FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON ... · Moujhuri Patra, Sirshendu Sekhar Ghosh...

International Journal of Research in Engineering, Technology and Science, Volume VI,

Special Issue, July 2016

www.ijrets.com, [email protected], ISSN 2454-1915

1

FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES

BASED ON DIFFERENT TIME OF RENDITION AND MOOD

Moujhuri Patra1,SirshenduSekhar Ghosh2, Soubhik Chakraborty3

1Department of Master of Computer Applications, NSEC Garia, Kolkata-700152, India 2Department of Computer Applications, NIT Jamshedpur, Jharkhand-831014, India

3Department 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/iD,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

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FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD

Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 2

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/iD

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].



[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.



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:


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

Higher Octave:


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



12

Figure: 5.1. Graph for lni vs. lnF

Raga Bivash

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

y = -1.82x + 4.6351R² = 0.6355

0.00

1.00

2.00

3.00

4.00

5.00

0.00 0.50 1.00 1.50 2.00 2.50

lnF

lni




Raga Bhimpalashree

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


y = -0.9664x + 3.3714R² = 0.1265

0.00

1.00

2.00

3.00

4.00

5.00

0.00 0.50 1.00 1.50 2.00

lnF

lni

y = -1.4122x + 3.9766R² = 0.4025

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.50 1.00 1.50 2.00 2.50

lnF

lni



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


y = -1.5871x + 4.6614R² = 0.6526

0.00

1.00

2.00

3.00

4.00

5.00

0.00 0.50 1.00 1.50 2.00 2.50

lnF

lni



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

Figure: 5.5. Graph for lni vs.lnF

y = -1.7196x + 4.5287R² = 0.6885

0.00

1.00

2.00

3.00

4.00

5.00

0.00 0.50 1.00 1.50 2.00 2.50

lnF

lni



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


R-squared (R2) 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 R2 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 R2 is not very high for all ragas.100R2, also called %

coefficient of determination, gives the percentage of variation in the response (here lnF)

y = -2.008x + 4.8359R² = 0.7624

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

lnF

lni



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.



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.



Author[s] brief Introduction

1Author 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.

2Author 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.

3Author 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.



Corresponding Address

1Dr. Moujhuri Patra

MCA Department, Netaji Subhash Engineering College (NSEC), Garia,

Kolkata-700152, West Bengal, India

E-Mail: [email protected]

Mobile: 09051450040

2Sirshendu Sekhar Ghosh

Department of Computer Applications, National Institute of Technology (NIT),

Jamshedpur-831014, Jharkhand, India


Mobile: 09955529117

3Dr. Soubhik Chakraborty

Department of Mathematics, Birla Institute of Technology (BIT), Mesra, Ranchi-835215,

Jharkhand, India


Mobile: 09835471223

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