### Friday, May 11, 2007

## Partitioning Melakarta Ragas into Equivalence Classes based on Sruti Bedham Relation

Overview:

In this post, I will examine the sruti bedham (aka graha bedham) relationship among ragas. I will show that ragas can be classified into different equivalence classes such that any two ragas belonging to the same class can be transformed into one another using graha bedham. To illustrate this, I have generated the equivalence classes for the melakarta ragas. I have used this classification to draw some conclusions and some interesting hypotheses.

Informal Treatment:

Sruti/Graha Bedham is the technique in which a new raga can be derived from the original raga, by changing the sruti (shadjam) . For example, if a singer is singing the raga mohanam (aro: S R2 G3 P D2) and he/she makes the rishabam (R2) of mohanam as the shadjam, then we get the madhyamavati raga (aro : S R2 M1 P N2 ). This is because (R2 G3 P D2 S) is equivalent to (S R2 M P N2) when R2 is assumed to be s. This is the essence of graha bedham.

Let me define a simple notation. Suppose A and B are two ragas, and it is possible to derive the raga B from A by assuming the swara j of raga A as shadjam, we denote this by A*I --> B. For example Mohanam*R --> Madhyamavati. This work hinges on three important facts which I will formally prove later. The two important facts are the following:

(1) Graha bedam relationship among ragas is symmetric: If the raga B can be derived from A through graha bedam then raga A can also be derived from B via graha bedham. That is, A*I-->B implies B*J--> A for some swara J. For example, we saw earlier how madhyamavati can be obtained from mohanam. Likewise, we can also obtain mohanam from madhyamavati by letting the nishadam of madhyamavati (N2) as shadjam.

(2) Graha bedam relation among ragas is transitive: If raga B can be derived from A and raga C can be derived from B then raga C can be derived from A. That is, A*I-->B and B*J--> C implies A*K-->C for some swara K. For example, madhyamavati can be derived from mohanam (by letting R of mohanam be the S) and hindolam can be derived from madhyamavati (by letting the R of madhyamavati be the S). This implies that hindolam can be derived from mohanam -- this is true. (by letting G of mohanam be the S).

(3) Graha bedam relation amon ragas is reflexive: This just says that the same raga can be derived from itself - which is trivially true, by not performing any sruti bedam and letting the shadjam be itself.

The above 3 facts imply that the sruti/graha bedham relation is an equivalence relation. This means that ragas can be partitioned into equivalence classes, such that any two ragas within the same equivalence class is related (they can be transformed into each other), but any two ragas from two different equivalence classes are not related. I wrote a program to find the equivalence classes across the melakarta ragas and this is the classification.

Equivalence Classes for Melakarthas:

Equivalence class of 6 ragas: (6x1 = 6)

(8,20,22,28,29,65): Thodi, Natabhairavi, Karaharapriya, Harikambodi, Shankarabharanam, Mechakalyani

Equivalence class of 4 ragas: (4x3) = 12

(9,35,56,66) : Dhenuka, Sulini, Shanmugapriya, Chitrambari

(10,23,26,64): Natakapriya, Gaurimanohari, Charukesi, Vachaspati

(14,21,58,71): Vakulabharanam, Kiravani, Hemavati, Kosalam

Equivalence class of 3 ragas: (3x6 = 18)

(2,19,53) : Ratnangi, Jhankaradvani, Gamanasramam

(3,54,55) : Ganamurthi, Viswambari, Shyamalangi

(7,17,63) : Senavati, Suryakantam, Lathangi

(15,57,72): Mayamalavagowla, Simhendramadyamam, Rasikapriya

(16,27,59): Chakravakam, Sarasangi, Dharmavati

(30,34,44): Naganandhini, Vagadeeshwari, Bavapriya

Equivalence class of 2 ragas: (2x10 = 20)

(1,51) : Kanakangi, Kamavardhini

(4,25) : Vanaspati, Mararanjani

(5,61) : Manavati, Kanthamani

(11,62): Kokilapriya, Rishabapriya

(13,69): Gayakapriya, Dhatuvardini

(18,43): Hatakambari, Gavambodhi

(24,32): Varunapriya, Ragavardhani

(33,60): Gangeyabhushani, Nitimati

(36,45): Chalanata, Subhapantuvarali

(46,70): Shadvidhamargini, Nasikabhushani

Equivalence class of 1 raga: (1x16 = 16)

