Two kinds of infinities and their relationship to Tala and Laya

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#1 Two kinds of infinities and their relationship to Tala and Laya
Let me state up front that this is at best half baked and also can be frustratingly theoretical and geeky. That should not dissuade you from reading further but just be forewarned. Members who are well versed in theoretical computer science may possibly find my way of describing this amusing.
I was refreshing my gyan I gathered many decades back in a theoretical computer science course that most decision problems in the world are not computable. Decision problems are of the type where the result is Yes or No for a given input. That can be a bit puzzling at first. How does one even know about all the problems of the world let alone be certain that one can not write a program to solve them. But the tactic they use to prove that is by reducing the problem to a well known mathematical fact about natural numbers and real numbers. Natural numbers are the ones like 1, 2, 3,.... infinity without any decimals and real numbers are the ones like 1.234565 with the number of decimal points being potentially infinity.
I would not go into the details of how computer scientists prove that most decision problems of the real world are not computable, it is actually quite straight forward, I can do that in a separate post if there is interest )
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So what does this have to do with Tala and Laya
Here is where the half baked part begins.
For this purpose, think of Tala as we know them. And Laya as a rhythmic composition laid out on time.
Akellaji has taught us that our Talaprasthara literature assigns serial numbers to talas. That is, all the talas of the world that one can potentially think of can be represented by their respective serial numbers. The formal way of stating this is each such serial number is a member of the set of all Natural Numbers(N).
Conclusion 1: There are infinite talas and talas are members of N.
Now Laya on the other hand is a more general rhythmic concept than tala. Laya can be thought of as a rhythmic composition laid out in the time axis. That is, a rhythmic composition is a sequence of rhythm alphabets arranged in time. Each such composition can only be represented by an infinite sequence of digits since you can extend the time involved in the composition to arbitrarily long ones. With a clever sleight of hand, we can generalize this to say that each laya composition is represented by a real number between 0 and 1 (R). ( just put a decimal point in front of that infinitely long set of digits to represent each composition). The formal way of stating this is each rhythmic composition is a member of the set of real numbers (R)
Conclusion 2: There are infinite rhythmic compositions (Laya) and rhythmic compositions are members of R
It is a well known fact in Mathematics that Natural numbers are Countably Infinite and Real numbers are Uncountably Infinite. The uncountably infinite is bigger than countably infinite ( let the audacity of that statement sink in, but it is true. )
This leads us to say
The number of talas are countably infinite
The number of rhythmic compositions are uncountably infinite
There are not enough talas to go around to account for all the rhythmic compositions.
Stating it differently,
Most rhythmic compositions are not representable by Talas

The above is just analogous to what I alluded to without elaboration that 'most problems are not computable'. It is the same proof.

While reading the above, you may think as a counter argument that a tala can be thought of as a ruler and you can cover the entire length of any rhythmic composition using repeated application of that ruler even if it takes infinite attempts. That is totally justified along practical grounds. That is how our current compositions are setup. But what the above says is that between two such compositions that are measurable with a tala there are infinite number of other compositions that you can come up with. That is just a restatement of the mathematical fact that there are infinite real numbers between two natural numbers.

I think the same argument can be applied to show that there are not enough ragas in the world to represent all the melodic compositions possible, ragas being countably infinite and melodic compositions being uncountably infinite
I was refreshing my gyan I gathered many decades back in a theoretical computer science course that most decision problems in the world are not computable. Decision problems are of the type where the result is Yes or No for a given input. That can be a bit puzzling at first. How does one even know about all the problems of the world let alone be certain that one can not write a program to solve them. But the tactic they use to prove that is by reducing the problem to a well known mathematical fact about natural numbers and real numbers. Natural numbers are the ones like 1, 2, 3,.... infinity without any decimals and real numbers are the ones like 1.234565 with the number of decimal points being potentially infinity.
I would not go into the details of how computer scientists prove that most decision problems of the real world are not computable, it is actually quite straight forward, I can do that in a separate post if there is interest )
===
So what does this have to do with Tala and Laya
Here is where the half baked part begins.
For this purpose, think of Tala as we know them. And Laya as a rhythmic composition laid out on time.
Akellaji has taught us that our Talaprasthara literature assigns serial numbers to talas. That is, all the talas of the world that one can potentially think of can be represented by their respective serial numbers. The formal way of stating this is each such serial number is a member of the set of all Natural Numbers(N).
Conclusion 1: There are infinite talas and talas are members of N.
Now Laya on the other hand is a more general rhythmic concept than tala. Laya can be thought of as a rhythmic composition laid out in the time axis. That is, a rhythmic composition is a sequence of rhythm alphabets arranged in time. Each such composition can only be represented by an infinite sequence of digits since you can extend the time involved in the composition to arbitrarily long ones. With a clever sleight of hand, we can generalize this to say that each laya composition is represented by a real number between 0 and 1 (R). ( just put a decimal point in front of that infinitely long set of digits to represent each composition). The formal way of stating this is each rhythmic composition is a member of the set of real numbers (R)
Conclusion 2: There are infinite rhythmic compositions (Laya) and rhythmic compositions are members of R
It is a well known fact in Mathematics that Natural numbers are Countably Infinite and Real numbers are Uncountably Infinite. The uncountably infinite is bigger than countably infinite ( let the audacity of that statement sink in, but it is true. )
This leads us to say
The number of talas are countably infinite
The number of rhythmic compositions are uncountably infinite
There are not enough talas to go around to account for all the rhythmic compositions.
