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Volume 11 Number 3, December 2010

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What do the fractals tell about a raga? A case study in raga Bhupali

by

Soubhik Chakraborty, Swarima Tewari and Gargi Akhoury

Birla Institute of Technology, Mesra, Ranchi, India

1. Introduction

The relative abundance, or the incidence frequency F, of notes of different acoustic frequency f in a musical composition (which note occurs how many times) is not fractal [1]. 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, however, can be 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 [2]. Here F gives the number of occurrences of a particular i value and should not be confused with the frequency F which gives the number of times a particular note occurs. The latter is not fractal but the former can be!

or, ln(F)=ln(c)-Dln(i)=C-Dln(i) where C=ln(c), another constant. …….(1)

The term fractal was coined by Benoit Mandelbrot in 1975 to describe shapes that are "self-similar" -- that is, shapes that look the same at different magnifications and we refer to his classic treatise [3] for an insight.. In our previous work [4], we investigated how fractals can be used to characterize the structure of a North Indian raga, namely, Malkauns.  Interestingly, notes of raga Malkauns can be created by simply making a change of scale in the notes of another North Indian raga, namely, Bhupali. This is explained in section 3. But while the melodic structure of the raga Malkauns, as evident by the note progression, clearly depicted a fractal nature previously, our present study indicates that in the case of Bhupali the fractal nature is not so prominent. The study is important to suggest that merely changing the scale does not create a new raga. A raga is created through its “chalan” or melodic movement and this is what is mathematically captured, at least to a reasonable degree, by a fractal analysis.

Appendix A gives the complete sequence of notes of  raga Bhupali taken from a standard text [5]. A raga is a melodic structure comprising of fixed notes and a set of rules characterizing a certain mood conveyed by performance [6]. Appendix B gives a very vice description of the chalan of raga Bhupali in the words of Rajan Parrikar, a recognized expert in Indian classical music. Parrikar has rightfully compared this raga with another North Indian raga Deshkar which uses the same notes (without any need of scale change) while describing its chalan. We reserve as our future work a fractal analysis of Deshkar as well.

It should be understood that musical data is chronological; table 1 gives the numbers representing pitches in different octaves which will be useful in understanding the note progression of the raga in question (appendix A).

Table 1: Numbers representing pitch of notes [6][7]:-

  C    Db   D   Eb    E    F    F#   G   Ab  A   Bb   B   

  S        r    R    g     G    M   m   P    d     D    n    N   (lower octave)

-12   -11 -10  -9     -8    -7   -6   -5   -4    -3   -2   -1 

       S    r    R   g     G    M   m   P   d   D  n    N   (middle octave)

  0    1   2    3     4      5    6    7   8   9  10   11 

    S       r       R      g      G        M       m      P       d        D      n         N   (higher octave)

   12     13     14     15    16        17       18    19     20       21     22       23

Abbreviations: 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. Sa, the tonic in Indian music, is taken at C. Corresponding Western notation is also provided. The terms “Sudh”, “Komal” and “Tibra” imply, respectively, natural, flat and sharp. We close this section giving some general feature of raga Bhupali:-

Raga: Bhupali

Thaat (a specific way of grouping ragas): Kalyan

Aroh (ascent):  S  R G P, D, S

Awaroh (descent): S, D P, G, R, S

Jati: Aurabh-Aurabh (5 distinct notes allowed in ascent and 5 in descent)

Vadi Swar (most important note): G

Samvadi Swar (second most important note): D

Anga: Poorvanga pradhan (first half more important)

Prakriti (nature): Restful

Pakad (catch): G, R, S, D, S R G, P G, D P G, R, S

Speciality: Meend (glide) from S to D (or S to D) and from P to G

Nyas Swar (Stay notes):  G, P and D

Time of rendition: First phase of night (6 PM to 9 PM)

2. Fractal Analysis

We calculate intervals i as the absolute values of differences in pitch of two successive notes. As an illustrative example, for the first six notes (instance 1-6), the pitch values are 0, -3, 0, 0, 2. 4. The i values are 3, 3, 0, 2 and 2 (after taking absolute value). The F values corresponding to these i values are F=2 for i=3 (the i value 3 occurs twice), F=1 for i=0(the i value 0 occurs once) and F= 2 for i=2 (the i value 2 occurs twice).  Following this technique a frequency distribution is found of the intervals giving F values corresponding to the i values for the entire sequence of i values. Table 2 of lnF versus lni is formed next, with the acknowledgement that calculations are made only for those values of F and i for which both lnF and lni are defined. Fig. 1 based on table 2 depicts the fractal nature graphically. In fig.1, y stands for lnF and x stands for lni. To the question why we are taking i and not other characteristics, we have the answer that musical notes are realized on the basis of distance from a fixed origin. This origin is called the tonic (“Sa” or “do”) which in our theoretical analysis is taken at natural C. The reason why we are not taking other features like note duration is because we are only analyzing the structure and not performance. We do not want the style of an artist to interfere with our analysis.