(6,) : Tanarupi

(12,): Rupavati

(31,): Yagapriya

(37,): Salagam

(38,); Jalarnavam

(39,): Jhalavaraali

(40,): Navaneetam

(41,): Pavani

(42,): Raghupriya

(47,): Suvarnangi

(48,): Divyambari

(49,): Davalambari

(50,): Namanarayani

(52,): Ramapriya

(67,): Sucharitra

(68,): Jothiswaroopini

You can view this larger file, that lists in detail, through what swaras the transformation between the ragas occur. This information will be very useful to a carnatic musician.

Observations and Analysis:

1. Observe that the biggest equivalence class is the the one with the 6 ragas and it includes the widely acknowledged six major ragas in carnatic music. (Actually bhairavi instead of natabharavi and kambodhi instead of harikambodhi are part of the 6. But they are very close janya ragas). This leads me to propose that a raga is more 'stable' if it can be transformed into several additional ragas by performing sruti bedam. Here, the 'stability' refers to how long one can explore the raga. The rationale behind this claim is as follows: For a raga to be explored, one needs to be able to perform alapanai anchoring on several of the ragas swaras. By anchoring on a swara, it can be viewed in some sense as changing the sruti to that note. The fact that a raga arises ('stability') out of this anchoring, as opposed to no raga arising ('instability') means one can anchor on that particular note, while performing the alapanai. I know I am hand waving a little here, but let me explore this is detail in another post.

2. There seems to be a correlation between the how common a raga is and the cardinality of its equivalence class. We arleady saw how the 6 major ragas formed the biggest equivalence class. It is worth noting that relatively major ragas like shanmugapriya, kiravani, charukesi, hemavati are all from the equivalence class of cardinality of 4. Note that several prathi madhyama ragas (ragas with M2) are the ones which are alone and they are not in practical use.

3. Two ragas that are surprisingly alone : jalavaraali (janaka raga of varaali which is in wide use) and jothiswaroopini.

4. These equivalence classes of ragas have an important consequence for an automated raga identifier. If a naive identifier worked by finding the swaras in the composition, supposing it found the notes corresponding to say kharaharapriya, then how can the identifier be sure its is not any of the other 5 ragas from the same class? Thus I claim, any automated raga identifier must first identify the sruti (the shadjamam) of the composition, before it can identify the raga.

Formal Treatment:

In this section, I will prove the 3 facts that have been used in the informal section.

This post is becoming very long and I am also getting bored, let me do this in the next post.

Related Work (2nd Nov 2007):

I came across this work, which is very much related to the above work.

In this post, I will examine the sruti bedham (aka graha bedham) relationship among ragas. I will show that ragas can be classified into different equivalence classes such that any two ragas belonging to the same class can be transformed into one another using graha bedham. To illustrate this, I have generated the equivalence classes for the melakarta ragas. I have used this classification to draw some conclusions and some interesting hypotheses.

Informal Treatment:

Sruti/Graha Bedham is the technique in which a new raga can be derived from the original raga, by changing the sruti (shadjam) . For example, if a singer is singing the raga mohanam (aro: S R2 G3 P D2) and he/she makes the rishabam (R2) of mohanam as the shadjam, then we get the madhyamavati raga (aro : S R2 M1 P N2 ). This is because (R2 G3 P D2 S) is equivalent to (S R2 M P N2) when R2 is assumed to be s. This is the essence of graha bedham.

Let me define a simple notation. Suppose A and B are two ragas, and it is possible to derive the raga B from A by assuming the swara j of raga A as shadjam, we denote this by A*I --> B. For example Mohanam*R --> Madhyamavati. This work hinges on three important facts which I will formally prove later. The two important facts are the following:

(1) Graha bedam relationship among ragas is symmetric: If the raga B can be derived from A through graha bedam then raga A can also be derived from B via graha bedham. That is, A*I-->B implies B*J--> A for some swara J. For example, we saw earlier how madhyamavati can be obtained from mohanam. Likewise, we can also obtain mohanam from madhyamavati by letting the nishadam of madhyamavati (N2) as shadjam.

