TALAPRASTHARA ( Combinatorics)

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#1
Dear member arunk, Hereunder I am furnishing the 1st instalment of the rarest topic, 'Talaprastara'. Please go through it and enjoy. amsharma.
TALAPRASTARA
Definition of prastara
Rhythmical forms are innumerable and they all are the derivatives of this element, Prastara, in which Talangas are permuted into all possible combinations in a systematic process to avoid repetition. Considering the number of different Talangas used in the process of permutation, the modes of permutation are divided into four kinds and they are:
1.Chaturangaprastara, in which only four Talangas, Druta, Laghu, Guru and Pluta are used in the process of permutation and this is restricted to permute the Talangas pertaining to Chaturashrajati only. This was used in and around 13th century.
2.Panchangaprastara, in which five Talangas, Anudruta, Druta, Laghu, Guru and Pluta are used in the process of permutation and this could be applied to all the Jatis including Chaturashrajati. Along with Chaturangaprastara this was also used in and around 16th century.
Note: Kakapada, being a fully unsounded Talanga, was prohibited from use in the olden days even though it was included in the Talangas of several Talas like Simhanandana etc. At least to serve the mathematical purpose, now, this has been included in Talaprastara. Thus, with the inclusion of Kakapada, the previous Chaturangaprastara and Panchangaprastara hereafter become the modern Panchangaprastara and modern Shadangaprastara respectively.
4.Samyuktangaprastara, in which, along with the usual six independent Talangas, combinations of one or more Talangas, written one above the other are also used in the process of this modern permutation.
Each one of these modes of permutation has two main divisions, namely
1.Akhandaprastara (synonyms: Nijaprastara and Sarvaprastara) in which the
figures should be permuted following stipulated rules and regulations,
2. Khandaprastara, in which, the figures should be permuted following the same
rules as in Akhandaprastara observing certain restrictions. Among them
Khandaprastara has two subdivisions and they are:
(a) Hinaprastara, in which, the process of permutation should be made devoid of a particular figure.
(b) Yuktaprastara, in which, the process of permutation should be made containing a particular number of a particular figure.
All these above divisions consist of two different processes of permutation, namely
(i) Anulomaprastara
(ii) Vilomaprastara.
Under all these above divisions various and innumerable permutations are derived on a mathematical basis and to obtain the details of all these permutations precisely, two keyfigures are prescribed, namely
1.Sankhya
2.Mahapatala.
These figures represent the total number of permutations and the total number of all the figures used in the respective permutations.
Even without adopting the laborious process of permutation there are very interesting, important, useful and easy methods to obtain the above two keyfigures, Sankhya and Mahapatala and to obtain the â€˜series of figures of a particular number of permutationâ€™, the â€˜serial number of a permutation containing a particular series of figuresâ€™ and the â€˜total number of all the different denominations of figures derived up to a particular number of permutationâ€™ which are named as â€˜Nashta, Uddishta and Kalitaâ€™ respectively. While Nashta and Uddishta can be answered with the help of Sankhya, Kalita can be answered with the help of both Sankhya and Mahapatala. All these details pertaining to â€˜Samyuktangaprastaraâ€™ are furnished here for an easy understanding.
To facilitate a better understanding of Talaprastara, numerals are used in place of Angas.
Samyuktangaprastara
In the process of this permutation, general numerals i.e., 1, 2, 3, 4, 5, 6 etc., are used. Let us recollect the unitâ€™s column, the tenthâ€™s column, the hundredâ€™s column etc., as is taught in general mathematics which are always counted and cited from right to left only. In this element of Prastara the figures of any permutation should always be cited and written from right towards left only for the purpose of permutation. The following rules and regulations are to be followed in writing all the possible permutations and combinations serially to avoid iteration.
AKHANDAPRASTARA
Process of permutation:
1. Write a figure of the required value of units in the 1st permutation.
2. If there is more than one figure in the permutation, the permutable figure at the extreme left should be permuted first.
3. If the figure in the extreme left is so minute that it cannot be permuted further the next permutable figure to its right only should be permuted.
4. To permute a figure, write its immediate lower figure below the above figure.
5. If there is a remainder, a corresponding figure should be written to the left of the figure already written, observing relevant restrictions. If two or more figures are to be written they should be written only in the decreasing order of value to the left of the figure already written.
6. If there are any figures to the left of the upper figure which is being permuted, the remainder value of the permuted figure should be added to the total of all the figures and a figure should be written to the left of the figure already written, observing relevant restrictions.
7. The figures to the right of the figure in permutation should always be brought down and written correspondingly.
8. Following the relevant rules and restrictions, the process of permutation should be continued until all the figures become so minute that none of them can further be permuted.
9. The total value of all the figures in each permutation should be the same as the figure of permutation in question.
Example: â€˜4â€™ of 4UnitsPermutation (4UP):
1st permutation: Write figure â€˜4â€™ in the unitsâ€™ column as per rule No.1 (    4 ).
2nd permutation: Write figure â€˜3â€™ in the unitsâ€™ column as per rule No.4 and the remainder â€˜1â€™ in the tensâ€™ column as per rule No.5 (   1 3 ).
3rd permutation: Write figure â€˜2â€™ in the unitsâ€™ column as per rules Nos.2, 3 & 4 and another figure â€˜2â€™ in the tensâ€™ column as per rule No.6 (   2 2 ).
4th permutation: Write figure â€˜1â€™ in the tensâ€™ column as per rules Nos.2 & 4, another figure â€˜1â€™ in the hundredsâ€™ column as per rule No.5 and figure â€˜2â€™ in the unitsâ€™ column as per rule No.7
(  1 1 2 ).
5th permutation: Write figure â€˜1â€™ in the unitsâ€™ column as per rules Nos.3 & 4 and figure â€˜3â€™ in the tensâ€™ column as per rule No.6 (   3 1 ).
6th permutation: Write figure â€˜2â€™ in the tensâ€™ column as per rule No.2 & 4, figure â€˜1â€™ in the hundredsâ€™ column as per rule No.5 and figure â€˜1â€™ in the unitsâ€™ column as per rule No.7
(  1 2 1 ).
