International Research Journal of Social Sciences____________________________________ ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 17 Modeling India’s National Anthem: A Statistical Approach Chakraborty Soubhik,Sarkar Saurabh, Tewari Swarima and Pal Mita Department of Applied Mathematics, Birla Institute of Technology Mesra, Ranchi-835215, INDIA Available online at: www.isca.in Received 05th September 2012, revised 08th September 2012, accepted 10th September 2012Abstract A major strength of statistics lies in modeling. Modeling a musical structure or performance is both an interesting and challenging endeavour given that the true model is not only complex but unknown even to the composer. On the other hand, although statistical models are neither perfect nor unbiased, it should be understood that i. we can at least make the data objective or nearly so ii. the true model may have multiple parameters and we usually do not have explicit knowledge about them nor we know how or in what functional way they enter the model and iii. doing a stochastic realisation of this deterministic true model (the decision process of the composer is deterministic as any musical sequence of notes is planned and not random) is within the scope of statistics including controlling the errors in the model. The present work highlights our maiden attempt to model the national anthem of India using a simple exponential smoothing. The fit is found to be explaining the note progression well enough with a smoothing factor 0.716404. Should such a model work well for the national anthem of any other country, it is of interest to see how the smoothing factor varies. Experimenting with other sophisticated models like Kalman filter where the smoothing factor is not fixed but varying is also of interest. Keywords: National anthem of India, simple exponential smoothing, information content, statisticsIntroduction India’s national anthem: Jana Gana Mana (Bengali: , Hindi: ) is the national anthem of India. Written in highly Sanskritised (Tatsama) Bengali, it is the first of five stanzas of a Brahmo hymn composed and scored by Nobel laureate Rabindranath Tagore. It was first sung in Calcutta Session of the Indian National Congress on 27 December 1911. Jana Gana Mana was officially adopted by the constituent assembly as the Indian national anthem on 24 January 1950. 27 December 2011 marked the completion of 100 years of Jana Gana Mana since it was sung for the first timeThe time allotted for it is 52 seconds. The present note gives an interesting simple exponential model fitted to the national anthem of India. The fit is found to be explaining the note progression well enough with smoothing factor Alpha 0.716404. Simple Exponential smoothing: A major strength of statistics lies in modeling. Modeling a musical structure or performance is both an interesting and challenging endeavour given that the true model is not only complex but unknown even to the composer. On the other hand, although statistical models are neither perfect nor unbiased, it should be understood that i. we can at least make the data objective or nearly so ii. the true model may have multiple parameters and we usually do not have explicit knowledge about them nor we know how or in what functional way they enter the model. Doing a stochastic realisation of this deterministic (the decision process of the composer is deterministic as any musical sequence of notes is planned and not random) true model is within the scope of statistics which includes controlling the errors in the model. Simple Exponential smoothing: is used to statistically model time series data for smoothing purpose or for prediction. Although it was Holt who proposed it first, it is Brown's simple exponential smoothing that is commonly used nowadays. Simple exponential smoothing is achieved by the model t+1 = + (1-) Ft, 01, to the data (t, ) where F is the predicted against Yand initially F = Y. Here is the smoothing