Experimental investigations of t¯anpur¯aacoustics

Rahul Pisharody and Anurag Gupta Department of Mechanical Engineering, Indian Institute of Technology Kanpur, 208016, . [email protected]

44 1 Summary shown in the bottom-most picture in the right side of Figure 1. The purpose of this brief note is to present 45 2 High-speed video camera recordings are used to ob- certain experimental results which elucidate the na- 46 3 serve dynamics of an actual t¯anpur¯astring. The tem- ture of t¯anpur¯asound while emphasizing the role of 47 4 poral evolution of the spectrum is obtained j¯ıv¯a. 48 5 by measuring the force during the string vibra- We use high-speed video camera recordings of the 49 6 tion. The characteristic sonorous of t¯anpur¯ais vibration of a single t¯anpur¯astring to capture the 50 7 attributed to not only the presence of a large num- string motion close to the and at the nut (see 51 8 ber of but also to the dominance of certain the videos provided as supplementary material). The 52 9 over the fundamental, the latter manifest- latter is used to measure the nut force and to subse- 53 10 ing itself as a certain cascading effect. The nature of quently plot 3-dimensional . The previ- 54 11 sound is shown to be strongly dependent on the ini- ous t¯anpur¯aexperimental measurements were based 55 12 tial plucking of the string. The stability either on the audio signals [7, 8] or the sensors placed 56 13 of the in-plane vertical motion of the string is also between the string and the nut [9]. Our experimen- 57 14 emphasised. tal setup provides a novel visualisation of string–j¯ıv¯a– 58 bridge interaction in an actual t¯anpur¯aand can be 59 used to further the scientific study of the musical in- 60 15 1 Introduction strument. In the present paper, we use the nut force 61 measurements to establish a cascading effect of en- 62 16 T¯anpur¯a(or tamb¯ur¯a)is an unfretted long-necked ergy transfer between lower and higher overtones, re- 63 17 lute, with four strings, used exclusively for provid- sulting in a strong presence of certain overtones, as 64 18 ing the drone in Indian classical . The purpose 19 of drone is to establish a firm basis for a 20 musical performance by constantly playing a particu- 21 lar note or a set of notes. The sound of a well tuned 22 t¯anpur¯a,and hence the resulting drone, is remark- 23 ably rich in overtones and creates a pleasant “melodic 24 background” for the performance [1, 2]. The t¯anpur¯a 25 drone is widely recognized for enhancing the musical- 26 ity of the r¯agabeing played by constantly reinstating 27 the notes which form the essence of the r¯aga[3]. The 28 distinctiveness of t¯anpur¯a’ssound is due to the unique 29 manner in which the strings interact with the sound- 30 board [1, 4]. The strings pass over a doubly-curved 31 bridge of finite width before reaching the board, see 32 Figure 1, as is the case with other Indian string instru- 33 ments such as sit¯arand rudra v¯ın. ¯a[4]. The curved 34 bridge provides a unilateral constraint to the vibrat- 35 ing string and, in doing so, becomes the source for an 36 rich sound of the musical instrument [5, 6]. 37 The characteristic feature of t¯anpur¯ais however the 38 cotton threads, known as j¯ıv¯a(meaning the “soul”), 39 which are inserted below the strings on the bridge, see Figure 1: (Colour online) A typical mir¯aj style 40 Figure 1. The correct placement of j¯ıv¯a,essential for t¯anpur¯a(left); (right top to bottom) side view of the 41 obtaining the required drone from a t¯anpur¯a,is when bridge with j¯ıv¯a,top view, close side view of the bridge 42 the string makes a grazing contact with the bridge with one string and j¯ıv¯a. 43 before going over the neck of the instrument [1, 4], as Pisharody and Gupta, p. 2