7th permutation: Write figure â€˜1â€™ in the tensâ€™ column as per rule No.3 & 4, figure â€˜2â€™ in the hundredsâ€™ column as per rule No.6 and figure â€˜1â€™ in the unitsâ€™ column as per rule No.7
(  2 1 1 ).
8th permutation: Write figure â€˜1â€™ in the hundredsâ€™ column as per rules Nos.2 & 4, figure â€˜1â€™ in the thousandsâ€™ column as per rule No.5 and figure â€˜1â€™ each in the unitsâ€™ and tensâ€™ columns as per rule No.7 (1 1 1 1 ).
As per rule No.8 none of the figures â€˜ 1 1 1 1 â€˜ of this above permutation can further be permuted as per rule No.9 each one of these â€˜8â€™ permutations carries the total value of â€˜4unitsâ€™ and hence this process of permutation is over with â€˜8permutationsâ€™ which are furnished hereunder.
401
1 302
2 203
1 1 204
3 105
1 2 106
2 1 107
1 1 1 108
Thus, the number of â€˜1â€™s, â€˜2â€™s, â€˜3â€™s & â€˜4â€™s are 12, 5, 2 & 1 respectively in these 8permutations arriving at the total of (12 + 5 + 2 + 1) 20 figures. In this Prastara, â€˜8â€™ (being the total number of permutations) is named as â€˜Sankhyaâ€™ and â€˜20â€™ (being the total number of the figures of all the denominations) is named as â€˜Mahapatalaâ€™. The same process should be repeated to permute figures further.
TALAPRASTARA
Definition of prastara
Rhythmical forms are innumerable and they all are the derivatives of this element, Prastara, in which Talangas are permuted into all possible combinations in a systematic process to avoid repetition. Considering the number of different Talangas used in the process of permutation, the modes of permutation are divided into four kinds and they are:
1.Chaturangaprastara, in which only four Talangas, Druta, Laghu, Guru and Pluta are used in the process of permutation and this is restricted to permute the Talangas pertaining to Chaturashrajati only. This was used in and around 13th century.
2.Panchangaprastara, in which five Talangas, Anudruta, Druta, Laghu, Guru and Pluta are used in the process of permutation and this could be applied to all the Jatis including Chaturashrajati. Along with Chaturangaprastara this was also used in and around 16th century.
Note: Kakapada, being a fully unsounded Talanga, was prohibited from use in the olden days even though it was included in the Talangas of several Talas like Simhanandana etc. At least to serve the mathematical purpose, now, this has been included in Talaprastara. Thus, with the inclusion of Kakapada, the previous Chaturangaprastara and Panchangaprastara hereafter become the modern Panchangaprastara and modern Shadangaprastara respectively.
4.Samyuktangaprastara, in which, along with the usual six independent Talangas, combinations of one or more Talangas, written one above the other are also used in the process of this modern permutation.
Each one of these modes of permutation has two main divisions, namely
1.Akhandaprastara (synonyms: Nijaprastara and Sarvaprastara) in which the
figures should be permuted following stipulated rules and regulations,
2. Khandaprastara, in which, the figures should be permuted following the same
rules as in Akhandaprastara observing certain restrictions. Among them
Khandaprastara has two subdivisions and they are:
(a) Hinaprastara, in which, the process of permutation should be made devoid of a particular figure.
(b) Yuktaprastara, in which, the process of permutation should be made containing a particular number of a particular figure.
All these above divisions consist of two different processes of permutation, namely
(i) Anulomaprastara
(ii) Vilomaprastara.
Under all these above divisions various and innumerable permutations are derived on a mathematical basis and to obtain the details of all these permutations precisely, two keyfigures are prescribed, namely
1.Sankhya
2.Mahapatala.
These figures represent the total number of permutations and the total number of all the figures used in the respective permutations.
Even without adopting the laborious process of permutation there are very interesting, important, useful and easy methods to obtain the above two keyfigures, Sankhya and Mahapatala and to obtain the â€˜series of figures of a particular number of permutationâ€™, the â€˜serial number of a permutation containing a particular series of figuresâ€™ and the â€˜total number of all the different denominations of figures derived up to a particular number of permutationâ€™ which are named as â€˜Nashta, Uddishta and Kalitaâ€™ respectively. While Nashta and Uddishta can be answered with the help of Sankhya, Kalita can be answered with the help of both Sankhya and Mahapatala. All these details pertaining to â€˜Samyuktangaprastaraâ€™ are furnished here for an easy understanding.
To facilitate a better understanding of Talaprastara, numerals are used in place of Angas.
Samyuktangaprastara
In the process of this permutation, general numerals i.e., 1, 2, 3, 4, 5, 6 etc., are used. Let us recollect the unitâ€™s column, the tenthâ€™s column, the hundredâ€™s column etc., as is taught in general mathematics which are always counted and cited from right to left only. In this element of Prastara the figures of any permutation should always be cited and written from right towards left only for the purpose of permutation. The following rules and regulations are to be followed in writing all the possible permutations and combinations serially to avoid iteration.
AKHANDAPRASTARA
Process of permutation:
1. Write a figure of the required value of units in the 1st permutation.
2. If there is more than one figure in the permutation, the permutable figure at the extreme left should be permuted first.
3. If the figure in the extreme left is so minute that it cannot be permuted further the next permutable figure to its right only should be permuted.
4. To permute a figure, write its immediate lower figure below the above figure.
5. If there is a remainder, a corresponding figure should be written to the left of the figure already written, observing relevant restrictions. If two or more figures are to be written they should be written only in the decreasing order of value to the left of the figure already written.
6. If there are any figures to the left of the upper figure which is being permuted, the remainder value of the permuted figure should be added to the total of all the figures and a figure should be written to the left of the figure already written, observing relevant restrictions.
7. The figures to the right of the figure in permutation should always be brought down and written correspondingly.
8. Following the relevant rules and restrictions, the process of permutation should be continued until all the figures become so minute that none of them can further be permuted.
9. The total value of all the figures in each permutation should be the same as the figure of permutation in question.
Example: â€˜4â€™ of 4UnitsPermutation (4UP):
1st permutation: Write figure â€˜4â€™ in the unitsâ€™ column as per rule No.1 (    4 ).
2nd permutation: Write figure â€˜3â€™ in the unitsâ€™ column as per rule No.4 and the remainder â€˜1â€™ in the tensâ€™ column as per rule No.5 (   1 3 ).
