Talaprasthara (Combinatorics)

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#1 Talaprasthara (Combinatorics)
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.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#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.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#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.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#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.

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#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.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#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.

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#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.

 Posts: 3424
 Joined: 07 Feb 2010, 21:41
 x 1
#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

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#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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#26
Dear member, sbala, You are correct. It is a bit difficult to render or follow Talas with unsequential Talangas. If at all you analyse things, generally, there will, definitely, be some background or other in arranging things by our ancestors. No doubt at all, they are very very clever. They have chosen only 7 Talas for a common man and they are very small in size when compared to bigger ones of 108 Talas. Even among them they have included only one Tala of odd number of Kriyas too which is Triputa. Even in choosing Jaatis they have given highest preference of 4 times to oddnumbers i.e., one even number of 4 and 4 oddnumbers of 3,5,7 & 9. It is very difficult to manage with an odd fellow. Isnâ€™t it? That is why they have selected 4 oddjaatis and only one evenjaati to strengthen us in Laya aspect. Even in Svara aspect they have created three very difficult notes, Saharanagandhara, Shuddhamadhyama and Kashikinishada, having extraordinary range of swinging unlike in any other music of the world. Even centuries back, even when computers or even calculators are not available, they have formulated this Prastara in Tala with precisive rules and regulations to make our music more effective and at the same time scientific too. In fact, while none of the chapters of the great Sangita Ratnakara is relevant with our present day music, only this Prastara is still relevant either with our present day Tala or Svarakalpana. See the greatness of our ancestors and always try to keep our culture in tact and pass it on to the posterity. Wishing you all the best, amsharma.
Last edited by msakella on 13 Jan 2007, 15:30, edited 1 time in total.