65 a distinguished feature of the t¯anpur¯a’sovertone rich the nut force (normalized with respect to string ten- 106 66 sound. In the absence of j¯ıv¯a,the overtones decay sion) is shown in Figure 2. One can clearly observe 107 67 faster, the fundamental remains the dominant fre- a high-frequency precursive , as reported earlier 108 68 quency, and there is no cascading effect. The ob- by Valette [9, 16]. The presence of precursive wave 109 69 served richness of overtones, their slow decay, and the validates the role of dispersion in t¯anpur¯astring vi- 110 70 energy cascade effect are in fact confirmation of the bration and hence of incorporating bending rigidity 111 71 findings in previous experimental [8, 9] and numerical even for small vibration . A video capture 112 72 [10, 11, 12, 13, 14, 15] work on t¯anpur¯a. of the string motion close to the nut is provided as a 113 73 We demonstrate the dependence of t¯anpur¯a’s supplement (video1.gif). The data was acquired from 114 74 sound, as evident from the spectrograms, on the ini- the video experiments using MATLAB’s (version 9.0, 115 75 tial plucking amplitude of the t¯anpur¯astring. We R2016) inbuilt image processing toolbox. The fre- 116 76 show that the effect of j¯ıv¯ain producing a desirable quency evolution of the nut force was plotted using the 117 77 sound is lost for high plucking amplitudes, as is ex- function, where the sampling frequency 118 78 pected from the actual practise of t¯anpur¯aplaying. was taken to be the same as in the video experiments. 119

79 We also establish that an arbitrary (out-of-plane) ini- We begin by comparing the spectrograms obtained 120 80 tial pluck of the string eventually stabilises into an with and without the j¯ıv¯a. The 3-dimensional spec- 121 81 in-plane vertical motion. None of the previous works trograms are shown in Figure 3. The presence of 122 82 have discussed the effect of initial plucking amplitude j¯ıv¯a,when positioned appropriately, not only brings 123 83 and the stability of the in-plane vertical motion of the out a richer set of overtones but is clearly marked 124 84 string. by a definite change in the pattern of how overtones 125 evolve over time as well as how they interact with 126 each other. The interaction among overtones is clearer 127 85 2 Experimental results in Figure 4 where the three important signatures of 128 t¯anpur¯asound are distinctively visible. First, there is 129 86 The experimental setup consisted of a t¯anpur¯a,with a characteristic reoccurring pattern of energy trans- 130 87 all but one string removed, two supports, to firmly fer leading to a cascading effect with higher overtones 131 88 hold the instrument, a high-speed camera, and DC giving way to immediately lower overtones. A car- 132 89 light sources. The string was plucked at the center toon of the effect is illustrated in Figure 5, where 133 90 using the index finger as is done in actual t¯anpur¯a the curves in green, red, violet, and blue represent 134 th th th th 91 playing. In doing so, the string experiences an initial the n , (n+1) , (n+2) , and (n+3) overtones, re- 135 92 vertical (in-plane) displacement of around 0.5 cm and spectively, for n ≥ 3. Second, in the presence of j¯ıv¯a, 136 93 a horizontal (out-of-plane) displacement of around 94 0.2 cm. The tension in the string was measured by 95 hanging a mass at the center of the string, observ- 96 ing the angle between the string on either side of the 97 mass, and using the principle of static equilibrium. 98 The string was consistently tuned to F-sharp (using 99 an electronic t¯anpur¯adrone), having a fundamental 100 frequency of around 92.8 Hz. The high-speed camera 101 was triggered manually and was adjusted to capture 102 10000 fps. The nut force was calculated as the verti- 103 cal component of the string tension at the nut using 104 the estimated tension value and the observed slope 105 of the string at the nut. The temporal evolution of (a)

0.04

0.03

0.02

0.01

0

−0.01

Normalized Transverse Force −0.02

−0.03

−0.04 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.2 Time (sec) (b)

Figure 2: Evolution of nut force, normalized with re- Figure 3: (Colour online) 3-dimensional nut force spect to string tension. spectrograms (a) without and (b) with the j¯ıv¯a. Pisharody and Gupta, p. 3