factor. This is the only parameter in the model that needs to be determined from the data. The smoothed statistic Ft+1 is a simple weighted average of the previous observation Yt and the previous smoothed statistic Ft. The term smoothing factor applied to here is something of a misnomer, as larger values of actually reduce the level of smoothing, and in the limiting case with = 1 the output series is just the same as the original series (with lag of one time unit). Simple exponential smoothing is easily applied and it produces a smoothed statistic as soon as two observations are available. Values of close to one have less of a smoothing effect and give greater weight to recent changes in the data, while values of closer to zero have a greater smoothing effect and are less responsive to recent changes. There is no formally correct procedure for choosing . Sometimes the statistician's judgment is used to choose an appropriate factor. Alternatively, a statistical technique may be used to optimize the value of . For example, the method of least squares might be used to determine the value of for which the sum of the quantities (F Y is minimized . International Research Journal of Social Sciences__________________________________________________ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 18 Research Methodology Time series is a series of observations in chronological order. Musical data can also be taken as a time series in which a note characterized by pitch Y is the entry corresponding to the argument time twhich may mean time of clock in actual performance or just the instance at which the note is realized.In our case, since we are modelling only the structure of India’s national anthem so the arguments will be simply the instances1-. The desired note sequence is given in table-1. Western Art Music readers should refer to table 2 where corresponding western notations are provided. The tonic (Sa in Indian music) is taken at natural C.The present note gives an interesting Simple Exponential Modeling fit of the national anthem of India. The fit is found to be well enough with smoothing factor Alpha 0.716404. Analyzing the structure of a musical piece helps in giving an approximate model that captures the note progression in general without bringing the style of a particular artist into play. On the other hand, performance analysis gives additional features like note duration and the pitch movements between notes etc.. In table-2, pitches of notes in three octaves are represented by corresponding integers, C of the middle octave being assigned the value 0.We are motivated by the works of Adiloglu, Noll and Obermayer. The letters S, R, G, M, P, D and N stand for Sa, Sudh Re, Sudh Ga, Sudh Ma, Pa, Sudh Dha and Sudh Ni respectively. The letters r, g, m, d, n represent Komal Re, Komal Ga, Tibra Ma, Komal Dha and Komal Ni respectively. Normal type indicates the note belongs to middle octave; italics implies that the note belongs to the octave just lower than the middle octave while a bold type indicates it belongs to the octave just higher than the middle octave. The terms Sudh, Komal and Tibra imply, respectively, natural, flat and sharp. Let us understand the national anthem of India with reference to the sequence of notes provided in table 1 before we move to the modeling part. Translation in detail 1: Line 1: Jana-Gana-mana-adhinayaka, jaya he Bharata-bhagya-vidhata Translation: O lord of our destiny you are the captain of our souls and of the people of Bharat (India). This line corresponds to notes 1-21 in that sequence in table 1. Line 2:Punjaba-Sindhu-Gujarata-Maratha, Dravida-Utkala-Banga Translation: Your name rouses the hearts of Punjab, Sindhu, Gujarat, Maratha, of the Dravida and Orissa and Bangla. This line corresponds to notes 22-44 in that sequence in table 1. Line 3:Vindhya-Himachala-Yamuna-Ganga Uchchala-Jaladhi-taranga Tava shubhaname jage Tava shubha asisa mage. Translation: Tava shubha name jage