Stating it differently,
Most rhythmic compositions are not representable by Talas

The above is just analogous to what I alluded to without elaboration that 'most problems are not computable'. It is the same proof.

While reading the above, you may think as a counter argument that a tala can be thought of as a ruler and you can cover the entire length of any rhythmic composition using repeated application of that ruler even if it takes infinite attempts. That is totally justified along practical grounds. That is how our current compositions are setup. But what the above says is that between two such compositions that are measurable with a tala there are infinite number of other compositions that you can come up with. That is just a restatement of the mathematical fact that there are infinite real numbers between two natural numbers.

I think the same argument can be applied to show that there are not enough ragas in the world to represent all the melodic compositions possible, ragas being countably infinite and melodic compositions being uncountably infinite
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#2 Re: Two kinds of infinities and their relationship to Tala and Laya
If you have an oldfashioned tala machine, you can dial in available/known talas, set the tempo, and leave the "ruler" (my father used to lecture me: a ruler is a king or queen; that straight thing with numbers on it is called a rule, and just occasionally I remember that ) repeating for as long as you want.
If you have a computer with a program like Bounce Metronome, you can set any "tala" you care to imagine, and more, and infinitely adjust it to give regular or irregular intervals or even combine them in cross rhythms. Serial numbers are irrelevant: I have always thought that the definition of a rhythmic cycle includes the fact that it is cyclic, that is, repeats. But maybe even that isn't even true.
Which leads us.... ???
If you have a computer with a program like Bounce Metronome, you can set any "tala" you care to imagine, and more, and infinitely adjust it to give regular or irregular intervals or even combine them in cross rhythms. Serial numbers are irrelevant: I have always thought that the definition of a rhythmic cycle includes the fact that it is cyclic, that is, repeats. But maybe even that isn't even true.
Which leads us.... ???
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#3 Re: Two kinds of infinities and their relationship to Tala and Laya
Nick, good thinking.
(This is still half baked so keep the challenge going, I am adapting the proof for 'most problems do not have programs to solve them' by creating a parallel 'Most laya sequences do not have talas to represent them' but I know I am stretching a bit, do follow my train of thought closely since some of it is a bit subtle but do keep me honest )
(For most of you who wonder what is the practical use of it in music, don't bother, there is none I can think of yet. Think of this as 'theoretical music' taken to the extreme. Just like how practicing computer scientists are not stymied by the fact that most problems are not computable, They go on and write computer programs for the problems they encounter)
For simplicity, let us define tala to be a sequences of '0's and '1's. 0 representing nishabda (silence) beat and 1 representing Sabda ( sounded ) beats. You can have other 'angas' as in real talas but they are not needed for this proof. The key is, each tala is finite sequence of 0s and 1s and if you convert that to a number, that is a Natural number. (The actual number may not be same as the serial number assigned by talaprasthara but that is understandable and not germane to this proof).
We can afford to ignore the specific details, the crux of the issue is a tala can be represented by a Natural number. That number itself may be go on till infinity but each number is a finite length.
e.g. A tala which has a sequence of 'Sounded' 'Silence' 'Silence wil be represented by binary '100' which is the natural number 4
Now, a rhythmic composition is again a sequence of Nishabda and Sabda beats arranged in time. We can think of that as a table: T1, T2, T3, T4 etc. time slots and against each one you have a '1' or '0' depending on whether that time slot is sounded or not. Theoretically (again only theoretically ), that table is possibly infinite in width since it can go to arbitrary length of time. One rhythmic composition is different from another one in that pattern of '1's and '0's in that table are different. Without loss of generality, we can say that each rhythmic composition is represented by an infinitely long sequence of '1's and '0's.
To bring it to mathematics, convert that infinitely long sequence of bits to a real number by putting a 'binary point' in front of that.
(binary point is the equivalent of decimal point we are all familiar with in binary, each one with a value of 2 power 1, 2 power 2 .... etc.).
Now each rhythmic composition in the world is represented by an arbitrary precision real number between 0 ( total silence) and 1( total sound). ( this 0 and 1 are the real numbers 0 and 1 which are made up on bits 0 and 1 to represent silence and sounds at each time slots. So all 0s is total silence and all 1s is total sound).
e.g a rhythmic composition whose sequence for T1 thru T10 is 1 0 0 will be represented by a real number .100 = .5
( the benefit of choosing real number for rhythmic composition is we can assume that each one goes on till infinity theoretically but practically if it stops after T1, T2 and T3, rest are 0s which does not change the value. So .100 = .1000000000000000 = .5
If we are ok up to this, then the proof is quite straightforward as I wrote in my previous post.
Talas are from Natural Numbers
Rhythmic Compositions are from Real Numbers.
Both the sets of Natural Numbers and Real Numbers are infinitely long, but we know the set of real numbers in uncountably infinite and set of natural numbers is countably infinite and so the number of possible real numbers is larger than the number of natural numbers. ( If you have a problem with that mind boggling result, take it up with mathematicians ).
That naturally leads to the conclusion
1) Number of rhythmic compositions are greater than the number of talas possible. In fact, it is infinitely greater.
2) As a corollary, most rhythmic compositions can not be represented by talas ( using our simpler definition of talas but should generalize to our actual def of tala )
This all boils down to an even simpler observation that a table of infinite width and infinite length is bigger than a table of finite width and infinite length
I still feel there may be something I am missing in my application of the proof from computer science to this but hey, I have never shied away from sharing half baked thoughts, why an exception now
Ragas and melodic compositions can be proven to have similar relationships. We have to include vakra ragas so you can have arbitrarily long raga signatures.
