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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
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].
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 3
[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.
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 4
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
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 5
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
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 6
Figure: 5.3. Graph for lni vs. lnF
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
Figure: 5.3. Graph for lni vs. lnF
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
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 7
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
Figure: 5.4. Graph for lni vs. lnF
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
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 8
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
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 9
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
Figure: 5.6. Graph for lni vs. lnF
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
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 10
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.
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 11
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.
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 12
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.
FRACTAL BEHAVIOUR ANALYSIS OF MUSICAL NOTES BASED ON DIFFERENT TIME OF RENDITION AND MOOD
Moujhuri Patra, Sirshendu Sekhar Ghosh and Soubhik Chakraborty 13
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
E-Mail: [email protected]
Mobile: 09955529117
3Dr. Soubhik Chakraborty
Department of Mathematics, Birla Institute of Technology (BIT), Mesra, Ranchi-835215,
Jharkhand, India
E-Mail: [email protected]
Mobile: 09835471223