Tava shubha asisa mageYour (the eternal charioteer of India) auspicious names echo in the hills of the Vindhyas and Himalyas, mingle in the music of Jamuna and Ganga and are chanted by the waves of the Indian Sea.This line corresponds to the notes 45-83 in that sequence in table 1.Line 4:Gahe tava jaya gatha Translation: They pray for your blessing and sing your praise (unknowingly or knowingly they chant your names and are blessed.) Line 4 corresponds to notes 84-92 in that sequence of table 1. Line 5:Jana-gana-mangala-dayaka jaya he Bharata-bhagya-vidhata Translation: The saving of all people waits in your hand, Oh You dispenser of India's destiny. Line 5 corresponds to notes 93-113 in that sequence of table 1. Line 6:Jaya he, jaya he, jaya he,Jaya jaya jaya, jaya he!Translation: Victory, victory, victory to You Victory Victory Victory to You (our God of destiny, because you are ruling us, and we are your servants. So you always win) This last line corresponds to notes 114-131 in that sequence of table 1. Next we provide the results of Single Exponential Smoothing obtained from Minitab 16. Results and Discussion Single Exponential Smoothing for C1 (Zero values of Yt exist; MAPE calculated only for non-zero Yt) Data C1 (C1 in Minitab represents pitch Yt), Length 131, Smoothing Constant, Alpha () = 0.716404 Accuracy Measures: MAPE (Mean Absolute Percent Error) = 35.3967, MAD (Mean Absolute Deviation) =1.2554, MSD(Mean Squared Deviation) = 3.6897, Table 3 gives the residuals. Table-1 Note sequence of India’s National Anthem Instance of note realization t Musical Note Pitch Y t 1. S 0 2. R 2 3. G 4 4. G 4 5. G 4 6. G 4 7. G 4 8. G 4 9. G 4 10. R 2 11. G 4 12. M 5 13. G 4 14. G 4 15. G 4 16. R 2 17. R 2 18. R 2 19.N -1 20. R 2 21. S 0 International Research Journal of Social Sciences__________________________________________________ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 19 22. S 0 23. S 0 24. P 7 25. P 7 26. P 7 27. P 7 28. P 7 29. P 7 30. P 7 31. P 7 32. P 7 33. m 6 34. D 9 35. P 7 36. M 5 37. M 5 38. M 5 39. G 4 40. G 4 41. G 4 42. R 2 43. M 5 44. G 4 45. G 4 46. G 4 47. G 4 48. G 4 49. G 4 50. R 2 51. P 7 52. P 7 53. P 7 54. M 5 55. M 5 56. M 5 57. G 4 58. G 4 59. G 4 60. R 2 61. R 2 62. R 2 63.N -1 64. R 2 65. S 0 66. G 4 67. G 4 68. G 4 69. G 4 70. G 4 71. G 4 72. R 2 73. G 4 74. M 5 75. G 4 76. M 5 77. P 7 78. P 7 79. M 5 80. G 4 81. R 2 82. M 5 83. G 4 84. G 4 85. G 4 86. G 4 87. R 2 88. R 2 89. R 2 90.N -1 91. R 2 92. S 0 93. P 7 94. P 7 95. P 7 96. P 7 97. P 7 98. P 7 99. P 7 100. P 7 101. P 7 102. m 6 103. D 9 104. P 7 105. M 5 106. M 5 107. M 5 108. G 4 109. G 4 110. G 4 111. R 2 112. M 5 113. G 4 114. N 11 115. N 11 116.S 12 117. D 9 118. D 9 119. N 11 120. P 7 121. P 7 122. D 9 123. S 0 124. S 0 125. R 2 126. R 2 127. G 4 128. G 4 129. R 2 130. G 4 131. M 5 International Research Journal of Social Sciences__________________________________________________ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 20 Table-2 Numbers representing pitches of musical notes in three octaves C Db D Eb E F F# G Ab A Bb B Western notation S r R g G M m P d D n N Notes (lower octave) -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Numbers for Pitch S r R g G M m P d D n N Notes (middle octave) 0 1 2 3 4 5 6 7 8 9 10 11 Numbers for Pitch S r R g G M m P d D n N Notes (higher octave) 12 13 14 15 16 17 18 19 20 21 22 23 Numbers for Pitch Table 3 Residual (Error) Analysis Time t C1 Smooth Predict Error(C1-Predict) 1 0 0.2065 0.7280 -0.72803 2 2 1.4914 0.2065 1.79353 3 4 3.2886 1.4914 2.50864 4 4 3.7982 3.2886 0.71144 5 4 3.9428 3.7982 0.20176 6 4 3.9838 3.9428 0.05722 7 4 3.9954 3.9838 0.01623 8 4 3.9987 3.9954 0.00460 9 4 3.9996 3.9987 0.00131 10 2 2.5671 3.9996 -1.99963 11 4 3.5936 2.5671 1.43291 12 5 4.6012 3.5936 1.40637 13 4 4.1705 4.6012 -0.60116 14 4 4.0483 4.1705 -0.17049 15 4 4.0137 4.0483 -0.04835 16 2 2.5711 4.0137 -2.01371 17 2 2.1620 2.5711 -0.57108 18 2 2.0459 2.1620 -0.16196 19 -1 -0.1362 