Concept korvai

[cmlover]

Let us simply accept
Chi. Ravikiran is just Chitravina Ravikiran and stop quibbling any further

[Ranganayaki]

Wasn't he called Gotuvadyam Ravikiran as a kid?

[sureshvv]

If you can explain in a simple way what we are talking about, it would be very helpful to have a meaningful discussion, at least one that has meaning to more than just a couple of us.

From the rest of your post, it is readily apparent that you lack the necessary humility to understand this arcane topic May be you can work on that aspect for a while and try again!

[Ranganayaki]

..

[Nick H]

Wasn't he called Gotuvadyam Ravikiran as a kid?

I doubt that Sri Ravikiran was ever a young goat.

These language quibbles can go on for ever and ever and ever and ever ]

[vasanthakokilam]

Nick

Just remembered that I have relatives who would (they still do ) snap up wedding invitations when they came in and read them through, just to find the English (mainly) or Tamil/Kannada/sampradaya mistakes in them.

Hmmm... I thought it was only my relatives who did that. On top of that, if they did not include the names of the right family members, that is a big scandal and it will come up in family conversations for decades.

[Pratyaksham Bala]

AhA!
A judicious digression to defuse a situation!

[Nick H]

kids and piglets --- all delightful creatures!

[Pratyaksham Bala]

Let us simply accept
Chi. Ravikiran is just Chitravina Ravikiran and stop quibbling any further

Let us also simply accept
Chi. Vasanthakokilam (Post #30) is just Ch.........i Vasanthakokilam and stop quibbling any further!

[sureshvv]

Another example of a rasikas thread that goes from the sublime to the ridiculous!

[vasanthakokilam]

PB, I see what you did there

[Ranganayaki]

Another example of a rasikas thread that goes from the sublime to the ridiculous!

According to my interpretation of P>Bala's post (#57) it's all your fault, Suresh!!

[msakella]

cmlover dear - As you are not acquainted with this topic, naturally, this number looks quite mysterious to you. Though it is so long of 20 digits the method to obtain it is of the very primary level to the people who are acquainted with this topic. As you wrote VK is smarter than many of us and I hope he knows this technique having learnt it when I taught this to one of our members long ago. However, now itself, I shall tell you this basic technique in a simple mathematical language though not in algebraic language. Its technical term is ‘Sankhya’ and I shall give it hereunder.

Units: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10
Sankhya: 1 - 2 - 4 - 8 - 16 - 32 - 64 - 128 - 256 - 512

The above is furnished for 10 by just doubling the previous figure. In the same process if you proceed up to 64-units you will get the present figure ‘92233720368547755808’. Try yourself, dear.

Just now I spoke to my friend, Chi. Vijaya Raghava, Bangalore (M-07259766899 - he is the elder brother of Chi. Vinay Sherva who some time back had demonstrated the ‘Simhanandana-tala of 128-units’ very ably and successfully at Bangalore) who could swallow this topic hardly within four days ( which took four decades for me) working from 7 am to 11 pm daily, to visit our forum and answer the questions of our members. Hereafter mostly he answers your questions relating to this topic and, whenever it is needed, I shall enter and do that. If is needed this matter could even be shifted to the thread ‘Talaprastara (Combinatorics)’ with a note in this thread.

ShrutiLaya dear - This number ‘92233720368547755808’ is the total number of permutations we obtain in the process of permutation of 64-units under Samyukthanga-prastara and, in our terminology, all the korvais carrying 64-units in the universe are the derivatives of this figure only.

Ranganayaki dear - The figure ‘92233720368547755808’ I have quoted is correct but I have wrongly furnished the digits as ‘19’ instead of ‘20’.
As a professional teacher, I mean that kids should be fed with easily digestible progressions only than with such complex progressions.
I already know the traditional method of dealing with all these figures even in the absence of any calculator or computer in those days but, being an un-educated person, I do not know the algebraic formula of the same. That is why I have requested my Bangalorian friend, as mentioned above, to answer your questions.

To tell the fact, this topic is not that difficult as people generally think. But, than going through all the pages in my three books written on this topic, it is far easier to learn it face to face as I, having already gone through all kinds or hurdles for four decades, very well know all the short-cuts to feed the aspirants very easily, quickly and efficiently. That is why my Bangaloraian friend could swallow it hardly within four days only which took four decades in my case.

I have already brought out three books on this topic which are available from The Karnatak Music Book Centre, Chennai. They are 1.Indian Genius in Talaprastara (English), 2.Talaprastara of Nisshanka Sharngadeva’s Sangita Ratnakara - a critical interpretation (English) and 3.Systematisation of Prastara Details of Deshi Talas (English). Interested persons can go through these books and consult me in case of any doubts.

In general, people are afraid even to talk of this complex topic and are used to, some how, run away even from my sight. But, as you all are very inquisitive to learn this topic I feel very happy of it and certainly help you in every respect. amsharma

[Ranganayaki]

Sri Akella, thank you for your message and the individual responses. I am determined to do my best to fully understand what you have said so far (at least). I follow how you obtain the long number. As everyone else has also certainly understood it now (that you have explained it), this number is expressed at "2 to the power of 63".