3rd permutation: Write figure â€˜2â€™ in the unitsâ€™ column as per rules Nos.2, 3 & 4 and another figure â€˜2â€™ in the tensâ€™ column as per rule No.6 (   2 2 ).
4th permutation: Write figure â€˜1â€™ in the tensâ€™ column as per rules Nos.2 & 4, another figure â€˜1â€™ in the hundredsâ€™ column as per rule No.5 and figure â€˜2â€™ in the unitsâ€™ column as per rule No.7
(  1 1 2 ).
5th permutation: Write figure â€˜1â€™ in the unitsâ€™ column as per rules Nos.3 & 4 and figure â€˜3â€™ in the tensâ€™ column as per rule No.6 (   3 1 ).
6th permutation: Write figure â€˜2â€™ in the tensâ€™ column as per rule No.2 & 4, figure â€˜1â€™ in the hundredsâ€™ column as per rule No.5 and figure â€˜1â€™ in the unitsâ€™ column as per rule No.7
(  1 2 1 ).
7th permutation: Write figure â€˜1â€™ in the tensâ€™ column as per rule No.3 & 4, figure â€˜2â€™ in the hundredsâ€™ column as per rule No.6 and figure â€˜1â€™ in the unitsâ€™ column as per rule No.7
(  2 1 1 ).
8th permutation: Write figure â€˜1â€™ in the hundredsâ€™ column as per rules Nos.2 & 4, figure â€˜1â€™ in the thousandsâ€™ column as per rule No.5 and figure â€˜1â€™ each in the unitsâ€™ and tensâ€™ columns as per rule No.7 (1 1 1 1 ).
As per rule No.8 none of the figures â€˜ 1 1 1 1 â€˜ of this above permutation can further be permuted as per rule No.9 each one of these â€˜8â€™ permutations carries the total value of â€˜4unitsâ€™ and hence this process of permutation is over with â€˜8permutationsâ€™ which are furnished hereunder.
401
1 302
2 203
1 1 204
3 105
1 2 106
2 1 107
1 1 1 108
Thus, the number of â€˜1â€™s, â€˜2â€™s, â€˜3â€™s & â€˜4â€™s are 12, 5, 2 & 1 respectively in these 8permutations arriving at the total of (12 + 5 + 2 + 1) 20 figures. In this Prastara, â€˜8â€™ (being the total number of permutations) is named as â€˜Sankhyaâ€™ and â€˜20â€™ (being the total number of the figures of all the denominations) is named as â€˜Mahapatalaâ€™. The same process should be repeated to permute figures further.
Last edited by msakella on 09 Jan 2007, 21:12, edited 1 time in total.
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#3
Dear moderator, coolkarni, Thanks a lot. You all are doing a great service to the cause of our Indian Music by creating such a nice forum and maintaining it properly. What I do is very little. Why because I am not doing any new thing at all like all of you but the usual thing what I daily do. No doubt your service is really greater. I am not writing this just to flatter all of you. Wishing you all the best at all times, amsharma.
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#4
Dear member, arunk, After understanding the process of Akhandaprastara of 4units in which we get a total of 8 permutations try to apply the same and permute the 5units, 6units, 7units, 8units and 9units. For the time being I shall tell you an easy method to know the total number of permutations which is just doubling the number in increasing order of value i.e., by permuting 1unit we will get only â€˜1â€™ permutation, for 2units â€˜2â€™ prastaras or permutations, for 3units â€˜4â€™ permutations, for 4units â€˜8â€™ permutations, for 5units â€˜16â€™ permutations, for 6units â€˜32â€™ permutations, for 7units â€˜64â€™ permutations,
for 8units â€˜128â€™ permutations and for 9units â€˜256â€™ permutations. By this you should understand that you would get â€˜256â€™ varieties of permutations by permuting 9units, which is Sankeernajaati in our music. Better to have a pencil and eraser than a pen and smallsquareruledpapers to write all these permutations without getting confused with the different digits of the permutations. Do it. amsharma.
for 8units â€˜128â€™ permutations and for 9units â€˜256â€™ permutations. By this you should understand that you would get â€˜256â€™ varieties of permutations by permuting 9units, which is Sankeernajaati in our music. Better to have a pencil and eraser than a pen and smallsquareruledpapers to write all these permutations without getting confused with the different digits of the permutations. Do it. amsharma.
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#5
Dear member, arunk, Hereunder I am furnishing a table in which 32 permutations, in total are obtained in the 6units permutation. I would like to tell you another easy method. While permuting the figure â€˜6â€™, by careful observation of the figures in the right extreme of each permutation of this table you can understand that while the first figure â€˜6â€™ and the immediate next lower figure occurs only once, the next immediate lower figure â€˜4â€™ occurs two times, the next immediate lower figure â€˜3â€™ occurs 4 times, the next immediate lower figure â€˜2â€™ occurs 8 times and the next immediate and the last figure â€˜1â€™ occurs 16 times making a total (1 + 1 + 2 + 4 + 8 + 16 =) 32. This should be applied to all.
6 01
1 5 02
2 4 03
1 1 4 04
3 3 05
1 2 3 06
2 1 3 07
1 1 1 3 08
4 2 09
1 3 2 10
2 2 2 11
1 1 2 2 12
3 1 2 13
1 2 1 2 14
2 1 1 2 15
1 1 1 1 2 16
5 1 17
1 4 1 18
2 3 1 19
1 1 3 1 20
3 2 1 21
1 2 2 1 22
2 1 2 1 23
1 1 1 2 1 24
4 1 1 25
1 3 1 1 26
2 2 1 1 27
1 1 2 1 1 28
3 1 1 1 29
1 2 1 1 1 30
2 1 1 1 1 31
1 1 1 1 1 1 32
From right to left the 1st column contains the serial number of the permutation, 2nd one is blank, from the 3rd to 8th columns they are from the 1st to 6th digits of the permutations. The process of permutation should always be made in this manner only. Proceed. amsharma.