2.0459 -3.04593 20 2 1.3942 -0.1362 2.13619 21 0 0.3954 1.3942 -1.39419 22 0 0.1121 0.3954 -0.39539 23 0 0.0318 0.1121 -0.11213 24 7 5.0238 0.0318 6.96820 25 7 6.4396 5.0238 1.97615 26 7 6.8411 6.4396 0.56043 27 7 6.9549 6.8411 0.15894 28 7 6.9872 6.9549 0.04507 29 7 6.9964 6.9872 0.01278 30 7 6.9990 6.9964 0.00363 31 7 6.9997 6.9990 0.00103 32 7 6.9999 6.9997 0.00029 33 6 6.2836 6.9999 -0.99992 34 9 8.2296 6.2836 2.71643 35 7 7.3487 8.2296 -1.22963 36 5 5.6661 7.3487 -2.34872 37 5 5.1889 5.6661 -0.66609 38 5 5.0536 5.1889 -0.18890 39 4 4.2988 5.0536 -1.05357 40 4 4.0847 4.2988 -0.29879 41 4 4.0240 4.0847 -0.08474 42 2 2.5740 4.0240 -2.02403 43 5 4.3120 2.5740 2.42599 44 4 4.0885 4.3120 -0.31200 45 4 4.0251 4.0885 -0.08848 46 4 4.0071 4.0251 -0.02509 47 4 4.0020 4.0071 -0.00712 48 4 4.0006 4.0020 -0.00202 49 4 4.0002 4.0006 -0.00057 50 2 2.5672 4.0002 -2.00016 51 7 5.7429 2.5672 4.43276 52 7 6.6435 5.7429 1.25711 53 7 6.8989 6.6435 0.35651 54 5 5.5385 6.8989 -1.89889 55 5 5.1527 5.5385 -0.53852 56 5 5.0433 5.1527 -0.15272 57 4 4.2959 5.0433 -1.04331 58 4 4.0839 4.2959 -0.29588 59 4 4.0238 4.0839 -0.08391 60 2 2.5739 4.0238 -2.02380 61 2 2.1628 2.5739 -0.57394 62 2 2.0462 2.1628 -0.16277 63 -1 -0.1361 2.0462 -3.04616 64 2 1.3942 -0.1361 2.13612 65 0 0.3954 1.3942 -1.39420 66 4 2.9777 0.3954 3.60461 67 4 3.7101 2.9777 1.02225 68 4 3.9178 3.7101 0.28991 69 4 3.9767 3.9178 0.08222 70 4 3.9934 3.9767 0.02332 71 4 3.9981 3.9934 0.00661 72 2 2.5667 3.9981 -1.99812 73 4 3.5935 2.5667 1.43334 74 5 4.6011 3.5935 1.40649 75 4 4.1705 4.6011 -0.60113 76 5 4.7648 4.1705 0.82952 77 7 6.3661 4.7648 2.23525 78 7 6.8202 6.3661 0.63391 79 5 5.5162 6.8202 -1.82023 80 4 4.4300 5.5162 -1.51621 81 2 2.6891 4.4300 -2.42999 82 5 4.3446 2.6891 2.31086 83 4 4.0977 4.3446 -0.34465 International Research Journal of Social Sciences__________________________________________________ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 21 84 4 4.0277 4.0977 -0.09774 85 4 4.0079 4.0277 -0.02772 86 4 4.0022 4.0079 -0.00786 87 2 2.5678 4.0022 -2.00223 88 2 2.1610 2.5678 -0.56782 89 2 2.0457 2.1610 -0.16103 90 -1 -0.1363 2.0457 -3.04567 91 2 1.3942 -0.1363 2.13626 92 0 0.3954 1.3942 -1.39417 93 7 5.1270 0.3954 6.60462 94 7 6.4688 5.1270 1.87304 95 7 6.8494 6.4688 0.53119 96 7 6.9573 6.8494 0.15064 97 7 6.9879 6.9573 0.04272 98 7 6.9966 6.9879 0.01212 99 7 6.9990 6.9966 0.00344 100 7 6.9997 6.9990 0.00097 101 7 6.9999 6.9997 0.00028 102 6 6.2836 6.9999 -0.99992 103 9 8.2296 6.2836 2.71643 104 7 7.3487 8.2296 -1.22963 105 5 5.6661 7.3487 -2.34872 106 5 5.1889 5.6661 -0.66609 107 5 5.0536 5.1889 -0.18890 108 4 4.2988 5.0536 -1.05357 109 4 4.0847 4.2988 -0.29879 110 4 4.0240 4.0847 -0.08474 111 2 2.5740 4.0240 -2.02403 112 5 4.3120 2.5740 2.42599 113 4 4.0885 4.3120 -0.31200 114 11 9.0399 4.0885 6.91152 115 11 10.4441 9.0399 1.96008 116 12 11.5588 10.4441 1.55587 117 9 9.7257 11.5588 -2.55876 118 9 9.2058 9.7257 -0.72565 119 11 10.4912 9.2058 1.79421 120 7 7.9901 10.4912 -3.49117 121 7 7.2808 7.9901 -0.99008 122 9 8.5124 7.2808 1.71922 123 0 2.4141 8.5124 -8.51244 124 0 0.6846 2.4141 -2.41409 125 2 1.6270 0.6846 1.31537 126 2 1.8942 1.6270 0.37303 127 4 3.4028 1.8942 2.10579 128 4 3.8306 3.4028 0.59719 129 2 2.5192 3.8306 -1.83064 130 4 3.5800 2.5192 1.48084 131 5 4.5973 3.5800 1.41996 Fig. 1 and Fig. 2 next give a graphical summary of the results of statistical analysis.  \n\n \n  \r    \n   !"# !#   #$%!& # \n \r   \r\n \r \r\r \r  \r\n #'\r# #$Figure-1 Single Exponential Smoothing Plot for C1 International Research Journal of Social Sciences__________________________________________________ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 22 ( (       \n  ( (  \r   ((( \n\n    ! \n \n ( ( "#$ \r"  \n \r## !