### 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