Talaprasthara (Combinatorics)

Tālam & Layam related topics
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msakella
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#76

Post by msakella » 12 Mar 2007, 20:51

Dear member, sbala, No deviation at all. Later you may forget to ask this. This is only to share our ideas with all our Indian brothers and sisters. Sometimes we may be wrong, I can’t say. Better to clear off our doubts at the earliest.
In the context of Pallavis - in Anuloma the lyric of the Pallavi should be sung in obverse order, in Viloma the lyric of the Pallavi should be sung in reverse order and in Pratiloma the lyric of the Pallavi should be sung in a single speed but the Tala should be rendered in three speeds. Even though, presently, I am unable to recollect, some more technical terms are there in this context about which I shall define them later if I can recollect them. amsharma.
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sbala
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#77

Post by sbala » 12 Mar 2007, 22:36

Dear Sharmaji,
Could you give some examples to illustrate these concepts? I don't understand how a lyric can be sung in reverse order and yet make sense.
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msakella
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#78

Post by msakella » 13 Mar 2007, 02:45

Dear member, sbala, I recollect that, many years back, I read some where about such Viloma Sahitya composed by our Maha Vaidyanatha Iyer, the great but, I do not remember the exact lyric as my main concentration has always been on Talaprastara for more than four decades. Also there are two kinds in singing Viloma like that, one kind is of singing lyric like ‘Vi-Ka-Ta-Ka-Vi’ and second one is of singing notation ‘RGMDNDMGR’ in Kalyani-raga. Among them Viloma Sahitya is still more difficult. While all the people remain under the control of language we come across a very few people under whom the language itself remains under their control. Such people only can do such incredible things. amsharma.
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sbala
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#79

Post by sbala » 14 Mar 2007, 15:16

Dear Sharmaji,
In another thread, you pointed out that thalams containing Samyuktangas should not be peformed as it contradicts the other requirements that the first kriya of any anga should be sounded. What is the purpose of Samyuktanga if it cannot be rendered?
Last edited by sbala on 14 Mar 2007, 15:18, edited 1 time in total.
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sbala
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#80

Post by sbala » 14 Mar 2007, 15:39

msakella wrote:In 2-7-1-1-12 the permutation ‘S’0’ becomes the 9th but not 11th. Please note.
Sharmaji, I have made the change.
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msakella
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#81

Post by msakella » 14 Mar 2007, 18:46

Dear member, sbala, For example, in 6-units’ permutation of Samyuktanga-prastara consisting of Sankhya of ‘32’, in Trisra-jati the 2nd, 3rd, 4th , 9th, 17th, 18th & 25th permutations,
in Chaturashra-jati the 1st, 2nd, 5th, 6th, 7th, 8th, 10th , 13th, 17th, 19th, 20th, 21st, 26th & 29th permutations, in Khanda-jati the
1st, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 13th, 18th, 19th, 20th, 21st, 25th, 26th & 29th permutations, in Mishra & Sankeerna-jatis the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 13th, 17th, 18th, 19th, 20th, 21st, 25th, 26th & 29th permutations carry Samyuktangas and these rhythmical forms should not be used as Talas but as mathematical forms in Svarakalpana. The rhythmical forms incorporating Samyuktangas can only be used in Svarakalpana for mathematical purpose but not as Talas. After having gone through the different processes of permutation, I hope, now, you can understand this. amsharma.
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sbala
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#82

Post by sbala » 14 Mar 2007, 20:06

Dear Sharmaji,
The reason why it cannot be used in thala is because the first kriya of every anga has to be sounded and in case of samyuktangas that rule will be violated. Is that right? I'm interested to know the application of talaprastara to kalpanaswaras. Is it too early or can we discuss the same?
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msakella
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#83