Please allow me to break down what you have said so far to ask questions.

What is the significance of "64'? Why are you specifically considering "64 units?"

This number ‘92233720368547755808’ is the total number of permutations we obtain in the process of permutation of 64-units under Samyukthanga-prastara and, in our terminology, all the korvais carrying 64-units in the universe are the derivatives of this figure only.

According to what you have written above, would it be reasonable to say (using the table of units and samkhya) that 4 is the total number of permutations we obtain in the process of permutation of 3 units?)

I do not follow your statement; All korvais carrying 64 units are derivatives of this figure. What do you mean by "derivatives"?

How does this connect to tala? What is a "korvai containing 64 units"? (if my question is too elementary, please let me know) -

[msakella]

Dear sister-member, Ranganayaki, By writing ‘1’ in Sankhya for the permutation of ‘1-unit’ and continuing the progression for each increasing unit by just doubling the figure in Sankhya in the table given by me if you proceed further up to 64-units you will at last get this figure ‘92233720368547755808’.

The total of the composed korvai which, in fact, is a rhythmical form bearing the following seriatim of units: 3 + 4 + 7 + 11 + 18 + 21 = 64-units

G,, - R,,, - G,R,SND - GRSNDPMGRSN - DN,R,, G | M,D,, MD,N,, - G, R, S | ND - R,S,NDP - D,P,MGR ||(Kamalambam)

--- as furnished by Chi. Ravikiran carries, in total, 64-units. This rhythmical form is one among the ‘92233720368547755808’ rhythmical forms pertaining to 64-units bearing a singinificant-serial-number of them. To find this number, to check it and also to cross-check it there are some procedures to follow as per the element of Prastara, which has unfortunately been kept at last and became the least element of Taladashapranas due to the sheer ignorance of our ancestors. Many are not aware that this is the origin of all rhythmical forms of the universe.

As per the available history of it, right from the author, Nisshanka Sharngadeva of Sangita Ratnakara, while all the authors of all our treatises, having no knowledge of it, successfully copied it from some other treatises of their times and furnished only contradictory information in their own treatises, only the authors of two treatises have kept mum without revealing the latent secrets of it. Thus, we have absolutely lost this topic since 13th century. You may believe it or not, only by the grace of the Almighty, I was able to retrieve it fully and wrote three books which are the originals in the annals of our music literature. Thus, the ignorance of our ancestors had swallowed four decades of my precious life. As I never like to repeat the same in case of any of my successors I have willingly spent my precious life of 4 decades, even sacrificing umpteen benefits as professional-violin-accompanist, to bring out three books with all the full details of this topic. amsharma

[Ponbhairavi]

Now that the situation has been defused jocularly by several people despite CMLs post 51, he will pl excuse me for this clarification.
The first person who felt uncomfortable with the usage of chi is NOT me..Pl see post 31. I did not do any quibbling. I made only one short post and I did not find fault with any person, I talked only in general of traditional wedding invitations where it is preferable not to shorten the words sowbhagyavathi and chiranjeevi as there is a possibility of justified uneasy feeling for some persons like mri_fan .
“kid” is not a shortened form of any foreign word. It is an English word Its meaning is goat of course . But the very same word also means “a young person esp a child “-; third meaning of the word kid in Merriam webster2004.
“ Sow” does not have a third or fourth meaning as “unmarried young girl”
I am aware that with the advent of computer English language has created and accepted several abbreviations in modern parlance and writing But it may be better to avoid such shortenings in respect of words of other language particularly in the traditional cultural Indian contexts.

[cmlover]

Thanks Sarmaji:
The explanation of the number as 2^63 is simple and elementary as the number of possibilities of a 64 length Korvai. The exact sequence number of this one can be easily derived reading it as a 64 digit binary number by coding (1 where there is a swaram and 0 where a swaram is absent) for the proposed sequence
G,, - R,,, - G,R,SND - GRSNDPMGRSN - DN,R,, G | M,D,, MD,N,, - G, R, S | ND - R,S,NDP - D,P,MGR

Good Ponbhairavi
for highlighting the hazards in the evolution of Tanglish (or is it Sanglish?)

[ShrutiLaya]

Why stop there? By having a 64 digit Octal number, we can encode all seven notes and silence, and get a "serial number" for the phrase! An entire krithi can be reduced to its serial number. But what, if anything, does it all mean?

- Sreenadh

[Nick H]

...third meaning of the word kid in Merriam webster2004.

Merriam-Webster is not an English dictionary! ]

[vasanthakokilam]

Why stop there? By having a 64 digit Octal number, we can encode all seven notes and silence, and get a "serial number" for the phrase! An entire krithi can be reduced to its serial number. But what, if anything, does it all mean?
- Sreenadh

Octal is not good enough, you need at least 12 and for full coverage 16. So hex will do But that is not the point, this is NOT about swaras, though I understand your intent is rhetorical.