(2) Graha bedam relation among ragas is transitive: If raga B can be derived from A and raga C can be derived from B then raga C can be derived from A. That is, A*I-->B and B*J--> C implies A*K-->C for some swara K. For example, madhyamavati can be derived from mohanam (by letting R of mohanam be the S) and hindolam can be derived from madhyamavati (by letting the R of madhyamavati be the S). This implies that hindolam can be derived from mohanam -- this is true. (by letting G of mohanam be the S).

(3) Graha bedam relation amon ragas is reflexive: This just says that the same raga can be derived from itself - which is trivially true, by not performing any sruti bedam and letting the shadjam be itself.

The above 3 facts imply that the sruti/graha bedham relation is an equivalence relation. This means that ragas can be partitioned into equivalence classes, such that any two ragas within the same equivalence class is related (they can be transformed into each other), but any two ragas from two different equivalence classes are not related. I wrote a program to find the equivalence classes across the melakarta ragas and this is the classification.

Equivalence Classes for Melakarthas:

Equivalence class of 6 ragas: (6x1 = 6)

(8,20,22,28,29,65): Thodi, Natabhairavi, Karaharapriya, Harikambodi, Shankarabharanam, Mechakalyani

Equivalence class of 4 ragas: (4x3) = 12

(9,35,56,66) : Dhenuka, Sulini, Shanmugapriya, Chitrambari

(10,23,26,64): Natakapriya, Gaurimanohari, Charukesi, Vachaspati

(14,21,58,71): Vakulabharanam, Kiravani, Hemavati, Kosalam

Equivalence class of 3 ragas: (3x6 = 18)

(2,19,53) : Ratnangi, Jhankaradvani, Gamanasramam

(3,54,55) : Ganamurthi, Viswambari, Shyamalangi

(7,17,63) : Senavati, Suryakantam, Lathangi

(15,57,72): Mayamalavagowla, Simhendramadyamam, Rasikapriya

(16,27,59): Chakravakam, Sarasangi, Dharmavati

(30,34,44): Naganandhini, Vagadeeshwari, Bavapriya

Equivalence class of 2 ragas: (2x10 = 20)

(1,51) : Kanakangi, Kamavardhini

(4,25) : Vanaspati, Mararanjani

(5,61) : Manavati, Kanthamani

(11,62): Kokilapriya, Rishabapriya

(13,69): Gayakapriya, Dhatuvardini

(18,43): Hatakambari, Gavambodhi

(24,32): Varunapriya, Ragavardhani

(33,60): Gangeyabhushani, Nitimati

(36,45): Chalanata, Subhapantuvarali

(46,70): Shadvidhamargini, Nasikabhushani

Equivalence class of 1 raga: (1x16 = 16)

(6,) : Tanarupi

(12,): Rupavati

(31,): Yagapriya

(37,): Salagam

(38,); Jalarnavam

(39,): Jhalavaraali

(40,): Navaneetam

(41,): Pavani

(42,): Raghupriya

(47,): Suvarnangi

(48,): Divyambari

(49,): Davalambari

(50,): Namanarayani

(52,): Ramapriya

(67,): Sucharitra

(68,): Jothiswaroopini

You can view this larger file, that lists in detail, through what swaras the transformation between the ragas occur. This information will be very useful to a carnatic musician.

Observations and Analysis:

1. Observe that the biggest equivalence class is the the one with the 6 ragas and it includes the widely acknowledged six major ragas in carnatic music. (Actually bhairavi instead of natabharavi and kambodhi instead of harikambodhi are part of the 6. But they are very close janya ragas). This leads me to propose that a raga is more 'stable' if it can be transformed into several additional ragas by performing sruti bedam. Here, the 'stability' refers to how long one can explore the raga. The rationale behind this claim is as follows: For a raga to be explored, one needs to be able to perform alapanai anchoring on several of the ragas swaras. By anchoring on a swara, it can be viewed in some sense as changing the sruti to that note. The fact that a raga arises ('stability') out of this anchoring, as opposed to no raga arising ('instability') means one can anchor on that particular note, while performing the alapanai. I know I am hand waving a little here, but let me explore this is detail in another post.

2. There seems to be a correlation between the how common a raga is and the cardinality of its equivalence class. We arleady saw how the 6 major ragas formed the biggest equivalence class. It is worth noting that relatively major ragas like shanmugapriya, kiravani, charukesi, hemavati are all from the equivalence class of cardinality of 4. Note that several prathi madhyama ragas (ragas with M2) are the ones which are alone and they are not in practical use.