(This is still half baked so keep the challenge going, I am adapting the proof for 'most problems do not have programs to solve them' by creating a parallel 'Most laya sequences do not have talas to represent them' but I know I am stretching a bit, do follow my train of thought closely since some of it is a bit subtle but do keep me honest )
(For most of you who wonder what is the practical use of it in music, don't bother, there is none I can think of yet. Think of this as 'theoretical music' taken to the extreme. Just like how practicing computer scientists are not stymied by the fact that most problems are not computable, They go on and write computer programs for the problems they encounter)
For simplicity, let us define tala to be a sequences of '0's and '1's. 0 representing nishabda (silence) beat and 1 representing Sabda ( sounded ) beats. You can have other 'angas' as in real talas but they are not needed for this proof. The key is, each tala is finite sequence of 0s and 1s and if you convert that to a number, that is a Natural number. (The actual number may not be same as the serial number assigned by talaprasthara but that is understandable and not germane to this proof).
We can afford to ignore the specific details, the crux of the issue is a tala can be represented by a Natural number. That number itself may be go on till infinity but each number is a finite length.
e.g. A tala which has a sequence of 'Sounded' 'Silence' 'Silence wil be represented by binary '100' which is the natural number 4
Now, a rhythmic composition is again a sequence of Nishabda and Sabda beats arranged in time. We can think of that as a table: T1, T2, T3, T4 etc. time slots and against each one you have a '1' or '0' depending on whether that time slot is sounded or not. Theoretically (again only theoretically ), that table is possibly infinite in width since it can go to arbitrary length of time. One rhythmic composition is different from another one in that pattern of '1's and '0's in that table are different. Without loss of generality, we can say that each rhythmic composition is represented by an infinitely long sequence of '1's and '0's.
To bring it to mathematics, convert that infinitely long sequence of bits to a real number by putting a 'binary point' in front of that.
(binary point is the equivalent of decimal point we are all familiar with in binary, each one with a value of 2 power 1, 2 power 2 .... etc.).
Now each rhythmic composition in the world is represented by an arbitrary precision real number between 0 ( total silence) and 1( total sound). ( this 0 and 1 are the real numbers 0 and 1 which are made up on bits 0 and 1 to represent silence and sounds at each time slots. So all 0s is total silence and all 1s is total sound).
e.g a rhythmic composition whose sequence for T1 thru T10 is 1 0 0 will be represented by a real number .100 = .5
( the benefit of choosing real number for rhythmic composition is we can assume that each one goes on till infinity theoretically but practically if it stops after T1, T2 and T3, rest are 0s which does not change the value. So .100 = .1000000000000000 = .5
If we are ok up to this, then the proof is quite straightforward as I wrote in my previous post.
Talas are from Natural Numbers
Rhythmic Compositions are from Real Numbers.
Both the sets of Natural Numbers and Real Numbers are infinitely long, but we know the set of real numbers in uncountably infinite and set of natural numbers is countably infinite and so the number of possible real numbers is larger than the number of natural numbers. ( If you have a problem with that mind boggling result, take it up with mathematicians ).
That naturally leads to the conclusion
1) Number of rhythmic compositions are greater than the number of talas possible. In fact, it is infinitely greater.
2) As a corollary, most rhythmic compositions can not be represented by talas ( using our simpler definition of talas but should generalize to our actual def of tala )
This all boils down to an even simpler observation that a table of infinite width and infinite length is bigger than a table of finite width and infinite length
I still feel there may be something I am missing in my application of the proof from computer science to this but hey, I have never shied away from sharing half baked thoughts, why an exception now
Ragas and melodic compositions can be proven to have similar relationships. We have to include vakra ragas so you can have arbitrarily long raga signatures.
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#4 Re: Two kinds of infinities and their relationship to Tala and Laya
In this analysis, you have focussed on talam as the agglomeration of multiple kriyas/angas. But what if talas are more abstract in nature? Would a tala not defined by any form of physical angas be countably or uncountably infinite?
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#5 Re: Two kinds of infinities and their relationship to Tala and Laya
Have we not been here before? There is no sounded or unsounded in rhythm. There are no claps waves or finger counts. There are no stamps of the foot on the floor, waves of the baton or beating with the sticks. There is just... rhythm. All that other stuff stuff is just ways of expressing or communicating it.
Remember the thing about time, which we might have discussed before? Sit and watch the movement of a clock's hands. Are you watching time? No: you are watching the movement of a machine which we find a useful indicator of "time" as it impacts our life. You cannot call the digits on a clock face "1" and the gaps between them "0." It doesn't work.
Whether it is a clock or a metronome, there is no one or zero. And even if we think we can see them, we are only seeing expressions of fragments. If you must "compute" all this, then I think you must escape the binary model of modern computing. You must look to some other model of computing.
Remember the thing about time, which we might have discussed before? Sit and watch the movement of a clock's hands. Are you watching time? No: you are watching the movement of a machine which we find a useful indicator of "time" as it impacts our life. You cannot call the digits on a clock face "1" and the gaps between them "0." It doesn't work.
Whether it is a clock or a metronome, there is no one or zero. And even if we think we can see them, we are only seeing expressions of fragments. If you must "compute" all this, then I think you must escape the binary model of modern computing. You must look to some other model of computing.
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#6 Re: Two kinds of infinities and their relationship to Tala and Laya
Nick, understood. I just picked Nishabda and Shabda so I have a variation. That is not the important part of the proof. You can pick up beat and down beat. We just need two things which we can represent as 0 or 1.