Your last question is valid since there is definitely more to it. The talaprasthara thread may have the full details. ( but I did not fully understand it due to lack of time at that time ).

( Those who feel uncomfortable that this thread is getting too theoretical, please bear with it. We thought of moving some of these posts to a different thread but decided against it for now. We started with Fibonacci and transitioned to combinatorial math, so theory is part of this thread. We still very much look forward to Ravikiran's upload which will add practical aspects to it. And all the side conversations are typical of our threads, just like real life.).

[mahavishnu]

Those who feel uncomfortable that this thread is getting too theoretical, please bear with it

For me, the thread is just getting started

I share Sreenadh's concern that the number by itself is not sufficient information to instantiate a "token" korvai. Having the total number of possible combinations (2^n-1) does not get around the "sequencing" problem. By which I mean, you could have a surrogate data set with a different number of swarams with kArvais/silences that will still have the same representation (9.22 X 10^18).

I am sure Sri Akella could clarify. I also wonder where is the aesthetics in all of this?

[ShrutiLaya]

The number 2^63 (=92233720368547755808 per Akella gari calculation) is simply the total number of combinations of 64 sound/silence combinations; As suggested by cmlover, a binary number of 64 digits will uniquely encode the sequence of a given combination; For instance, Number 13 = 1101 means silence for the first 60 beats, two strokes, one beat silence, and one stroke. My point/question is, so what? Does this give us any more information, or does it lead us to any greater insight, than the original? vk suggests that the answer may be found in the other thread on Talaprastaras. One day, it shall all be made clear ..

- Sreenadh

[vasanthakokilam]

>One day, it shall all be made clear ..

Well, one day, it shall become clear... if one spends enough dedicated time to the subject. This kind of drive-by Q&A and conversations are definitely instigators for that.

There are two aspects to this: One is to learn what is in the treatises and the second is to challenge it, the why and how of it, the aesthetics of it etc. We need to keep them separate. The first part allows that knowledge to propagate, even if you personally think it is all superfluous and not practically useful, if you believe such propagation is useful in itself. The second part allows for intelligent discourse on the topic. I definitely encourage all the people to spend a few days understanding it and getting a feel for the rationale for what our forefathers were trying to do and discuss it here.

[ShrutiLaya]

Just for the record, powers of two (up to 84) can be looked up from http://en.wikipedia.org/wiki/Power_of_t ... ers_of_two ; According to this source, 2^63 is actually 9,223,372,036,854,775,808 so Akella gari calculation appears to be in error (he has an extra digit 5 after the 77)

- Sreenadh

[VijayR]

Agree with mahavishnu; the thread is only just getting warmed up!

A minor comment: 2^63 is only 19 digits long and equals 9223372036854775808. There is an extra 5 (4 places from the least significant digit) in the number that Sri. Akella garu provided.

Edit: Sreenadh, you beat me to it!

[cmlover]

Right!
The number of digits in decimal system for 2^63 = Log(2^63) = 63*Log(2) ~ 19

[Ranganayaki]

Merriam-Webster is not an English dictionary! ]

First of all, Nick, very cute, LOL - made me - to read this in the midst f all the back and forth!!!

Haven't we had this discussion before, in another context?

Ponbhairavi,

The first person who felt uncomfortable with the usage of chi is NOT me..Pl see post 31. I did not do any quibbling. I made only one short post and I did not find fault with any person,

Ponbhairavi, I don't remember this part of the discussion too well, I can't go to the previous page without losing this one.. I think I was just addressing you to clarify the Tamil/Sanskrit part.. I will have to look it up.

In talking about the fault finding attitude, I was not referring to you but just remembering how valid or not I felt the attitude of nit-picking on the words was, and it was really these relatives of mine I was thinking about. And I was just expressing my disagreement from the standpoint of an old memory. Some of them are in fact still my favorite relatives . The Chi-Sow thing here sounded SO familiar! Please don't take it personally.

Oh, I've landed myself in so much trouble in this thread!

I did not mean that "sow" as abbreviation IS a homonym of "sow" meaning "female pig". It is not. I just talked about an attitude we can adopt because "sow" is a common abbreviation among Indian people. And it has been accepted into the Indian parlance. All that said, I must say that I think Sowbhagyavati and Chiranjeevi are much nicer and convey much more meaning than an abbreviation can. Mr. conveys all the meaning of Mister. But to me "Sow." conveys only the meaning of "Miss" and not the beauty and auspiciousness of Sowbhagyavati. For that reason, I would prefer not shortening it.

[Ranganayaki]

If you can explain in a simple way what we are talking about, it would be very helpful to have a meaningful discussion, at least one that has meaning to more than just a couple of us.

From the rest of your post, it is readily apparent that you lack the necessary humility to understand this arcane topic May be you can work on that aspect for a while and try again!