3. Two ragas that are surprisingly alone : jalavaraali (janaka raga of varaali which is in wide use) and jothiswaroopini.

4. These equivalence classes of ragas have an important consequence for an automated raga identifier. If a naive identifier worked by finding the swaras in the composition, supposing it found the notes corresponding to say kharaharapriya, then how can the identifier be sure its is not any of the other 5 ragas from the same class? Thus I claim, any automated raga identifier must first identify the sruti (the shadjamam) of the composition, before it can identify the raga.

Formal Treatment:

In this section, I will prove the 3 facts that have been used in the informal section.

This post is becoming very long and I am also getting bored, let me do this in the next post.

Related Work (2nd Nov 2007):

I came across this work, which is very much related to the above work.

Labels: equivalence class, graha bedham, melakartha, raga, sruti bedham

Comments:

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I think I finally came across a posting that i might have understood regarding the concept of sruti bedham. i like the way you explain it in words rather than JUST math. the first time i heard sruti bedham was sanjay subramaniam trying his hand at it. I might be totally mistaken, but i remember him singing bahudari and another ragam, which i dont know if its possible..

Thanks Ramya.

Bahudari :

Aro: S G3 M1 P D2 N2 S

Ava: S N2 P M1 G3 S

Since this raga is assymetric (there is no "D2" is the descent), it reduces the possibilities of sruti bedham.

However, if we assume "D2" in the descent, then there are 3 ragas that arise:

1. M as S --> Vilasini

2. P as S --> Sriranjani

3. N as S --> Saranga Tarangini

Bahudari :

Aro: S G3 M1 P D2 N2 S

Ava: S N2 P M1 G3 S

Since this raga is assymetric (there is no "D2" is the descent), it reduces the possibilities of sruti bedham.

However, if we assume "D2" in the descent, then there are 3 ragas that arise:

1. M as S --> Vilasini

2. P as S --> Sriranjani

3. N as S --> Saranga Tarangini

Bonjour, nvijayanand.blogspot.com!

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Superb write-up. Love the equivalence treatment on the melas, the observations are credit-worthy too. You have no idea how elated I feel to find a fellow musician who can explain things mathematically this way (i have a penchant for the two too)

Very detailed and an interesting post! Was looking for something on the internet and accidentally came across this one:) About the automated raga identifier - absolutely perfect. For years I always heard nee paadhi naan paadhi kaNNae as sarasaangi instead of chakaravagam. That was because I never heard the first few seconds of valachi and hence thought it was sa,,sa,,sa, n2r2sn2d1pm1 pa,,da1,,:) What a beautiful transition that from pa,,ni1 da2,,ga2 MA1,,ga2 ri1,, sa ...> ma,, ma,, ma, g2pm1g2r1sn1 sa,, ri,,! I always hear margazhi poove as shudha danyasi. It is just about how our mind sees it. The identifier might take Pa as the shadjam and tell you it is HindoLam. Hence an automated identifier cannot actually identify the shadjam -one would have to feed that initially, then it can do its job:) Keep writing more such stuff!

-Divya Girish

-Divya Girish

Excellent examples, Divya! Thanks!

Another song where IR does interesting graha bedam is the "vandhal vandhal rajakumari" in the charanam -- I've not analysed it completely -- but the transitions are really beautiful there.

btw, are you the Divya from X1F1 PSBB??

Another song where IR does interesting graha bedam is the "vandhal vandhal rajakumari" in the charanam -- I've not analysed it completely -- but the transitions are really beautiful there.

btw, are you the Divya from X1F1 PSBB??