Post by msakella » 15 Mar 2007, 03:16

Dear member, sbala, You caught the point, while rendering any Samyuktanga the established rule has to be violated. That is why Samyuktangas should never be rendered at all.
Any thing must have relevancy with the present day life. Now, if is the time to discuss about this relevancy of this topic, Talaprastara, in our present day life. Even though many are not at all aware of it, Talaprastara has great relevancy with our Svarakalpana. Please go through the contents of the pages from x to xv of the Prologue of my book, Talaprastara Ratnakara and sing all the 25 varieties of 16-units furnished there along with the rhythmical form, its authentic serial number, Jati and notation of the rhythmical form to be used in our Svarakalpana. amsharma.
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ram
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#84

Post by ram » 16 Mar 2007, 12:03

Dear Sri msakella,

This thread is so educative and that too on a topic that is very hard to find even in many of the books. I don't know how I did not go through this for all the time this thread has been in existence now. I had bought two of your books from Karnatic Music Book Center as soon as I landed in Chennai a few months back but then have not had a chance to go through them till now. I think the time has come to take them out and also read this thread in detail. I hope it is ok to trouble you with any questions I might have.

Thank you so much for your efforts to educate us.

Regards,
Ram
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msakella
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#85

Post by msakella » 16 Mar 2007, 13:25

Dear member, ram, Glad to know that you are interested in learning Talaprastara. Welcome. As I have taken the propagation of the easy methods in singing Svarakalpana and Ragalapana and Talaprastara as a mission of life, I am always ready to help you in every manner in learning this topic, Talaprastara with pleasure. No trouble at all. I have written 3 books on Talaprastara and they all are available in Karnatic Music Book Centre, Chennai. You take the 3rd book also. Mr. sbala, the Chennai-based brother-member of our forum had started learning this topic and covered much of it. You can take his g-mail address from this forum profiles and contact him to learn from him in person what he had already learnt.
amsharma.
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ram
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#86

Post by ram » 16 Mar 2007, 14:06

Thank you Sri msakella.

In fact I still recall the day I came to Chennai a few months back. I had gone to Karnatic Music Book Center and asked him if he has books on tala and tala numbers. I had tried going through a version of "Sangeetha Ratnakara" I had bought earlier but it was not very explanatory and was confusing too. He showed me a few that talked about the basic 35 talas. When I asked him if there is some treatise on talas, he showed me two books and said this is very good research material of very advanced nature. I bought them immediately but never got a chance to sit down and go through the material in detail. Little did I know back then that I would get a chance to interact with the author of the books himself in a helpful and educative forum like this !!

(When I read from cover of one of the books that you were a violin vidwan, the first thought that came to my mind was that many percussion artists do not even seem to care about knowing details of talas and tala numbers and how strange it was that a violin vidwan was researching this topic and publishing on the same)
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sbala
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#87

Post by sbala » 16 Mar 2007, 14:18

Ram,
Welcome to the thread! I'm sure with your knowledge of music, you can ask deeper questions and contribute more .
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msakella
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#88

Post by msakella » 16 Mar 2007, 18:48

Dear member, ram, Felt glad to go through your last post and to note the contents. As per a saying in Sanskrit - ‘Kshanashah Kanashachaiva Vidya Dhanamaarjayet’- one should always remain in the process of acquiring knowledge and money at the rate of one atom each second. Thus, knowledge should always be acquired and be distributed at all times by all. Then only our community flourishes. The Almighty had driven me into this rarest topic, Talaprastara of our Indian music and also blessed me with all the required devotion, patience and perseveration along with all the latent secrets of it which have never been brought out by any author of any century. But, it took four decades of my life and I don’t want to let any other brother/sister Indian struggle for his life to get it like me. That is why I have taken the propagation of this topic as a mission of my life. You people all must learn this rarest topic fully to pass on our cultural heritage to the posterity. Wishing you all the best, amsharma.
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sbala
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#89

Post by sbala » 17 Mar 2007, 17:02

Dear Sharmaji,
I have attempted to give a logical explanation of why the method of computation of Sankhya works as some people might be wondering about proxies and why we need to add them. Let me know if this explanation is right or if I need to modify anything.