Having said that, I think you are not thinking about Nishabda, Shabda properly especially in the laya context. Silence and sound are indeed two aspects of rhythm
Melam72, If a tala by any definition can be represented by a finite set of digits or bits, then that would fall under countably infinite.
Having said that, I think you are not thinking about Nishabda, Shabda properly especially in the laya context. Silence and sound are indeed two aspects of rhythm
Melam72, If a tala by any definition can be represented by a finite set of digits or bits, then that would fall under countably infinite.
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#7 Re: Two kinds of infinities and their relationship to Tala and Laya
I have that covered already, in my assertion that you can'tYou can pick up beat and down beat.
represent as 0 or 1.
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#8 Re: Two kinds of infinities and their relationship to Tala and Laya
Nick, I don't understand. Any interesting tala is a pattern, a pattern can not all be the same then it ceases to be a pattern. So you need some variation. I am just picking the minimum that is needed for a variation which is 'two different things' and I am choosing to model mathematically as 0 and 1.
But if you say tala is nothing but a count of beats and not a pattern, that is fine, that is just a degenerate case of a pattern, it makes the job easy. I do not even need to code it as a natural number, the count is its natural number.
The reason to code this a natural number is to capture the fact that its length is bounded where as an arbitrary precision real number's length is not bounded. And the fact there can be infinite number of real numbers between two natural numbers.
But if you say tala is nothing but a count of beats and not a pattern, that is fine, that is just a degenerate case of a pattern, it makes the job easy. I do not even need to code it as a natural number, the count is its natural number.
The reason to code this a natural number is to capture the fact that its length is bounded where as an arbitrary precision real number's length is not bounded. And the fact there can be infinite number of real numbers between two natural numbers.
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#9 Re: Two kinds of infinities and their relationship to Tala and Laya
My desk and corner of t he room is a mess. An absolute tip. Let me arbitrarily assign zeros and ones to the items and use that to come up with a number: it might be a measure of messiness! But actually, it would be what... a arbitrary number. It wouldn't have any meaning, except perhaps to seed a call to rand().
So this is my problem: you are ascribing ones and zeros to parts of a rhythmic pattern and trying to make sense of the resultant numeric value. According to my countertheory, this ascribing of binary value is a fallacy, as there is no binary nature in the rhythm: there is an infinite number of potential subdivisions, none of which have a value of "zero," even if there appears to be "nothing" happening, ie silence, at that specific moment.
At the same time, it occurs to me that, of course, music can be digitised. We are listening to that all the time now. Each sample has a numeric value, and each numeric value sounds different. But you can't add them all up. At least, if you do, the result has no meaning. And even digital music is not really binary: it is just that our machines use binary representations of the numbers because they can only work that way.
An upbeat or a downbeat, a clap or a wave, is not a one or a zero: it is not a something or a nothing. It is not binary!
I feel that you would have to look to other computing models.
So this is my problem: you are ascribing ones and zeros to parts of a rhythmic pattern and trying to make sense of the resultant numeric value. According to my countertheory, this ascribing of binary value is a fallacy, as there is no binary nature in the rhythm: there is an infinite number of potential subdivisions, none of which have a value of "zero," even if there appears to be "nothing" happening, ie silence, at that specific moment.
At the same time, it occurs to me that, of course, music can be digitised. We are listening to that all the time now. Each sample has a numeric value, and each numeric value sounds different. But you can't add them all up. At least, if you do, the result has no meaning. And even digital music is not really binary: it is just that our machines use binary representations of the numbers because they can only work that way.
An upbeat or a downbeat, a clap or a wave, is not a one or a zero: it is not a something or a nothing. It is not binary!
I feel that you would have to look to other computing models.
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#10 Re: Two kinds of infinities and their relationship to Tala and Laya
This is especially true, and I have to grant it to Nick. We observe talam as kriyas, but talam is not made up of kriyas. Talam is an uncountably infinite ocean from where musicians open small taps into our dimension. Thus, any items at quantifying talam is essentially a waste of one's effort. Talam has no beginning or end like the story of Brahma and Vishnu trying to find the beginning of end of the Siva Lingam, it is essentially futile. Hence, talam is the pita  necessary for the natural production of music. However, things like alapanas, arguably the simplest form of manodharmam, have little kalapramanam, and, like Mother's love, is omnipresent and infinite. Thus, we may conclude that
SRUTHI MATHA
and
LAYA PITHA
SRUTHI MATHA
and
LAYA PITHA
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#11 Re: Two kinds of infinities and their relationship to Tala and Laya
One observation first. Melam and Nick, I am not sure if you two are saying the same thing. I will leave that aside for now.
Nick, While I see what you are saying in general, I think there is still a problem of definition of what we are talking about. I agree with your examples of degree of messiness and the digitized music. That is what gives me the impression we are not in sync on the crux of the issue. And it is not even about binary, tertiary, octal, digital or hex. Those things are not really germane to this, I had to pick a coding scheme and binary is the most general.
Just for illustration, outside of music, to illustrate what I am talking about, let me go back to how the proof for 'there are not enough programs to go around solving all the problems' Not for the proof itself but to illustrate the coding scheme. Let us say you write a computer program and it is contained in a file. Let us say It is a sequence of ASCII characters. Now what is its natural number equivalent for these purposes. Just write out the ASCII in binary form for each character. You will have a really long sequence of 0s and 1s. Treat that long sequence as a number. That will be a natural number but a number of finite length. It may be huge, really really huge.. does not matter. So by that coding scheme, every program in the world, does not matter what language it is written in and what it actual does or does not, has a corresponding natural number.