137 the fundamental decays faster than many of the over- 138 tones so much so that it is completely overshadowed 139 after a short initial span of time. This is in contrast 140 to the situation without j¯ıv¯awhere the fundamental 141 remains the dominant frequency and the contribution psd 142 of the higher overtones remains low in comparison. 143 The coupling of various modes, and therefore of the 144 overtones, is also present in this case due to the wrap- 145 ping/unwrapping motion of the string on the bridge

146 [5, 6]. With j¯ıv¯a,the interaction of the string with Time 147 the bridge becomes more complex as it leads to a 148 more impactful collision of the string on the bridge. Figure 5: (Colour online) A cartoon of the reoccurring 149 This is clearly visible in the video recordings, pro- pattern in the evolution of psd for various overtones 150 vided as supplementary files, of the string interacting of a t¯anpur¯astring. 151 with the bridge in both the cases (see video2.gif and

152 video3.gif). Third, the presence of j¯ıv¯aclearly slows −4 fundamental 153 −5 first overtone down the decay of overtones thereby adding to the second overtone −6 third overtone 154 richness of t¯anpur¯asound. Finally, we report the psd fourth overtone −7 fifth overtone 155 evolution as obtained from the numerical simulation sixth overtone −8 seventh overtone 156 based on the recently proposed penalty based models eighth overtone −9

log(P) (dB/Hz) ninth overtone 157 [10, 11, 12, 13, 14, 15], see Figure 6 (the details of the tenth overtone −10 eleventh overtone 158 twelfth overtone numerical model and the choice of parameters can be −11 thirteenth overtone fourteenth overtone 159 −12 found is a recent thesis [17]). The numerical model is 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (sec) 160 clearly able to capture the cascading effect, the domi- 161 nance over the fundamental, as well as the slow decay Figure 6: (Colour online) Numerical simulation of psd 162 of the overtones. evolution for various overtones with the j¯ıv¯ausing a 163 The t¯anpur¯a drone is very sensitive to the ini- penalty based model. 164 tial plucking amplitude of the string. In fact, it is 165 commonly said among the musicians that a t¯anpur¯a 166 should be played such that the strings should not 167 know that they have been plucked. To support this 168 argument, we obtain spectrograms when the pluck- 169 ing amplitude is 2.5 cm (Figure 7(a)) and 1 cm (Fig-

−2 fundamental first overtone −3 second overtone third overtone −4 fourth overtone fifth overtone −5 sixth overtone seventh overtone (a) −6 eighth overtone

log(P) (dB/Hz) ninth overtone −7 tenth overtone eleventh overtone twelfth overtone −8 thirteenth overtone fourteenth overtone −9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (sec) (a)

−2 fundamental first overtone −3 second overtone third overtone −4 fourth overtone fifth overtone −5 sixth overtone seventh overtone −6 eighth overtone

log(P) (dB/Hz) ninth overtone −7 tenth overtone eleventh overtone (b) twelfth overtone −8 thirteenth overtone fourteenth overtone −9 Figure 7: (Colour online) 3-dimensional nut force 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (sec) spectrograms with initial plucking amplitudes of (a) (b) 2.5 cm and (b) 1 cm. Figure 4: (Colour online) Evolution of power (psd) for various overtones in the nut force ure 7(b)), in comparison to 0.5 cm used to generate 170 spectrogram (a) without and (b) with the j¯ıv¯a. the plot in Figure 4(b). The higher plucking ampli- 171 Pisharody and Gupta, p. 4

Supplementary material 199

(i) video1.gif: string motion close to the nut. 200 (ii) video2.gif: string motion close to the bridge with- 201 out j¯ıv¯a. 202 (iii) video3.gif: string motion close to the bridge with 203 j¯ıv¯a. 204

References 205 (a) [1] B. C. Deva. Tonal structure of tambura. The Journal 206 of the Music Academy, Madras, 27:89–112, 1956. 207

[2] A. Ranade. : A phenomenon. In Proceedings 208 of the ITC-SRA Workshop on Tanpura, pages 40–48, 209 1997. 210