Hahaha I dont know whether to laugh or cry listening to such a genius composition. Cannot believe I hadnt heard this song after the pallavi till now. Thanks for making me listen to vandhal vandhal rajakumari now. Genius genius!! He has made a shift from S to D2 so beautifully. Will analyze this song more. You can dedicate a separate post for this song alone - there is just so much stuff in this single composition:)

Yes I am the same Divya :)

Yes I am the same Divya :)

About VandhaL raja kumari charanam. Let us say it is based on Shankarabharanam and if we are talking about griha bedam,logically D2 becomes S'. So, it has to be Natabhairavi. Let us forget the raga bhavam etc - considering just the notes and logic. Later Natabhairavi's P becomes the R2 of Karaharapriya , but for just a small part. From there he moves to Gowrimanohari by introducing N2 suddenly! Then P of gowrimanohari(ie kharaharapriya with n2 introduced in place of n1) becomes S of Charukesi, which is Gowrimanohari's equivalent now.. and ends with P of Charukesi which is the G2 of the base raga Shankarabharanam! Then goes g1 m1 pa; papa; pa da pa pa daga g2 m1 d2 p d2 p m1 g2 r2 m1 g2 r2:) Let me know your views.

-Divya

-Divya

Great post Vijay. I learnt about shruthibedham a few years ago from Ilaiyaraaja's songs.

Divya: Regarding the ongoing discussion on Vandhaal Vandhaal Rajakumaari, thanks for the analysis. Interesting to know so many things going on in the charanam. I could identify charukesi in the 3rd and 4th lines and probably natabhairavi in the 1st 2 lines. Can you tell me the exact lines where the Gowrimanohari comes? I just thought there could be just 1 grahabedham to Natabhairavi and then the Ga2 changes to Ga3 to change to Charukesi and then returns back to Shankarabharanam via another Graha bedham ( Graha bedam relationship among ragas is symmetric :) )

One reason why there is a big change when the pallavi returns is that Shankarabharanam's Sa is not there in the Grahabedham to Charukesi. So when it returns back to Shankarabharanam, we notice a sea change.

Vijay: One thing I feel is that Grahabedham need not necessarily use the same set of notes. There could be 1 or 2 note deviations. Basically the Sa should change while continuing to use almost the same notes. This way, the listener will not feel any drastic difference when Grahabedham happens.

I had earlier cited 4 examples of shruthibedam in film music in my blog. Do check out those: http://rameshonmusic.blogspot.com/search/label/shruthibedham and drop in your comments.

I am also composing couple of tracks where I tried to use Grahabedham. I will share that later.

Divya: Regarding the ongoing discussion on Vandhaal Vandhaal Rajakumaari, thanks for the analysis. Interesting to know so many things going on in the charanam. I could identify charukesi in the 3rd and 4th lines and probably natabhairavi in the 1st 2 lines. Can you tell me the exact lines where the Gowrimanohari comes? I just thought there could be just 1 grahabedham to Natabhairavi and then the Ga2 changes to Ga3 to change to Charukesi and then returns back to Shankarabharanam via another Graha bedham ( Graha bedam relationship among ragas is symmetric :) )

One reason why there is a big change when the pallavi returns is that Shankarabharanam's Sa is not there in the Grahabedham to Charukesi. So when it returns back to Shankarabharanam, we notice a sea change.

Vijay: One thing I feel is that Grahabedham need not necessarily use the same set of notes. There could be 1 or 2 note deviations. Basically the Sa should change while continuing to use almost the same notes. This way, the listener will not feel any drastic difference when Grahabedham happens.

I had earlier cited 4 examples of shruthibedam in film music in my blog. Do check out those: http://rameshonmusic.blogspot.com/search/label/shruthibedham and drop in your comments.

I am also composing couple of tracks where I tried to use Grahabedham. I will share that later.

Gowrimanohari comes when he adds N2 in place of N1 in karaharapriya scale - r2,g1r2mg1r2ss, sr2 g1 pppp, r2m1pn2n2...s d2,ppp(in shankarabharam scale it is - g,mgpmgrrrg mddd,gp dr1r1r1 r2, n2 d2d2d2, where the raga's integrity collapses in the end. so never mind that). Yes you are right. He has deviated a bit. Charukesi is not shankarabharanams equivalent. It is just a wild imagination I feel. Such a coincidence that nabhairavi's P and charukesi's P coincide(forget the G) so when it ends it becomes G2 of parent shankarabharam. Note: forget the bhavam of notes of the said ragas. consider just the notes. and im using 12 swarasthana and not 16 here. so hope there is no confusion.

So nice to see you here in this forum, Divya! Hope you are doing well!

I cannot disagree with your fine analysis here -- as you know, however, there is also a subjective element here since the shadjam (sruti) is not very explicit in some places.

