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..
That is a good read. Keep it up mate!
ReplyDeleteThat is mind boggling information for those who are not aware of our music's greatness. I guess that is why you are blogging your mind off and blocking the readers' minds. :D
ReplyDeleteVarali's Ma according to me is not touching any anya swara or a different note. I see it as utilising its entire space wherein I view each swara or note having its own box of space. So, according to me, in Varali's case, the Ma has the freedom to move till the edge of its box almost touching the boundary of Pa but not exactly touching it and so giving it a unique identity. :)
Thanks Vijay..
ReplyDeleteBut then Varali Ghamakam goes up and up and enters into the frequency above Prati Madhyamam and below Panchamam. Now, the interesting thing you brought out is the freedom.
When we talk freedom, are we saying each note has a range of frequencies where it can go up to? But strictly , every note should have a fixed frequency isnt it?
Like shankara bharanam ghamakam to ri starts at ri and touches ga and comes back, which is OK, as we are still talking about fixed frequency.. touching other notes and coming back .. but somehow, varali and asaveri kind of ragas , i think falls in a fuzzy understanding system and not as per documentation of fixed frequency of notes as given in some text books..
No doubt that makes carnatic a subject of extreme intelligence and research..
Hey btw, I am thinking that this is probably the reason the Old school of thoughts ssay that Guru Should not teach varali to Students, as it may end the Guru Shishya relationship. I see how this can happen :-) Guru says , varali is this... and shishya says, i dont get this as this is not purely prati madhyamam. Guru says Oh! Well. you got to do it this way and it is prati madhyamam and the conflicts starts there :-) So the old School says, Students should learn this by listening alone. Leaving the students to develop there own intelligence on the subject :-) this adds up
Vasu,
ReplyDeleteWanted to add some of my random notes, not in any specific order. More discussions welcome :)
1. When the mELakartha system took its roots, it clearly did not need mELas to be krama sampUrNa. Nominal requirement was that it should have all the 7 notes (ascent and descent combined). That's why rAgas such as kAmbhOji and nATa were considered to be mELas. Actually in some rare cases, even rAgas that did not have all the 7 notes were considered mELas too (for eg bhoopAla)
2. Even when Venkatamakhi proposed his 72 mELa scheme, he did not require the mELas to have krama sampUrNatva. But he did illustrate the combinations of R, G , D , N (which you have elaborated well in your post)
3. Although Venkatamakhi proposed the 72 mELa scheme, he describes only 20 of them (that were current at his time) and leaves the rest for the imagination of musicians to come later.
3. Sometime after Venkatamakhi, an asampoorNa mELa list was created to fill up the 72 mELa table. Rules were exactly as before - A raga could be a mELa if it could be a representative of the rAgas taking that note and if it has all 7 notes (not necessarily in order). This scheme took the aesthetic aspects of the rAga in coming up with new rAgas to fill up the mELa table.
4. Sometime after this (possibly) came up the current krama sampoorNa scheme (which we now refer to as the kanakAngi scheme) where all the 7 notes were to occur in both ascent and descent in order.
5. When this scheme came up, lot of rAgas which were supposed to be mELas had to be turned into the children of the newly created mELas. For examples - varALi became a janya of jhAla varALi, and nATa became a janya of chalanATa, kAmbhOji a janya of harikAmbhOji and vasantabhairavi a janya of vakuLabharana.
IMHO, although krama sampUrNa scheme yielded some new rAgas which were brought to life by Tyagaraja (primarily), it also did a dis-service by pushing back some beautiful asampUrNa mELas.
6. Coming to considering vivAdi notes while coming up with mELa schemes - Since two of the most ancient mELas (nATa and varALi) are vivAdi so to say, I think it should not be a contentious issue at all.
Vasu...Very nice arithmetic twist to a melodious subject :-) Never thought this way...
ReplyDeleteI will post my "Serious" thoughts about this subject later.
Keep blogging!!! - Vishnu
good!
ReplyDeleteThanks Vishnu..
