A Comparative Study between Properties of Carnatic Raagas and Emotions of Devotional Songs Using Bidirectional Associative Memories (BAM)

A Comparative Study between Properties of Carnatic Raagas and Emotions of Devotional Songs Using Bidirectional Associative Memories (BAM)

A Comparative Study between Properties of Carnatic Raagas and Emotions of Devotional Songs Using Bidirectional Associative Memories (BAM)

A. Victor Devadoss1,

Head and Associate Professor1,

PG & Research Department of Mathematics, Loyola College, Chennai-34, India

S. Aseervatham2

Ph.D. Research scholar2

PG & Research Department of Mathematics, Loyola College, Chennai-34, India

Art is a means of expressing emotions to the fullest. One element that is common among all performing arts is their ability to express the emotions which make up different slices of life. Emotions characterize life as well as art. Music is one such language of emotions. In this paper, we propose an idea of how the properties of Carnatic raagas (Indian Classical Music) relate with the emotions while listening Tamil Christian devotional songs. This paper consists of four sections. Section 1 describes some basic ideas of Indian Classical Music and few previous works on it. The proposed method from the field of fuzzy logic is demonstrated in Section

2. Section 3 provides the description of the problem and hidden pattern. The conclusion of this paper is given in the final section

Key words Carnatic music, Neuronal dynamical system, Bivalent additive BAM, Synaptic connection matrices.

1. In 2004, Shivani Yardi et. al analysis show strong correspondence between the pitch patterns of each raga and its emotion and traditional performance time with the use of the Harmonic Network [1]. Parag Chordia and Alex Rae results suggest that raagas do consistently elicit specific emotions that are linked to musical properties and the responses from people did not differ significantly for enculturated and non- enculturated listeners, suggesting that musical rather than cultural factors are dominant [2]. Gopala Krishna

   Koduri and Bipin Indurkhya results show that ragas are quite useful as a first step in a different direction towards content-based music recommendation. Along the way, a total of 750 subjective emotional responses to tunes composed in popular raagas of Carnatic music are empirically investigated to find out the long speculated relation between raagas and rasas (emotion clusters) [3]. Considering all of these works, we have brought out the new foundation to carnatic raagas in devotional songs and the proposed BAM model of fuzzy logic also suited to bring out best result.

   1. Carnatic Music

      Carnatic music (Sanskrit: Karnataka samgita) is a system of music commonly associated with the southern part of the Indian subcontinent, with its area roughly confined to four modern states of India: Andhra Pradesh, Karnataka, Kerala, and Tamil Nadu. It is one of two main sub-genres of Indian classical music that evolved from ancient Hindu traditions; the other sub-genre being Hindustani music, which emerged as a distinct form because of Persian and Islamic influences in North India. In contrast to Hindustani music, the main emphasis in Carnatic music is on vocal music; most compositions are written to be sung, and even when played on instruments, they are meant to be performed in gayaki (singing) style.

      Although there are stylistic differences, the basic elements of sruti (the relative musical pitch), swara (the musical sound of a single note), raaga (the mode or melodic formulæ), and tala (the rhythmic cycles) form the foundation of improvisation and composition in both Carnatic and Hindustani music. Here A raga, the nucleus of Indian classical music, is a melodic structure with fixed notes and a set of rules characterizing a certain mood endorsed through

      performance. The notes in the raga are called swars. The seven swars Sa, Re, Ga, Ma, Pa, Dha and Ni in Indian music correspond to C, D, E, F, G, A, B in Western music. Actually there are 12 notes in Western music with Sharps and Flats notes. A sharp raises a note by a half-step which is denoted by #. A flat lowers a note by a half step which is denoted by b. Therefore, these 12 notes are C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab, A, A#/Bb and B then same C having twice of frequency than C. The equivalent notes in Indian classical music are S, R1, R2/G1, R3/G2, G3, M1, M2, P, D1, D2/ N1, D3/N2, N3 and S. The pictorial diagram is given in figure [1]. There are 72 Melakarta raagas and lot of Janya raagas within each one of Melakarta raagas.

      C# / Db D# / Eb F# / Gb G# / Ab A# / Bb

      R1

      R2

      R3 / G2

      M2

      D1

      D2

      D3 / N2

      S

      / G1

      G3

      M1

      P

      / N1

      N3

      S

      C D E F G A B C

      Figure 1. Note names in Harmonium

   2. Devotional Songs

      The most of the Christian devotional songs used to compose with some criteria that includes welcome ride (Entrance Hymn), Responsorial Psalm, Song for the presentation of the gifts (Offertory Hymn), Communion Hymn and finally Concluding ride (Thanks giving song). Composers may use any raagas for these songs in which we mentioned above criteria. Here, we have taken only nine raagas with accidental notes in a sample space. Finally we decided to take only three dominated raagas from each criterion among these nine raagas.