Suresh, I get what you mean, re-reading this line. "Meaningful" seems to be the word that jars here, I do agree. In that light, I do feel it can be interpreted as talking down to Sri Akella!! It now sounds to me as though I believe this is a meaningless discussion, and I apologize. I meant "have a discussion that had meaning to more than a couple", "make the discussion accessible to more people". I am so sorry. Was this my worst line, that you selected for the punch? But I don't want to look at the rest of my post and find more things to feel bad about, so just a blanket apology if I have annoyed anyone.

[chitravina ravikiran]

I apologise in advance for this long post!

I had taken time off to go over Akella garu's book, Indian Genius in Talaprastara and it is doubtless a monumental effort. I have only glanced at some parts of it so far but intend to puruse it in greater depth as and when time permits. Based on this, I will, from time to time share some observations/demystify some parts of it.

1. Tala-prastara (permutation) is a fascinating subject of empirical ramifications.

2. It is essentially not only a calculation tool but also a predictive one (what are the possible permutations using various parts of the tala such as anudhrta, dhrta, laghu, guru, plutam etc, which serial number can give what kind of part-permutation or viceversa).

3. A smaller parallel is the 72 melakarta scheme wherein we can detect the notes using serial number or viceversa.

4. A much larger parallel is seen in the number of possible ragas (parent as well as derivative) using 4, 5, 6 or 7 notes including vakra prayogas (zig-zag patterns). The book, Sangita Chandrika (pub in 1902 by Manicka Mudaliar) comes up with over 7.2 million ragas using permutation and combination. Almost any normal raga discovered by a musician can fit into one of these 7.2 million. And one can create a method to serialise them based on the number and type of notes, order, zig-zaggedness and various other complex factors.

5. The exercise can be of fascinating academic interest but when one deals with these numbers in reality, it's almost like naming stars as X 235897583 or X988734390. What personalises ragas or talas is aesthetic beauty churned out by explorations of great musicians or composers. That said, prastara is a subject that we surely need to acquaint ourselves with if only to take pride in the founding geniuses of this all encompassing concept.

6. While tala-prastara is a a huge subject, I noticed a significant point, which Akella ji can corraborate - the books including his make it clear that the focus is only on the external count of the talas (based on angas). It has nothing to do with internal units or their developments within - those do not alter the tala or beget a new tala. For instance, when we talk about possible permutations for 64 units and come up with

9223372036854775808,

we are only talking about a tala of 64 whole units wherein parts such as dhrta/laghu can be permuted in various ways. We are not referring to the 64 units embedded in say, Adi tala in 2 kalais here.

7. And this begs some question, which Akella garu would surely be able to shed some light upon...

(a) What ramifications does prastara have with respect to korvais that we come up with? For instance, the korvai that I mentioned in post #1 can as well fit into Adi tala, Tishra Mathyam, or Khanda Jhampa (all 8 units) or even Chaturashra Eka (which have different serial numbers, based on prastara - the order in which the parts are displayed).

(b) I also was struck by the numbers Akella garu himself has provided for some of the 35 talas in P 114 of his book.

(i) How can the serial number be 41 for Adi tala as well as Tishra Mathya?
(ii) How can it be the same for Mishra Eka, Khanda Rupaka (both given as 1), Sankeerna Eka and Mishra Rupaka (again 1)?

From what I gleaned, this would be impossible. But am I missing something here?

8. To get back to the initial topic, any korvai deals with internal divisions within a tala. It neither gets affected by nor affects anga-prastara (in talas of same units). The two are quite distinct paths.

9. Practical music is mostly based on aksharas within a tala and not its angas.

[Ranganayaki]

And so I move on, ready to put my foot in my mouth one more time. I'm quite nervous now.

Sri Akella, I think I understand all that you have said here. I have been reading online.
http://www.berklee.edu/bt/211/lesson.html
https://canvas.instructure.com/courses/179456


I have a question, a doubt I would like clarified, I do not mean to challenge (as Vasanthakokilam nicely distinguished).

I understand that these are really permutations, positional changes. Then the number of permutations of a small number, say, 3 units should be 6 and not 4.. and by extending that thought, the number of permutations of 64 units should be a MUCH bigger number than 2 to the 63rd power. I asked you this question yesterday too (about the permutations of 3 units), but you answered in more general terms. Can anyone tell me if there is anything wrong with my reasoning? There must be because no one else has pointed it out ! My idea of the right number is 64!. [Sri Akella, I don't know if you are serious about your inabilities you mention, or if it is purely rhetorical, but in case you do not know what 64! is, it is understood as 64 X 63 X 62 X 61.......... 3 X 2 X 1 (all the way) ]

One of these two websites above lists permutation numbers for up to 12 units and the number it listed for 12 units is 479,001,600. I checked and it is equal to 12!. In your table, it reaches 2048. Please resolve this for me, you or anyone else.