Table 2: ln F and lni calculations for raga Bhupali (refer to appendix A)

                                               Fig. 1: Graph of lnF versus lni in raga Bhupali

 3. Discussion

In our previous work [4], we investigated how fractals can be used to characterize the structure of a North Indian raga, namely, Malkauns. A summary of the previous results are as follows:-

As the Malkauns note sequence progressed, the graph of lnF versus lni captured a linear trend gradually depicting a fractal nature with four sequential segments of the total notes giving R² values .828, .921, .995 and .978. The fractional dimension D took values in close proximity, namely, 2.57, 2.351, 2.528 and 2.367 respectively with mean 2.454 and standard deviation 0.111. Since the four segments correspond to the same raga sequence or the same musical piece, this cloxe proximity logically related the fractal dimension to the raga concerned.

Now it is interesting to note that notes of raga Malkauns are created by simply changing the scale of raga Bhupali. If the sudh Ga in raga Bhupali is taken as Sa, we get the notes of raga Malkauns. But while the melodic structure of the raga Malkauns, as evident by the note progression, clearly depicted a fractal nature [4] previously, our presnt study indicates that in the case of raga Bhupali the fractal nature is not so prominent (fig. 1). The value of R² is 0.8473 which is not very high. This means only 84.73% of the variation in the response is captured by the model. Or, in other words, the fractal nature is explaing about 85% of the nature of the raga note progression which is clearly worse than what we got in raga Malkauns. Using equation (1) and comparing with fig. 1 we get the fractal dimension of the present Bhupali note sequence as 2.8065. The study is important to suggest that merely changing the scale does not create a new raga. A raga is created through its “chalan” or melodic movement and this is what is mathematically captured, at least to a reasonable degree, by a fractal analysis.

4. Concluding remarks

Comparing our present findings with the previous work [4], we make some important conjectures before closing this paper.

1. Not all ragas have a clear cut fractal nature.

2. Merely a change of scale does not create a new raga. A raga depends on its chalan.

3. The fractal nature definitely captures some aspects of the chalan of a raga.

Of course, the second conjecture is not new to musicians and musicologists. But the first and the last conjectures will certainly draw their interest too.

However, there is a limitation of the present line of investigaion. The note progression only gives the note combinations of a raga. For a more complete idea of the melodic movement of the raga, not only the note combinations but how they are to be rendered is important. For the “how” part, the transitory and non-transitory pitch movements between the notes becomes crucial. For a statistical analysis of such pitch movements between notes, we refer to [8] where a performance analysis of the raga Ahir Bhairav has been statistically analysed. The “mood” of the raga is a psychophysical aspect [9].

As future work, as indicated earlier, we would undertake  a similar study in raga Deshkar which uses the same notes as used by raga Bhupali. We would also like to know how the fractional dimension D changes if we consider a raga with more than five (say six or seven) notes.  In an independent analytical study on some songs of Ravindranath Tagore, Choudhury and Ray [10] have used fractals to good effect.                              [Concluded]

Appendix A: Note sequence of raga Bhupali [5]

Appendix B: We reproduce, with permission (http://www.parrikar.org/raga-central/bhoopali), a very vice description of chalan of raga Bhupali in the words of Rajan Parrikar, a recognized expert in Indian classical music. In Parrikar’s notation, a single quote after a note indicates the note being in the lower octave, a double quote refers to the higher octave, an absence of any quote implies  the middle octave. Parrikar has rightfully compares this raga with another raga Deshkar which uses the same notes (without any need of scale change) while describing its chalan. We reserve as our future work a fractal analysis of Deshkar as well.

“Bhoopali is a Kalyan-anga raga whereas Deshkar is a Bilawal-anga raga; their respective characteristics can be inferred from this proposition. It must be underscored that this is a statement not of historical chronology but of the relevance of specific melodic groupings – ragangas, in our terminology – attending the orthogenesis of ragas and of their continual presence in the Indian musical imagination.

Let us first examine Bhoopali. The definitive tonal sentences are:

S, S (S)D’ S R G, G R S (S)D’ S
The nyasa on G and the grace of S on D are points of note.