2. Bidirectionality, forward and backward information flow, is introduced in neural networks to produce two- way associative search for stored stimulus-response associations (Ai,Bi).

   A group of neurons forms a field. Neural networks contain many fields of neurons. Fx denotes a neuron field which contains n neurons and Fy denotes a neuron field which contains p neurons.

   Neuronal Dynamical Systems The neuronal dynamical system is described by a system of first order differential equations that govern the time evaluation of the neuronal activations or membrane potentials.

   Xi gi X ,Y ,…, Yj hj X ,Y ,…

   Where xi and yj denote respectively the activation time function of the ith neuron in Fx and the jth neuron in Fy. The over dot denotes time differentiation, gi and hj are functions of X, Y etc., where X(t) = (x1(t),..,xn(t), Y(t) = (y1(t),..,yn(t)) define the state of the neuronal

   dynamical system at time t. Additive Bivalent Models describe asynchronous and stochastic behaviour. At each moment each neuron can randomly decide whether to change state, or whether to omit a new signal given its current activation. The BAM is a non- adaptive, additive, bivalent neural network.

   1. Bivalent Additive BAM

      In neural literature, the discrete version of the earlier equations is often referred to as the Bidirectional Associative Memories or BAMs. A discrete additive BAM with threshold signal functions, arbitrary thresholds and inputs, an arbitrary but a constant synaptic connection matrix M and discrete time steps K are defined by the equations.

      i j j ij i

      i j j ij i

      p

      xk 1 S yk m I

      j

      j i i ij j

      j i i ij j

      n

      yk 1 S yk m I

      i

      Where, mij M, Si and Sj are the signal functions. They represent binary or bipolr threshold functions. For arbitrary real-valued thresholds U = (U1,U2,Un) for Fx neurons and V = (V1,V2,Vn) for Fy neurons. The threshold binary signal functions corresponds neurons.

   2. Synaptic Connection Matrices

      Let us suppose that the field Fx with n neurons is synoptically connected to the field Fy with p neurons. Let mij be a synapse where the axon from the ith neuron in F terminates, mij can be positive, negative or zero.

      The synaptic matrix M is a n p matrix of real numbers whose entries are the synaptic efficacies mij. The matrix M describes the forward projections from the neuronal

      field Fx to the neuronal field Fy. Similarly, MT, a p n

      synaptic matrix and describes the backward projections

      Fy to Fx.

   3. Unidirectional Networks

      These kinds of networks occur when a neuron synoptically interconnects to itself. The matrix N is

      n n square matrix.

   4. Bidirectional Networks

      A network is said to be a bidirectional network if M = NT and N = MT.

   5. Bidirectional Associative Memories

      When the activation dynamics of the neuronal fields Fx and Fy lead to the overall stable behaviour, the bi- directional networks are called as Bi-directional Associative Memories or BAM. A unidirectional network also defines a BAM if M is symmetric

      i.e. M = MT.

   6. Additive Activation Models

      An additive activation model is defined by a system of n+p coupled first-order differential equations that interconnects the fields Fx and Fy through the constant synaptic matrices M and N described earlier. Si(xi) and Sj(yj) denote respectively the signal function of the ith neuron in the field Fx and the signal function of the jth neuron in the field Fy. Discrete additive activation

      models correspond to neurons with threshold signal functions. The neurons can assume only two values ON and OFF. ON represents the signal value +1 and OFF represents 0 or -1 (-1 when the representation is bipolar). The bipolar version of these equations yield the signal value -1 when xi<Ui or yj<Vj.

      The bivalent signal functions allow us to model complex asynchronous state-change patterns. At any moment different neurons can decided whether to compare their activation to their threshold. An each moment any of the 2n subsets of Fx neurons or the 2p subsets of Fy neurons can decide to change state. Each neuron may randomly decide whether to check the threshold conditions in the equations given above.

      At each moment each neuron defines a random variable that can assume the value ON (+1) or OFF (0 or -1). The network is often assumed to be deterministic and state changes are synchronous i.e. an entire field of neurons is updated at a time. In case of simple asynchrony only one neuron makes a state change decision at a time. When the subsets represent the entire fields Fx and Fy synchronous state change results. In a real life problem the entries of the constant synaptic matrix M depends upon the investigators

      feelings. The synaptic matrix is given a weight age according to their feelings. If xFx and yFy the forward projections from Fx to Fy is defined by the

      matrix M.:{F(xi,yj)} = (mij)=M, 1<i<n, 1<j<p. the backward projection is defined by the matrix M T.