(I have modified this table 3 or 4 times and, somehow, this time I am able to type this figures of the table nearer to the original table. Sorry for the inconvenience caused in this connection)
6 01
1 5 02
2 4 03
1 1 4 04
3 3 05
1 2 3 06
2 1 3 07
1 1 1 3 08
4 2 09
1 3 2 10
2 2 2 11
1 1 2 2 12
3 1 2 13
1 2 1 2 14
2 1 1 2 15
1 1 1 1 2 16
5 1 17
1 4 1 18
2 3 1 19
1 1 3 1 20
3 2 1 21
1 2 2 1 22
2 1 2 1 23
1 1 1 2 1 24
4 1 1 25
1 3 1 1 26
2 2 1 1 27
1 1 2 1 1 28
3 1 1 1 29
1 2 1 1 1 30
2 1 1 1 1 31
1 1 1 1 1 1 32
From right to left the 1st column contains the serial number of the permutation, 2nd one is blank, from the 3rd to 8th columns they are from the 1st to 6th digits of the permutations. The process of permutation should always be made in this manner only. Proceed. amsharma.
(I have modified this table 3 or 4 times and, somehow, this time I am able to type this figures of the table nearer to the original table. Sorry for the inconvenience caused in this connection)
Last edited by msakella on 09 Jan 2007, 12:37, edited 1 time in total.
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#6
Dear MsAkella Sir,
This method of permutations is very interesting. This is what I got for 5 units.
1. 5
2. 14
3. 23
4. 113
5. 32
6. 122
7. 212
8. 1112
9. 41
10. 131
11. 221
12. 1121
13. 311
14. 1211
15. 2111
16. 11111
In programming, there is a similar problem called countchange program where you have to count the number of ways of giving change for a particular amount assuming you have coins of certain denominations. That program is used to demonstrate the power of recursive thinking. I'm interested in knowing who framed the rules for arriving at these permutations?
The reason I'm asking this is it seems like you are giving some special importance to the serial number or the order of the permutations. Is the serial number important? Or to put it in another way, is the order of the generated permutations important to you?
I don't know much about music but if you replace the numbers with the angas that fit those numbers, you should get all the permissible thalams. And if anyone comes up with a new thalam, it should satisfy these constraints to be classifed as a thalam. But, in that case, whatever that person came up with should be one of the permutations. Just want to make sure I'm understanding this right as I have no musical background.
This method of permutations is very interesting. This is what I got for 5 units.
1. 5
2. 14
3. 23
4. 113
5. 32
6. 122
7. 212
8. 1112
9. 41
10. 131
11. 221
12. 1121
13. 311
14. 1211
15. 2111
16. 11111
In programming, there is a similar problem called countchange program where you have to count the number of ways of giving change for a particular amount assuming you have coins of certain denominations. That program is used to demonstrate the power of recursive thinking. I'm interested in knowing who framed the rules for arriving at these permutations?
The reason I'm asking this is it seems like you are giving some special importance to the serial number or the order of the permutations. Is the serial number important? Or to put it in another way, is the order of the generated permutations important to you?
I don't know much about music but if you replace the numbers with the angas that fit those numbers, you should get all the permissible thalams. And if anyone comes up with a new thalam, it should satisfy these constraints to be classifed as a thalam. But, in that case, whatever that person came up with should be one of the permutations. Just want to make sure I'm understanding this right as I have no musical background.
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#8
Dear member, sbala, The rules of permutation were framed centuries ago but have never been defined in any century by any author without any ambiguity at all. That is why I am compelled to frame the rules which are applicable to all the modes of permutations i.e., for the Old Chaturanga and Panchangaprastaras, modern Panchanga and Shadangaprastaras and the modern Samyuktangaprastara.
We have to make the process of permutation basing upon certain rules and restrictions furnished in the topic and after finishing it we have to write the serial numbers at the right extreme of each and every permutation. The form of the permutation has an interesting and indispensable link with its specific serial number about which you will come across later.
To make the matters easier to the aspirant I always start teaching this topic with numerals only. Later on, after understanding all the details of the topic the aspirant himself/herself will become able to replace all these numerals with Talangas. All these permutations we can use in Svarakalpana and the permutations devoid of Samyuktangas can also be rendered as Talas. To understand this Samyuktangaprastara, in particular, one need not be a musician. No doubt you are able to understand the topic well. Please go through the modified previous posts again and proceed further. Wishing you all the best, amsharma.
We have to make the process of permutation basing upon certain rules and restrictions furnished in the topic and after finishing it we have to write the serial numbers at the right extreme of each and every permutation. The form of the permutation has an interesting and indispensable link with its specific serial number about which you will come across later.
To make the matters easier to the aspirant I always start teaching this topic with numerals only. Later on, after understanding all the details of the topic the aspirant himself/herself will become able to replace all these numerals with Talangas. All these permutations we can use in Svarakalpana and the permutations devoid of Samyuktangas can also be rendered as Talas. To understand this Samyuktangaprastara, in particular, one need not be a musician. No doubt you are able to understand the topic well. Please go through the modified previous posts again and proceed further. Wishing you all the best, amsharma.
Last edited by msakella on 09 Jan 2007, 12:01, edited 1 time in total.
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#10
Dear MsAkella Sir,
Thanks for the explanation. I was wondering why you were writing the serial number to the right of the permutations. It seemed odd and that's why I put it to the left (the way we usually number line items). Now it seems like you have a strong reason for doing so. I will wait till we reach that point.
Thanks for the explanation. I was wondering why you were writing the serial number to the right of the permutations. It seemed odd and that's why I put it to the left (the way we usually number line items). Now it seems like you have a strong reason for doing so. I will wait till we reach that point.
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#11
dear msakellagaru,
i went through the information presented so far and i think i do follow the permutation scheme. It is quite attractive in its logical simplicity! Being a software programmer, i am tempted to want to write a program for it, but that would have to wait a bit longer .
I did have one doubt. I had thought that the total # of units would typically refer to total # of aksharas in a tala (as in 8 aksharas/units for Adi 1kalai). So a permutation of 9 say would give us all possible talas whose total unit count = 9, and so would include e.g. khanDa jAti tripuTa, miSra jAti rUpaka, etc. But you seem to imply differently:
(PS: i think you may have implied a possibly different meaing for "akshara" in some earlier posts  but i am not sure. In any case, here I am referring is of course as referred to by the 8 units in Adi, where each such unit takes up 4 subunits in catusra gati, 9 subunits in sankIrna gati etc.)