All this said, if the purpose of this is to find the serial number of one particular permutation, what is the use of that? What is the use of saying (for example) that Ravikiran's work already appeared somewhere else? His work remains original. What is the aesthetic purpose of knowing these serial numbers? Still, if there is a quick, methodical way of arriving at this serial number and if it was truly codified centuries ago without the help of computing aids, it is an amazing feat.

[msakella]

Chi. Ravikiran dear, I am extremely happy of this development which forced you glance into this highly interested science of Talaprastara.

To tell the fact, in music, I did never find such an absorbing topic having more precision than Talaprastara in music. More over, most unfortunately, it has been proved beyond any doubt that this most interesting and invaluable topic has absolutely been vanished due to the sheer negligence and ignorance of the egocentric authors of our treatises since last 8 or 9 centuries and, very pitiably, each and every musician is very busily engaged in earning money, fame, honours and titles and none did care to retrieve this lost topic and give it to the posterity until I, only by the grace of the Almighty, could decipher it fully and bring it in three books which altogether consumed the lion’s share of forty years of my life. Nobody can even imagine how much hardship I had undergone, how much discouragement I had faced and how much I have sacrificed in my life to do this all unlike any other person on earth.

I shall give you answers to all your questions one after the other. At the first instance, if you don’t mind, please modify anudhrta, dhrta & Tishra Mathyam as anudrta, drta & Trisra Mathyam.

Yes, as you wrote, this element focuses only on the external count of units of the rhythmical form but not the internal units or fractional-units or their developments like nadai or gati or kalai or marga.

Prastara is the process of permuting a particular number of units into different kinds.For example 4-units could be permuted as - 4, 1-3, 2-2, 1-1-2, 3-1, 1-2-1, 2-1-1 & 1-1-1-1 – in all, into 8 different forms altogether but each carrying 4-units only. Thus, 64-units has ‘9223372036854775808’ varieties of rhythmical forms among which some can be used as Talas basing upon the established norms to use them as Talas and some cannot be.

Any where in the universe 8 x 8 = 64 and thus, being mathematics, any one who works out accordingly will certainly get the same serial-numbers of the 35 Talas. As you do not have the required acquaintance with this topic you felt struck on finding these serial-numbers.

As you do not have the required acquaintance with this topic you are able to see the common serial-number ‘41’ only but not the difference of the total units of 8 & 7 of Adi & Trisra-mathyam respectively.
In the same manner you are unable to see the required difference even in respect of others Talas you have mentioned obviously revealing that your are certainly missing the required in this respect.

You are writing ‘practical music is mostly based on Aksharas within a tala and not its angas’ but, in the absence of the required knowledge of Prastara you are unable to understand that this topic is also based on Aksharas or units within a Tala but not its Angas. That is the pity. Most unfortunately, no musician is aware that this topic is the only origin of all the rhythmical forms of the universe. amsharma

Chi. Sow. Ranganayaki dear - In the process of permutation of 12-units we get ‘2048’ only but not ‘479,001,600’ as mentioned in that website.

From 25-02-1995 to 02-03-1995 (6 days) one workshop on Talaprastara, sponsored by the South Central Zone Cultural Centre, Nagpur was held at Hyderabad under my Director-ship. For this 25 participants, Teachers, Lecturers and Professors from the Music Colleges and Departments of Andhra Pradesh, Tamilnadu, Karnataka, Kerala, Maharashtra and Madhya-pradesh are invited to participate including Prof. N.Ramanathan from Taminadu and Dr.Sachi Devi from Karnatanaka. In the end of it, in the valedictory function, all the participants spoke and unanimously expressed that all the Doctorates they already have must be conferred upon me for bringing out this amazing topic, Talaprastara. Such amazing topic this Talaprastara is. amsharma

[VijayR]

I understand that these are really permutations, positional changes. Then the number of permutations of a small number, say, 3 units should be 6 and not 4.. and by extending that thought, the number of permutations of 64 units should be a MUCH bigger number than 2 to the 63rd power. I asked you this question yesterday too (about the permutations of 3 units), but you answered in more general terms. Can anyone tell me if there is anything wrong with my reasoning? There must be because no one else has pointed it out ! My idea of the right number is 64!. [Sri Akella, I don't know if you are serious about your inabilities you mention, or if it is purely rhetorical, but in case you do not know what 64! is, it is understood as 64 X 63 X 62 X 61.......... 3 X 2 X 1 (all the way) ]

Ranganayaki, these are not really permutations (which have a different meaning in formal mathematics). What is really being done here is called "sampling with replacement". Think of this in the following way (sorry if this is a confusing explanation):

You are given a small box containing two pens, one of which is a red pen (R) and the other is a blue pen (B). You also have a piece of paper with 3 empty squares to color. You proceed with the coloring by first dipping into the box and selecting (sampling) one of the pens. Use it to color the first square on the paper and put (replace) the pen back in the box. Repeat for the second square, and then for the third square. At the end, you have a Red-Blue sequence on your piece of paper.