      :{F(yj,xi)} = (mji) = M T, 1<i<n, 1<j<p.

   7. Bidirectional Stability

   All BAM state changes lead a fixed-point stability. This property holds for synchronous as well as synchronous state changes.

   A BAM system (Fx, Fy, M) is bidirectionally stable if all inputs coverage to fixed point equilibrium. Bidirectional stability is a dynamic equilibrium. The same signal information flows back and forth in a bidirectional fixed point.

   Let us suppose that A denotes a binary n-vector and B denotes a binary p-vector. Let A be initial input to the BAM system. Then the BAM equilibrates a bi- directional fixed point (Ai,Bj) as

   A M B

   A MT B

   p

   p

   A M B

   .

   x A x S ( yk )m I

   A MT B

   1. i j j ji i j

      . n

      …

      Af M Bf

      Af M B

      Af M B

      y A y S ( yk )m I T

      f

      f

   2. j i i ij j i

   Where A, A and B, B represents intermediate or transient signal state vectors between A and Af , B and Bf respectively. The fixed point of a bidirectional system is time dependent. The fixed point for the initial input vectors can be attained at different times which are illustrated later. Based on the synaptic matrix M which is developed by the investigators feelings, the time at which bidirectional stability is attained also varies accordingly.

3. The raaga roots for the taken ten ragas in Carnatic notes are given below

   Ra1 SivaRanjani – S R2 G2 P D2 S

   Ra2 Natabhairavi – S R2 G2 M1 P D1 N2 S

   Ra3 Kiravani – S R2 G2 M1 P D1 N3 S

   Ra4 Karaharapriya – S R2 G2 M1 P D2 N2 S

   Ra5 Harikambhoji – S R2 G3 M1 P D2 N2 S

   Ra6- DhiraShankarabharanam-S R2 G3 M1 P D2 N3 S

   Ra7 Mohanam – S R2 G3 P D2 S

   Ra8 Hamsadwani – S R2 G3 P N3 S

   Ra9 MechaKalyani – S R2 G3 M2 P D2 N3 S Accidental note – Notes other than raaga root notes

   From the Table [1] given below, we take only the following dominated raagas attributes in domain and the emotions of five different criteria of Christian devotional songs in co-domain.

   E7 Accompany and solemnize the communion of the faithful

   E8 To join in the procession without the distraction of carrying hymnals or other worship aids

   E9 Give thanks for the whole work of salvation Now we have given the relations with weights by a bipartite graph among these attributes from the expert opinions and theoretical aspects.

   P1 P2 P3 P7

   . . .

   . . .

   E1 E2 E3 E9

   Figure 2.

   As the data is an unsupervised one and involves lot of uncertainties we are given a dynamical system in matrix M and its reverse M T with weighted values.

   1	3	3	1	3	3	3	3	0
   1 3		3	2	3	3	3	3 0
   0	3	3	1	3	3	3	3	0
   M 2 1		1	3	2	2	2	1 3
   1	1	2	3	2	2	1	2	3
   2	1	1	3	1	1	1	1	3
   3	1	1	2	0	0	0	0	0

   td>

   2

   1	3	3	1	3	3	3	3	0
   1 3		3	2	3	3	3	3 0
   0	3	3	1	3	3	3	3	0
   M 2 1		1	3	2	2	2	1 3
   1	1	3	2	2	1	2	3
   2	1	1	3	1	1	1	1	3
   3	1	1	2	0	0	0	0	0

   P1 Devotion of the Listeners

   P2 Feelings of grandeur

   P3 Elevated to Ecstasy

   P4 Large scope for Compositions

   P5 Mellifluous and smooth

   1 1 0 2 1 2 3

   3 3 3 1 1 1 1

   P6 Laid back majestic presentations

   P7 Usually sing at a beginning of performance

   3 3 3 1 2 1

   1 2 1 3 3 3

   1

   2

   Where P1, P2

   and P3

   are from Natabhairavi, P4, P5

   and

   M T 3 3 3 2 2 1 0

   P6 from Dhira Sangarabharanam and P7 from	3	3	3	2	2	1	0
   Hamsadhwani.	3	3	3	2	1	1	0
   3 3			3	1	2	1	0
   E1 To greet with pleasure and hospitality 0 0			0	3	3	3	0

   P6 from Dhira Sangarabharanam and P7 from	3	3	3	2	2	1	0
   Hamsadhwani.	3	3	3	2	1	1	0
   3 3			3	1	2	1	0
   E1 To greet with pleasure and hospitality 0 0			0	3	3	3	0

   E2 An aid to prayer

   E3 Personal devotion and private meditation

   E4 Acts of self surrender and oblation

   E5 To receive or accept with satisfaction

   E6 To meet, receive or acknowledge in a worshiped way

   Let the initial vector X1 = (1 0 0 0 0 0 0) in BAM. The effect of X1 on the dynamical system M as follows.