Logical Justification of Sankhya computation.

Number of permutations of N units= Sum of the (Number of permutations ending with a particular anga) ,for all the angas that are part of the prastara.

Number of permutations of N units ending with a particular anga=Number of permutations of (N - the value of the anga) using all the angas which will correspond to the Sankhya of (N- value of anga) units.

This is the reason why we sum the relevant proxies for Sankhya to get the total number of permutations.
Last edited by sbala on 17 Mar 2007, 17:03, edited 1 time in total.
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msakella
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#90

Post by msakella » 17 Mar 2007, 21:02

Dear member, sbala, I am now 70 years of age and nowadays I do not want to think much about things and to retain them in my mind. But, as I always want to give away the knowledge blessed by the Almighty to my brothers and sisters until my last breath, I am trying my level best to do the same. At the same time, as I am not much educated like you all, at this juncture I cannot and do not want to try to understand more than what I need. Please don’t mistake me. In respect of these applications, as a highly educated person, you can try all kinds of logics to make it easier, understandable and digestible to the aspirants as you yourself should propagate this topic. amsharma.
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sbala
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#91

Post by sbala » 17 Mar 2007, 21:25

No problem Sharmaji. I'm just trying to present the same information in a different format so that others who come from a similar background like me may read this thread later and understand. Though, I have to admit, you have presented the information in the simplest fashion possible. Infact, the way you have presented the information in this forum is much easier to understand than even the book. In that sense, we are all lucky!
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msakella
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#92

Post by msakella » 18 Mar 2007, 01:37

Chi. sbala dear, I feel extremely glad to know that you have started trying to distribute what you have already acquired in respect of Talaprastara in your own way just like a true soldier
of Talaprastara. While I am the senior most soldier, you are the junior soldier and our another brother-member, ram is the junior-most soldier. Irrespective of the capacity we all are the soldiers of Talaprastara Army. Just like in which way our entire Indian community is safe under the most vigilant brother-Indian-soldiers of our country we, the soldiers of Talaprastara Army also should always remain vigilant in enlightening our Indian brothers and sisters in this aspect and also to strive hard to keep it intact and pass on this great Indian culture to posterity. We all have to work hard in letter and spirit. Then only our great Indian culture remains alive perennially and survives. OK. Wishing you all the best, amsharma.
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sbala
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#93

Post by sbala » 18 Mar 2007, 15:50

Logical Proof for the method of computing Sankhya.

As can be seen from the process, the Sankhya for a given numberof units can be expressed as a sum of the Sankhya of lower number of units. The lower units to be chosen would vary from one pratara to another and depends on whether it is Akhanda or Khanda. For instance, if we want the details of the Druta-Meru prastaras for 0, 1 and 2 drutas.

To find the number of permutations of n units containing 0 drutas, it is clear that we need worry about only laghu, guru and kakapada ending permutations with no drutas which are the 2nd, 4th, 6th and 8th proxies. We should not count the first proxy as it denotes the number of druta ending permutations that contains 0 drutas which does not make sense. This is the reason why we sum the 2nd, 4th, 6th and 8th proxies for Duruta meru with 0 drutas in Pancanga Khanda Prastara.

Druta Meru for 1 Druta
Here, we have to find the number of permutations of n units using one Druta. All the permutations have to end with one of the possible angas. If we look at the number of druta ending permutations, that is the same as number of permutations of (n-value of druta) using 0 drutas. This is the reason we look at the first proxy in the lower line for this permutation and not on the same line.

Similarly, for the laghu ending permutations, we have to look at the number of permutations of 2nd proxy (n-value of laghu) units using 1 druta. This is on the same line and is nothing but the 2nd proxy.