As you can see, because of the replacement, the process is completely independent for each square on the paper. For the first square, you can have 2 possibilities (R or B). For each of these, the second square can have 2 possibilities, resulting in a total of 4 possibilities so far (RR, RB, BR, and BB). For each of these four possibilities, the third square can have 2 more possibilities, resulting in a total of 8 (RRR, RRB, RBR, RBB, BRR, BRB, BBR, and BBB). That gives you the total of 8 possibilities (i.e., 2x2x2 = 2^3 = 8 ).

If R denotes a unit of silence, and B denotes a swaram, then any korvai that is 3 units long has to fall into one of the 8 possibilities below: (silence, silence, silence), (silence, silence, swaram), (silence, swaram, silence), (silence, swaram, swaram), (swaram, silence, silence)m (swaram, silence, swaram), (swaram, swaram, silence), and (swaram, swaram, swaram). You can extend the same up to 63.

If R is denoted as a 0 and B is denoted as a 1 (traditional Boolean notation), then this becomes fairly simple Boolean number theory. It is simply the number of Boolean/binary numbers that can be represented using 3 digits, which is 8 (the process is the same; each digit can be either a 0 or 1, independent of the other digits).

[Nick H]

If R denotes a unit of silence, and B denotes a swaram, then any korvai that is 3 units long has to fall into one of the 8 possibilities below: (silence, silence, silence), (silence, silence, swaram), (silence, swaram, silence), (silence, swaram, swaram), (swaram, silence, silence)m (swaram, silence, swaram), (swaram, swaram, silence), and (swaram, swaram, swaram). You can extend the same up to 63.

So rhythmic (not tonal) combinations are binary: silent or sound, 0 or 1, thus we end up with powers of two for the possible combinations of any given number of units?

[ShrutiLaya]

Prastara is the process of permuting a particular number of units into different kinds.For example 4-units could be permuted as - 4, 1-3, 2-2, 1-1-2, 3-1, 1-2-1, 2-1-1 & 1-1-1-1 – in all, into 8 different forms altogether but each carrying 4-units only. Thus, 64-units has ‘9223372036854775808’ varieties of rhythmical forms among which some can be used as Talas basing upon the established norms to use them as Talas and some cannot be.

Part of the problem arises from the fact that Sarma garu is using a commonly accepted mathematical term "permuting" to mean something completely different. In Number theory, the technical term for what he is getting at is "composition" (see http://en.wikipedia.org/wiki/Compositio ... _theory%29) "a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. "

The proof that a composition of N units contains 2^(N-1) parts is quite interesting. Taking the number 4, for instance, write it as follows

1 ? 1 ? 1 ? 1

Now replacing each question mark by either a plus or a dash gives one composition - eg.

1 + 1 - 1+1 = (2-2)

So all the compositions can be obtained by systematically replacing the ? by + or - and clearly, there are 2^(n-1) ways of doing this (2 choices at each of n-1 positions)

- Sreenadh

[VijayR]

As an addendum to my previous post, you can also look at it another way (might be more intuitive). [Edit: this is exactly the composition that Sreenadh refers to in his previous post as well]

This is related to the the number of ways of assigning "n" indistinguishable objects to "k" distinguishable boxes, with the constraint that no box should be empty. The answer to that is C(n-1, k-1) = (n-1)! / ((k-1)! * (n-k-2)!).

The exact problem that Akella garu refers to is the number of ways that "n" aksharas can be split into 1 box + the no. of ways it can be split into 2 boxes + the no. of ways it can be split into 3 boxes + ... + the no. of ways it can be split into n boxes.

This is given by: C(n-1, 0) + C(n-1, 1) + C(n-2, 2) + ... + C(n-1, n-1), which is well known to be 2^(n-1).

Therefore, for n = 4 (same example given by Akella garu in post #81, this comes out to:

C(3, 0) = 1 (this is the - 4 that he mentions)
C(3, 1) = 3 (these are the 1-1-2, 1-2-1, and 2-1-1 that he mentions)
C(3, 2) = 3 (these are the 2-2, 3-1, and 1-3 that he mentions)
C(3, 3) = 1 (this is the 1-1-1-1 that he mentions).

For n = 64, it comes out to 2^63 which is the long 19 digit number listed.

[VijayR]

So rhythmic (not tonal) combinations are binary: silent or sound, 0 or 1, thus we end up with powers of two for the possible combinations of any given number of units?

Yes, you're right.

[msakella]

Dear brother-member, VijayR, I have seen your last post also but somehow missed to reply. I am extremely sorry for this, dear.

