International Journal of Research in Engineering, Technology and Science, Volume VI, Special Issue, July 2016 www.ijrets.com, editor@ijrets.com, ISSN 2454-1915 FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD Moujhuri Patra 1,SirshenduSekhar Ghosh 2, Soubhik Chakraborty 3 1 Department of Master of Computer Applications, NSEC Garia, Kolkata-700152, India 2 Department of Computer Applications, NIT Jamshedpur, Jharkhand-831014, India 3 Department of Mathematics, BIT Mesra, Ranchi, Jharkhand-835215, India ABSTRACT: The present paper finds some interesting facts about the fractal dimension of different ragas with respect to different time of rendering and different mood/nature. It is observed that some ragas do have a prominent fractal nature but this totally depends on the nature of the raga (such as restless or restful/serious). In contrast, when we consider the time of raga rendering, there is no such dimensional difference between a morning raga and a night raga. The implication of this finding is that while distinguishing between the ragas based on the fractal dimension it is better to take their mood as the basis of distinguishing characteristics rather than the appropriate time of their rendition. The time theory of ragas is itself controversial and many musicians do not support it. Thus this paper gives at least one scientific basis to explain the disagreement among musicians. Keywords: Fractals, Melodic Movement, Musical Notes, Raga. [1] INTRODUCTION Music is an emotional and experimental form of art. We can characterize an Indian Music by several mathematical and statistical techniques. Recently there is a great interest of modern science interacting with this highly emotional and experiential phenomenon of music. Music is organized sound that is capable of conveying emotion; hence melody has to be ordered successions of musical notes and it is of interest to investigate if the successions depict a fractal nature. Successions are fractal if the incidence frequency F and the interval between successive notes i in a musical piece bear the relation: F = c/i D,where D is the fractional dimension and c is a constant of proportionality [1]. In this paper we try to investigate the self-similar nature of the musical notes based on Ragas with different time of rendition. Here we explore the application of fractals in music. One direction of research could be to investigate whether a musical succession of digital notes depicts a fractal nature or not. Another interest can be if we can mathematically characterize the difference between the musical component of ragas (with different moods) and it is found that fractal nature is more prominent in both the restless ragas compared to the restful ragas. It is observed that some ragas do have a prominent fractal nature but that this totally depends on the nature of the raga (such as restless or restful/serious). In contrast, in this paper when we 1
consider the time of raga rendering, there is no such dimensional difference between a morning raga and a night raga. The implication of this finding is that while distinguishing between the ragas based on the fractal dimension it is better to take their mood as the basis of distinguishing characteristics rather than the appropriate time of their rendition. Here we take six Indian Ragas with different time of rendition.we take Ragas Jounpuri, Bivash and Bhimpalashree in one set(morning ragas), Puriya, Desh and Kafi are taken as other set (night ragas). The paper is organized as follows: Section 2 gives a brief outline on fractal and music. Section 3 describes Musical feature of Ragas. Section 4 gives the methodology that we use. Section 5 gives the experimental results followed by a discussion. Finally, Section 6 draws the conclusion. [2] FRACTAL AND MUSIC Fractals are geometric shapes with interesting properties that set them apart from normal Euclidean shapes. The first interesting property is that of self-similar nature. Another property of fractal is a non-integer dimension which is related to the concept of self-similarity. The term fractal was coined by Benoit Mandelbrot in 1975[2] to describe shapes that are self-similar that is, shapes that look the same at different