for. Table 5 : Explanation of variance Total Variance Explained Component Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative % 1 4.736 33.828 33.828 4.736 33.828 33.828 3.326 23.756 23.756 2 3.349 23.918 57.746 3.349 23.918 57.746 3.029 21.638 45.395 3 2.520 18.001 75.747 2.520 18.001 75.747 2.962 21.158 66.553 4 1.520 10.858 86.605 1.520 10.858 86.605 2.548 18.203 84.755 5 1.018 7.269 93.874 1.018 7.269 93.874 1.277 9.119 93.874 6 .546 3.902 97.776 7 .311 2.224 100.000 8 2.801E016 2.001E015 100.000 9 1.792E016 1.280E015 100.000 10 1.948E017 1.392E016 100.000 11 -5.120E017 -3.657E016 100.000 12 -7.063E017 -5.045E016 100.000 13 -2.223E016 -1.588E015 100.000 14 -4.431E016 -3.165E015 100.000 Extraction Method: Principal Component Analysis. The above table shows all the factors extractable from the analysis along with their eigenvalues, the percentage of variance attributable to each factor, and the cumulative variance of the factor and the previous factors. Notice that the first factor accounts for 33.828 % of the variance, the second 23.918 % of the variance, the third 18.001%, the fourth 10.858% and the fifth 7.269%. All the remaining factors are not significant. Fig 9: Scree plot of eigenvalues The scree plot is a graph of the eigenvalues against all the factors. The graph is useful for determining how many factors to retain. For the first five factors the Eigen values are greater than 1 and explains significant amount of variance. Hence we take the first five factors in account for further analysis. Table 6: Component factor in rotated component matrix Rotated Component Matrix a Component/ Factor 1 2 3 4 5 Brightness -.813 Tristimulus1 .933 Tristimulus2 -.538 -.829