   X1M = (1 3 3 1 3 3 3 3 0)

   (1 1 1 1 1 1 1 1 0) = Y1

   Table 1. Domination of Carnatic raagas in Christian devotional songs

   S. No	Ra-Raga	Ra1	Ra2	Ra3	Ra4	Ra5	Ra6	Ra7	Ra8	Ra9	Accidental notes	Dominated Raaga
   	Song
   I		Entrance Hymn										Hamsadwani
   1	Archanai Malaraaga						Y
   2	Azhaikirar Yesu Aandavar						Y
   3	Anbai Kondaaduvom Irai						Y
   4	Irai Yesu Azhaippaetru								Y
   5	Sammadhamae Iraivaa			Y
   6	Thamizhaal Un Pugazh								Y
   7	Idhayangal Malarattumae								Y
   8	Isaiyil Swaram Saerththu								Y
   9	Dhinamdhorum Dhinamdhorum		Y
   10	Iraivan Nammai Azhaikkinraarae							Y
   11	Varam Kaettu Varukin	Y
   Average		0.09	0.09	0.09	0	0	0.27	0.09	0.36	0	0
   II		Responsorial Hymn										Natabhairavi
   12	Aandavarae En Aanmaavin		Y								Y
   13	Inba Kanavondru Naan		Y								Y
   14	Amaithiyin Thoodhanaai Ennayae		Y								Y
   15	Aandavari Naan Poatriduvaen									Y
   16	Yesuvae Enniraivaa Vumadhu					Y
   17	Irai Samookamaai Naangal					Y
   18	Nirandharam Nirandharam		Y								Y
   19	Nilayillaa Vulaghu Nijamillaa		Y
   20	Oru Kodi Paadalkal Naan	Y
   21	Ungalukku Samaadhaanam						Y
   Average		0.1	0.5	0	0	0.2	0.1	0	0	0.1	0.4
   III		Offertory Hymn										Dhira Sangarabharanam
   22	Ponnum Porulumillai Ennidaththil		Y								Y
   23	Anbin Paliyaai Aerppaai Vummai	Y
   24	Yennayae Muzhuvadhum							Y
   25	Yellaam Tharukinroam Thanthaai							Y
   26	Arppanam Arppanam Arppanamae				Y
   27	Idhayam Paadum Iniya						Y
   28	Archai Malaraai Vanthaen						Y				Y
   29	Naanae Oru Kaanikkai						Y
   30	Idho Umadhu Adimai Iraivaa		Y
   31/p>	Yeduththukkollum Aandavarae En						Y
   Average		0.1	0.2	0	0.1	0	0.4	0.2	0	0	0.2
   IV		Communion Hymn										Natabhairavi with Accidental Notes
   32	Uravu Ondru Ulagil Thedi						Y				Y
   33	Ovvoru Pagirvum Punitha		Y								Y
   34	Yaezhisai Naadhanae Iraivaa		Y								Y
   35	Thiruvirundhu Thiruvizhaa		Y								Y
   36	Iraivanin Vaanaga Virundhu		Y								Y
   37	Unnodu Naan Virundhunna						Y				Y
   38	Yenil Vaarum En Yesuvae		Y								Y
   39	Unnil Naan Ondraaga Uyirae					Y
   40	Yennodu Nee Pesa Vandhaai					Y
   41	Thedum Anbu Theivam					Y
   42	Theivamae Vaarum Yennilae Naan			Y							Y
   43	Yen Vaazhvil Yesuvae Ennaalum			Y							Y
   44	Vaazhvai Alikkum Vallavaa						Y
   Average		0	0.38	0.15	0	0.23	0.23	0	0	0	0.69
   V		Thanks Giving Song										Dhira Sangarabharanam
   45	Ummai Potrukindrom Ummai						Y
   46	Nandreeyaal Thuthi Paadu						Y
   47	Nandri Koori Paaduvom Nalla						Y
   48	Ummai Thedi Vanthom sumai						Y
   49	Yennil Vantha Naadhanukku		Y								Y
   50	Alayolir Arunanai Aninthidumaam						Y
   51	Yesuvin Pinnaal Naanum Selvaen						Y
   52	Ootruth Thanneerae Endhan						Y
   53	Thanthaani Thuthppomae						Y
   54	Nandrikal Pala Koori							Y
   55	Oyaadha Karunaiyin Iraivanae						Y
   Average		0	0.09	0	0	0	0.81	0.09	0	0	0.09
   Membership (Net Average)		0.058	0.234	0.048	0.02	0.086	0.362	0.094	0.072	0.02	0.276	0.362 (Max)