magnifications and we refer to his classic treatise for an insight. Mandelbrot s fractal geometry has provided a new qualitative and quantitative approach for the understanding of the complex shapes of nature. The calculation of fractal dimension is an important way to classify objects that exhibit fractal characteristics. The relative abundance or the incidence frequency F, of notes of different acoustic frequency f in a musical composition is not fractal. Unplanned striking of the keys in a piano or a harmonium will not create music. Music is organized sound that conveys emotion; hence melody has to be ordered successions of musical notes. These successions are fractal if the incidence frequency F of the interval between successive notes i in a musical piece bear the relation: F=c/i D where, D is the fractional dimension and c is a constant of proportionality or, ln(f)=ln(c)-dln(i)=c-dln(i) where C=ln(c), another constant. Voss and Clark [1] determined that music exhibits 1/f -power spectra at low frequencies. This fact allows us to consider music as a time series and analyze the fractal dimension of a particular piece of music. Bigerelle and Lost [3] found the global D to be an invariant for different types of music. In another work, D in the music of Mozart and Bach was calculated. Hsu and Hsu[4] discussed the application of D to music in detail and for a work of Bach, found D to be 2.418[5]. Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 2
[3] MUSICAL FEATURES OF RAGAS In this present paper, we have two sets of ragas,jounpuri, Bivash and Bhimpalashree in one set(morning ragas), Puriya, Desh and Kafi are in another set(night ragas).we take Jounpuri, Bivash and Bhimpalashree from Asavari, Bhairavi and KafiThaat respectively. The best time for singing all these Ragas in morning/day time.jounpuri raga uses the swaraskomal G, Komal D whereas Bhimpalashree uses the swaraskomal G and Komal Ni. Amomg all evening/night ragas we took Kafi, Desh and Puriya from Kafi,Khambaz and MarwaThaat respectively, uses the swaras like KomalRi, TivraMadhyam,Shuddha and Komal Ni. Abbreviations: The letters S, R, G, M, P, D and N stand for Sa, Sudh Re, Sudh Ga, Sudh Ma, Pa, SudhDha and Sudh Ni respectively. The letters r, g, m, d, n represent Komal Re, Komal Ga, Tibra Ma, KomalDha 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. Sa the tonic in Indian music, is taken at C. Corresponding Western notation is also provided. (See Table 4.1) The terms Sudh, Komal and Tibra imply, respectively, natural, flat and sharp. [4] METHODOLOGY We take six different ragas from differentthaat and calculate intervals i as the absolute values of differences in pitch of two successive notes. For each sequence of notes, a frequency distribution is found of the intervals. Accordingly four tables of ln F versus ln i are formed, one for each raga. Calculations are made only for those values of F and i for which bothln F andln i are defined. The note sequences are taken from a standard text [6] and not from any audio recording. There are some obvious advantages and disadvantages for doing so. If we go for audio recordings, it is not always necessary that the same raga performed by different artists (or even the same artist on different occasions) will exhibit the same fractal nature. Even if we analyze a single recording of an artist, it is not easy to say which part of the fractal nature is attributable to the raga itself and which part to the style. In a structure analysis, the style of the artist does not interfere with our analysis whereby the fractal nature can be studied for its presence (with dimension) in the raga structure itself in a general sense [7]. The technique is to assign the number 0 to C (where the tonic Sa or S is taken), 1 to the next note Db (Komal Re or r) and so on (Table 4.1)[8]. On the disadvantage side, we miss information on note duration and pitch movements between the notes which we could get in audio recordings. Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 3