But the key point is, the number may be huge but each one can be represented by a finite set of bits.
That is the domain of programs.
Problems of the real world on the other hand, can be modeled as decision problems. ( go with me on this for now ). That is, for each input there is an Yes/No result. Without loss of generality, we can say that each problem can be modeled by its output which is then an infinite sequence of '0's and '1's.
Each problem will be a different sequence of 0s and 1s
That is the domain of problems.
As noted above, each program is represented by a finite sequence of bits and each problem is represented by an infinite sequence of bits. We can map the first one to Natural numbers and the second one to real numbers. Real numbers are uncountably infinite and natural numbers are countably infinite and so the Real number set is much bigger than the natural number set.
That is the line of argument for the proof that there are not enough programs to go around
===
Back to Tala and Laya, I am making a clear distinction. Tala is discrete and it is a finite count. It is a rhythmic signature. It represents a pulse of emphasis in music but it is not the pulse itself. I do not assume anything else about talas. And it is not any abstract concept, it is the ame concept as Talas that we use in CM. The important aspect is, it is coded as a finite number.
Laya on the other hand is the actual pulses in the music and that is what we call as Rhythmic composition. That is, laya is the representation of the rhythmic structure of a musical composition. It is not same as Tala. Tala exists outside of compositions where as Laya is inherent in compositions.
If you have different definitions of 'laya' in mind, that is fine, but for this purpose go with the above definition and treat it just as a label.
It exhibits itself in time and it is discrete and it is the pulse of music. Here we do not need to define Pulse anything more than 'that which gives us the impression of changing emphasis in music'. Since each rhythmic composition is laid out on time, each one can only be represented by an infinite sequence of bits, representing what the emphasis/pulse value is for each unit of time (the unit of time or how small the time is, does not matter. Only thing for this discussion is the commonly accepted semantics of rhythm that it is discrete and not continuous. If you disagree on that point, you are operating at a different level of abstractness which is fine but for this purpose go with what we normally perceive in the practice of music. I am just stating the obvious here just in case the lack of sync is in this aspect.
Now the parallels are established. Tala is coded as a natural number and Laya is coded as a real number, hence talas are countably infinite and laya is uncountably infinite. Hence the conclusion that the set of Laya is much bigger than the set of Talas which leads us to state that 'Most layas can not be represented by talas since there are not enough talas to go around'.
Nick, While I see what you are saying in general, I think there is still a problem of definition of what we are talking about. I agree with your examples of degree of messiness and the digitized music. That is what gives me the impression we are not in sync on the crux of the issue. And it is not even about binary, tertiary, octal, digital or hex. Those things are not really germane to this, I had to pick a coding scheme and binary is the most general.
Just for illustration, outside of music, to illustrate what I am talking about, let me go back to how the proof for 'there are not enough programs to go around solving all the problems' Not for the proof itself but to illustrate the coding scheme. Let us say you write a computer program and it is contained in a file. Let us say It is a sequence of ASCII characters. Now what is its natural number equivalent for these purposes. Just write out the ASCII in binary form for each character. You will have a really long sequence of 0s and 1s. Treat that long sequence as a number. That will be a natural number but a number of finite length. It may be huge, really really huge.. does not matter. So by that coding scheme, every program in the world, does not matter what language it is written in and what it actual does or does not, has a corresponding natural number.
But the key point is, the number may be huge but each one can be represented by a finite set of bits.
That is the domain of programs.
Problems of the real world on the other hand, can be modeled as decision problems. ( go with me on this for now ). That is, for each input there is an Yes/No result. Without loss of generality, we can say that each problem can be modeled by its output which is then an infinite sequence of '0's and '1's.
Each problem will be a different sequence of 0s and 1s
That is the domain of problems.
As noted above, each program is represented by a finite sequence of bits and each problem is represented by an infinite sequence of bits. We can map the first one to Natural numbers and the second one to real numbers. Real numbers are uncountably infinite and natural numbers are countably infinite and so the Real number set is much bigger than the natural number set.
That is the line of argument for the proof that there are not enough programs to go around
===
Back to Tala and Laya, I am making a clear distinction. Tala is discrete and it is a finite count. It is a rhythmic signature. It represents a pulse of emphasis in music but it is not the pulse itself. I do not assume anything else about talas. And it is not any abstract concept, it is the ame concept as Talas that we use in CM. The important aspect is, it is coded as a finite number.
Laya on the other hand is the actual pulses in the music and that is what we call as Rhythmic composition. That is, laya is the representation of the rhythmic structure of a musical composition. It is not same as Tala. Tala exists outside of compositions where as Laya is inherent in compositions.
If you have different definitions of 'laya' in mind, that is fine, but for this purpose go with the above definition and treat it just as a label.
It exhibits itself in time and it is discrete and it is the pulse of music. Here we do not need to define Pulse anything more than 'that which gives us the impression of changing emphasis in music'. Since each rhythmic composition is laid out on time, each one can only be represented by an infinite sequence of bits, representing what the emphasis/pulse value is for each unit of time (the unit of time or how small the time is, does not matter. Only thing for this discussion is the commonly accepted semantics of rhythm that it is discrete and not continuous. If you disagree on that point, you are operating at a different level of abstractness which is fine but for this purpose go with what we normally perceive in the practice of music. I am just stating the obvious here just in case the lack of sync is in this aspect.
Now the parallels are established. Tala is coded as a natural number and Laya is coded as a real number, hence talas are countably infinite and laya is uncountably infinite. Hence the conclusion that the set of Laya is much bigger than the set of Talas which leads us to state that 'Most layas can not be represented by talas since there are not enough talas to go around'.