I have toiled my self to the saturation point in digging out this element, Prastara when neither calculator nor computer are available. More over, I was so poor in mathematics that I could score only 11% in my SSLC examination in 1951 and successfully failed. Only by the grace of the Almighty, I could retrieve the lost topic, Prastara with great difficulty. In this process, in formulating flawless rules and regulations for writing figures for different tables and also for the short-cut-methods which have never been furnished by any author of any century, I have calculated mostly up to 5 or 6 digits maximum. Only very few times I have proceeded up to 19 digits. But, I am not sure of the correctness in furnishing such large figures due to my old age and other constraints for which I can’t help. amsharma

[cmlover]

Nice to have this interesting introduction to Math 101.
Let us move on to more interesting practical application of 'Talaprastara' and to issues adverted to by Chi. Ravikiran.
Technically the division of an integer into integer components falls under the 'Theory of Partitions'.
The partition number itself is interesting whose calculation is not so simple; an area to which our Ramanujan had made a monumental contribution which finds application in Quantum Theory.
The total number of permutations of the partitions however of a number 'n' is simply 2^(n-1).
These partitions can be used to construct interesting korvais and played/sung effectively.
Our ancestors, not being aware of the maths enumerated these (talaprastara?) at times erroneously.
Now we can cast them in a sound mathematical foundation thanking our all-time mathematical genius Ramanujan!
For an inkling on partion theory see
http://www.virtuescience.com/partitions.html

With the able assistance of our Sarmaji/Ravikiran/.. let us formalize the theory of 'Construction of Korvais' which will be a valuable scientific contribution to our CM Theory..

[ShrutiLaya]

In the interests of clarity and to keep to accepted mathematical terminology - what is being discussed here is not a partition, it is a composition. In a Partition, the order is not important eg. 1-2-1 is considered the same as 2-1-1; However, these are different Compositions

- Sreenadh

[chitravina ravikiran]

In the interests of clarity

In the same vein, let us also note that prastara discussions in the treatises and in this discussion by Akella garu is about tala Compositions and not korvais, which have more to do with the internal elements. So, all discussions about serial numbers of talas etc that Akella garu and I have been having - while certainly of academic interest - are not connected with the korvais and principles thereof.

Prastara aims to give serial numbers of talas and not of korvais. Korvais is a wholly different subject and the principles of akshara-prastara have to be employed to study this.

Akella garu:

Thank you so much for your patient answers. However, both of us missed out a vital (unintended) incorrectness in my question, 7 (b) (i).

(i) How can the serial number be 41 for Adi tala as well as Tishra Mathya?

The fact is Adi and Tishra Mathya do not have the same number. Adi is 41 and Trisra Mathya is 21. It was Chatushra Mathya which had 41, which makes that question invalid. Given that, I am even more curious about your answer. Can you please clarify again about all my questions 7 (a) and 7 (b) (ii)?

[vasanthakokilam]

Prastara aims to give serial numbers of talas and not of korvais. Korvais is a wholly different subject and the principles of akshara-prastara have to be employed to study this.

Yes, that was my understanding from before and hence I was surprised that Akella ji mentioned about serial numbers in the context of your Korvai.

But, coming to think of it, 'Akshara-prastara' can also be studied using the same 'composition' technique ( as discussed by Vijay and Sreenadh above as a mathematical term ) since the structural aspect of a korvail is about spliting N Aksharas into various groupings. So it should indeed be possible to talk of Korvai serial numbers along the same 'composition'al lines as Talaprasthara.

As an example, a 64 akshara Korvai (for, say, Adi ), will have the same set of serial numbers as a tala with an avarthana count of 64.

If talaprasthara has a defined Anga model ( i.e. the angas defining the groupings ) and has rules and restrictions on what combinations of angas are permitted, then we will have cases that serial numbers that are not valid in talaprasthara are valid in aksharaprasthara, since aksharaprasthara ( korvai ) may not have the same rules and restrictions.

Akellaji had written before about certain talas that are not allowed by talaprasthara because it violates some talaprasthara rules ( like the one example of the tala used by Sri. Shyama Sastri in a tala competition ). I am keying in on that recollection to theorize that not all serial numbers are valid talas, as per talaprasthara.

( I am using Aksharaprasthara in a generic form, I really do not know if our treatises talk about it with that name )

[cmlover]

Interesting!
While serial numbering of talas (talaprastara) is just academic
construction of Korvais is eminently practical. If there is already info on 'aksharaprastara' (as VK terms it) do please share, if not let us collaborate and work on it....

[vasanthakokilam]

>'aksharaprastara' (as VK terms it)

Ravikrian called it that and I was piggybacking on it

[msakella]

Chi. Ravikiran dear, Clarifications:
7 (a): Unless you go deep into Prastara you can never understand that all these korvais are only of the mathematical base and, thus, each and every korvai certainly comes under Prastara only. We can fit any korvai in any Tala. VK’s Aksharaprastara also comes under this Prastara.

(b) (i) Unless you get the needed acquaintance with this topic you cannot understand this, dear. (ii) Unless you get the needed acquaintance with this topic you cannot understand this, dear.