Arun
i went through the information presented so far and i think i do follow the permutation scheme. It is quite attractive in its logical simplicity! Being a software programmer, i am tempted to want to write a program for it, but that would have to wait a bit longer .
I did have one doubt. I had thought that the total # of units would typically refer to total # of aksharas in a tala (as in 8 aksharas/units for Adi 1kalai). So a permutation of 9 say would give us all possible talas whose total unit count = 9, and so would include e.g. khanDa jAti tripuTa, miSra jAti rUpaka, etc. But you seem to imply differently:
Am i misunderstanding you?By this you should understand that you would get â€˜256â€™ varieties of permutations by permuting 9units, which is Sankeernajaati in our music.
(PS: i think you may have implied a possibly different meaing for "akshara" in some earlier posts  but i am not sure. In any case, here I am referring is of course as referred to by the 8 units in Adi, where each such unit takes up 4 subunits in catusra gati, 9 subunits in sankIrna gati etc.)
Arun
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#12
Dear member, arunk, Yes, no doubt, being a software programmer you can very easily write a programme and get any thing out of them within minutes. Please wait for a little more time and you can do it as you like. In the same manner, nearly 15 years back, one of my disciples, as a software programmer like you, brought out, in print, 32,768 varieties of rhythmical forms pertaining to 16units in which all the rhythmical forms pertaining to all the jaatis of 3, 4, 5, 6, 7, 8, 9, 10, 12, 16 are included in a large book containing nearly 300 pages. 3 or 4 years back I have donated this voluminous book to the library of The Music Academy, Chennai through Prof. N.Ramanathan and the present Secretary Shri Shrivatsa along with one copy of all my six books.
I did all these mathematical calculations at the time we are not having computer or even calculator. You can think of my misery I experienced at that time in 1960s.
1kalai consists of 2 aksharas while executing. We should bring out the kala into picture when we want to render some thing like Svarakalpana or a composition or some such thing. Presently we should call it as Amsha or Unit and Aditala consists of 8 Kriyas or Amshas or Units. In respect of 9units you will get 256 varieties of permutations and as you wrote Khandajaatitriputa (522) the 81st permutation and Mishrajaatirupaka (27) the 3rd permutation of them could be rendered as Talas as they abide by the established rules I have already furnished previously in one of my posts. But, the rhythmical forms like the 5th (36), 9th (45), 10th (135), 26th (1314), 35th (243) etc., etc., which include Samyuktangas while transliterating the respective rhythmical form, should not be rendered as they do not abide by these rules.
What you wrote in your last para is right. Aditala of 8units consists of 8x4=12 aksharas or subunits in Chaturashragati and 8x9=72 aksharas or subunits in Sankeernagati. Akshara means which is spelt out in a Unit or Kriya or Amsha. amsharma.
I did all these mathematical calculations at the time we are not having computer or even calculator. You can think of my misery I experienced at that time in 1960s.
1kalai consists of 2 aksharas while executing. We should bring out the kala into picture when we want to render some thing like Svarakalpana or a composition or some such thing. Presently we should call it as Amsha or Unit and Aditala consists of 8 Kriyas or Amshas or Units. In respect of 9units you will get 256 varieties of permutations and as you wrote Khandajaatitriputa (522) the 81st permutation and Mishrajaatirupaka (27) the 3rd permutation of them could be rendered as Talas as they abide by the established rules I have already furnished previously in one of my posts. But, the rhythmical forms like the 5th (36), 9th (45), 10th (135), 26th (1314), 35th (243) etc., etc., which include Samyuktangas while transliterating the respective rhythmical form, should not be rendered as they do not abide by these rules.
What you wrote in your last para is right. Aditala of 8units consists of 8x4=12 aksharas or subunits in Chaturashragati and 8x9=72 aksharas or subunits in Sankeernagati. Akshara means which is spelt out in a Unit or Kriya or Amsha. amsharma.
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#13
Hello everybody learning the topic, Talaprastara!!!
In fact, this is a very complicated topic, which needs much IQ and more perseverance. It is always desirable to teach this topic in person. But, for the first time I am trying to teach this through this forum by writing the relevant material in my posts. If you write in your post up to which point you could get into, I shall be able to help you and also to proceed further. I, myself, do not know at which point I shall be compelled to stop. In such case, in consultation with the learners, I prefer to conduct workshops at different centers to a small group of aspirants gathered from nearby places in near future to impart the remaining subject. amsharma
In fact, this is a very complicated topic, which needs much IQ and more perseverance. It is always desirable to teach this topic in person. But, for the first time I am trying to teach this through this forum by writing the relevant material in my posts. If you write in your post up to which point you could get into, I shall be able to help you and also to proceed further. I, myself, do not know at which point I shall be compelled to stop. In such case, in consultation with the learners, I prefer to conduct workshops at different centers to a small group of aspirants gathered from nearby places in near future to impart the remaining subject. amsharma
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#15
Dear member, rajumds, Ours is a combined effort. Some of our friends at first started this forum with a noble thought of enlightening and promoting our society in our great cultural heritage and it is working successfully. Standing upon this noble base I am trying to contribute my mite. Thatâ€™s all. I shall definitely do as you wrote and give gap of 2 or 3 days. No hurry at all. Thank you for your kind suggestion. amsharma.
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#16
MsAkella Sir,
Just so that I'm clear with some of the terms and the permutation scheme.
1. Sankhya is the total number of permutations for n units and it will always be 2^(n1).
2. Mahapatala is the total number of parts and it will be (n+1)2^n2
I'm assuming that the total number of units in our thala system will always be an integer and the parts also have to be integer values. Is this right as I got a bit confused with the 13.5 units in the other thread?
Since there are so many permutations available to explore, what is the significance of the 35 thalams? Why were only these patterns chosen as important? It would be great if you can address some of these questions in the next lessons.
Just so that I'm clear with some of the terms and the permutation scheme.
1. Sankhya is the total number of permutations for n units and it will always be 2^(n1).
2. Mahapatala is the total number of parts and it will be (n+1)2^n2
I'm assuming that the total number of units in our thala system will always be an integer and the parts also have to be integer values. Is this right as I got a bit confused with the 13.5 units in the other thread?
Since there are so many permutations available to explore, what is the significance of the 35 thalams? Why were only these patterns chosen as important? It would be great if you can address some of these questions in the next lessons.