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#27
3rd part:
NASHTA, UDDISHTA & KALITA
In order to avoid the Himalayan task of the process of permutation, for which a human life is too short, some interesting, important and easy methods are furnished in this Prastara to obtain
(a) The series of figures of a particular serial number of permutation,
(b) The serial number of a particular prastara containing a particular series of figures,
(c)The details of all the figures of different denominations obtained up to a
particular number of permutation with the help of few figures.
They are named as â€˜Nashta, Uddishta and Kalitaâ€™ respectively and are elaborated hereunder along with the relevant rules and examples.
NASHTA
The easy method of obtaining the series of figures of a permutation, containing a particular serial number with the help of Sankhya is called â€˜Nashtaâ€™.
Rules for answering Nashta:
1. Write the figures of Sankhya up to the required units of permutation.
2. Subtract the questionnumber from the Sankhya and later subtract the respective proxies consecutively from the remainder.
3. If the first proxy is not subtracted from the remainder, write itâ€™s figure and repeat the process of subtraction of the relevant proxies of the â€˜unsubtractedproxyâ€™ from the remainder. For the subtraction of the 1st proxy, consider the â€˜nonsubtractionâ€™ of the 2nd proxy. For the consecutive subtraction of the 1st & 2nd proxies, consider the â€˜nonsubtractionâ€™ of the 3rd proxy. For the consecutive subtraction of the 1st, 2nd & 3rd proxies, consider the â€˜nonsubtractionâ€™ of the 4th proxy. For the consecutive subtraction of the 1st, 2nd, 3rd & 4th proxies, consider the â€˜nonsubtractionâ€™ of the 5th proxy  thus always considering the â€˜nonsubtractionâ€™ of the last proxy, write its corresponding figure and leave away its proxy. Repeat the process of subtraction by subtracting the relevant proxies of the â€˜unsubtractedproxyâ€™. In the absence of the remainder no figure needs to be left off.
4. Figures should always be written by subtracting the existing respective
proxies observing relevant rules and restrictions.
5. To answer the Nashta of Akhandaprastara, one or more figures of the
least denomination should be written to fillin the deficit value of the
permutation, when neither the remainder nor the proxy or both do not exist.
Note: 1. The figures of a permutation should always be cited from right to left only.
Examples:
1. What is the series of figures pertaining to the 1st permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜31â€™ (321).
From this above remainder, as the 1st, 2nd, 3rd, 4th and 5th proxies can consecutively be subtracted (3116=158=74=32=11=0) consider that the 6th proxy cannot be subtracted and write its figure â€˜6â€™ as per rule No.3.
Thus this permutation contains â€˜6â€™ in its series of figures.
2. What is the series of figures pertaining to the 2nd permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜30â€™ (322).
From the above remainder, as the 1st, 2nd, 3rd and 4th proxies can consecutively be subtracted (3016=148=64=22=0) and considering that the 5th proxy cannot be subtracted, write its figure â€˜5â€™ as per rule No.3.
Till now, out of the total value of â€˜6unitsâ€™ of the permutation, 5units are derived and to fillin the deficit value of 1unit of the permutation write one figure â€˜1â€™ in the left extreme of the permutation as per rule No.5.
Thus this permutation contains â€˜1 5â€™ in its series of figures.
3. What is the series of figures pertaining to the 3rd permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜29â€™ (323).
From the above remainder, as the 1st, 2nd and 3rd proxies could consecutively be subtracted (2916=138=54=1) and considering that the 4th proxy cannot be subtracted , write its figure â€˜4â€™ leaving off its proxy â€˜2â€™ as per rule No.3 to continue the process.
Now, as per rule 3, the process of Nashta starts afresh basing upon the above leftoff proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ can be subtracted (11=0)and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ as per rule No.3.
Thus this permutation contains â€˜2 4â€™ in its series of figures.
4. What is the series of figures pertaining to the 6th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜26â€™ (326).
From the above remainder, as the 1st and 2nd proxies can consecutively be subtracted (2616=108=2) and considering that the 3rd proxy cannot be subtracted , write its figure â€˜3â€™ leaving off its proxy â€˜4â€™ as per rule No. 3 to continue the process.
Now, as per rule No. 3, the process of Nashta starts again from the beginning basing upon the above leftoff proxy â€˜4â€™ and from the last remainder â€˜2â€™, as its 1st proxy â€˜2â€™ can be subtracted (22=0) and considering that the 2nd proxy cannot be subtracted , write its figure â€˜2â€™ as per rule No. 3.
Till now, out of the total value of â€˜6unitsâ€™ of the permutation, 5units are derived and to fillin the deficit value of 1unit of the permutation write one figure â€˜1â€™ in the extreme left of the permutation as per rule No.5.
Thus this permutation contains â€˜1 2 3â€™ in its series of figures.
5. What is the series of figures pertaining to the 7th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜25â€™ (327).
From the above remainder, as the 1st and 2nd proxies can consecutively be subtracted (2516=98=1) and considering that the 3rd proxy cannot be subtracted, write its figure â€˜3â€™ leaving off its proxy â€˜4â€™ as per rule No. 3 to continue the process.
Later, as per rule No. 3, the process of Nashta starts again basing upon the above leftoff proxy â€˜4â€™. Accordingly, from the last remainder â€˜1â€™, as its 1st proxy â€˜2â€™ cannot be subtracted, write the figure â€˜1â€™ of the 1st proxy itself and continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ cannot be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted ,write its figure â€˜2â€™ as per rule No. 3.
Thus this permutation contains â€˜2 1 3â€™ in its series of figures.
6. What is the series of figures pertaining to the 14th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No. 2 the remainder is â€˜18â€™ (3214).
From the above remainder, as the 1st proxy could only be subtracted (1816=2) consider that the 2nd proxy could not be subtracted and write its figure â€˜2â€™ leaving off its proxy â€˜8â€™ as per rule No. 3 to continue the process.
Later, as per rule No. 3, the process of Nashta starts again basing upon the above leftoff proxy â€˜8â€™ and from the last remainder â€˜2â€™, its 1st proxy â€˜4â€™ cannot be subtracted, being higher than the remainder, write the figure â€˜1â€™ of the same 1st proxy itself and continue the process as per rule No.3.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜4â€™ and from the last remainder â€˜2â€™, as its 1st proxy â€˜2â€™ can be subtracted (22=0)and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ as per rule No. 3.
Till now, out of the total value of â€˜6unitsâ€™ of the permutation, in total, 5units are derived and to fillin the deficit value of 1unit of the permutation write one figure â€˜1â€™ in the extreme left of the permutation as per rule No.5.
Thus this permutation contains â€˜1 2 1 2â€™ in its series of figures.
7. What is the series of figures pertaining to the 19th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜13â€™ (3219).
From the above remainder, as the 1st proxy â€˜16â€™ cannot be subtracted write the figure â€˜1â€™ of the same 1st proxy itself and continue the process as per rule No.3.
Later, as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜16â€™ and from the above remainder â€˜13â€™, as its 1st and 2nd proxies can be subtracted consecutively (138=54=1) and considering that the 3rd proxy cannot be subtracted, write its figure â€˜3â€™ leaving off its proxy â€˜2â€™ as per rule No. 3 to continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last leftoff proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its `1st proxy â€˜1â€™ can be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ as per rule No. 3.
Thus this permutation contains â€˜2 3 1â€™ in its series of figures.
8. What is the series of figures pertaining to the 23rd permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜9â€™ (3223).
From the above remainder, as the 1st proxy â€˜16â€™ could not be subtracted write the figure â€˜1â€™ of the same 1st proxy itself as per rule No.3 and continue the process.
Later, as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜16â€™ and from the remainder â€˜9â€™, as its 1st proxy â€˜8â€™ can be subtracted (98=1) and considering that the 2nd proxy cannot be subtracted , write its figure â€˜2â€™ leaving off its proxy â€˜4â€™ as per rule No. 3 to continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last leftoff proxy â€˜4â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜2â€™ cannot be subtracted, write the figure â€˜1â€™ of the 1st proxy as per rule No.3 and continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ can now be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted write its figure â€˜2â€™ as per rule No. 3.
Thus this permutation contains â€˜2 1 2 1â€™ in its series of figures.
9. What is the series of figures pertaining to the 27th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜5â€™ (3227).
From the above remainder, as the 1st and 2nd proxies, â€˜16 & 8â€™ respectively, cannot be subtracted write figures â€˜1 1â€™ representing the same 1st proxy itself as per rule No. 3 and continue the process.
Now, as per rule No. 3, the process of Nashta starts again basing upon the undeducted last proxy â€˜8â€™ and from the above remainder â€˜5â€™, as its 1st proxy â€˜4â€™ can be subtracted (54=1) and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ leaving off its proxy â€˜2â€™ to continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last leftoff proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ can be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2.â€™
Thus this permutation contains â€˜2 2 1 1â€™ in its series of figures.
NASHTA, UDDISHTA & KALITA
In order to avoid the Himalayan task of the process of permutation, for which a human life is too short, some interesting, important and easy methods are furnished in this Prastara to obtain
(a) The series of figures of a particular serial number of permutation,
(b) The serial number of a particular prastara containing a particular series of figures,
(c)The details of all the figures of different denominations obtained up to a
particular number of permutation with the help of few figures.
They are named as â€˜Nashta, Uddishta and Kalitaâ€™ respectively and are elaborated hereunder along with the relevant rules and examples.
NASHTA
The easy method of obtaining the series of figures of a permutation, containing a particular serial number with the help of Sankhya is called â€˜Nashtaâ€™.
Rules for answering Nashta:
1. Write the figures of Sankhya up to the required units of permutation.
2. Subtract the questionnumber from the Sankhya and later subtract the respective proxies consecutively from the remainder.
3. If the first proxy is not subtracted from the remainder, write itâ€™s figure and repeat the process of subtraction of the relevant proxies of the â€˜unsubtractedproxyâ€™ from the remainder. For the subtraction of the 1st proxy, consider the â€˜nonsubtractionâ€™ of the 2nd proxy. For the consecutive subtraction of the 1st & 2nd proxies, consider the â€˜nonsubtractionâ€™ of the 3rd proxy. For the consecutive subtraction of the 1st, 2nd & 3rd proxies, consider the â€˜nonsubtractionâ€™ of the 4th proxy. For the consecutive subtraction of the 1st, 2nd, 3rd & 4th proxies, consider the â€˜nonsubtractionâ€™ of the 5th proxy  thus always considering the â€˜nonsubtractionâ€™ of the last proxy, write its corresponding figure and leave away its proxy. Repeat the process of subtraction by subtracting the relevant proxies of the â€˜unsubtractedproxyâ€™. In the absence of the remainder no figure needs to be left off.
4. Figures should always be written by subtracting the existing respective
proxies observing relevant rules and restrictions.
5. To answer the Nashta of Akhandaprastara, one or more figures of the
least denomination should be written to fillin the deficit value of the
permutation, when neither the remainder nor the proxy or both do not exist.
Note: 1. The figures of a permutation should always be cited from right to left only.
Examples:
1. What is the series of figures pertaining to the 1st permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜31â€™ (321).
From this above remainder, as the 1st, 2nd, 3rd, 4th and 5th proxies can consecutively be subtracted (3116=158=74=32=11=0) consider that the 6th proxy cannot be subtracted and write its figure â€˜6â€™ as per rule No.3.
Thus this permutation contains â€˜6â€™ in its series of figures.
2. What is the series of figures pertaining to the 2nd permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜30â€™ (322).
From the above remainder, as the 1st, 2nd, 3rd and 4th proxies can consecutively be subtracted (3016=148=64=22=0) and considering that the 5th proxy cannot be subtracted, write its figure â€˜5â€™ as per rule No.3.
Till now, out of the total value of â€˜6unitsâ€™ of the permutation, 5units are derived and to fillin the deficit value of 1unit of the permutation write one figure â€˜1â€™ in the left extreme of the permutation as per rule No.5.
Thus this permutation contains â€˜1 5â€™ in its series of figures.
3. What is the series of figures pertaining to the 3rd permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜29â€™ (323).
From the above remainder, as the 1st, 2nd and 3rd proxies could consecutively be subtracted (2916=138=54=1) and considering that the 4th proxy cannot be subtracted , write its figure â€˜4â€™ leaving off its proxy â€˜2â€™ as per rule No.3 to continue the process.
Now, as per rule 3, the process of Nashta starts afresh basing upon the above leftoff proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ can be subtracted (11=0)and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ as per rule No.3.
Thus this permutation contains â€˜2 4â€™ in its series of figures.
4. What is the series of figures pertaining to the 6th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜26â€™ (326).
From the above remainder, as the 1st and 2nd proxies can consecutively be subtracted (2616=108=2) and considering that the 3rd proxy cannot be subtracted , write its figure â€˜3â€™ leaving off its proxy â€˜4â€™ as per rule No. 3 to continue the process.
Now, as per rule No. 3, the process of Nashta starts again from the beginning basing upon the above leftoff proxy â€˜4â€™ and from the last remainder â€˜2â€™, as its 1st proxy â€˜2â€™ can be subtracted (22=0) and considering that the 2nd proxy cannot be subtracted , write its figure â€˜2â€™ as per rule No. 3.
Till now, out of the total value of â€˜6unitsâ€™ of the permutation, 5units are derived and to fillin the deficit value of 1unit of the permutation write one figure â€˜1â€™ in the extreme left of the permutation as per rule No.5.
Thus this permutation contains â€˜1 2 3â€™ in its series of figures.
5. What is the series of figures pertaining to the 7th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜25â€™ (327).
From the above remainder, as the 1st and 2nd proxies can consecutively be subtracted (2516=98=1) and considering that the 3rd proxy cannot be subtracted, write its figure â€˜3â€™ leaving off its proxy â€˜4â€™ as per rule No. 3 to continue the process.
Later, as per rule No. 3, the process of Nashta starts again basing upon the above leftoff proxy â€˜4â€™. Accordingly, from the last remainder â€˜1â€™, as its 1st proxy â€˜2â€™ cannot be subtracted, write the figure â€˜1â€™ of the 1st proxy itself and continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ cannot be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted ,write its figure â€˜2â€™ as per rule No. 3.
Thus this permutation contains â€˜2 1 3â€™ in its series of figures.
6. What is the series of figures pertaining to the 14th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No. 2 the remainder is â€˜18â€™ (3214).
From the above remainder, as the 1st proxy could only be subtracted (1816=2) consider that the 2nd proxy could not be subtracted and write its figure â€˜2â€™ leaving off its proxy â€˜8â€™ as per rule No. 3 to continue the process.
Later, as per rule No. 3, the process of Nashta starts again basing upon the above leftoff proxy â€˜8â€™ and from the last remainder â€˜2â€™, its 1st proxy â€˜4â€™ cannot be subtracted, being higher than the remainder, write the figure â€˜1â€™ of the same 1st proxy itself and continue the process as per rule No.3.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜4â€™ and from the last remainder â€˜2â€™, as its 1st proxy â€˜2â€™ can be subtracted (22=0)and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ as per rule No. 3.
Till now, out of the total value of â€˜6unitsâ€™ of the permutation, in total, 5units are derived and to fillin the deficit value of 1unit of the permutation write one figure â€˜1â€™ in the extreme left of the permutation as per rule No.5.
Thus this permutation contains â€˜1 2 1 2â€™ in its series of figures.
7. What is the series of figures pertaining to the 19th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜13â€™ (3219).
From the above remainder, as the 1st proxy â€˜16â€™ cannot be subtracted write the figure â€˜1â€™ of the same 1st proxy itself and continue the process as per rule No.3.
Later, as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜16â€™ and from the above remainder â€˜13â€™, as its 1st and 2nd proxies can be subtracted consecutively (138=54=1) and considering that the 3rd proxy cannot be subtracted, write its figure â€˜3â€™ leaving off its proxy â€˜2â€™ as per rule No. 3 to continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last leftoff proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its `1st proxy â€˜1â€™ can be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ as per rule No. 3.
Thus this permutation contains â€˜2 3 1â€™ in its series of figures.
8. What is the series of figures pertaining to the 23rd permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜9â€™ (3223).
From the above remainder, as the 1st proxy â€˜16â€™ could not be subtracted write the figure â€˜1â€™ of the same 1st proxy itself as per rule No.3 and continue the process.
Later, as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜16â€™ and from the remainder â€˜9â€™, as its 1st proxy â€˜8â€™ can be subtracted (98=1) and considering that the 2nd proxy cannot be subtracted , write its figure â€˜2â€™ leaving off its proxy â€˜4â€™ as per rule No. 3 to continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last leftoff proxy â€˜4â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜2â€™ cannot be subtracted, write the figure â€˜1â€™ of the 1st proxy as per rule No.3 and continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last undeducted proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ can now be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted write its figure â€˜2â€™ as per rule No. 3.
Thus this permutation contains â€˜2 1 2 1â€™ in its series of figures.
9. What is the series of figures pertaining to the 27th permutation of 6UP?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
As per rule No.2 the remainder is â€˜5â€™ (3227).
From the above remainder, as the 1st and 2nd proxies, â€˜16 & 8â€™ respectively, cannot be subtracted write figures â€˜1 1â€™ representing the same 1st proxy itself as per rule No. 3 and continue the process.
Now, as per rule No. 3, the process of Nashta starts again basing upon the undeducted last proxy â€˜8â€™ and from the above remainder â€˜5â€™, as its 1st proxy â€˜4â€™ can be subtracted (54=1) and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2â€™ leaving off its proxy â€˜2â€™ to continue the process.
Now, again as per rule No. 3, the process of Nashta starts again basing upon the last leftoff proxy â€˜2â€™ and from the last remainder â€˜1â€™, as its 1st proxy â€˜1â€™ can be subtracted (11=0) and considering that the 2nd proxy cannot be subtracted, write its figure â€˜2.â€™
Thus this permutation contains â€˜2 2 1 1â€™ in its series of figures.
Last edited by msakella on 15 Jan 2007, 14:17, edited 1 time in total.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#28
Hello brothers and sisters of our forum!!!
I wish you all a very happy Sankranthi / Pongal. I hereby request all the aspirants who have started learning Talaprastara through our forum to kindly give me their email addresses and telephone numbers (both landline and cell) to enable me to invite you to the workshops on â€˜Talaprastaraâ€™ to be arranged at different places like
Chennai, Bangalore, Mumbai, Delhi etc., at the time of my visits to these places. You all can send these details to my email address â€˜[email protected]â€™. Wishing you all the best, amsharma.
I wish you all a very happy Sankranthi / Pongal. I hereby request all the aspirants who have started learning Talaprastara through our forum to kindly give me their email addresses and telephone numbers (both landline and cell) to enable me to invite you to the workshops on â€˜Talaprastaraâ€™ to be arranged at different places like
Chennai, Bangalore, Mumbai, Delhi etc., at the time of my visits to these places. You all can send these details to my email address â€˜[email protected]â€™. Wishing you all the best, amsharma.