Table 4.1: Numbers representing pitch of notes C Db D Eb E F F# G Ab A Bb B Lower Octave: S r R g G M m P d D n N 12-11 -10-9 -8-7 -6-5 -4-3 -2-1 Middle Octave: S r R g G M m P d D n N 0 1 2 3 4 5 6 7 8 9 10 11 Higher Octave: S r R g G M m P d D n N 12 13 14 15 16 17 18 19 20 21 22 23 [5] EXPERIMENTAL RESULT ANALYSIS AND DISCUSSION Our experimental results are summarized in Tables 5.1-5.6 and corresponding Figures 5.1-5.6 for two sets. One set describes day ragas i.e., Jounpuri, Bivash and Bhimpalashree and the other set describes night ragas i.e., Puriya, Desh and Kafi. Raga Jounpuri Table 5.1: Data for Raga Jounpuri i F lni lnf 0 8 2.08 1 32 0.00 3.47 2 90 0.69 4.50 3 33 1.10 3.50 4 10 1.39 2.30 5 6 1.61 1.79 6 7 1 1.95 0.00 8 9 10 11 Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 4
lnf FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD 12 5.00 4.00 3.00 2.00 1.00 0.00 0.00 0.50 1.00 1.50 2.00 2.50 y = -1.82x + 4.6351 lni R² = 0.6355 Raga Bivash Figure: 5.1. Graph for lni vs. lnf Table 5.2: Data for Raga Bivash i F lni lnf 0 16 2.77 1 57 0.00 4.04 2 1 0.69 0.00 3 71 1.10 4.26 4 29 1.39 3.37 5 3 1.61 1.10 6 3 1.79 1.10 7 8 9 10 11 12 Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 5
lnf lnf FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD 5.00 4.00 3.00 2.00 1.00 Raga Bhimpalashree 0.00 0.00 0.50 1.00 1.50 2.00 y = -0.9664x + 3.3714 lni R² = 0.1265 Figure: 5.3. Graph for lni vs. lnf Table 5.3: Data for Raga Bhimpalashree i F lni lnf 0 8 2.08 1 16 0.00 2.77 2 126 0.69 4.84 3 7 1.10 1.95 4 11 1.39 2.40 5 11 1.61 2.40 6 0 1.79 7 1 1.95 0.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0.00 0.50 1.00 1.50 2.00 2.50 y = -1.4122x + 3.9766 lni R² = 0.4025 Figure: 5.3. Graph for lni vs. lnf Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 6
lnf FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD Raga Puriya Table 5.4: Data for Raga Puriya i F lni lnf 0 4 1.39 1 69 0.00 4.23 2 35 0.69 3.56 3 24 1.10 3.18 4 20 1.39 3.00 5 17 1.61 2.83 6 10 1.79 2.30 7 1 1.95 0.00 8 9 10 5.00 4.00 3.00 2.00 1.00 0.00 0.00 0.50 1.00 1.50 2.00 2.50 y = -1.5871x + 4.6614 lni R² = 0.6526 Figure: 5.4. Graph for lni vs. lnf Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 7
lnf FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD Raga Desh Table 5.5: Data for Raga Desh i F lni lnf 0 23 3.14 1 32 0.00 3.47 2 85 0.69 4.44 3 19 1.10 2.94 4 13 1.39 2.56 5 7 1.61 1.95 6 7 8 1 2.08 0.00 9 10 5.00 4.00 3.00 2.00 1.00 0.00 0.00 0.50 1.00 1.50 2.00 2.50 y = -1.7196x + 4.5287 lni R² = 0.6885 Figure: 5.5. Graph for lni vs.lnf Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 8
lnf FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD Raga Kafi Table 5.6: Data for Raga Kafi i F lni lnf 0 6 1.79 1 48 0.00 3.87 2 74 0.69 4.30 3 25 1.10 3.22 4 21 1.39 3.04 5 4 1.61 1.39 6 1 1.79 0.00 7 8 9 10 11 1 2.40 0.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 y = -2.008x + 4.8359 lni R² = 0.7624 Figure: 5.6. Graph for lni vs. lnf R-squared (R 2 ) is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression. R-square can take on any value between 0 and 1, with a value closer to 1 indicating that a greater proportion of variance is accounted for by the model. For example, an R-square value of 0.8234 means that the fit explains 82.34% of the total variation in the data about the average. By inspection of R 2 value can result whether a raga is fractal natured or not. Finding means a low fractal dimension. The results on the six ragas are indeed very interesting. The ragas Jounpuri,Bivash and Bhimpalashree are depicting fractal nature with low dimension i.e., 0.635,0.126 and 0.402 respectively. Whereas the night ragas like Puriya, Desh and Puriya have the dimension 0.705,0.688 and 0.6526 respectively. The finding are not so prominent as R 2 is not very high for all ragas.100r 2, also called % coefficient of determination, gives the percentage of variation in the response (here lnf) Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 9