ReplyDeleteHey Hamsanandi,
That was an elaborate post.. I just want to bring back the original intention of the post.
1) To Prove mathematically the Number of Melakartha Ragams
2) To see if there are more melakartha ragams possible.
While I proved the first point, The second was a little bit tricky and I intended to put down my thoughts on that.
My interest here is particularly the Madhyamam of Varali. As it is not the same frequency of Prati Madhyamam and it is slightly above the Prati madhyamam and below Panchamam, Does it qualify to be a third Ma?? I am not challenging the decisions of the Creators of these schemes and neither am I suggesting that we shud have a third Ma, I am just writing as a pure theoretical point of discussion.. So If Varali Madhyamam is called as a komala madhyamam or some such thing.. Will it be theoretically correct, assuming the rules that we applied when the Other notes were named. If we did so, will the human mind be able to differentiate between the notes? If so then what were the reasons these were not taken ??
Hamsanandi Explained a lot and gave us a lot of insight into the pre Melakartha Era..where ragas were mixed etc.. where the parent child relation of the raagas were also different. he also explained about certain post Melakartha Era facts, which was very very good read. Thanks for the info :-)
Although, I was mostly concerned about post melakartha era. where the rules were decided. I wanted to play with those exact rules and try to question more...
Also why Vivadi notes are considered for Melakartha Ragams??? I appreciate you said they were there in the Ancient scheme.. but with the new scheme, why shud we have the phantom notes in the Melakartha?? Is it because having both Ris together in some ragam will make it difficult to decide which one shud be the parent?? Some how I am not too much convinced about these vivadi notes being a part of the PArent scales :-).. We already have cases while both notes come in some ragas but one going up and other coming down.. Like in Bhairavi.. Going up is Chatushruti Dhaivatam and COming down is shudha dhaivatham.. So With Arohanam being to Karaharapriya and Avarohanam being to Natabhairavi, we decided that parent should be natabhairavi, based on sangathis and prayogams.. Although , purely speaking, Natabhairavi does not have chatushruti dhaivatam.. so really we are kind of tweeking the point here already. So if We have both Da together, than it could be a Janya of either the samporna raga of D1 or Sampoorna raga of D2 ??
Example:
we tweek karaharapriya ( added D1 and removed n2)
S R2 G2 M1 P D1 D2 S
in current scheme, this would be
S R2 G2 M1 P D1 N1 S | S N1 D1 P M1 G2 R2 S
which is Jhankara dhwani
So could we have scrapped JhankaraDhwani out of Melakartha and added it as a child of either
KaraharaPriya ( based on Chatu Dhaivatam) or
Natabhairavi ( based on Shudha Dhaivatam)
But again, What is the basis of the Ragam being parent?
1) Notes??? *or*
2) Bhava ?? *or*
3) Prayogams / Sangathees ??
It looks like current pattern of Parent and Child is based one or all of the above in one raga or the other..
( May be learnt experts know all these out of their head and may have a hearty laugh at my ignorance :-) .. but then I have to write what I know and may be this way people can join in some interesting/lively discussions and be able to learn from each other ):-)
but just my thought. my two aNas
Hmm Good discussion. But I think there are no more Melakarthas possible than the 72 as our wise ancient masters have already analysed and arrived at a more appropriate delineation of the scales. Now coming to the komal ma of varali, excuse me, it is not komal as it is already close to are a little tivra to Pratimadhyama. Komal Ma being the suddha Ma . Now, see as there are individualities with each person within him/her self, there can be individual traits in ragas which could be the derivatives of a particular parent scale. So a varali's Madhyama need not be made into a big issue and therefore there need not be further thought on bringing out the 3rd Madhyama. In that case we need to bring out the other swaras between Rishabha, Gandhara etc as there are 22 sruthis in each octave. So let us put it this way, the Madhyama of Varali has a special characteristic or individuality of its own than its parent scale. Obviously we are not duplicates of our parents. We can be born with or develop our own unique traits and characters that could be different from our parents though we are still their children. Similarly ragas do. :)
ReplyDeleteHey Vijay,
ReplyDeleteWhen i said komal ma, I dint mean komal ma of Hindustani music. I just meant some other note.