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#29
Msakella Sir,
I have furnished the 4th, 5th and 8th perumutations of the 6 unit permutation and the 4th and 11th permutations of 7UP. Please verify when you have time.
4th Permutation of 6UP.
1. Deduct the serial number from the Sankhya. 324=28
2. Start deducting the proxies. 281684=0. 4th proxy is not deducted. Therefore, write 4.
3. Deficit is 2. Has to be filled with the lowest denomination units possible ie 1 1
4. Therefore, the 4th permutation is 114.
5th permutation of 6UP.
1. 325=27
2. 27168=3. Undeducted proxy=3
3. Start the process from the 3rd proxy.
4. 321=0. Undeducted proxy=3.
5. 5th permutation is 33.
8th permutation of 6UP
1. 328=24
2. 24168=0. 3rd proxy undeducted.
3. deficit is 3. to be filled as 111.
4. 8th permutation is 1113
4th permutation of 7UP
1. 644=60
2. 60321684=0. Undeducted proxy=5
3. Deficit=2.
4. 4th permutation is 115
11th permutation of 7UP
1. 6411=53
2. 533216=5. Undeducted proxy=3.
3. 54=1. Undeducted proxy=2.
4. 11=0. Undeducted proxy=2.
5. 11th permutation of 7UP is 223
1.
I have furnished the 4th, 5th and 8th perumutations of the 6 unit permutation and the 4th and 11th permutations of 7UP. Please verify when you have time.
4th Permutation of 6UP.
1. Deduct the serial number from the Sankhya. 324=28
2. Start deducting the proxies. 281684=0. 4th proxy is not deducted. Therefore, write 4.
3. Deficit is 2. Has to be filled with the lowest denomination units possible ie 1 1
4. Therefore, the 4th permutation is 114.
5th permutation of 6UP.
1. 325=27
2. 27168=3. Undeducted proxy=3
3. Start the process from the 3rd proxy.
4. 321=0. Undeducted proxy=3.
5. 5th permutation is 33.
8th permutation of 6UP
1. 328=24
2. 24168=0. 3rd proxy undeducted.
3. deficit is 3. to be filled as 111.
4. 8th permutation is 1113
4th permutation of 7UP
1. 644=60
2. 60321684=0. Undeducted proxy=5
3. Deficit=2.
4. 4th permutation is 115
11th permutation of 7UP
1. 6411=53
2. 533216=5. Undeducted proxy=3.
3. 54=1. Undeducted proxy=2.
4. 11=0. Undeducted proxy=2.
5. 11th permutation of 7UP is 223
1.
Last edited by sbala on 16 Jan 2007, 12:16, edited 1 time in total.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#30
Member, sbala dear, Hearty congrats. Extremely well. You got it.
Get a smallsquareruledbook, please make the permutations up to 9units, at the least, and do the Nashta to many of them. Regular practice is a must for music. Never forget it. You have very successfully got it. That is why I am posting the next two parts, Uddishta & Kalita. Go through them successfully and report back. I wish you should become the teacher of Talaprastara in Chennai as early as possible to propagate this rare topic of our Indian culture. Wishing you all the best, amsharma.
Get a smallsquareruledbook, please make the permutations up to 9units, at the least, and do the Nashta to many of them. Regular practice is a must for music. Never forget it. You have very successfully got it. That is why I am posting the next two parts, Uddishta & Kalita. Go through them successfully and report back. I wish you should become the teacher of Talaprastara in Chennai as early as possible to propagate this rare topic of our Indian culture. Wishing you all the best, amsharma.
Last edited by msakella on 19 Jan 2007, 21:31, edited 1 time in total.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#31
4th part:
UDDISHTA
Uddishta is the easy method to obtain the serial number of a permutation containing a particular series of figures, based upon the figures of Sankhya. However the rules are the same as in Nashta.
(a) As per the rules governing Nashta the undeducted proxies always derive the least figure i.e. â€˜1â€™ and the successfully deducted proxies always derive other higher figures. In a series of figures of a permutation, take out the relevant proxies of the figures which are successfully subtracted which in turn derive the â€˜AdhikangajanitaSankhyaâ€™ (AS, i.e; the figure from which the higher Angas/figures are derived). Later, subtract the AS from the Sankhya and the remainder is the serial number.
E X A M P L E S:
1. What is the serial number of the 6UP containing â€˜6â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the figure â€˜6â€™ of the permutation it should be presumed that this is derived by the consecutive subtraction of five proxies i.e., 1st proxy â€˜16â€™, 2nd proxy â€˜8â€™, 3rd proxy â€˜4â€™, 4th proxy â€˜2â€™ and 5th proxy â€˜1â€™ and the nonsubtraction of the 6th proxy â€˜1â€™ as per rule No. 3.
Thus the AS is â€˜31â€™ (16+8+4+2+1) and as per rule No. (a) the remainder â€˜1â€™(3231) is the serial number.
2. What is the serial number of the 6UP containing â€˜1 1 4â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜4â€™ of the permutation it should be presumed that this is derived by the consecutive subtraction of three proxies i.e., 1st proxy â€˜16â€™, 2nd proxy â€˜8â€™ and 3rd proxy â€˜4â€™ and the nonsubtraction of the 4th proxy â€˜2â€™ as per rule No. 3.
By the next two least figures â€˜1 1â€™ of the permutation it should always be presumed that they are written only to fillin the deficit value of the permutation as per rule No. 5.
Thus the AS is â€˜28â€™ (16+8+4) and as per rule No.(a) the remainder â€˜4â€™ (3228) is the serial number.
3. What is the serial number of the 6UP containing â€˜3 3â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜3â€™ of the permutation it should be presumed that this figure â€˜3â€™ is derived by the consecutive subtraction of the 1st proxy â€˜16â€™ and 2nd proxy â€˜8â€™ and the nonsubtraction of the 3rd proxy after which its proxy â€˜4â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next figure â€˜3â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜4â€™ the process is continued and this figure â€˜3â€™ is also derived by the consecutive subtraction of the 1st proxy â€˜2â€™ and 2nd proxy â€˜1â€™ and the nonsubtraction of the 3rd proxy â€˜1â€™ as per rule No. 3.
Thus the AS is â€˜27â€™ (16+8+2+1) and as per rule No.(a) the remainder â€˜5â€™ (3227) is the serial number.
4. What is the serial number of the 6UP containing â€˜2 2 2â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜2â€™ of the permutation it should be presumed that this figure is derived by the subtraction of the 1st proxy â€˜16â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜8â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next figure â€˜2â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜8â€™ the process is continued and again this figure â€˜2â€™ also is derived in the same way by the subtraction of the 1st proxy â€˜4â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜2â€™ has to be leftoff as per rule No. 3 to continue the process.
Again, by the last figure â€˜2â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜2â€™ the process is continued and again this figure â€˜2â€™ also is derived in the same way by the subtraction of the 1st proxy â€˜1â€™ and the nonsubtraction of the 2nd proxy as per rule No. 3.
Thus the AS is â€˜21â€™ (16+4+1) and as per rule No. (a) the remainder is â€˜11â€™ (3221) is the serial number.
5. What is the serial number of the 6UP containing â€˜2 1 1 2â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜2â€™ of the permutation it should be presumed that it is derived by the subtraction of the 1st proxy â€˜16â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜8â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next two figures â€˜1 1â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜8â€™ the process is continued and these figures â€˜1 1â€™ are derived by the consecutive nonsubtraction of the 1st proxy â€˜4â€™ of the above leftoff proxy â€˜8â€™ and again the 1st proxy â€˜2â€™ of the last undeducted proxy â€˜4â€™ as per rule No. 3 to continue the process.
Now, by the last figure â€˜2â€™ of the permutation it should be presumed that, basing upon the undeducted last proxy â€˜2â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜1â€™ and the nonsubtraction of the 2nd proxy as per rule No. 3.
Thus the AS is â€˜17â€™ (16+1) and as per rule No.(a) the remainder â€˜15â€™ (3217) is the serial number.
6. What is the serial number of the 6UP containing â€˜3 2 1â€™ in the series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜1â€™ of the permutation it should be presumed that this figure is derived by the nonsubtraction of the 1st proxy â€˜16â€™ as per rule No.3 and the process is continued.
Later, by the next figure â€˜2â€™ of the permutation it should be presumed that, basing upon the last undeducted proxy â€˜16â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜8â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜4â€™ has to be leftoff as per rule No. 3 to continue the process.
Now, by the last figure â€˜3â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜4â€™ the process is continued and this figure â€˜3â€™ is derived by the consecutive subtraction of the 1st proxy â€˜2â€™, and the 2nd proxy â€˜1â€™ and the nonsubtraction of the 3rd proxy â€˜1â€™ as per rule No. 3.
Thus the AS is â€˜11â€™ ( 8+2+1) and as per rule No.(a) the remainder â€˜21â€™ (3211) is the serial number.
7. What is the serial number of the 6UP containing â€˜2 1 2 1â€™ in the series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figureâ€™1â€™ of the permutation it should be presumed that this figure is derived by the nonsubtraction of the 1st proxy â€˜16â€™ as per rule No.3 and the process is continued.
Later, by the next figure â€˜2â€™ of the permutation it should be presumed that, basing upon the last undeducted proxy â€˜16â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜8â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜4â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next figure â€˜1â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜4â€™ the process is continued and this figure â€˜1â€™ is derived by the nonsubtraction of the 1st proxy â€˜2â€™ as per rule No. 3 and the process is continued.