Similarly, the same logic can be extended to 2-DCP as well.
Last edited by sbala on 18 Mar 2007, 15:51, edited 1 time in total.
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msakella
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#94

Post by msakella » 19 Mar 2007, 06:57

Dear member, sbala, I am damn tired of my life after having spent four decades of my life on this very heavy strenuous research on this topic, Talaprastara. You do not know that, at last, I have reached a saturating point of research of this topic which was ending every time I try to think with my head with terrible heat and loosing the vision of my eyes for some minutes. There I was compelled to stop. That is why I always think that the Almighty blessed me with every thing of it but permitted me to solve 99.9% of this topic and this saturating point lies beyond that. I shall explain this to you when we meet in person, if you remind me. So, you please proceed with all this logical applications of it. I am already exhausted myself to the saturating point and cannot further concentrate upon any such thing. Try to proceed with the Nashta, Uddishta and Kalita of Khanda-prastara also with the help of my book. Wishing you all the best, amsharma.
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sbala
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#95

Post by sbala » 19 Mar 2007, 09:27

Dear Sharmaji,
I will try to cover the book as soon as possible. As I said earlier, I'm posting the logical explanations only for other members who might read this in the future, to understand the topic from a mathematical perspective and not for any other reason.
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ram
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#96

Post by ram » 19 Mar 2007, 13:24

sbala,

Though Sri msakella has explained everything in a easy-to-understand manner, please continue with your logical explanations too, if possible. When I was going through the process of permutations the other day, I was trying to derive a formula for sankhya and mahapatala calculation and just when I had done that and went a few posts ahead, I found you had already posted the same. I could have readily used your formula.
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sbala
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#97

Post by sbala » 19 Mar 2007, 14:10

Ram,
I gave the formula only for Samyuktanga. You could derive them for the other prastaras. My only fear is Math drives people away and that's why I stick to wordy explanations.
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listener
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#98

Post by listener » 26 Mar 2007, 05:18

All,

I just finished going through this topic...

For those (engineers/mathematicians)well acquainted with the binary system and the language of computation, I found that the Nashta, Uddhishta and Kalita computation for the Samyuktanga prastara can be easily summerized:

Samyuktanga Nashta/Uddhishta computation:

To obtain the constituent figures of the 'm'th permutation of 'n' units:

Write m-1 in binary form. Append the requisite number of zeros to make it a (n-1)-bit number. Consider the resulting representation as a string. Reverse it (eg., 00110101 -> 10101100). From the resulting string, form another string, by replacing every '0' in the original string with a '1' and every '1' in the original string with a '+1'. Append a '1' to the left of this string. This string will contain the angas separated by a '+' sign.

Eg.,

41st permutation of units:

41-1 = 40 = 101000 in binary ---> write in 8-1 = 7 bits ---> 0101000 ---> reverse ---> 0001010 ---> 111+11+11---> append '1' ---> 1111+11+11 ---> 4+2+2

the Uddhishta process is exactly the reverse of this.

Kalita:

Number of occurrences of figure 'k' in a prastara containing permutations of 'n' units can be summarized with a formula as follows:

N(n,k) = 2^(n-k) + (n-k-1)*2^(n-k-2), if k < n
= 1 if k=n
Last edited by listener on 26 Mar 2007, 05:26, edited 1 time in total.
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listener
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#99

Post by listener » 26 Mar 2007, 05:23

sbala wrote:Ram,
I gave the formula only for Samyuktanga. You could derive them for the other prastaras.
Unfortunately, I think exact formulas (even Sankhya and Mahapatala) cannot be obtained for the other prastaras (Panchanga and Shadanga) this would require us to factorize polynomials of degree 6. Will explain why if someone here is interested in knowing...

I think Sharmaji's 'Recursive' approach is the only way...
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msakella
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#100

Post by msakella » 26 Mar 2007, 06:03

Dear brother-members, sbala, ram, listener, Today I feel extremely happy and am able to forget the hardship I have experienced for four decades (from 1963 to 2003) of my life to get this Talaprastara out. When I happen to come to Chennai in near future we shall all meet and I shall explain the very important details of Talaprastara which can only be explained easily in person. Wishing you all the best, amsharma.
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