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#12 Re: Two kinds of infinities and their relationship to Tala and Laya
I am playing the roll of the jester here, the dumbo joker who insists that stuff makes sense to him.
And it isn't too hard. Unlike you and many other fellow members, I have neither formal qualification nor even a feel for numbers or number science. That may mean one of two things: first is that I really don't know what I am talking about , second is that I don't get drawn in to what may or may not be a nice intellectual process or theory.
If you start ascribing values to individual elements, let's say in digital files, whether ascii text, executables, music or images, then we are moving into the area of encryption and data integrity, which are closely related, and which, to me, seem to exercise even the greatest mathematical minds! We ascribe a value to a file, as a simple example, to compare to the same file before we downloaded it. Then we know it is the same (is it called a hash code? I forget! Something of that sort). But the value that we end up with is, whilst it guarantees the integrity of the file, is nothing to do with its contents.
Similarly, you may ascribe "values" to either the rhythmic and/or melodic components of music. You will perhaps have a method of uniquely identifying the container of that music, but you won't have said anything valid about the music. You might decide to change track and invent digital music, even, except that's been done already
Marchers may chant left, right, left, right, left, right or on, off, on, off, on, off or even one, zero, one, zero, one, zero. But the only advantage of chanting something numeric would be knowing how many steps they had taken, and that would be its only meaning.
And it isn't too hard. Unlike you and many other fellow members, I have neither formal qualification nor even a feel for numbers or number science. That may mean one of two things: first is that I really don't know what I am talking about , second is that I don't get drawn in to what may or may not be a nice intellectual process or theory.
If you start ascribing values to individual elements, let's say in digital files, whether ascii text, executables, music or images, then we are moving into the area of encryption and data integrity, which are closely related, and which, to me, seem to exercise even the greatest mathematical minds! We ascribe a value to a file, as a simple example, to compare to the same file before we downloaded it. Then we know it is the same (is it called a hash code? I forget! Something of that sort). But the value that we end up with is, whilst it guarantees the integrity of the file, is nothing to do with its contents.
Similarly, you may ascribe "values" to either the rhythmic and/or melodic components of music. You will perhaps have a method of uniquely identifying the container of that music, but you won't have said anything valid about the music. You might decide to change track and invent digital music, even, except that's been done already
Marchers may chant left, right, left, right, left, right or on, off, on, off, on, off or even one, zero, one, zero, one, zero. But the only advantage of chanting something numeric would be knowing how many steps they had taken, and that would be its only meaning.
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#13 Re: Two kinds of infinities and their relationship to Tala and Laya
Nick, your hash/checksum example is right on. That is one way to assign a numerical value to the contents of a file.
(Beyond checking for data integrity, the same domain of study includes the fascinating topics of convolutional error correction Just as an aside, they are not as mindbogglingly complex as one may think or as it may seem at first encounter, it just requires focused study over a period of time.)
Just to be sure, for our current purposes we do not need any of that. There is just that overlap that both involve some kind of coding, and in this case it can be as elementary as we can make it.
It is not about reinventing digital music as you say later, but I am glad you talked about creating unique natural number signatures which is indeed germane to the discussion and only to that extent.
>You will perhaps have a method of uniquely identifying the container of that music, but you won't have said anything valid about the music.
Very true. But we need to be precise about what it is and what it is not. What it captures about the tala has to be valid ( there is no escape from that ) but it need not capture all aspects of the tala. Back to your statement, yes, it does not say anything about its musical semantics or aesthetics. The interesting thing is It need not say anything about such things to prove the statements we are making.
When discussing computer programs and decision problems, we did not really say much about them. It is not even about the details of problem solving or computer programming. That is the counter intuitive part of it.We are saying something so universal and substantial ( which some may even characterize as fatalistic) but to prove those we do not need to know the details of its internals. We operate at a meta level. How else can you even approach proving a statement like 'Most problems in the world are not computable'.
The statement 'Most laya expressions of music can not be represented by a tala' is a meta statement in that vein.
Thanks Melam and Nick for engaging with me on this, this level of discussion itself is more than what can be expected. It is quite abstract and it would not be out of norm to skip right past it. So it is cool you thought it is worthwhile enough to discuss it. I am still open for any attack on my proof or method of proof as it applies to Tala. Mine is not the last word, of course.
Even if it has any merit, I don't claim to know any practical significance of it. One encouraging thing at a very minuscule level is, in computer science that statement and proof belongs to the 'algorithmic complexity' specialty. Even if that statement by itself is just a curiosity other closely associated results are monumentally important to computer science (some monumentally important results in that field are not yet proven though most people believe them to be true ).
So by comparison there may be some interesting associated results that may have significance to something like 'rhythmic/melodic complexity theory'. I am totally speculating now.
(Beyond checking for data integrity, the same domain of study includes the fascinating topics of convolutional error correction Just as an aside, they are not as mindbogglingly complex as one may think or as it may seem at first encounter, it just requires focused study over a period of time.)
Just to be sure, for our current purposes we do not need any of that. There is just that overlap that both involve some kind of coding, and in this case it can be as elementary as we can make it.
It is not about reinventing digital music as you say later, but I am glad you talked about creating unique natural number signatures which is indeed germane to the discussion and only to that extent.
>You will perhaps have a method of uniquely identifying the container of that music, but you won't have said anything valid about the music.