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#17
Member, sbala dear, I cannot understand your signs or technical terms. I am not that much educated like all of you. I am only an old Matriculate. But, having interest in the English language and in typing it I could gather some of it by reading some of the English novels etc., etc. Please donâ€™t think I am well versed in English like you. Mine is only a Telugisedenglish. Please donâ€™t mind it. But, as I have taken the propagation of the knowledge of music as my mission I am trying to put the things together as I can in this unfamiliar language. I shall try to explain the subject in my terms only. Of course, if I can meet you in person, I can learn these things from you and try to explain in your terms. Until such time you have to bear with me please. OK.
As you wrote Sankhya is the total number of permutations and Mahapatala is the total number all the figures of all denominations put together. As you also wrote the total number of units in our Tala system will always be an integer and the parts also have to be integer values.
Always each and every part of the rhythmical form is full and it will not and should not allow fractions like Â¼ or Â½ or Â¾ at all. That is why the rhythmical form carrying 13.5 units, created recently by Adityamohan is not even a rhythmical form at all. If you learn this topic at least up to some extent you can also declare that this is neither a Tala nor even a rhythmical form. Even Sharabhanandanatala, having Samyutangas, may not be suitable to render as a Tala but, no doubt, it is a rhythmical form without containing any of the fractions, Â¼ or Â½ or Â¾. But this Shivapalatalam is not even a rhythmical form at all as it carries Â½ unit. Always each and every number is full like the figures 19 or 128 or 4567 but not 19Â¼, 128Â½, 4567Â¾ and among all these infinite number of permutations in the universe there will never be a fraction of a figure by itself. We have to make these fractions for mathematical purposes.
In the good olden days the Talas are very lengthier and as they all are very difficult to remember all the Talangas of them and also to execute them properly these smaller Talas are introduced to make the matters easier to a common man. Even in these Talas, all the composers preferred to compose their compositions in much smaller Talas only. That is why we will find all the compositions of Saint Thyagaraja are composed either in Adi or Rupaka or Triputa but not in any other Talas at all.
I always try to answer any of your questions. Never hesitate to ask any question. OK. Wishing you all the best. amsharma.
As you wrote Sankhya is the total number of permutations and Mahapatala is the total number all the figures of all denominations put together. As you also wrote the total number of units in our Tala system will always be an integer and the parts also have to be integer values.
Always each and every part of the rhythmical form is full and it will not and should not allow fractions like Â¼ or Â½ or Â¾ at all. That is why the rhythmical form carrying 13.5 units, created recently by Adityamohan is not even a rhythmical form at all. If you learn this topic at least up to some extent you can also declare that this is neither a Tala nor even a rhythmical form. Even Sharabhanandanatala, having Samyutangas, may not be suitable to render as a Tala but, no doubt, it is a rhythmical form without containing any of the fractions, Â¼ or Â½ or Â¾. But this Shivapalatalam is not even a rhythmical form at all as it carries Â½ unit. Always each and every number is full like the figures 19 or 128 or 4567 but not 19Â¼, 128Â½, 4567Â¾ and among all these infinite number of permutations in the universe there will never be a fraction of a figure by itself. We have to make these fractions for mathematical purposes.
In the good olden days the Talas are very lengthier and as they all are very difficult to remember all the Talangas of them and also to execute them properly these smaller Talas are introduced to make the matters easier to a common man. Even in these Talas, all the composers preferred to compose their compositions in much smaller Talas only. That is why we will find all the compositions of Saint Thyagaraja are composed either in Adi or Rupaka or Triputa but not in any other Talas at all.
I always try to answer any of your questions. Never hesitate to ask any question. OK. Wishing you all the best. amsharma.
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#18
Thanks Msakella Sir. Your explanations are very clear and your English is also better than mine. I don't think I would ever be able to explain a complex topic in such detail using simple terms. I got a bit excited and tried to derive the formula for Sankhya and Mahapatala. Anyway, I'm looking forward to learning more.
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#19
2nd part:
Drawing the tables of Prastara
There are very interesting and easy methods to obtain full details of all the permutations without adopting the process of permutation and they are furnished hereunder.
In the table of Akhandaprastara three horizontal lines are required to be drawn and among them the figures of the top horizontal line, Amshashreni indicates the value of the permutation in terms of numerals, gradually increasing in order of value. The middle horizontal line, Sankhyashreni indicates the figures of the respective Sankhya and the lowest horizontal line, Mahapatalashreni indicates the figures of the respective Mahapatala.
Writing the figures of the tables
In this Prastara there are some very interesting and easy methods to obtain the details of all the permutations with the help of some figures without going through the process of permutation. The method of writing these figures is furnished here.
Generalrule: As per the generalrule, figure â€˜1â€™ should always be written to the left extreme of the 1st house, containing figures pertaining to â€˜Sankhyaâ€™ of Akhandaprastara.
While writing the figures of the table, the total of the figures of all the preceding houses, which are hereafter called â€˜Proxiesâ€™, should be written.
1.AKHANDAPRASTARA
Keyrules:
1.Sankhya: As per the generalrule always write the figure â€˜1â€™ in the left extreme of the 1st house, containing figures pertaining to â€˜Sankhyaâ€™. Later while writing the figures of â€˜Sankhyaâ€™, write the total of all the existing proxies, including the figure â€˜1â€™ written to the extreme left.
2.Mahapatala: Write the total of all the existing proxies of both Sankhya & Mahapatalaâ€™ including the figure â€˜1â€™ in the extreme left of â€˜Sankhyaâ€™, even if its corresponding figure of Mahapatala does not exist.
TABLE OF AKHANDA PRASTHARA
1 2 3 4 5 6 7 8 9
1 1 2 4 8 16 32 64 128 256
1 3 8 20 48 112 256 576 1280
In the above table, the figures in the top horizontal line, Amshashreni indicates the value of the permutation in terms of figures of units i.e., 1UP, 2UP, 3UP, 4UP and so on, the figures in the middle horizontal line, Sankhyashreni denotes the value of â€˜Sankhyaâ€™ and the figures in the bottom horizontal line, Mahapatalashreni denotes the value of â€˜Mahapatalaâ€™. For example, as per the figures â€˜6, 32 & 112â€™ in the top, middle and bottom houses respectively it should be presumed that, the process of permutation of â€˜6unitsâ€™ results in â€˜32â€™ permutations and â€˜112â€™ figures of different denominations which are its Sankhya and Mahapatala respectively.