Now, by the last figure â€˜2â€™ of the permutation it should be presumed that, basing upon the last undeducted proxy â€˜2â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜1â€™ and the nonsubtraction of the 2nd proxy as per rule No.3.
Thus the AS is â€˜9â€™ (8+1) and as per rule No.(a) the remainder â€˜23â€™ (329) is the serial number.
5th part:
KALITA
Kalita is the method for obtaining the full particulars of all the figures/ Angas used in the process of permutation up to the required prastara in question avoiding the laborious process of permutation.
E X A M P L E S:
1.Give the particulars of all the Figures of 6UP (6unitspermutation).
In Samyuktangaprastara the figures of all the preceding houses (including the figure â€˜1â€™ written in the extreme left as per the generalkeyrule) are the proxies of which the total of all forms the Sankhya.
According to the general table of permutations it should be analysed that as per the 
1st proxy there is/are â€˜16â€™ 1UEPs with â€˜48â€™ other figures
2nd ,, â€˜8â€™ 2UEPs â€˜20â€™ ,,
3rd ,, â€˜4â€™ 3UEPs â€˜8â€™ ,,
4th ,, â€˜2â€™ 4UEPs â€˜3â€™ ,,
5th ,, â€˜1â€™ 5UEPs â€˜1â€™ ,,
6th ,, â€˜1â€™ 6UEPs â€˜1â€™ ,,
(Note: UEP (unitending permutation) i.e., permutation ending with a particular figure)
 thus arriving at the total of â€˜32â€™ permutations along with â€˜112â€™ figures.
These â€˜32â€™ permutations starting from the figure â€˜6â€™ in the 1st permutation should be analysed in the following manner.
(a) As per the Sankhya â€˜1â€™ of the corresponding sixth proxy of 6UEP (6unitsending permutation), it should be understood that there is only one 6UEP with a single figure 6 as its corresponding Mahapatala does not exist.
(b) According to the analysis, as per the Sankhya â€˜1â€™ and its Mahapatala â€˜1â€™ of the corresponding fifth proxy pertaining to the 5UEP, it should be understood that, apart from the â€˜5â€™ in the right extreme end of this 5UEP, there is another figure â€˜1â€™ also in this permutation as per the Sankhya â€˜1â€™ of its corresponding first proxy of 1UEP  thus totaling to two (1+1) figures in this 5UEP. Here it is also interesting to know that there are only two figures of â€˜5â€™ in all these 32 permutations.
(c) According to the brief analysis, as per the Sankhya â€˜2â€™ and its Mahapatala â€˜3â€™ of the corresponding fourth proxy pertaining to the 4UEP, it should be understood that, apart from the two figures of 4 which are in the right extreme end of these two 4UEPs, there are â€˜3â€™ other figures also  thus totaling to five (2+3) figures in all these two 4UEPs. Here it is also interesting to know that there are only five figures of 4 in all these 32 permutations. According to the detailed analysis it should also be understood that, in these two 4UEPs, there are two figures of 4 (2/4) as per the Sankhya â€˜2â€™, one figure of 2 (1/2) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding second proxy of 2UEP and two figures of 1 (2/1) as per the SMfigure (combined figure of Sankhya and Mahapatala (1+1) â€˜2â€™ of the corresponding first proxy of 1UEP of the same 4UEP proxy  thus totaling to five (2+3) figures (2/4, 1/2 & 2/1=5 figures).
(d) According to the brief analysis as per the Sankhya â€˜4â€™ and its Mahapatala â€˜8â€™ of the corresponding third proxy pertaining to the 3UEP proxy, it should be understood that, apart from the four figures of 3 which are in the right extreme end of these four 3UEPs, there are â€˜8â€™ other figures also thus totaling to (4+8) â€˜12â€™ figures in all these four 3UEPs. Here it is also interesting to know that there are only four figures of 3 in all these 32 permutations. According to the detailed analysis it should also be understood that, in these four 3UEPs, there are four figures of 3 (4/3) as per the Sankhya â€˜4â€™, one figure of â€˜3â€™ (1/3) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding third proxy of 3UEP, two figures of 2 (2/2) as per the SMfigure (1+1) â€˜2â€™ of the corresponding second proxy of 2UEP and five figures of 1 (5/1) as per the SMfigure (2+3=) â€˜5â€™ of the corresponding first proxy of 1UEP of the same 3UEP proxy  thus totaling to twelve (4+8) figures (4/3, 1/3, 2/2 & 5/1=12 figures).
(e) According to the brief analysis as per the Sankhya â€˜8â€™ and its Mahapatala â€˜20â€™ of the corresponding second proxy pertaining to the 2UEP, it should be understood that, apart from the eight figures of 2 which are in the right extreme end of these eight 2UEPs, there are 20 other figures also thus totaling to (8+20) â€˜28â€™ figures in all these eight 2UEPs. Here it is also interesting to know that there are only eight figures of 2 in all these 32 permutations. According to the detailed analysis it should also be understood that, in these eight 2UEPs, there are eight figures of 2 (8/2) as per the Sankhya â€˜8â€™, one figure of 4 (1/4) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding fourth proxy of 4UEP, two figures of 3 (2/3) as per the SMfigure (1+1) â€˜2â€™ of the corresponding third proxy of 3UEP, five figures of 2 (5/2) as per the SMfigure (2+3=) â€˜5â€™ of the corresponding second proxy of 2UEP and twelve figures of 1 (12/1) as per the SMfigure (4+8=) â€˜12â€™ of the corresponding first proxy of 1UEP of the same 2UEP proxy  thus totaling to (8+20=) â€˜28â€™ figures (8/2, 1/4, 2/3, 5/2 12/1=28 figures).
(f) According to the brief analysis as per the Sankhya â€˜16â€™ and its Mahapatala â€˜48â€™ of the corresponding first proxy pertaining to the 1UEP, it should be understood that, apart from the sixteen figures of 1 which are in the right extreme end of these â€˜16â€™ 1UEPs, there are â€˜48â€™ other figures also thus totaling to (16+48) â€˜64â€™ figures in all these â€˜16â€™ 1UEPs. Here it is also interesting to know that there are only sixteen figures of 1 in all these 32 permutations. According to the detailed analysis it should also be understood that in these sixteen 1UEPs, there are sixteen figures of 1 (16/1) as per the Sankhya â€˜16â€™, one figure of 5 (1/5) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding fifth proxy of 5UEP, two figures of 4 (2/4) as per the SMfigure (1+1) â€˜2â€™ of the corresponding fourth proxy of 4UEP, five figures of 3 (5/3) as per the SMfigure (2+3) â€˜5â€™ of the corresponding third proxy of 3UEP, twelve figures of 2 (12/2) as per the SMfigure (4+8) â€˜12â€™ of the corresponding second proxy of 2UEP and twentyeight figures of 1 (28/1) as per the SMfigure (8+20) â€˜28â€™ of the corresponding first proxy of the 1UEP proxy of the same 1UEP proxy  thus totaling to â€˜64â€™ (16+48) figures (16/1, 1/5, 2/4, 5/3, 12/2 & 28/1=64 figures) and, at last, arriving at a grand total of (1+2+5+12+28+64=) â€˜112â€™ figures in the grand total of 32 permutations.
UDDISHTA
Uddishta is the easy method to obtain the serial number of a permutation containing a particular series of figures, based upon the figures of Sankhya. However the rules are the same as in Nashta.
(a) As per the rules governing Nashta the undeducted proxies always derive the least figure i.e. â€˜1â€™ and the successfully deducted proxies always derive other higher figures. In a series of figures of a permutation, take out the relevant proxies of the figures which are successfully subtracted which in turn derive the â€˜AdhikangajanitaSankhyaâ€™ (AS, i.e; the figure from which the higher Angas/figures are derived). Later, subtract the AS from the Sankhya and the remainder is the serial number.
E X A M P L E S:
1. What is the serial number of the 6UP containing â€˜6â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the figure â€˜6â€™ of the permutation it should be presumed that this is derived by the consecutive subtraction of five proxies i.e., 1st proxy â€˜16â€™, 2nd proxy â€˜8â€™, 3rd proxy â€˜4â€™, 4th proxy â€˜2â€™ and 5th proxy â€˜1â€™ and the nonsubtraction of the 6th proxy â€˜1â€™ as per rule No. 3.
Thus the AS is â€˜31â€™ (16+8+4+2+1) and as per rule No. (a) the remainder â€˜1â€™(3231) is the serial number.
2. What is the serial number of the 6UP containing â€˜1 1 4â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜4â€™ of the permutation it should be presumed that this is derived by the consecutive subtraction of three proxies i.e., 1st proxy â€˜16â€™, 2nd proxy â€˜8â€™ and 3rd proxy â€˜4â€™ and the nonsubtraction of the 4th proxy â€˜2â€™ as per rule No. 3.
By the next two least figures â€˜1 1â€™ of the permutation it should always be presumed that they are written only to fillin the deficit value of the permutation as per rule No. 5.
Thus the AS is â€˜28â€™ (16+8+4) and as per rule No.(a) the remainder â€˜4â€™ (3228) is the serial number.
3. What is the serial number of the 6UP containing â€˜3 3â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜3â€™ of the permutation it should be presumed that this figure â€˜3â€™ is derived by the consecutive subtraction of the 1st proxy â€˜16â€™ and 2nd proxy â€˜8â€™ and the nonsubtraction of the 3rd proxy after which its proxy â€˜4â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next figure â€˜3â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜4â€™ the process is continued and this figure â€˜3â€™ is also derived by the consecutive subtraction of the 1st proxy â€˜2â€™ and 2nd proxy â€˜1â€™ and the nonsubtraction of the 3rd proxy â€˜1â€™ as per rule No. 3.
Thus the AS is â€˜27â€™ (16+8+2+1) and as per rule No.(a) the remainder â€˜5â€™ (3227) is the serial number.
4. What is the serial number of the 6UP containing â€˜2 2 2â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜2â€™ of the permutation it should be presumed that this figure is derived by the subtraction of the 1st proxy â€˜16â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜8â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next figure â€˜2â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜8â€™ the process is continued and again this figure â€˜2â€™ also is derived in the same way by the subtraction of the 1st proxy â€˜4â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜2â€™ has to be leftoff as per rule No. 3 to continue the process.