Very true. But we need to be precise about what it is and what it is not. What it captures about the tala has to be valid ( there is no escape from that ) but it need not capture all aspects of the tala. Back to your statement, yes, it does not say anything about its musical semantics or aesthetics. The interesting thing is It need not say anything about such things to prove the statements we are making.
When discussing computer programs and decision problems, we did not really say much about them. It is not even about the details of problem solving or computer programming. That is the counter intuitive part of it.We are saying something so universal and substantial ( which some may even characterize as fatalistic) but to prove those we do not need to know the details of its internals. We operate at a meta level. How else can you even approach proving a statement like 'Most problems in the world are not computable'.
The statement 'Most laya expressions of music can not be represented by a tala' is a meta statement in that vein.
Thanks Melam and Nick for engaging with me on this, this level of discussion itself is more than what can be expected. It is quite abstract and it would not be out of norm to skip right past it. So it is cool you thought it is worthwhile enough to discuss it. I am still open for any attack on my proof or method of proof as it applies to Tala. Mine is not the last word, of course.
Even if it has any merit, I don't claim to know any practical significance of it. One encouraging thing at a very minuscule level is, in computer science that statement and proof belongs to the 'algorithmic complexity' specialty. Even if that statement by itself is just a curiosity other closely associated results are monumentally important to computer science (some monumentally important results in that field are not yet proven though most people believe them to be true ).
So by comparison there may be some interesting associated results that may have significance to something like 'rhythmic/melodic complexity theory'. I am totally speculating now.
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#14 Re: Two kinds of infinities and their relationship to Tala and Laya
Something I would also like to point out is the fact that recorded music is a kalanchiyam (mixture) of melody and laya. It is impossible to record laya as a manifestation alone the same way melody cannot be isolated; it is the aural manifestation of laya which enables us to perceive it.
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#15 Re: Two kinds of infinities and their relationship to Tala and Laya
Well, VK, I know you like a good mental problem, but I do think you are doing a bit of a Rube Goldberg on this. And I still have a doubt: Rube Goldberg machines, however absurd the process, work. I am still unconvinced that your initial premise of ascribing value actually does. Maybe you'll be able to develop the machine and prove me wrong.
I have not looked at it for ages, but the Bounce Metronome program indicates the levels of complexity that can be handled by software. Unfortunately, it is not FOSS, so you cannot peruse the code. Of course, there are Linux "drum machines" which you would be able to. I don't know if this would contribute anything at all, but hey... bedtime reading
I have not looked at it for ages, but the Bounce Metronome program indicates the levels of complexity that can be handled by software. Unfortunately, it is not FOSS, so you cannot peruse the code. Of course, there are Linux "drum machines" which you would be able to. I don't know if this would contribute anything at all, but hey... bedtime reading
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#16 Re: Two kinds of infinities and their relationship to Tala and Laya
melam, what you say is quite true from the total musical aesthetics and perception. But for the purposes here we can treat laya as the written down rhythmic syllables of the mridangists when they play 'for the song' as opposed to the tala. CM percussionists are trained to play to the song and what that means is they are accentuating the rhythmic patterns that exist in the song. In that limited sense, they are indeed isolating the rhythmic composition out of the melodic composition. But a more direct approach to writing down the rhythmic part of a song is to notice that the ',', ';' etc that are used in CM notation are indeed about rhythm rather than melody. And then replace the swaras with out any karvai by a rhythmic symbol. Then write it out totally devoid of any swara names, what you will end up is the rhythmic aspects of the song. In fact, it is a great way to study the structural differences in the compositions of the trinities. It is a wonderful exercise even for a rasika who is interested in enjoying the rhythmic aspects deeply in addition to the CM composition as a whole.
Mridangists start off the tani avarthanam sometimes using the laya established by the singer. Not sure how prevalent that practice is but when it is done you will know it. They are not 'playing' the song on the mridangam of course but what gives us the resemblance to the song is they are duplicating as close as possible the laya of the song without consideration to the swaras.
In western music notation, that is the technique they use for rhythmic compositions and much more formally. If you consider the bar and rhythm signature at the left of the staff as equivalent of our tala(they are not exactly the same, but they play the same role,) then the actual notes on the staff consisting of quarter notes, half notes etc. represent the laya. That is in the horizontal direction which is the time axis. The vertical placement is what conveys the melodic aspects. If you leave that out and sort of flatten it, then what results is the pure rhythmic representation of the song which is what I am calling Laya in this context. The staff notation makes it quite explicit and let you visually see the compression and rarefaction of the symbols and silences that are packed into a bar. They represent the rhythmic layout which is laya ( pun intended )
Anyway, this is all fairly well known and not anything new, but I thought I will write this in this context since this is a better way to distinguish between tala and laya though both are in the rhythmic realm.
Mridangists start off the tani avarthanam sometimes using the laya established by the singer. Not sure how prevalent that practice is but when it is done you will know it. They are not 'playing' the song on the mridangam of course but what gives us the resemblance to the song is they are duplicating as close as possible the laya of the song without consideration to the swaras.
In western music notation, that is the technique they use for rhythmic compositions and much more formally. If you consider the bar and rhythm signature at the left of the staff as equivalent of our tala(they are not exactly the same, but they play the same role,) then the actual notes on the staff consisting of quarter notes, half notes etc. represent the laya. That is in the horizontal direction which is the time axis. The vertical placement is what conveys the melodic aspects. If you leave that out and sort of flatten it, then what results is the pure rhythmic representation of the song which is what I am calling Laya in this context. The staff notation makes it quite explicit and let you visually see the compression and rarefaction of the symbols and silences that are packed into a bar. They represent the rhythmic layout which is laya ( pun intended )
Anyway, this is all fairly well known and not anything new, but I thought I will write this in this context since this is a better way to distinguish between tala and laya though both are in the rhythmic realm.