Drawing the tables of Prastara
There are very interesting and easy methods to obtain full details of all the permutations without adopting the process of permutation and they are furnished hereunder.
In the table of Akhandaprastara three horizontal lines are required to be drawn and among them the figures of the top horizontal line, Amshashreni indicates the value of the permutation in terms of numerals, gradually increasing in order of value. The middle horizontal line, Sankhyashreni indicates the figures of the respective Sankhya and the lowest horizontal line, Mahapatalashreni indicates the figures of the respective Mahapatala.
Writing the figures of the tables
In this Prastara there are some very interesting and easy methods to obtain the details of all the permutations with the help of some figures without going through the process of permutation. The method of writing these figures is furnished here.
Generalrule: As per the generalrule, figure â€˜1â€™ should always be written to the left extreme of the 1st house, containing figures pertaining to â€˜Sankhyaâ€™ of Akhandaprastara.
While writing the figures of the table, the total of the figures of all the preceding houses, which are hereafter called â€˜Proxiesâ€™, should be written.
1.AKHANDAPRASTARA
Keyrules:
1.Sankhya: As per the generalrule always write the figure â€˜1â€™ in the left extreme of the 1st house, containing figures pertaining to â€˜Sankhyaâ€™. Later while writing the figures of â€˜Sankhyaâ€™, write the total of all the existing proxies, including the figure â€˜1â€™ written to the extreme left.
2.Mahapatala: Write the total of all the existing proxies of both Sankhya & Mahapatalaâ€™ including the figure â€˜1â€™ in the extreme left of â€˜Sankhyaâ€™, even if its corresponding figure of Mahapatala does not exist.
TABLE OF AKHANDA PRASTHARA
1 2 3 4 5 6 7 8 9
1 1 2 4 8 16 32 64 128 256
1 3 8 20 48 112 256 576 1280
In the above table, the figures in the top horizontal line, Amshashreni indicates the value of the permutation in terms of figures of units i.e., 1UP, 2UP, 3UP, 4UP and so on, the figures in the middle horizontal line, Sankhyashreni denotes the value of â€˜Sankhyaâ€™ and the figures in the bottom horizontal line, Mahapatalashreni denotes the value of â€˜Mahapatalaâ€™. For example, as per the figures â€˜6, 32 & 112â€™ in the top, middle and bottom houses respectively it should be presumed that, the process of permutation of â€˜6unitsâ€™ results in â€˜32â€™ permutations and â€˜112â€™ figures of different denominations which are its Sankhya and Mahapatala respectively.
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#20
Msakella Sir,
I'm now able to understand the reason why we do not see longer thalams. But even among the shorter thalams, there should be 128 combinations for a 8 unit thalam. Why did the 422 pattern (Adi?) become so popular as opposed to say 242? Is that because it has some interesting properties that the other patterns don't have? Also, you said many thalas are no longer rendered, as it had to be simplified for the common man. Does the common man here refer to the rasika or music students/performers?
I'm now able to understand the reason why we do not see longer thalams. But even among the shorter thalams, there should be 128 combinations for a 8 unit thalam. Why did the 422 pattern (Adi?) become so popular as opposed to say 242? Is that because it has some interesting properties that the other patterns don't have? Also, you said many thalas are no longer rendered, as it had to be simplified for the common man. Does the common man here refer to the rasika or music students/performers?
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#21
Explanation: Example â€“ 6units permutation:
(To make it easy, even though the doubling of the numbers of Sankhya i.e., 1, 2, 4, 8, 16, 32 and so on is suggested at the first instance, in fact, the following rules have to be followed in writing the figures of this table)
SANKHYA: As per the Generalrule figure â€˜1â€™ should always be written to the left extreme of the 1st house containing figures pertaining to â€˜Sankhyashreniâ€™ of Akhandaprastara. Later while writing the figures of â€˜Sankhyaâ€™, write the total of all the existing proxies, including the figure â€˜1â€™ written to the extreme left. Accordingly, the total of the figures of all the preceding Proxies should be written in each house of Sankhya. Thus, figure 1 should be written in the 1st house pertaining to 1unit permutation, 1+1=2 should be written in the 2nd house pertaining to the 2units permutation, 2+1+1=4 should be written in the 3rd house pertaining to the 3units permutation, 4+2+1+1=8 should be written in the 4th house pertaining to the 4units permutation, 8+4+2+1+1=16 should be written in the 5th house pertaining to the 5units permutation, 16+8+4+2+1+1=32 should be written in the 6th house pertaining to the 6units permutation and so on.
MAHAPATALA: Write the total of all the existing proxies of both Sankhya & Mahapatalaâ€™ including the figure â€˜1â€™ in the extreme left of â€˜Sankhyaâ€™, even if its corresponding figure of Mahapatala does not exist. Accordingly, as per the figure 1 of Sankhya written as per the Generalrule write it in the 1st house of Mahapatala, write 1+1 + 1=3 in the 2nd house, write 2+3
+ 1+1 + 1=8 in the 3rd house, write 4+8 + 2+3 + 1+1 + 1=20 in the 4th house, write 8+20 + 4+8 + 2+3 + 1+1 + 1=48 in the 5th house, write 16+48 + 8+20 + 4+8 + 2+3 + 1+1 + 1=112 and so on.
