During high school days when I first learnt about the permutations and combinations in my Mathematics,The first thing that occurred to me is the set of notes in Carnatic music and the number of ragams that can be generated out of it. This looked like a classic example to me. Those days I used to jump into solving and used to get unusually high number of ragams .. Then I left it and did not think I should be looking into it. While I said I left it, I really did not leave it.. I then decided, i should first write down the rules of forming Melakartha Ragams if I should really solve this !!
the result is what you will read here.
I wanted to see if I can mathematically arrive at the total number of ragams/ ragas/ scales we can have with 17 notes.
So the problem definition is given as :
Find the number of ragas. let us be a bit more clear and define this mathematically with definitions/rules and exceptions?
Definition : Melakartha Ragam/Raga/Scale : Series of 8 notes, where the first and 8th note are essentially same, but are in different octaves. Also the important rule is that these notes come down the way they go up. that is same notes appear in the aarohanam ( upscale ) and avarohanam (downscale).Definition from Wikipedia says
Rules for Melakarta ( from Wikipedia)
Ragas must contain the following characteristics to be considered Melakarta.
- They are sampurna ragas - they contain all seven swaras (notes) of the octave in both ascending and descending scale[1][2]
- They are krama sampurna ragas - that is the sequence is strictly ascending and descending in the scales, without any jumps or Vakra Prayogams
- The upper shadjam is included in the raga scale[2] (ragas like Punnagavarali and Chenchurutti are not melakarta as they end with nishadham)
- The ascending and descending scales must have the same notes [2]
Definition and Specification of Notes. Total number of notes : 17 including the first note from the second octave. These notes are as follows.
1. Sa ( shadjyamam)
2. R1 ( Shudha Rishabham)
3. R2 ( Chatushruti Rishabham)
4. R3 ( shatshruti Rishabham)
5. Ga1 ( Shudha Gandharam)
6. Ga2 ( Sadharana Gandharam)
7. Ga3 ( antara gandharam)
8. Ma1 ( shudha Madhyamam)
9. Ma2 ( prati madhyamam)
10. P ( Panchamam)
11. Da1( Shudha Dhaivatam)
12. Da2(chatushruti Dhaivatam)
13. D3( Shatshruti Dhaivatam)
14. Ni1 ( shudha Nishadam)
15. Ni2(Kaishiki Nishadam)
16. Ni3( Kakali Nishadam)
17. S ( Shadjyamam top : of next octave)
Rules :
1. Total 8 spaces
2. 1st space is always taken by Sa
3. 2nd space is taken by any one of the three notes from { R1,R2,R3}
4. 3rd space is taken by any one of the three notes from { G1,G2,G3}
5. 4th space is taken by any one of the two from { M1,M2}
6. 5th space is always taken by Pa
7. 6th space is taken by any one of the three notes from { D1,D2,D3}
8. 7th space is taken by any one of the three notes from { N1,N2,N3}
9. 8th space is always taken by top Sa
Exceptions: Permutations of R and G
1. R1 can go with G1,G2 or G3
2. R2 can go Only with G2 or G3 ( not G1 as R2 and G1 are the same notes essentially . I dont want to digress into the topic of why they are named different when they are actually the same notes... Scroll Down to the Vivaadi Raga section for explanation on this ..)
3. R3 can go only with G3 ( not G1 and G2 because G1 is R2 and Going up, R2 can never come after R3 , and G2 is Same as R3 for the reason in point 2 above.
Permutations of D and Ni
1. D1 can go with N1,N2 or N3
2. D2 can go Only with N2 or N3 ( not N1 as D2 and N1 are the same notes essentially )
3. D3 can go only with N3 ( not N1 and N2 because N1 is D2 and Going up, D2 can never come after D3 , and N2 is Same as D3 for the reason in point 2 above.
Solution: Looking at the Permutations of R and G together .. If we assume one of the three notes R1,R2 and R3 can occupy the second position, then the number of Permutations of R and G together is 6 as:
>> R1 can appear with 3 versions of G
>> R2 can appear with 2 versions of G
>> R3 can appear with 1 version of G
So this can happen in 6 ways
Now that the positions of R and G are fixed, Let us move on to Ma.. 4th position Ma can be occupied by either one of M1 or M2. So its 2P1 ways which is 2 again The permutations of D and N is identical to R and G permutations.. so the number of permutations again is 6 for the D and Ni positions.
First position can be occupied in 1 way, 5th can be occupied in 1 way and 8th can be occupied on 1 way.. so the total permutations is :
============
1*6*2*1*6*1 = 72
============
Another way to get to the solution is :
For the Positions of Ri and Ga, If no rules were applied, then the number of permutations would have been 3*3 = 9
but applying rules,
- R2 can not come with G1 ( both are same notes)
- R3 can not come with G1( G1 = R2 .. so since R2 can not come after R3, this combination can not happen
- R3 can not come with G2 ( both are same notes)
Identical reasoning for the positions of Da and Ni.
For position of Ma there are 2 possibilities.
So total possibilities are
1*6*2*1*6*1 = 72 || QED ||
I stressed on saying permutations here because the order was important.
So total number of Melakartha Ragas is 72.
Vivaadi Ragas
Interesting point here is, while doing the melakartha , the creator of the scheme decided that the vivadi note should be considered in the Melakartha Scheme and can be a Janaka Raga ( Parent Raga ).. By Vivadi , I mean when the 2 notes of the same jaathi come together. That is when R1 and R2 come together or G1 and G2 come together. In Carnatic Scale, it is NEVER pronounced as R R when R1 and R2 come together. R2 is then pronounced as G and that G ends up becoming shudha gandharam. So technically Frequency of the Shudha Gandharam note is the same as Chatushruti Rishabham note. This is what I was talking about earlier in this blog when I said they are same.This could be a perfectly valid reason , because the definition of Melakartha , and by having the vivadi notes concept in picture, they fit the bill. But.. Should the Vivadi Notes be qualified as proper notes and should they be considered for melakartha scheme is again something I wouldn't want to comment. It probably is possible that if they are not , then the Melakartha will be reduced to 2*2*2*2*2 = 32 and the remaining 40 ragas ( that are formed out of vivadi swaras) will have to be made as Janya Ragas ( child ragas ) of the main 32 ragas.. Which could be an interesting thing to do.. but is a completely different paradigm of thought. I wouldn't want to go there :-)
Ambiguity in Some notes
So after all that discussion on the notes and the schemes and the scales etc.. Those who have learnt carnatic music, especially those who are violinists or some other instrument players of carnatic music, will know that they actually play some notes which are not in any of these 16 frequencies ( i dint say 17 because 17th in this blog is the first note in upper octave) .. So what are they?? Carnatic classical music and also Hindustani Classical music defines a unique 22 shruti concept .. Read about it here ..
I can straight away give examples of these ragas ..Varali for example.. The scale is
S G1 R1 G1 M2 P D1 N3 S | S N3 D1 P M2 G1 R1 S
This ragam is Janya ( Child ) of Jaala Varali which is 39th Melakartha Ragam.
Although the Documents and text books read M2, the Instrumentalists know that it is actually slightly higher than M2 and Slightly less than Pa. So really should these notes have their own names ? and should it then have been added into the Melakartha scheme? If theoretically we did, then will it be sounding the same?? Will human mind be able to differentiate between the prati madhyamam and the frequency after that and before the Pa ( Panchamam) ??
All of these for the pure pleasure of theoretical study..