So ok Teevra Ma :-)
Yea quite rightly pointed out,That is where I was getting at.. Slowly walking towards the 22 shruti Avenue.. :-)
Since Ma of varali is from that 22 shruti.. thaz why all these discussions came in.. and I am kind of convinced that crossing Prati Madhyamam frequency is not prati madhyamam..
\I am copying a very interesting post by a guy called Raja from the Website :
http://jtauber.com/blog/2006/11/17/why_a-sharp_is_not_b-flat/
he says :
As noted above, all Indian notes except Sa and Pa, can move a microtone or two depending on the raaga. This practice can be traced back to the ancient concept of Shruthi, where an octave is divided into 22 shruthis. Each of the 10 notes (i.e. excluding Sa and Pa) have two variants. For Eg. Komal Rishab (r) can be either the normal r or r+. The pitch difference between these two notes is small, but the use of one instead of the other, guarantees a simple ratio with the other notes used in that Raaga. That is the reason, why r is used in some Raagas and r+ is some others. It has been observed that the shruthis are such that, between two adjacent saptasvara notes there is a ratio of 256/243 and between shudh and komal notes the ratio is 25/24. It is this simple ratio that makes the music pleasing, natural and sweet. The 22 shruthis, their frequencies and equivalents are given in the table 3. An octave can be divided into 1200 cents. Scales are usually defined in cents so that for any base pitch, the scale can be calculated. The difference in cents between two frequencies, f1 and f2, is given as log(f1/f2) * 1200 * log(2) or approximately log(f1/f2) * 3986.3137
Table 3 - Shruthi
Shruthi Names || Western Eq.|| Abbreviation Hz. Adjucent Ratio Cents Ratio
Shadja C S 261.63 0 1
Ekasruti Rishabha r 273.38 256/243 76.03 256/245
Dvisruti Rishabha Db r+ 279.07 111.73 16/15
Trisruti Rishabha R- 290.7 25/24 182.40 10/9
Chatusruti Rishabha D R 294.33 203.91 9/8
Shudha Gandhara g 310.08 256/243 294.13 32/27
Sadharana Gandhara Eb g+ 313.96 315.64 6/5
Antara Gandhara E G 327.03 25/24 386.31 5/4
Chyuta Madhyama Gandhara G+ 331.12 407.82 81/64
Shudha Madhyama F m 348.84 256/243 498.04 4/3
Tivra Sudha Madhyama m+ 353.20 519.55 27/20
Prati Madhyama F# M 367.92 25/24 590.22 45/32
Chyuta Panchama Madhyama M+ 372.52 611.73 729/512
Panchama G P 392.45 256/243 701.95 3/2
Ekasruti Dhaivata d 413.44 256/243 792.18 128/81
Dvisruti Dhaivata Ab d+ 418.61 813.68 8/5
Trisruti Dhaivata A D- 436.05 25/24 884.36 5/3
Chatusruti Dhaivata D 441.50 905.86 27/16
Shudha Nishada n 465.12 25/24 996.09 16/9
Kaishiki Nishada Bb n+ 470.93 1017.60 9/5
Kakali Nishada B N 490.56 256/243 1088.29 15/8
Chyuta Shadja Nishada N+ 496.69 1109.78 243/128
Tara Shadja C S' 523.26 25/24 1200 2
=============================================
back to my post :p
So we are looking at Chyuta panchama Madhyama for the varali scale isnt it?
So i think out of these discussions, it is kind of making some cloud clear that the Notes have Range of frequencies because the 22 shrutis may not be differentiable to the ear or be pleasant to the ear .. so R1 for example ends up getting a range.. son on and so forth untill a note is reached which is perfectly identifiable ...
So Conclusion: Present Version of the Carnatic Notes ( 12) Do not have a fixed frequency so to say???