If we analyse the Sankhya32 and Mahapatala112 of 6units permutation we will come to understand that each group of figures of Sankhya & Mahapatala together are applicable in three ways. For example, the 1st proxy and its Mahapatala (figure of the 1st preceding house) gives the details of figure â€˜1sâ€™ and accordingly, Sankhya16 and Mahapatala48 indicate that (a) from the bottom to top there are 16 permutations (32nd to 17th permutation) ending with figure â€˜1â€™ in the right extreme of each permutation, (b) apart from these sixteen â€˜1sâ€™ there are 48 other figures also pertaining to different denominations and (c) the â€˜SaMafigure (the 1st letters of Sankhya and Mahapatala), which is the grandtotal of the figures (16+48=) 64 of Sankhya & Mahapatala, put together, is the total number of â€˜1sâ€™ in all the 32 permutations. In the same manner, 2nd proxy â€˜8â€™ and its Mahapatala â€˜20â€™ indicate that (a) from the bottom to top there are 8 permutations (from 16th tom 9th permutation) ending with figure â€˜2â€™ in the right extreme of each permutation, (b) apart from these eight â€˜2sâ€™ there are 20 other figures also and (c) the â€˜SaMafigureâ€™ (8+20=) 28 is the total number of â€˜2sâ€™in all the 32 permutations and thus, 3rd proxy along with its Mahapatala gives the details of â€˜3sâ€™, 4th proxy along with its Mahapatala gives the details of â€˜4sâ€™, 5th proxy along with its Mahapatala gives the details of â€˜5sâ€™ and the 6th proxy along with its Mahapatala gives the details of â€˜6sâ€™. This should be applied to all. amsharma.
(To make it easy, even though the doubling of the numbers of Sankhya i.e., 1, 2, 4, 8, 16, 32 and so on is suggested at the first instance, in fact, the following rules have to be followed in writing the figures of this table)
SANKHYA: As per the Generalrule figure â€˜1â€™ should always be written to the left extreme of the 1st house containing figures pertaining to â€˜Sankhyashreniâ€™ of Akhandaprastara. Later while writing the figures of â€˜Sankhyaâ€™, write the total of all the existing proxies, including the figure â€˜1â€™ written to the extreme left. Accordingly, the total of the figures of all the preceding Proxies should be written in each house of Sankhya. Thus, figure 1 should be written in the 1st house pertaining to 1unit permutation, 1+1=2 should be written in the 2nd house pertaining to the 2units permutation, 2+1+1=4 should be written in the 3rd house pertaining to the 3units permutation, 4+2+1+1=8 should be written in the 4th house pertaining to the 4units permutation, 8+4+2+1+1=16 should be written in the 5th house pertaining to the 5units permutation, 16+8+4+2+1+1=32 should be written in the 6th house pertaining to the 6units permutation and so on.
MAHAPATALA: Write the total of all the existing proxies of both Sankhya & Mahapatalaâ€™ including the figure â€˜1â€™ in the extreme left of â€˜Sankhyaâ€™, even if its corresponding figure of Mahapatala does not exist. Accordingly, as per the figure 1 of Sankhya written as per the Generalrule write it in the 1st house of Mahapatala, write 1+1 + 1=3 in the 2nd house, write 2+3
+ 1+1 + 1=8 in the 3rd house, write 4+8 + 2+3 + 1+1 + 1=20 in the 4th house, write 8+20 + 4+8 + 2+3 + 1+1 + 1=48 in the 5th house, write 16+48 + 8+20 + 4+8 + 2+3 + 1+1 + 1=112 and so on.
If we analyse the Sankhya32 and Mahapatala112 of 6units permutation we will come to understand that each group of figures of Sankhya & Mahapatala together are applicable in three ways. For example, the 1st proxy and its Mahapatala (figure of the 1st preceding house) gives the details of figure â€˜1sâ€™ and accordingly, Sankhya16 and Mahapatala48 indicate that (a) from the bottom to top there are 16 permutations (32nd to 17th permutation) ending with figure â€˜1â€™ in the right extreme of each permutation, (b) apart from these sixteen â€˜1sâ€™ there are 48 other figures also pertaining to different denominations and (c) the â€˜SaMafigure (the 1st letters of Sankhya and Mahapatala), which is the grandtotal of the figures (16+48=) 64 of Sankhya & Mahapatala, put together, is the total number of â€˜1sâ€™ in all the 32 permutations. In the same manner, 2nd proxy â€˜8â€™ and its Mahapatala â€˜20â€™ indicate that (a) from the bottom to top there are 8 permutations (from 16th tom 9th permutation) ending with figure â€˜2â€™ in the right extreme of each permutation, (b) apart from these eight â€˜2sâ€™ there are 20 other figures also and (c) the â€˜SaMafigureâ€™ (8+20=) 28 is the total number of â€˜2sâ€™in all the 32 permutations and thus, 3rd proxy along with its Mahapatala gives the details of â€˜3sâ€™, 4th proxy along with its Mahapatala gives the details of â€˜4sâ€™, 5th proxy along with its Mahapatala gives the details of â€˜5sâ€™ and the 6th proxy along with its Mahapatala gives the details of â€˜6sâ€™. This should be applied to all. amsharma.
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#22
Dear member, sbala, In pattern 242 of Adi there is every possibility of confusion as we may forget to render 2 Drutas consecutively many a time. That is why our ancestors have chosen 422 but not 242 fortunately enough. In creating confusion it effects either Rasika or student or performer in the same manner. Isnâ€™t it? amsharma.
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#23
Guru Akella  those of us who have degrees and the like are no match for your godgiven brilliance and intelligence. Not to speak of your patience and wonderful teaching style.msakella wrote:Member, sbala dear, I cannot understand your signs or technical terms. I am not that much educated like all of you. I am only an old Matriculate.
Please continue to educate us not just in knowledge, but also vinaya and humility.
My namaskarams to you.
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#24
Member, Chi.Jayaram dear, Thank you very much for your kind compliments. â€˜Yadyad Vibhoothi Matsatvamâ€™ â€“ all the good things what I have are only the blessings of the Almighty and the bad things are what I have fondly earned. So, entire credit must go to the Almighty and I am HIS Sevak. Thatâ€™s all. By Godâ€™s grace I have taken the propagation of our music as my mission of my life. Along with the knowledge of Music what the Almighty had proposed to give you will come out from me and you have to bear with me. OK. Wishing you all the best, amsharma.
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#25
MsAkella Sir,
I think you have explained why I was having a problem with Dhruva thalam which starts with a laghu and ends with a laghu. There were times when I forgot which Laghu I was on. In general, thalams that start and end with the same thala anga are more difficult to render(performer) as well as follow(for the rasika). Is this correct?
I think you have explained why I was having a problem with Dhruva thalam which starts with a laghu and ends with a laghu. There were times when I forgot which Laghu I was on. In general, thalams that start and end with the same thala anga are more difficult to render(performer) as well as follow(for the rasika). Is this correct?
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