Again, by the last figure â€˜2â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜2â€™ the process is continued and again this figure â€˜2â€™ also is derived in the same way by the subtraction of the 1st proxy â€˜1â€™ and the nonsubtraction of the 2nd proxy as per rule No. 3.
Thus the AS is â€˜21â€™ (16+4+1) and as per rule No. (a) the remainder is â€˜11â€™ (3221) is the serial number.
5. What is the serial number of the 6UP containing â€˜2 1 1 2â€™ in its series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜2â€™ of the permutation it should be presumed that it is derived by the subtraction of the 1st proxy â€˜16â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜8â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next two figures â€˜1 1â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜8â€™ the process is continued and these figures â€˜1 1â€™ are derived by the consecutive nonsubtraction of the 1st proxy â€˜4â€™ of the above leftoff proxy â€˜8â€™ and again the 1st proxy â€˜2â€™ of the last undeducted proxy â€˜4â€™ as per rule No. 3 to continue the process.
Now, by the last figure â€˜2â€™ of the permutation it should be presumed that, basing upon the undeducted last proxy â€˜2â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜1â€™ and the nonsubtraction of the 2nd proxy as per rule No. 3.
Thus the AS is â€˜17â€™ (16+1) and as per rule No.(a) the remainder â€˜15â€™ (3217) is the serial number.
6. What is the serial number of the 6UP containing â€˜3 2 1â€™ in the series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figure â€˜1â€™ of the permutation it should be presumed that this figure is derived by the nonsubtraction of the 1st proxy â€˜16â€™ as per rule No.3 and the process is continued.
Later, by the next figure â€˜2â€™ of the permutation it should be presumed that, basing upon the last undeducted proxy â€˜16â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜8â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜4â€™ has to be leftoff as per rule No. 3 to continue the process.
Now, by the last figure â€˜3â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜4â€™ the process is continued and this figure â€˜3â€™ is derived by the consecutive subtraction of the 1st proxy â€˜2â€™, and the 2nd proxy â€˜1â€™ and the nonsubtraction of the 3rd proxy â€˜1â€™ as per rule No. 3.
Thus the AS is â€˜11â€™ ( 8+2+1) and as per rule No.(a) the remainder â€˜21â€™ (3211) is the serial number.
7. What is the serial number of the 6UP containing â€˜2 1 2 1â€™ in the series of figures?
As per rule No.1 write the figures of Sankhya up to the required units of permutation.
By the first figureâ€™1â€™ of the permutation it should be presumed that this figure is derived by the nonsubtraction of the 1st proxy â€˜16â€™ as per rule No.3 and the process is continued.
Later, by the next figure â€˜2â€™ of the permutation it should be presumed that, basing upon the last undeducted proxy â€˜16â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜8â€™ and the nonsubtraction of the 2nd proxy after which its proxy â€˜4â€™ has to be leftoff as per rule No. 3 to continue the process.
Later, by the next figure â€˜1â€™ of the permutation it should be presumed that, basing upon the above leftoff proxy â€˜4â€™ the process is continued and this figure â€˜1â€™ is derived by the nonsubtraction of the 1st proxy â€˜2â€™ as per rule No. 3 and the process is continued.
Now, by the last figure â€˜2â€™ of the permutation it should be presumed that, basing upon the last undeducted proxy â€˜2â€™ the process is continued and this figure â€˜2â€™ is derived by the subtraction of the 1st proxy â€˜1â€™ and the nonsubtraction of the 2nd proxy as per rule No.3.
Thus the AS is â€˜9â€™ (8+1) and as per rule No.(a) the remainder â€˜23â€™ (329) is the serial number.
5th part:
KALITA
Kalita is the method for obtaining the full particulars of all the figures/ Angas used in the process of permutation up to the required prastara in question avoiding the laborious process of permutation.
E X A M P L E S:
1.Give the particulars of all the Figures of 6UP (6unitspermutation).
In Samyuktangaprastara the figures of all the preceding houses (including the figure â€˜1â€™ written in the extreme left as per the generalkeyrule) are the proxies of which the total of all forms the Sankhya.
According to the general table of permutations it should be analysed that as per the 
1st proxy there is/are â€˜16â€™ 1UEPs with â€˜48â€™ other figures
2nd ,, â€˜8â€™ 2UEPs â€˜20â€™ ,,
3rd ,, â€˜4â€™ 3UEPs â€˜8â€™ ,,
4th ,, â€˜2â€™ 4UEPs â€˜3â€™ ,,
5th ,, â€˜1â€™ 5UEPs â€˜1â€™ ,,
6th ,, â€˜1â€™ 6UEPs â€˜1â€™ ,,
(Note: UEP (unitending permutation) i.e., permutation ending with a particular figure)
 thus arriving at the total of â€˜32â€™ permutations along with â€˜112â€™ figures.
These â€˜32â€™ permutations starting from the figure â€˜6â€™ in the 1st permutation should be analysed in the following manner.
(a) As per the Sankhya â€˜1â€™ of the corresponding sixth proxy of 6UEP (6unitsending permutation), it should be understood that there is only one 6UEP with a single figure 6 as its corresponding Mahapatala does not exist.
(b) According to the analysis, as per the Sankhya â€˜1â€™ and its Mahapatala â€˜1â€™ of the corresponding fifth proxy pertaining to the 5UEP, it should be understood that, apart from the â€˜5â€™ in the right extreme end of this 5UEP, there is another figure â€˜1â€™ also in this permutation as per the Sankhya â€˜1â€™ of its corresponding first proxy of 1UEP  thus totaling to two (1+1) figures in this 5UEP. Here it is also interesting to know that there are only two figures of â€˜5â€™ in all these 32 permutations.
(c) According to the brief analysis, as per the Sankhya â€˜2â€™ and its Mahapatala â€˜3â€™ of the corresponding fourth proxy pertaining to the 4UEP, it should be understood that, apart from the two figures of 4 which are in the right extreme end of these two 4UEPs, there are â€˜3â€™ other figures also  thus totaling to five (2+3) figures in all these two 4UEPs. Here it is also interesting to know that there are only five figures of 4 in all these 32 permutations. According to the detailed analysis it should also be understood that, in these two 4UEPs, there are two figures of 4 (2/4) as per the Sankhya â€˜2â€™, one figure of 2 (1/2) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding second proxy of 2UEP and two figures of 1 (2/1) as per the SMfigure (combined figure of Sankhya and Mahapatala (1+1) â€˜2â€™ of the corresponding first proxy of 1UEP of the same 4UEP proxy  thus totaling to five (2+3) figures (2/4, 1/2 & 2/1=5 figures).
(d) According to the brief analysis as per the Sankhya â€˜4â€™ and its Mahapatala â€˜8â€™ of the corresponding third proxy pertaining to the 3UEP proxy, it should be understood that, apart from the four figures of 3 which are in the right extreme end of these four 3UEPs, there are â€˜8â€™ other figures also thus totaling to (4+8) â€˜12â€™ figures in all these four 3UEPs. Here it is also interesting to know that there are only four figures of 3 in all these 32 permutations. According to the detailed analysis it should also be understood that, in these four 3UEPs, there are four figures of 3 (4/3) as per the Sankhya â€˜4â€™, one figure of â€˜3â€™ (1/3) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding third proxy of 3UEP, two figures of 2 (2/2) as per the SMfigure (1+1) â€˜2â€™ of the corresponding second proxy of 2UEP and five figures of 1 (5/1) as per the SMfigure (2+3=) â€˜5â€™ of the corresponding first proxy of 1UEP of the same 3UEP proxy  thus totaling to twelve (4+8) figures (4/3, 1/3, 2/2 & 5/1=12 figures).
(e) According to the brief analysis as per the Sankhya â€˜8â€™ and its Mahapatala â€˜20â€™ of the corresponding second proxy pertaining to the 2UEP, it should be understood that, apart from the eight figures of 2 which are in the right extreme end of these eight 2UEPs, there are 20 other figures also thus totaling to (8+20) â€˜28â€™ figures in all these eight 2UEPs. Here it is also interesting to know that there are only eight figures of 2 in all these 32 permutations. According to the detailed analysis it should also be understood that, in these eight 2UEPs, there are eight figures of 2 (8/2) as per the Sankhya â€˜8â€™, one figure of 4 (1/4) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding fourth proxy of 4UEP, two figures of 3 (2/3) as per the SMfigure (1+1) â€˜2â€™ of the corresponding third proxy of 3UEP, five figures of 2 (5/2) as per the SMfigure (2+3=) â€˜5â€™ of the corresponding second proxy of 2UEP and twelve figures of 1 (12/1) as per the SMfigure (4+8=) â€˜12â€™ of the corresponding first proxy of 1UEP of the same 2UEP proxy  thus totaling to (8+20=) â€˜28â€™ figures (8/2, 1/4, 2/3, 5/2 12/1=28 figures).
(f) According to the brief analysis as per the Sankhya â€˜16â€™ and its Mahapatala â€˜48â€™ of the corresponding first proxy pertaining to the 1UEP, it should be understood that, apart from the sixteen figures of 1 which are in the right extreme end of these â€˜16â€™ 1UEPs, there are â€˜48â€™ other figures also thus totaling to (16+48) â€˜64â€™ figures in all these â€˜16â€™ 1UEPs. Here it is also interesting to know that there are only sixteen figures of 1 in all these 32 permutations. According to the detailed analysis it should also be understood that in these sixteen 1UEPs, there are sixteen figures of 1 (16/1) as per the Sankhya â€˜16â€™, one figure of 5 (1/5) as per the Sankhya â€˜1â€™ (no Mahapatala) of the corresponding fifth proxy of 5UEP, two figures of 4 (2/4) as per the SMfigure (1+1) â€˜2â€™ of the corresponding fourth proxy of 4UEP, five figures of 3 (5/3) as per the SMfigure (2+3) â€˜5â€™ of the corresponding third proxy of 3UEP, twelve figures of 2 (12/2) as per the SMfigure (4+8) â€˜12â€™ of the corresponding second proxy of 2UEP and twentyeight figures of 1 (28/1) as per the SMfigure (8+20) â€˜28â€™ of the corresponding first proxy of the 1UEP proxy of the same 1UEP proxy  thus totaling to â€˜64â€™ (16+48) figures (16/1, 1/5, 2/4, 5/3, 12/2 & 28/1=64 figures) and, at last, arriving at a grand total of (1+2+5+12+28+64=) â€˜112â€™ figures in the grand total of 32 permutations.