The Question of VIVADI Ragas being in the Melakartha still remains :-)
I am trying to stick to the original intention :-)
I think, if we made these distinctions based on the 22 sruthi model, then we will loose the freedom of gamakas in our music and we better play piano kind of music. So our ancient wise masters said lets deal with 12 sruthis to be able to play the music Indian classical music with its unique feature, the gamakas. I believe that this is the real reason. I love to have my gamakas for my ragas and don't want to enter the 22 sruthi frequencies, except for theoretical knowledge and purposes, as it disables our freedom with ragas and notes. :)
ReplyDeletegr88 info sir...:-)
ReplyDeleteDear Vasu,
ReplyDeleteIn response to yr logic for 72 melakartas...
good work. I have some reservation regarding
the reasons though...
U said
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)
Had a discussion on same topic with one Sri S.Parthasarathy, musicologist and who belonged to one of the Thyagaraja shishya paramparas about 25 yrs ago...unfortunately he is no more. He had demonstrated how R2 and G1 are not the same by playing ganamurte on violin...
Okay, if you are not convinced, look at the table given by Mr.Raja...
Take a look at the frequencies of the 22 shrutis given by Mr.Raja
Chatusruti Rishabha D R 294.33 203.91 9/8
Shudha Gandhara g 310.08 256/243 294.13 32/27
The first one is R2 and the second one is G1. They are clearly different notes!
R2 and G1 cannot come together not because they are same but because they are too close to be
differentiated by the human ear. I could play the two notes for you on violin and show..
Only rule 3- R3 (shatshruti r) can come only with antara gandhara is because R3 = G1.
Iam not an expert in this subject.
you may want to see this website, very technical
where this is the only thing I understood.(http://www.associatepublisher.com/e/0/%C5%9Aruti_(music).html
varali probably has the Chyuta Panchama Madhyama.
The best person to ask about these things would be Chitraveena Ravikiran...he is not only an expert, he is very articulate.
Thanks for creating a lively discussion,
Jyothi
Wow,
ReplyDeleteJyoti.. You have made this discussion even more lively..
So what happens to the Music books that tell that R2 = G1 and R3 = G2 etc??
ALso I understand by the table given by Raja that the frequencies are different. for R2 and G1, which I am kind of pinching myself to beleive .. But what about R3 and G2. The table doesnt list ShatShruti Rishabham frequency.. So is R3 = G2? It also doesnt list shatshruti dhaivatam...
So This brings out an interesting topic of what the definition of vivadi notes and vivadi ragas are? According to you , its clearly not the case when both rishabhas come together??
Seriously That table that i published out of Mr. Raja ' post has been an why opener for myself.. I will think more on this and try to channelise the thoughts more into the definitions.
Thanks Dr.Jyoti for your inputs..
Vasu,
ReplyDeleteUnfortunately there is no proper citation or accuracy maintained in classical music books or websites(which are worse) and that's why these confusions.
Alright, one more brain teaser: if there are three rishabhas before chatushruthi rishabhas, which one is the shuddha rishabha?
Latantapriya, S R1 M1 P D1 S, S D1 P M1 R1 S is considered a janyam of kanakaangi and not that of mayamalavagowla. why?
Jyothi
Jyoti,
ReplyDeletei think we both agree on the fact that the rules I have put is correct, otherwise we will end up more ragams than 72. i think the rules that are atleastin the practice of carnatic music is that R3 and G2 are the ssame.and the R2 and G1 are the same. u my be technically right in the practicality of the notes and the frequencies aspects of it, which makes me think that what we are taught is not the same as what the actualy fact is.. which may be true again, because the teaching faculty of music, atleast practicing music probably decided that it will be almost impossible to learn music if all the 22 notes were exposed.. and the documentation of the practical learning wud have become deadly..so the limitation wud have come into the picture and then the idea wud have been that these concepts would be introduced at a much later stage after the basics would have been cleared.
This may be of interest to you all...author clearly shows how swaras and melas are just useful abstractions but raga is beyond that.
ReplyDeletehttp://www.musicresearch.in/categorydetails.php?imgid=166