\r \r % \n\r"  \r \r\rFigure-2 Residual Plots for C1 Interpretations from figures 1 and 2: The random pattern of the residuals (figure-2) together with the closeness of smoothed data with the observed one (figure-1) justifies the Simple Exponential Smoothing. A detailed discussion of the findings is given next. MAPE (Mean Absolute Percent Error) - measures the accuracy of fitted time series values. It expresses accuracy as a percentage. MAD (Mean Absolute Deviation) - measures the accuracy of fitted time series values. It expresses accuracy in the same units as the data, which helps conceptualize the amount of error. MSD (Mean Squared Deviation) - measures the accuracy of fitted time series values. MSD is always computed using the same denominator (the number of forecasts) regardless of the model, so one can compare MSD values across models and hence compare the accuracy of two different models. For all three measures, smaller values generally indicate a better fitting model. In case we fit other models to the same data, it is of interest to compare the corresponding MAPE, MAD and MSD values. This is reserved as a rewarding future work. The normal probability graph plots the residuals versus their expected values when the distribution is normal. The residuals from the analysis should be normally distributed. In practice, for data with a large number of observations, moderate departures from normality do not seriously affect the results. The normal probability plot of the residuals should roughly follow a straight line. One can use this plot to look for the following: This pattern... Indicates... Not a straight line Nonnormality Curve in the tails Skewness A point far away from the line An outlier Changing slope An unidentified variable As is clear from figure-2, the plot roughly follows a straight line with one outlier: The next graph plots the residuals versus the fitted values. The residuals should be scattered randomly about zero. One can use this plot to look for the following: This pattern... Indicates... Fanning or uneven spreading of residuals across fitted values Non-constant variance Curvilinear A missing higher-order term A point far away from zero An outlier As is clear from figure-2, the points are randomly scattered: A histogram of the residuals shows the distribution of the residuals for all observations. One can use the histogram as an exploratory tool to learn about the following characteristics of the data: i. Typical values, spread or variation, and shape, ii. Unusual values in the data This pattern... Indicates... Long tails Skewness A bar far away from the other bars An outlier International Research Journal of Social Sciences__________________________________________________ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 23 The histogram of the residuals should be bell-shaped. One can use this plot to look for the following: Because the appearance of the histogram can change depending on the number of intervals used to group the data, use the normal probability plot and goodness-of-fit tests to assess whether the residuals are normal. We have already given the normal probability plot for residuals. The graph residuals versus order plots the residuals in the order of the corresponding observations. The plot is useful when the order of the observations may influence the results, which can occur when data are collected in a time sequence (as in our case) or in some other sequence, such as geographic area. This plot can be particularly helpful in a designed experiment in which the runs are not randomized. The residuals in the plot should fluctuate in a random pattern around the center line as in figure-2. One can examine the plot to see if any correlation exists between error terms that are near each other. Correlation among residuals may be signified by: i. An ascending or descending trend in the residuals, ii. Rapid changes in signs of adjacent residuals Remark: Simple exponential smoothing is useful when i. there is no trend ii. there is no seasonal variation iii. there is no missing value iv. we want short term forecast. For the sake of completeness we show figure-3 which clearly indicates an absence of linear trend. There was no missing value in table-1. The melody phrase {G, G, G, R, R, R, , R S} of length 9 (as there are nine notes in it) comes thrice which is maximum for any melody phrase in the entire sequence and this leads to a melody significance of 9x3=27. Table 4 gives the information content (IC) of the notes. This measures the surprise a note carries when it appears. Higher the probability, lower the information content and hence lower the surprise. The formula is IC (note) = -log{Probability(note)}.  \n\n \n     \n\n  !"# !#   )%!& #&\r ! \r \r\r*#!+!#$$# ,-\n. /Figure-3 No linear trend in India’s national anthem Table-4 Probability of the notes and their information content Sl. No. Note No. of Occurrence=f Probability=f/131 Information Content 1 S 9 0.06873.8635 2 R 22 0.16792.5740 3 G 43 0.32821.6072 4 M 17 0.12982.9460 5 m 2 0.01536.0334 6 P 27 0.20612.2785 7 D 5 0.03824.7115 8 N 6 0.04584.4485 International Research Journal of Social Sciences__________________________________________________ISSN 2319–3565Vol. 1(2), 1-7, October (2012) I Res. J. Social Sci. International Science Congress Association 24 Conclusion Simple Exponential Model fitted to the national anthem of India is found to be explaining the note progression well enough with smoothing factor 0.716404. Possibly this is the first attempt to model India’s national anthem statistically. One advantage with simple exponential modeling is that the inner mechanism is fairly straightforward and as such can be easily understood by a general audience. Should such a model work well for the national anthem of any other country, it is of interest to see how the smoothing factor varies. Experimenting with more sophisticated smoothing techniques like Kalman filter where the smoothing factor dynamically changes is also within the scope of our research and is reserved as a rewarding future work, given that the three measures MAPE, MAD and MSD given by the Minitab package are not very informative by themselves but they are helpful in comparing the fits obtained by using different methods. Remark: According to the first author (who, apart from being a statistician, is also a harmonium player) India’s national anthem is based on the raga Yaman Kalyan. But unlike a raga rendition where notes are rendered with meend (glide), in the national anthem the notes are rendered flat, i.e., in straight transition. Therefore the raga mood does not build, neither it is necessary. In fact, because of the flat rendition of the notes, India’s national anthem sounds very melodious if rendered properly by an orchestra. AcknowledgementThis research is a part of an ongoing UGC Major Research Project in our department. We thank the University Grants Commission (UGC) for the sponsorship (Sanction Letter No.F.N.5-412/2010 (HRP); draft no.392889 dated 28.02.2011). References 1.http://en.wikipedia.org/wiki/Jana_Gana_Mana (2012)2.Holt C. C., Forecasting Trends and Seasonal by Exponentially Weighted Averages". Office of Naval Research Memorandum 52, (1957) reprinted in Holt, C. C. (January–March 2004). Forecasting Trends and Seasonal Exponentially Weighted Averages, International Journal of Forecasting 20(1), 5–10 (2004) 3.Brown, R. G., Smoothing Forecasting and Prediction of Discrete Time Series. Englewood Cliffs, NJ: Prentice-Hall (1963)4.http://en.wikipedia.org/wiki/Exponential_smoothing (2012)5.Chakraborty, S., Ranganayakulu, R., Chauhan, S., Solanki, S. S. and Mahto, K. A Statistical Analysis of Raga Ahir Bhairav. Journal of Music and Meaning, Vol. 8, sec. 4, http://www.musicandmeaning.net/issues/show Article.php?artID=8.4 (2009)6.Adiloglu K., Noll T. and Obermayer K., A Paradigmatic Approach to Extract the melodic Structure of a Musical Piece, Journal of New Music Research,35(3), 221-236 (2006) 7.Chakraborty, S., Krishnapryia, K., Loveleen, Chauhan, S., Solanki, S. S. and Mahto, K., Melody, Revisited: Tips from Indian Music Theory, International Journal of Computational Cognition, 8(3), 26-32 (2010)