[3] E. C. Carterette, K. Vaughn, and N. A. Jairazb- 211 hoy. Perceptual, acoustical and musical aspects of the 212 tamb¯ur¯adrone. Music Perception, 7:75–108, 1989. 213

[4] C. V. Raman. On some Indian stringed instruments. 214 Proceedings of the Indian Association for the Culti- 215 vation of Science, 7:29–33, 1921. 216 (b) [5] R. Burridge, J. Kappraff, and C. Morshedi. The 217 Figure 8: Out-of-plane motion of the t¯anpur¯astring, string, a vibrating string with a one-sided inelastic 218 with and without j¯ıv¯a,for (a) an in-plane initial pluck- constraint. SIAM Journal of Applied Mathematics, 219 ing and (b) an out-of-plane initial plucking of the 42:1231–1251, 1982. 220 string. [6] A. K. Mandal and P. Wahi. Natural , 221 mode shapes and modal interactions for strings vi- 222 brating against an obstacle: Relevance to sitar and 223 . Journal of Sound and Vibration, 338:42–59, 224 172 tudes are given by pulling the string vertically at the 2015. 225 173 center using a thread and then burning the thread. 174 The desired pattern, and hence the desired drone, dis- [7] H. V. Modak. Physics of tanpura: Some investiga- 226 175 appears as the amplitude goes above 0.5 cm. We have tions. In Proceedings of the ITC-SRA Workshop on 227 Tanpura, pages 8–15, 1997. 228 176 observed this conclusion to remain valid for different 177 plucking positions on the string. [8] R. Sengupta, B. M. Banerjee, S. Sengupta, and 229

178 Finally, we present some results on the out-of-plane D. Nag. Tonal qualities of the Indian tanpura. In 230 Proceedings of the Stockholm Music Acoustic Confer- 231 179 motion of the t¯anpur¯astring. The camera is now ence, pages 333–342, 1983. 232 180 placed in the direction of the wire and a fixed point 181 on the string is marked and then tracked during the [9] C. Valette, C. Cuesta, M. Castellengo, and 233 182 vibratory motion. Figure 8(a) plots the locus of the C. Besnainou. The tampura bridge as a precursive 234 wave generator. Acustica, 74:201–208, 1991. 235 183 point when the initial pluck is an in-plane triangle 184 with a peak amplitude of 1 cm. Figure 8(b) plots the [10] M. van Walstijn and V. Chatziioannou. Numeri- 236 185 locus of the point when the initial pluck is as given cal simulation of tanpura string vibrations. In Pro- 237 186 in the actual playing of the instrument. In all the ceedings of the International Symposium on Musical 238 , Le Mans, pages 609–614, 2014. 239 187 cases we note that the in-plane (vertical) motion of the 188 string is stable with the point on the string eventually [11] S. Bilbao, A. Torin, and V. Chatziioannou. Numerical 240 189 coming back to the vertical plane. The presence of modeling of collisions in musical instruments. Acta 241 190 j¯ıv¯ahas no noticeable effect on the stability of the Acustica united with Acustica, 101:155–173, 2015. 242

191 string. Our conclusion remains valid even when we [12] V. Chatziioannou and M. van Walstijn. Energy con- 243 192 varied the plucking position on the string. serving schemes for the simulation of musical instru- 244 ment contact dynamics. Journal of Sound and Vibra- 245 tion, 339:262–279, 2015. 246

193 Acknowledgement [13] M. van Walstijn, J. Bridges, and S. Mehes. A real- 247 time synthesis oriented tanpura model. In Proceed- 248 th 194 We are grateful to Prof. Venkitanarayanan and his ings of the 19 International Conference on Digi- 249 195 laboratory staff for helping us with the experiments. tal Audio Effects (DAFx-16), Brno, pages 175–182, 250 196 We are also thankful to Prof. Shakti Singh Gupta and 2016. 251

197 one of the referees for their constructive comments on [14] J. Bridges and M. van Walstijn. Modal based tan- 252 198 the manuscript. pura simulation: Combining tension modulation and 253 Pisharody and Gupta, p. 5

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