S (G)R G, G R P G, P R G, S R, R G R, S R S (S)’ S
The tonal activity is centred on G. Another important nyasa swara is R.

G R G P, P G D P (P)G, G P R G, G R, S R S (S)D’ S
The G-D coupling and the arohi nyasa on P are illustrated.

G P (S”)D, (S”)D, S”, S” (S”)D S” R”, R” S”
This represents a typical uttaranga launch.

To summarize, the nyasa locations in Bhoopali are S, R, G and P. Tonal activity revolves around G. The G-D coupling and arohi nyasa on P are points of note. The raga swaroopa unravels in the poorvanga region. Tonal clusters such as S R S (S)D’ S or S R (S)D’ S serve as delimiters during elaboration. It should now be obvious that Bhoopali’s simple aroha-avarohana masks its non-linearity. The perceptive mind will also see in Bhoopali the shadow of raganga Kalyan. The nyasa swaras and formulation of tonal contours derive from Kalyan minus madhyam and nishad, which is why some vidwans refer to Bhoopali as “Bhoop Kalyan” or “Ma-Ni-varjit-Kalyan.” En passant, the P-G and the S-D arcs may occasionally create an abhasa of m and N, respectively.

Let us turn to Deshkar. The definitive tonal sentences are:

P, P G P D, D, P, P D G P
This is an uttaranga-pradhana raga. The tonal activity is concentrated on D. The avarohi nyasa on P and the D-G coupling attending Deshkar are gestures obverse of those plied in Bhoopali.

P D G P (S”)D, (S”)D, S”, D R” S”, (S”)D, D, P, P D G P
Another raganga-vachaka sangati.

P D G P G R S R (S)D’ S, S G P D, D, P
The rishab is alpa; some musicians render it langhan (i.e. skip it) during alapi, others acknowledge its presence without rendering it deergha. Whereas in Bhoopali R is an important nyasa sthana.

To summarize, the nyasa locations in Deshkar are P, D and S” (tar shadaj). The D-G coupling and avarohi nyasa on P are points of note. The raga swaroopa unravels in the uttaranga region. A little reflection reveals the hand of raganga Bilawal lurking below the Deshkar surface; the dominant D and the D-G sangati may be laid at Bilawal’s door.

The behaviours of Bhoopali and Deshkar are, as established above, driven by entirely different genetic imperatives despite their sharing a common scale, a striking illustration of the conceptual power of raga. Assigning them to two different thats also points to Pandit Vishnu Narayan Bhatkhande’s insight into the nature of raga.”

A glossary of technical terms associated with Indian classical music can be found in http://www.culturalindia.net/indian-music/music-glossary.html

Acknowledgement

The first author thanks Rajan Parrikar for the permission to reproduce some materials from his archive.

References

[1] F. R. Voss and J. Clarke, 1/f Noise in Music and Speech, Nature (London) 258, 1975, p. 317-318

[2]K. J. Hsu and A. J. Hsu, Fractal geometry of music, Proc. Natl. Acad. Sc. USA, Vol. 87, 1990,  p. 938-941

[3] B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman N. Y. 1977

[4] S. Chakraborty, S. Sundareisan, S. Kashyap and M. Pal, A Simple Fractal Analysis of a North Indian Raga, International Journal of Computational Cognition, (to appear)

[5] Dutta, D., Sangeet Tattwa (Pratham Khanda), Brati Prakashani, 5^th ed, 2006

      ( Bengali)

[6] Chakraborty, S, K. Krishnapriya, Loveleen, S. Chauhan and S. S. Solanki,  Analyzing the melodic  structure of a North Indian raga: A statistical approach, Electronic Musicological Review, Vol. XII, 2009

[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, 2006, Vol. 35,

      no. 3, 221-236

[8] S. Chakraborty, R. Ranganayakulu, S. Chauhan, S. S. Solanki and K. Mahto, A Statistical  Analysis of Raga Ahir Bhairav, Journal of Music and Meaning, Vol. 8, sec. 4, Winter 2009 http://www.musicandmeaning.net/issues/showArticle.php?artID=8.4

[9] Roederer, J., The physics and psychophysics of music (4th ed.): an introduction, Springer Pub. Co. Inc., 2008(see its review by the first author published in Computing Reviews, Nov. 4, 2009) http://www.reviews.com/widgets/reviewer.cfm?reviewer_id=123180&count=19

[10] Choudhury, M.; and Ray, P.; In Search of Musicality: Can Fractals Show Us The  Way?; Proceedings of the 1st World Congress on Lateral Computing (WCLC-2004)