explained by the predictor (here lni) through the model (here a straight line). Irrespective of the time of rendition of ragas the results of D is less than 1 so there is no prominent value that can differ the dimension of morning and night ragas. [6] CONCLUSION In a recent paper it has been argued that fractal dimension is related with the chalan (melodic movement) of the raga which suggests that some ragas do have a prominent fractal nature but that this totally depends on the nature of the raga (such as restless or restful/serious). Our earlier study confirms that fractals do provide interesting mathematical properties that may be related to the melodic movement (in this case, whether restful or restless) of a raga where the restless ragas have high fractal dimension than the restful ragas[9]. In contrast, when we consider the time of raga rendering, there is no such dimensional difference between a morning raga and a night raga. So we can conclude that the implication of this finding is that while distinguishing between the ragas based on the fractal dimension it is better to take their mood as the basis of distinguishing characteristics rather than the appropriate time of their rendition. Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 10
REFERENCES [1] R. F. Voss and J. Clarke, 1/f Noise in Music and Speech, Nature, 258, 317 318,1975 [2] B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, N. Y., 1977. [3] M. Bigerelle and A. Lost, Fractal Dimension and Classification of Music,Chaos, Solitons, and Fractals, 11,2179 2192, 2000. [4] K. J. Hsu and A. J. Hsu, Fractal geometry of music, Proc. Natl.Acad. Sc.,USA, Vol.87, 938-941,1990. [5] J. Hemenway, Fractal Dimensions in the Music of Mozart and Bach. The Nonlinear Journal, 86 90, 2000. [6] D. Dutta. SangeetTattwa (prathamkhanda), BratiPrakashani, 5th ed,2006 (Bengali) [7] K. Adiloglu, T. Noll and K. Obermayer, A Paradigmatic Approach to Extract the melodic Structure of a Musical Piece, Journal of New Music Research, Vol. 35(3), 221-236, 2006. [8] S. Chakraborty, S. Tewari and G. Akhoury, What do the fractals tell about a raga? A case study in raga Bhupali,, Consciousness, literature and the arts, Vol. 11, No. 3, 1-9, 2010. [9] M. Patra, S. Chakraborty, Analyzing the Digital Note Progression of ragas within a thaat using fractal, International Journal of Advanced Computer and Mathematical Sciences. ISSN 2230-9624. Vol 4, Issue2, pp. 148-153, 2013. Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 11
Author[s] brief Introduction 1 Author Dr. Moujhuri Patra is now working as an Assistant Professor in MCA Department, Netaji Subhash Engineering College(NSEC), Techno India,Garia, Kolkata. She did her M.Tech from BIT,Mesra,Ranchi,India in the field of Computer Science and has awarded her Ph.D. degree in last year from the same Institute.Her research interest is Music Analysis with Statistics, Artificial Neural Networkand Fractal geometry. She has published a bookbased on her research area along with several other research papers. 2 Author SirshenduSekhar Ghoshhas submitted his Ph.D. thesis from Department of Computer Science and Engineering, Birla Institute of Technology (BIT), Mesra, Ranchi, Jharkhand, India in 2015. He has completed his M.Tech in Information Technology from Indian Institute of Engineering Science and Technology (IIEST), Shibpur, West Bengal, India in 2010.Currently he is working as a Faculty in the Department of Computer Applications at National Institute of Technology (NIT), Jamshedpur, Jharkhand, India. His research interests are Internet Technology and Web Mining. 3 Author Dr.SoubhikChakrabortyis currently a Professor in the Department of Mathematics, BIT Mesra,Ranchi,India.His research interests are algorithm analysis and music analysisin which he has published two books,two research monograms and several papers.he has guided several Ph.D.Scholars in both the areas. He is a recipientof several awards and has been the PI of a UGC Major Research Project. Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 12
Corresponding Address 1 Dr. Moujhuri Patra MCA Department, Netaji Subhash Engineering College (NSEC), Garia, Kolkata-700152, West Bengal, India E-Mail: moujhuri@gmail.com Mobile: 09051450040 2 Sirshendu Sekhar Ghosh Department of Computer Applications, National Institute of Technology (NIT), Jamshedpur-831014, Jharkhand, India E-Mail: ssghosh.ca@nitjsr.ac.in Mobile: 09955529117 3 Dr. Soubhik Chakraborty Department of Mathematics, Birla Institute of Technology (BIT), Mesra, Ranchi-835215, Jharkhand, India E-Mail: soubhikc@yahoo.co.in Mobile: 09835471223 Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 13