   Y1M T = (20 19 19 14 14 11 -1)

   (1 1 1 1 1 1 0) = X2

   X2M = (5 12 13 13 14 14 13 13 9)

   (1 1 1 1 1 1 1 1 1) = Y2

   Y2M T = (20 19 19 17 17 14 -1)

   (1 1 1 1 1 1 0) = X3 (= X2)

   X3M = (5 12 13 13 14 14 13 13 9)

   (1 1 1 1 1 1 1 1 1) = Y3 (=Y2)

   Hence the limit point is ((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1

   1)).

   The set of all limit points corresponding to the different input vectors

   No	Input Vector	Limit points
   1	(1 0 0 0 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   2	(0 1 0 0 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   3	(0 0 1 0 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   4	(0 0 0 1 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   5	(0 0 0 0 1 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   6	(0 0 0 0 0 1 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   7	(0 0 0 0 0 0 1)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))

   No	Input Vector	Limit points
   1	(1 0 0 0 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   2	(0 1 0 0 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   3	(0 0 1 0 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   4	(0 0 0 1 0 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   5	(0 0 0 0 1 0 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   6	(0 0 0 0 0 1 0)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))
   7	(0 0 0 0 0 0 1)	((1 1 1 1 1 1 0),(1 1 1 1 1 1 1 1 1))

Table 2.

1. Gopala Krishna Koduri and Bipin Indurkhya, A Behavioral Study of Emotions in South Indian Classical Music and its Implications in Music Recommendation Systems, SAPMIA10, ACM, October 29, 2010, Firenze, Italy.

2. Soubhik Chakraborty, Rayalla Ranganayakulu, Shivee Chauhan, Sandeep Singh Solanki and Kartik Mahto, A Statistical Analysis of Raga Ahir Bhairav, The Journal of Music and Meaning (JMM), Winter 2009, vol.8. pp. 1-9.

3. Bart Kosko, Adaptive Bidirectional Associative Memories, APPLIED OPTICS, 1 December 1987, Vol.26, No.23, pp. 4947-4960.

4. Bart Kosko, Bidirectional Associative Memories, IEEE Transactions on Systems, Man and Cybernetics,

   We have analysed the relation between properties of Carnatic raagas and emotions of Christian devotional songs where the fuzzy model BAM helped us to get the best result. The result shows that the attributes P1, P2, P3, P4, P5 and P6 from Properties of Carnatic ragas and

   January/February 1988, VOL. 18, No. 1, pp. 49-60.

5. A. Victor Devadoss and A. Felix, A New Bidirectional Associative Neutrosophic Cognitive Dynamical System approach to study Youth Violence, International Journal of Computer Applications (IJCA), Volume 53-No.11, Sptember 2012, pp. 32-36.

E1, E2, E3, E4, E5, E6, E7, E8 and E9 from Emotions of

Devotional songs are in ON state except P7 from the set of all limit points for each input vector and we also got the same output of limit points. So the raagas Natabhairavi, Dhira Sangarabharanam and Hamsadwani properties ultimately provide good emotions which are to greet with pleasure and hospitality, an aid to prayer, personal devotion and private meditation, acts of self surrender and oblation, to receive or accept with satisfaction, to meet, receive or acknowledge in a worshiped way, accompany and solemnize the communion of the faithful, to join in the procession without the distraction of carrying hymnals or other worship aids, give thanks for the whole work of salvation in Christian devotional songs.

So the composers can use these raagas to make devotion and peaceful life.

This research is supported by UGC scheme MANF. Award Letter ID: F1-17.1/2011/MANF-CHR-TAM-

7467 / (SA-III/Website).

1. Shivani Yardi and Elaine Chew, Giving Ragas the Time of Day: Linking structure, emotion and performance time in North Indian Classical Music using the Harmonic Network", Proceedings of the 8th International Conference on Music Perception & Cognition (ICMPC), Evanston, IL, USA, 2004, pp.705-708.

2. Parag Chordia and Alex Rae, Understanding Emotion in Raag: An Empirical Study of Listener Responses, International Conference on Mathematical Computing (ICMC), Atlanta, 2007.