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#32
Thanks Msakella Sir. I already got the squared ruled note book yesterday and had my first exposure to a Nagaswaram concert at Krishna Gana Sabha. I do like teaching but I'm a rank beginner in music to teach a topic that I presume even some of the musicians might not understand. I can definitely try this topic on some of the programmers in my team

 Posts: 976
 Joined: 04 Dec 2006, 13:56
 x 4
 x 1
#34
Thanks Akella sir for the pains you are taking in publishing rare and interesting topics. they are long enough and take time to absorb and assimilate. Thinking about the time you might have spent in acquiring such vast knowledge i am wondering whether by being a mere performer we can reach such Alpine heights of knowledge and wisdom. Thanks once again sir

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#35
Dear member, sbala, My computer has gone to hospital and just returned today after incurring a bill for Rs.14,000/. S0, I was unable to contact you. Very happy to know that you are able to go through Nashta successfully. Please proceed. You learn it fully and then you can try to propagate. Keep me informed about your progress and I shall post the necessary material. With best wishes, amsharma.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#36
Dear member, mridhangam, As an aged person knowing a few more details than our youngsters, I feel it as my duty to enlighten my brothers and sisters who are enthusiastic to learn it. It is not pains taking to me at all. More over it gives me great pleasure.
â€˜Mookam karothi vaachaalamâ€™ the Almighty had decided to bring this rare chapter out through this body. HE can do any thing in this universe. Thus, HE had chosen a fool (me) who successfully failed twice in SSLC, to bring this chapter out. This is HIS greatness not mine at all. If HE proposes anybody can do wonders. In the same manner I did. Thatâ€™s all. Thanking you for your kind appreciation, amsharma.
â€˜Mookam karothi vaachaalamâ€™ the Almighty had decided to bring this rare chapter out through this body. HE can do any thing in this universe. Thus, HE had chosen a fool (me) who successfully failed twice in SSLC, to bring this chapter out. This is HIS greatness not mine at all. If HE proposes anybody can do wonders. In the same manner I did. Thatâ€™s all. Thanking you for your kind appreciation, amsharma.

 Posts: 13684
 Joined: 02 Feb 2010, 22:26
 x 808
 x 12
#37
A review of our Sri Akella M Sarama's lecdem in included in this article:
http://www.hindu.com/fr/2007/01/19/stor ... 960300.htm
http://www.hindu.com/fr/2007/01/19/stor ... 960300.htm

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#39
Msakella Sir,
I think I understood the Uddishta process. I have given the details of a a few permutations below.
6 UP, Figure 3 1 1 1
Undeducted figures = 16,8,4
deducted figures =2,1
Sum of deducted figures=3
Serial Number = 323=29.
7UP, Figure 3 4
Deducted Figures  32,16,8 and 2,1
Sum of deducted figures=59
Serial Number=64=59=5.
7UP, Figure 2 3 2
Deducted figures  32,8,4,1
Serial Number = 6445=19
Please verify and if this is fine, I will try to proceed to Kalita
I think I understood the Uddishta process. I have given the details of a a few permutations below.
6 UP, Figure 3 1 1 1
Undeducted figures = 16,8,4
deducted figures =2,1
Sum of deducted figures=3
Serial Number = 323=29.
7UP, Figure 3 4
Deducted Figures  32,16,8 and 2,1
Sum of deducted figures=59
Serial Number=64=59=5.
7UP, Figure 2 3 2
Deducted figures  32,8,4,1
Serial Number = 6445=19
Please verify and if this is fine, I will try to proceed to Kalita

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#41
Dear Msakella Sir,
I have a clarification in the kalita section. Shouldn't section d say 12 figures of 3 in all the 32 permutations? Similarly in sections e and f, should we not have 28 figures of 2 in 32 permutations and 64 figures of 1 in the 32 permutations?
I have a clarification in the kalita section. Shouldn't section d say 12 figures of 3 in all the 32 permutations? Similarly in sections e and f, should we not have 28 figures of 2 in 32 permutations and 64 figures of 1 in the 32 permutations?