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#17 Re: Two kinds of infinities and their relationship to Tala and Laya
There is a somewhat different but inconspicuously related item that is less abstract than the above. This is something that directly applies to us, rasikas.
We have to start at a technical level first but stay with me, it is related to music.
In the theoretical computing world there is an unsolved conjecture ( known as 'P = NP' problem ). At a lay person level, it can be described like this.
There are two aspects to solving a problem.
1) Coming up with a method to produce the solution to the problem
2) Verifying if that is indeed a solution.
The class of problems called 'P' is where items 1 and 2 can be done easily (There is a technical definition of what 'easy' is, we do not need to go there )
That is, both producing the solution and verifying the solution are easy. For example, given a list of carnatic musicians, sorting them by alphabetical order and verifying if the sorted list is indeed sorted properly can be done easily. They all take some time which is proportional to the number of items in the list. That is considered easy. So both aspects are easy.
The class of problems called 'NP' where item 1 is hard but item 2 is easy. That is, producing a solution is not easy ( it can take a long time ) but verifying if it is a solution is easy. Example is Sudoku. As the square gets bigger, the time it takes to solve gets exponentially harder. But once it is solved, someone can verify it is solved correctly.
Of course there are problems that are both hard to solve and hard to verify. Chess is one such thing.
The unsolved problem is the question 'Is P = NP'. That is, is there a clever method to do Sudoku just as easy as verifying the solution? There is not one currently but we do not know if it just does not exist or we have not found it yet.
Most people think the conjecture 'P = NP' is false. That is, there are problems that are hard to do even if it is easy to verify.
Coming to music,
One expert in this field had a serious of metaphors about what is the equivalent if P is indeed equal to NP.
One of them is 'Listening and appreciating Mozart will make you a Mozart'
it is a metaphorical description by using the equivalence
Becoming Composers and musicians are hard. Coming up with a piece originally can take an inordinate amount of time, sometimes no composition at all
Becoming a rasika is easy. Once there is a composition or performance, it is much easier to verify if it conforms to the raga, tala, aesthetics etc.
Of course we know all that intuitively and do not need any great proof for the obvious statement, most of us believe that there is no way to become a Thyagaraja by listening to Thyagaraja.
I thought it is interesting that the rasika vs composer contrast is akin to verifying a solution vs producing a solution.
We have to start at a technical level first but stay with me, it is related to music.
In the theoretical computing world there is an unsolved conjecture ( known as 'P = NP' problem ). At a lay person level, it can be described like this.
There are two aspects to solving a problem.
1) Coming up with a method to produce the solution to the problem
2) Verifying if that is indeed a solution.
The class of problems called 'P' is where items 1 and 2 can be done easily (There is a technical definition of what 'easy' is, we do not need to go there )
That is, both producing the solution and verifying the solution are easy. For example, given a list of carnatic musicians, sorting them by alphabetical order and verifying if the sorted list is indeed sorted properly can be done easily. They all take some time which is proportional to the number of items in the list. That is considered easy. So both aspects are easy.
The class of problems called 'NP' where item 1 is hard but item 2 is easy. That is, producing a solution is not easy ( it can take a long time ) but verifying if it is a solution is easy. Example is Sudoku. As the square gets bigger, the time it takes to solve gets exponentially harder. But once it is solved, someone can verify it is solved correctly.
Of course there are problems that are both hard to solve and hard to verify. Chess is one such thing.
The unsolved problem is the question 'Is P = NP'. That is, is there a clever method to do Sudoku just as easy as verifying the solution? There is not one currently but we do not know if it just does not exist or we have not found it yet.
Most people think the conjecture 'P = NP' is false. That is, there are problems that are hard to do even if it is easy to verify.
Coming to music,
One expert in this field had a serious of metaphors about what is the equivalent if P is indeed equal to NP.
One of them is 'Listening and appreciating Mozart will make you a Mozart'
it is a metaphorical description by using the equivalence
Becoming Composers and musicians are hard. Coming up with a piece originally can take an inordinate amount of time, sometimes no composition at all
Becoming a rasika is easy. Once there is a composition or performance, it is much easier to verify if it conforms to the raga, tala, aesthetics etc.
Of course we know all that intuitively and do not need any great proof for the obvious statement, most of us believe that there is no way to become a Thyagaraja by listening to Thyagaraja.
I thought it is interesting that the rasika vs composer contrast is akin to verifying a solution vs producing a solution.
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#18 Re: Two kinds of infinities and their relationship to Tala and Laya
Before we get to the issues of easy or not : this reminds of me of another thing we discussed way back  that it still requires a conscious being to verify it  i.e. machines even if they attempt to produce a proof of a theorem  cannot themselves verify if the proof is correct!vasanthakokilam wrote: ↑20 Jun 2017, 02:24Becoming a rasika is easy. Once there is a composition or performance, it is much easier to verify if it conforms to the raga, tala, aesthetics etc.
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#19 Re: Two kinds of infinities and their relationship to Tala and Laya
True but the division is not along conscious beings versus machines. It is by the nature of the problems. There are basically 4 classes
Easy to solve, Easy to verify
Hard to solve, Easy to verify
Hard to solve, Hard to verify ( there are several grades of them )
Unsolvable
Easy to solve, Easy to verify
Hard to solve, Easy to verify
Hard to solve, Hard to verify ( there are several grades of them )
Unsolvable
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