In my opinion, there is one sruti bedham here : Shankarabharanam to Kharaharapriya and back -- but I agree the nishadham of "karaharapriya" is changed, so it is really Gourimanohari.

In more detail (all swarams are with respect to the original shankabaranam scale)

(a) Start off with shankarabharanam

(b) Charanam, we hear: d d- d- d d d- d d pd- p m...

But what is this???...lets wait....

(c) then it clearly changes to gourimanohari as you say...r2,g1r2mg1r2ss, sr2 g1 pppp, r2m1pn2n2...s d2,ppp...

Since the (b) portion is so short and the swaras of the (b) portion are legal gourimanohari swaras, I am tending to assume that the start of the charanam (b) is also gourimanohari....it is just a subjective call.

Now coming to Mohan. First, thanks for your comments. Second, I really don't know if it is ok to change a note; but, one thing I can say is that it would make sruti bedham extremely complicated to analyse...:)

Finally, I saw your post...the examples are very nice indeed. I also remember IR used a lot of sruti bedham in "how to name it" -- shanmughapriya to srotaswini (g m p d ni s r) comes to my mind. Here again it is not a classical sruti bedham, in that, not all notes of shanmughapriya are used -- only a subset is.

I cannot disagree with your fine analysis here -- as you know, however, there is also a subjective element here since the shadjam (sruti) is not very explicit in some places.

In my opinion, there is one sruti bedham here : Shankarabharanam to Kharaharapriya and back -- but I agree the nishadham of "karaharapriya" is changed, so it is really Gourimanohari.

In more detail (all swarams are with respect to the original shankabaranam scale)

(a) Start off with shankarabharanam

(b) Charanam, we hear: d d- d- d d d- d d pd- p m...

But what is this???...lets wait....

(c) then it clearly changes to gourimanohari as you say...r2,g1r2mg1r2ss, sr2 g1 pppp, r2m1pn2n2...s d2,ppp...

Since the (b) portion is so short and the swaras of the (b) portion are legal gourimanohari swaras, I am tending to assume that the start of the charanam (b) is also gourimanohari....it is just a subjective call.

Now coming to Mohan. First, thanks for your comments. Second, I really don't know if it is ok to change a note; but, one thing I can say is that it would make sruti bedham extremely complicated to analyse...:)

Finally, I saw your post...the examples are very nice indeed. I also remember IR used a lot of sruti bedham in "how to name it" -- shanmughapriya to srotaswini (g m p d ni s r) comes to my mind. Here again it is not a classical sruti bedham, in that, not all notes of shanmughapriya are used -- only a subset is.

Mohan -- typo -- Shanmughapriya to Srotaswini:

G M D N R G (Shanmugapruya) = S G M P N S (srotaswini)

G M D N R G (Shanmugapruya) = S G M P N S (srotaswini)

Thank you Divya and Vijay for the explanations.

To me it is difficult to differentiate Gowrimanohari and Charukesi since the Shadjamam is not clear. As Divya mistook 'Nee paadhi' as Sarasaangi instead of Chakravagam, I earlier mistook 'Kannil Thagam' as Chakravagam instead of Sarasaangi.

BTW Vijay, my name is Ramesh and not Mohan :)

To me it is difficult to differentiate Gowrimanohari and Charukesi since the Shadjamam is not clear. As Divya mistook 'Nee paadhi' as Sarasaangi instead of Chakravagam, I earlier mistook 'Kannil Thagam' as Chakravagam instead of Sarasaangi.

BTW Vijay, my name is Ramesh and not Mohan :)

Oops sorry!

actually ramesh --> mohan is an instance of two names belonging to the same equivalence class in the name-bedham relation...

I better stop here, lest people suspect an instance of buddhi-bedham

actually ramesh --> mohan is an instance of two names belonging to the same equivalence class in the name-bedham relation...

I better stop here, lest people suspect an instance of buddhi-bedham

Vijay and Divya,

Listen to this video where Ilaiyaraaja sings and demonstrates grahabedam:

http://www.youtube.com/watch?v=U9UWshVmJqw#t=240s

Regards

Ramesh

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Listen to this video where Ilaiyaraaja sings and demonstrates grahabedam:

http://www.youtube.com/watch?v=U9UWshVmJqw#t=240s

Regards

Ramesh

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