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#42
Dear member, sbala, What you say is correct. You got this also. Very nice. Congrats. The same has already been furnished in the brief analysis I have given. But, along with it, another detailed analysis is also furnished. If you go through it again carefully and patiently you can very easily understand the detailed analysis also. Eagerly awaiting your positive response, amsharma.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#43
6th part:
Rules for the process of permutation of Talangas:
1.The value of any prastara should be cited in number of the least (inferior) Angas only occurring in a particular mode of permutation and for the value of such prastara, minimum possible number of Angas only should be written in the 1st prastara, carefully observing the relevant rules and restrictions, if any,. If two or more Angas are to be written in the 1st prastara they should be written anticlockwise only in descending order of value from right to left of the 1st prastara.
2.If there is more than one Anga in the prastara, the permutable Anga at the extreme left should at first be permuted.
3.If the Angas in the left extreme are so minute that they cannot further be permuted, the immediate next permutable Anga to their right only should be permuted.
4.To permute an Anga write down the immediate lower Anga in value than of the upper Anga in the same column.
5.After writing down the lower Anga than of the upper Anga, for the remainder, a suitable independent Anga should be written to the left of the Anga already written. For this remainder, if two or more Angas are to be written they should be written anticlockwise only in descending order of value to the left of the Anga already written.
6.If there are Angas to the left of the upper Anga the remainder value of the permuted Anga should be added to the total value of all the left side Angas and a suitable Anga should be written to the left of the Anga already written and if two or more Angas are to be written for this total value they should be written anticlockwise to the left of the Anga already written in descending order of value.
7.The Angas, if any, which are at the right of the Anga be permuted, should be brought down and written as they are in the same columns.
8.Carefully observing the relevant rules and restrictions, if any, the process of permutation should be continued until all the Angas become so minute that none of them could further be permuted.
9.Among all the prastaras the total value of all the Angas of each prastara should always be one and the same.
List of the first Talangas to be written in the processes of permutation
Unit Panchanga Shadanga
Value 2 3 â€“ 4 â€“ 5 â€“ 7 â€“ 9
01 0 U U U U U
02 l 0 0 0 0 0
03 0l l U0 U0 U0 U0
04 S Ul l 00 00 00
05 0S 0l Ul l U00 U00
06 Ś S 0l Ul 000 000
07 0Ś US U0l 0l l U000
08 + 0S S U0l Ul 0000
09 0+ Ś US 00l 0l l
10 l+ UŚ 0S S U0l Ul
(Important note: The Talangas written above in the respective Jaatis of Shadangaprastara should be written one above the other while replacing the respective figures of Samyuktangaprastara with the respective Talangas of the different Jaatis)
Examples:
Pamchangaprastara: (restricted to Chaturashrajati having the least Anga, Druta)
1unit: 0 01
2units: l 01
0 0 02
3units: 0 l 01
l 0 02
0 0 0 03
4unirts: S 01
1 l 02
0 0 1 03
0 1 0 04
1 0 0 05
0 0 0 0 06
5units: 0 S 01
0 1 1 02
1 0 1 03
0 0 0 1 04
S 0 05
1 1 0 06
0 0 1 0 07
0 1 0 0 08
1 0 0 0 09
0 0 0 0 0 10
6units: Ś 01
1 S 02
0 0 S 03
S 1 04
1 1 1 05
0 0 1 1 06
0 1 0 1 07
1 0 0 1 08
0 0 0 0 1 09
0 S 0 10
0 1 1 0 11
1 0 1 0 12
0 0 0 1 0 13
S 0 0 14
1 1 0 0 15
0 0 1 0 0 16
0 1 0 0 0 17
1 0 0 0 0 18
0 0 0 0 0 0 19
Shadangaprastara: (Trisrajaati):
1unit: U 01
2units: 0 01
U U 02
3units: l 01
U 0 02
0 U 03
U U U 04
4units: U l 01
0 0 02
U U 0 03
U 0 U 04
0 U U 05
U U U U 06
5units: 0 l 01
U U l 02
l 0 03
U 0 0 04
0 U 0 05
U U U 0 06
U l U 07
0 0 U 08
U U 0 U 09
l U U 10
U 0 U U 11
0 U U U 12
U U U U U 13
Shadangaprastara: (Chaturashrajaati):
1unit: U 01
2units: 0 01
U U 02
3units: U 0 01
0 U 02
U U U 03
4units: l 01
0 0 02
U U 0 03
U 0 U 04
0 U U 05
U U U U 06
Shadangaprastara: (Khandajaati):
1unit: U 01
2units: 0 01
U U 02
3units: U 0 01
0 U 02
U U U 03
4units: 0 0 01
U U 0 02
U 0 U 03
0 U U 04
U U U U 05
5units: l 01
U 0 0 02
0 U 0 03
U U U 0 04
0 0 U 05
U U 0 U 06
U 0 U U 07
0 U U U 08
U U U U U 09
In the same manner, the process of permutation of the Talangas, Laghu, Guru, Pluta and Kakapada should carefully be made always keeping the respective value of Jaati in mind.
7th part:
TABLE OF GENERALPERMUTATIONS
Amsha
Shreni: 1  2  3  4  5  6  7  8  9
1  1  1 2 4 8 16 32 64 128 256
1 3 6 20 48 112 256 576 1280
2  1  1 2 3 6 10 19 33 61 108
1 3 7 16 34 72 147 299 596
3  1  1 2 4 7 13 25 46 86 162
1 3 8 19 43 95 204 431 900
4  1  1 2 3 6 10 18 31 56 98
1 3 7 16 34 71 143 286 562
5  1  1 2 3 5 9 15 26 44 75
1 3 7 15 31 62 122 235 447
7  1  1 2 3 5 8 13 22 36 60
1 3 7 15 30 58 110 205 378
9  1  1 2 3 5 8 13 21 34 56
1 3 7 15 30 58 109 201 366
In this above â€˜Table of General Permutationsâ€™ the figures in the Amshashreni at the top are the units of permutations, the figures in the left side from top to bottom are the figures representing the modes of permutations, 1 pertaining to Samyuktangasarvajati, 2 of Panchanga,
3 of Shadangatrisrajati, 4 of Shadangachaturashrajati, 5 of Shadangakhandajati, 7 of Shadangamishrajati and 9 of Shadangasankeernajati and the respective figures in the upper line consists of â€˜Sankhyaâ€™ and the lower line â€˜Mahapatalaâ€™ of these above modes of permutations.
Explanation: Finding out the proxies:
(Importantnote: For the purpose of Prastara the Talangas should always be cited from right to left only)
As was furnished in the beginning, in the modern period from when Kakapada has also been included in the process of permutation, mainly, there are three processes of permutation and they are 1.Panchangaprastara 2.Shadangaprastara and 3. Samyuktangaprastara. In writing the figures of the respective tables and in getting the answers for Nashta, Uddishta and Kalita the Proxies (the figures of the respective preceding houses only) play a very important role. There is an easy method to find out the respective proxies of these three processes of permutation. In Panchangaprastara in which the five Talangas, Druta, Laghu, Guru, Pluta and Kakapada are used in the process of permutation, the Druta is the lowest denomination of the five Talangas and the unitvalue of the permutation should always be cited only in terms of Drutas i.e., Ekadrutaprastara, Dvidrutaprastara, Tridrutaprastara, Chaturdrutaprastara, Panchadrutaprastara, Shaddrutaprastara and so on wherein the first Talanga is 0, l, 0l, S, Ś and so on respectively but not Drutaprastara, Laghuprastara, Laghudrutaprastara, Guruprastara, Gurudrutaprastara, Plutaprastara and so on. In the same manner, also in Shadangaprastara in which the six Talangas, Anudruta, Druta, Laghu, Guru, Pluta and Kakapada are used in the process of permutation, the Anudruta is the lowest denomination of the six Talangas and the unitvalue of the permutation should always be cited only in terms of Anudrutas i.e., Ekaanudrutaprastara, Dvianudrutaprastara, Trianudrutaprastara, Chaturanudrutaprastara, Panchaanudrutaprastara, Shadanudrutaprastara and so on wherein the first Talanga is U, 0, l for Trisrajati or U0 for other Jatis, Ul for Trisrajati or l Chaturashrajati or 00 for other Jatis, 0l for Trisrajati or Ul for Chaturashrajati or U00 for other Jatis, S for Trisrajati or 0l for Chaturashrajati or 000 for other Jatis and so on respectively but not Anudrutaprastara, Drutaprastara, Laghuprastara for Trisrajati or Drutaanudrutaprastara for other Jatis and so on. In the same manner, also in Samyuktangaprastara (in which, at the first instance, numerals are used in the process of permutation) Ekaamsha or 1unit is the lowest denomination and the unitvalue of the permutation should always be cited in terms of Amshas i.e., Ekaamshaprastara, Dviamshaprastara, Triamshaprastara, Chaturamshaprastara, Panchaamshaprastara, Shadamshaprastara and so on wherein the first Talanga is Anudruta, Druta, Laghu for Trisrajati or Drutavirama (Samyuktanga of Virama written above the Druta together) for other Jatis, Samyuktanga of Laghuvirama for Trisrajati Laghu for Chaturashrajati or Samyuktanga of Dvidruta for other Jatis, Samyuktanga of Laghudruta for Trisrajati or Samyuktanga of Laghuvirama for Chaturashrajati or Samyuktanga of Dvidrutavirama for other Jatis, Guru for Trisrajati or Samyuktanga of Laghudruta for Chaturashrajati or Samyuktanga of Tridruta for other Jatis and so on respectively but not Anudrutaprastara, Drutaprastara, Laghuprastara for Trisrajati or Drutaviramaprastara for other Jatis and so on. Thus, the respective proxies are: 1, 2, 3, 4, 5, 6, 7, 8, 9 etc., serially representing the required units of permutation in Samyuktangaprastara, 1, 2, 4, 6 & 8 for Panchangaprastara representing the respective unitvalue of the five Talangas, Druta, Laghu, Guru, Pluta and Kakapada in terms of respective Drutas. and in Shadangaprastara 1, 2, 3, 6, 9 & 12 for Trisrajati, 1, 2, 4, 8, 12 & 16 for Chaturashrajati, 1, 2, 5, 10 15 & 20 for Khandajati, 1, 2, 7, 14, 21 & 28 for Mishrajati and 1, 2, 9, 18, 27 & 36 for Sankeernajati representing the respective unitvalue of the six Talangas, Anudruta, Druta, Laghu, Guru, Pluta and Kakapada in terms of respective Anudrutas. By careful analysis one can easily understand that, while 1st & 2nd proxies are common to all, later all the numbers serially are the proxies for Samyuktangaprastara, in Panchangaprastara three consecutive multiples of 2 from the figure 4 i.e., 4, 6 & 8 are the respective proxies and in Shadangaprastara the four consecutive multiples of the respective Jatiunits i.e., 3, 6, 9 & 12 for Trisrajati, 4, 8, 12 & 16 for Chaturashrajati, 5, 10, 15 & 20 for Khandajati, 7, 14, 21 & 28 for Mishrajati and 9, 18, 27 & 36 for Sankeernajati are the respective proxies. By all the above it is also very important to note that, always, while â€˜1â€™ represents Samyuktangaprastara, â€˜2â€™ represents Panchangaprastara, â€˜3â€™ represents Trisrajatishadangaprastara, â€˜4â€™ represents Chaturashrajatishadangaprastara, â€˜5â€™ represents Khandajatishadangaprastara, â€˜7â€™ represents Mishrajatishadangaprastara and â€˜9â€™ represents Sankeenajatishadangaprastara.
Actually, in the tables of the respective figures, the respective proxies should always be observed from right to left only.
(AAnudruta, DDruta, LLaghu, GGuru, PPluta, KKakapada)
Samyuktanga: 1  2  3  4  5  6 
D L G P K
Panchanga: 1  2  4  6  8
Shadanga: A D L G P K
 Trisrajati: 1  2  3  6  9  12
 Chaturashrajati: 1  2  4  8  12  16
 Khandajati: 1  2  5  10  15  20