In the absence of the required knowledge you are unable to follow me. I am not all talking about any Tala-compositions, dear. I may not be well educated like all you people. But, every one among you must mind that I, unlike any other human being on earth, have toiled myself for 40 long years of my precious life to bring out this rarest topic which has never been brought out fully by any author of our so called treatises and you all are becoming ready to define it within few minutes. In which way it is justifiable, dear? Unless you go deep into it none of you can understand this in your lifetime. I am talking about the rhythmical compositions only. This is not of academic interest only, dear. In the absence of the required knowledge you all are unable to recognise the relation between the rhythmical form and korvai. I can prove what I write at any time and place, dear. amsharma

[cmlover]

Sarmaji
I am totally lost since I certainly do not have 40 years to spare unless the Divine in his infinite mercy will spare me!
Good luck with you young guys!

[chitravina ravikiran]

CML,

1. Akshara prastara is a term that can be used to denote practical compositions of music (melody/rhythm) based on number of aksharas (units within a tala) that one is dealing with. The principles of prastara are same. The excellent Foreword by Dr R Sathyanarayana in Akealla garu's book talks about how all types of prastaras owe a lot to chandah-prastara (based on how short/long syllables could be arranged in various ways).

2. Even in theory, Akshara prastara can get far more sophisticated and involved than anga-prastara or chanda-prastara because:

(a) in chanda, we are only dealing with short/long syllables - in music parlance, this can translate into units of 1 or 2. In music, once you talk of sustained syllables/notes beyond 2 units, the number of ways of expressing each possibility can get exponentially higher.
(b) in anga-prastara: we deal with only permissible angas of the tala of various units and the rest are combinations and as it has well been established, certain rules have been laid out to make them valid prastaras.

3. If we were to also bring in melody into this equation, the possibilities become literally endless, even within one raga. Then imagine the ramifications of 100s of ragas.

4. That said in music, aesthetics come into play and at the end of the day, an exercise in akshara-prastaras could sound like computer-generated sequences unless one has the musical sense to hone in on arrangements that are aurally and emotionally satisfying.

5. An anology to 3 is: we can theoretically say X number of stars/planets are within a cluster and any particular star discovered by an observer could be 'already there'. But in practice, those stars/planets supporting life as we know it are of more interest to us.

On a personal note: very few artistes venture into even akshara prastara - in most concerts they focus on vistara. But I have been fascinated by this topic from my childhood thanks to my father's genius in making me do a lot of practical exercises in musically expressing something as simple as even a 7 or 15 in various ways when we used to do kalpana swaras. Certain tavil artistes have been excellent exponents of prastara. But more on this later!

Akella garu:

Words cannot express the admiration we all have for your dedicated efforts. I do intend to read your book in detail. Meantime, since you already have the expertise in the field, we may trouble you for certain clarifications now and then! I am not contradicting your statement at all:

that all these korvais are only of the mathematical base and, thus, each and every korvai certainly comes under Prastara only. We can fit any korvai in any Tala

It is almost self-evident. I was merely talking in terms of practical ramifications.

[cmlover]

Thx Ravikiran
I can follow your line beautifully. The math is accessible to all of us. However the indefinable 'aesthetics' is accessible to only a few blessed souls like you! Though I can drive my automobile well there is always the curiosity to know how it works. Same is true more in trying to understand the mechanics of CM as a Rasika since our system is totally scientific...

[msakella]

Chi. Ravikran dear, Thank you all for your affectionate admiration towards me and I am always ready to ably serve our community to the maximum without expecting anything in turn.

While there are fixed number of rhythmical ramifications for a particular number of units there are umpteen korvais for each rhythmical form of them. That is the nicety of music. amsharma

[Ranganayaki]

Think of this in the following way (sorry if this is a confusing explanation):
.

VijayR, thank you for that lovely explanation, it wasn't confusing at all.

[vasanthakokilam]

all types of prastaras owe a lot to chandah-prastara (based on how short/long syllables could be arranged in various ways).

This is quite significant. One thing that is not explicitly talked about is the relationship between musical emphasis/stress and tala 'beats'. I am now curious if there is a correspondence between Chandah-prasthara and where automatic emphasis occurs in phrases. One can observe that if a short is followed by a long, the emphasis/stress naturally falls on the beginning of the long.

If this is true and if there are more such things, then there is an intriguing possibility of relating tala prasthara as an intersection of akshara prasthara and chandah-prasthara, on the notion that the thala 'beats' are aligned with where emphasis/stress occurs ( for the most part ) in non-syncopated musical phrases.

Then it may be possible to say something like Serialnumber(s) in aksharaprasthara and serial number in chandah-prasthara(s) matche with a serial number in talaprasthara. This will then help us provide a formal answer to the oft asked question: 'why a certain song is in Adi as opposed to trisra matya or khanda jhampa'.

(The chandah-prasthara has correspondence to lyrical aspects as well, as we see in tamil veNpA grammar ( kuRil, neDil based ) and similar lyrical prastharas in Kannada poetry. This makes it the 4th dimension of a musical composition, thus integrating the rhythmical aspects with the lyrical aspects. The melodic aspects, the 5th dimension, rides on top of this rhythmic-lyrical structure)