 Mishrajati: 1  2  7  14  21  28
 Sankeernajati: 1  2  9  18  27  36
Rules for the process of permutation of Talangas:
1.The value of any prastara should be cited in number of the least (inferior) Angas only occurring in a particular mode of permutation and for the value of such prastara, minimum possible number of Angas only should be written in the 1st prastara, carefully observing the relevant rules and restrictions, if any,. If two or more Angas are to be written in the 1st prastara they should be written anticlockwise only in descending order of value from right to left of the 1st prastara.
2.If there is more than one Anga in the prastara, the permutable Anga at the extreme left should at first be permuted.
3.If the Angas in the left extreme are so minute that they cannot further be permuted, the immediate next permutable Anga to their right only should be permuted.
4.To permute an Anga write down the immediate lower Anga in value than of the upper Anga in the same column.
5.After writing down the lower Anga than of the upper Anga, for the remainder, a suitable independent Anga should be written to the left of the Anga already written. For this remainder, if two or more Angas are to be written they should be written anticlockwise only in descending order of value to the left of the Anga already written.
6.If there are Angas to the left of the upper Anga the remainder value of the permuted Anga should be added to the total value of all the left side Angas and a suitable Anga should be written to the left of the Anga already written and if two or more Angas are to be written for this total value they should be written anticlockwise to the left of the Anga already written in descending order of value.
7.The Angas, if any, which are at the right of the Anga be permuted, should be brought down and written as they are in the same columns.
8.Carefully observing the relevant rules and restrictions, if any, the process of permutation should be continued until all the Angas become so minute that none of them could further be permuted.
9.Among all the prastaras the total value of all the Angas of each prastara should always be one and the same.
List of the first Talangas to be written in the processes of permutation
Unit Panchanga Shadanga
Value 2 3 â€“ 4 â€“ 5 â€“ 7 â€“ 9
01 0 U U U U U
02 l 0 0 0 0 0
03 0l l U0 U0 U0 U0
04 S Ul l 00 00 00
05 0S 0l Ul l U00 U00
06 Ś S 0l Ul 000 000
07 0Ś US U0l 0l l U000
08 + 0S S U0l Ul 0000
09 0+ Ś US 00l 0l l
10 l+ UŚ 0S S U0l Ul
(Important note: The Talangas written above in the respective Jaatis of Shadangaprastara should be written one above the other while replacing the respective figures of Samyuktangaprastara with the respective Talangas of the different Jaatis)
Examples:
Pamchangaprastara: (restricted to Chaturashrajati having the least Anga, Druta)
1unit: 0 01
2units: l 01
0 0 02
3units: 0 l 01
l 0 02
0 0 0 03
4unirts: S 01
1 l 02
0 0 1 03
0 1 0 04
1 0 0 05
0 0 0 0 06
5units: 0 S 01
0 1 1 02
1 0 1 03
0 0 0 1 04
S 0 05
1 1 0 06
0 0 1 0 07
0 1 0 0 08
1 0 0 0 09
0 0 0 0 0 10
6units: Ś 01
1 S 02
0 0 S 03
S 1 04
1 1 1 05
0 0 1 1 06
0 1 0 1 07
1 0 0 1 08
0 0 0 0 1 09
0 S 0 10
0 1 1 0 11
1 0 1 0 12
0 0 0 1 0 13
S 0 0 14
1 1 0 0 15
0 0 1 0 0 16
0 1 0 0 0 17
1 0 0 0 0 18
0 0 0 0 0 0 19
Shadangaprastara: (Trisrajaati):
1unit: U 01
2units: 0 01
U U 02
3units: l 01
U 0 02
0 U 03
U U U 04
4units: U l 01
0 0 02
U U 0 03
U 0 U 04
0 U U 05
U U U U 06
5units: 0 l 01
U U l 02
l 0 03
U 0 0 04
0 U 0 05
U U U 0 06
U l U 07
0 0 U 08
U U 0 U 09
l U U 10
U 0 U U 11
0 U U U 12
U U U U U 13
Shadangaprastara: (Chaturashrajaati):
1unit: U 01
2units: 0 01
U U 02
3units: U 0 01
0 U 02
U U U 03
4units: l 01
0 0 02
U U 0 03
U 0 U 04
0 U U 05
U U U U 06
Shadangaprastara: (Khandajaati):
1unit: U 01
2units: 0 01
U U 02
3units: U 0 01
0 U 02
U U U 03
4units: 0 0 01
U U 0 02
U 0 U 03
0 U U 04
U U U U 05
5units: l 01
U 0 0 02
0 U 0 03
U U U 0 04
0 0 U 05
U U 0 U 06
U 0 U U 07
0 U U U 08
U U U U U 09
In the same manner, the process of permutation of the Talangas, Laghu, Guru, Pluta and Kakapada should carefully be made always keeping the respective value of Jaati in mind.
7th part:
TABLE OF GENERALPERMUTATIONS
Amsha
Shreni: 1  2  3  4  5  6  7  8  9
1  1  1 2 4 8 16 32 64 128 256
1 3 6 20 48 112 256 576 1280
2  1  1 2 3 6 10 19 33 61 108
1 3 7 16 34 72 147 299 596
3  1  1 2 4 7 13 25 46 86 162
1 3 8 19 43 95 204 431 900
4  1  1 2 3 6 10 18 31 56 98
1 3 7 16 34 71 143 286 562
5  1  1 2 3 5 9 15 26 44 75
1 3 7 15 31 62 122 235 447
7  1  1 2 3 5 8 13 22 36 60
1 3 7 15 30 58 110 205 378
9  1  1 2 3 5 8 13 21 34 56
1 3 7 15 30 58 109 201 366
In this above â€˜Table of General Permutationsâ€™ the figures in the Amshashreni at the top are the units of permutations, the figures in the left side from top to bottom are the figures representing the modes of permutations, 1 pertaining to Samyuktangasarvajati, 2 of Panchanga,
3 of Shadangatrisrajati, 4 of Shadangachaturashrajati, 5 of Shadangakhandajati, 7 of Shadangamishrajati and 9 of Shadangasankeernajati and the respective figures in the upper line consists of â€˜Sankhyaâ€™ and the lower line â€˜Mahapatalaâ€™ of these above modes of permutations.
Explanation: Finding out the proxies:
(Importantnote: For the purpose of Prastara the Talangas should always be cited from right to left only)
As was furnished in the beginning, in the modern period from when Kakapada has also been included in the process of permutation, mainly, there are three processes of permutation and they are 1.Panchangaprastara 2.Shadangaprastara and 3. Samyuktangaprastara. In writing the figures of the respective tables and in getting the answers for Nashta, Uddishta and Kalita the Proxies (the figures of the respective preceding houses only) play a very important role. There is an easy method to find out the respective proxies of these three processes of permutation. In Panchangaprastara in which the five Talangas, Druta, Laghu, Guru, Pluta and Kakapada are used in the process of permutation, the Druta is the lowest denomination of the five Talangas and the unitvalue of the permutation should always be cited only in terms of Drutas i.e., Ekadrutaprastara, Dvidrutaprastara, Tridrutaprastara, Chaturdrutaprastara, Panchadrutaprastara, Shaddrutaprastara and so on wherein the first Talanga is 0, l, 0l, S, Ś and so on respectively but not Drutaprastara, Laghuprastara, Laghudrutaprastara, Guruprastara, Gurudrutaprastara, Plutaprastara and so on. In the same manner, also in Shadangaprastara in which the six Talangas, Anudruta, Druta, Laghu, Guru, Pluta and Kakapada are used in the process of permutation, the Anudruta is the lowest denomination of the six Talangas and the unitvalue of the permutation should always be cited only in terms of Anudrutas i.e., Ekaanudrutaprastara, Dvianudrutaprastara, Trianudrutaprastara, Chaturanudrutaprastara, Panchaanudrutaprastara, Shadanudrutaprastara and so on wherein the first Talanga is U, 0, l for Trisrajati or U0 for other Jatis, Ul for Trisrajati or l Chaturashrajati or 00 for other Jatis, 0l for Trisrajati or Ul for Chaturashrajati or U00 for other Jatis, S for Trisrajati or 0l for Chaturashrajati or 000 for other Jatis and so on respectively but not Anudrutaprastara, Drutaprastara, Laghuprastara for Trisrajati or Drutaanudrutaprastara for other Jatis and so on. In the same manner, also in Samyuktangaprastara (in which, at the first instance, numerals are used in the process of permutation) Ekaamsha or 1unit is the lowest denomination and the unitvalue of the permutation should always be cited in terms of Amshas i.e., Ekaamshaprastara, Dviamshaprastara, Triamshaprastara, Chaturamshaprastara, Panchaamshaprastara, Shadamshaprastara and so on wherein the first Talanga is Anudruta, Druta, Laghu for Trisrajati or Drutavirama (Samyuktanga of Virama written above the Druta together) for other Jatis, Samyuktanga of Laghuvirama for Trisrajati Laghu for Chaturashrajati or Samyuktanga of Dvidruta for other Jatis, Samyuktanga of Laghudruta for Trisrajati or Samyuktanga of Laghuvirama for Chaturashrajati or Samyuktanga of Dvidrutavirama for other Jatis, Guru for Trisrajati or Samyuktanga of Laghudruta for Chaturashrajati or Samyuktanga of Tridruta for other Jatis and so on respectively but not Anudrutaprastara, Drutaprastara, Laghuprastara for Trisrajati or Drutaviramaprastara for other Jatis and so on. Thus, the respective proxies are: 1, 2, 3, 4, 5, 6, 7, 8, 9 etc., serially representing the required units of permutation in Samyuktangaprastara, 1, 2, 4, 6 & 8 for Panchangaprastara representing the respective unitvalue of the five Talangas, Druta, Laghu, Guru, Pluta and Kakapada in terms of respective Drutas. and in Shadangaprastara 1, 2, 3, 6, 9 & 12 for Trisrajati, 1, 2, 4, 8, 12 & 16 for Chaturashrajati, 1, 2, 5, 10 15 & 20 for Khandajati, 1, 2, 7, 14, 21 & 28 for Mishrajati and 1, 2, 9, 18, 27 & 36 for Sankeernajati representing the respective unitvalue of the six Talangas, Anudruta, Druta, Laghu, Guru, Pluta and Kakapada in terms of respective Anudrutas. By careful analysis one can easily understand that, while 1st & 2nd proxies are common to all, later all the numbers serially are the proxies for Samyuktangaprastara, in Panchangaprastara three consecutive multiples of 2 from the figure 4 i.e., 4, 6 & 8 are the respective proxies and in Shadangaprastara the four consecutive multiples of the respective Jatiunits i.e., 3, 6, 9 & 12 for Trisrajati, 4, 8, 12 & 16 for Chaturashrajati, 5, 10, 15 & 20 for Khandajati, 7, 14, 21 & 28 for Mishrajati and 9, 18, 27 & 36 for Sankeernajati are the respective proxies. By all the above it is also very important to note that, always, while â€˜1â€™ represents Samyuktangaprastara, â€˜2â€™ represents Panchangaprastara, â€˜3â€™ represents Trisrajatishadangaprastara, â€˜4â€™ represents Chaturashrajatishadangaprastara, â€˜5â€™ represents Khandajatishadangaprastara, â€˜7â€™ represents Mishrajatishadangaprastara and â€˜9â€™ represents Sankeenajatishadangaprastara.
Actually, in the tables of the respective figures, the respective proxies should always be observed from right to left only.
(AAnudruta, DDruta, LLaghu, GGuru, PPluta, KKakapada)
Samyuktanga: 1  2  3  4  5  6 
D L G P K
Panchanga: 1  2  4  6  8
Shadanga: A D L G P K
 Trisrajati: 1  2  3  6  9  12
 Chaturashrajati: 1  2  4  8  12  16
 Khandajati: 1  2  5  10  15  20
 Mishrajati: 1  2  7  14  21  28
 Sankeernajati: 1  2  9  18  27  36
Last edited by msakella on 25 Jan 2007, 00:44, edited 1 time in total.

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#44
Dear Msakella Sir,
I have given the Kalita for the 5UP below. Please let me know if this is right. I can then proceed to the next post.
As per Sankhya of 5, there should be a total of 16 permutations.
1. 5th proxy. S=1, M=0 S+M=1
1 figure of 5UEP
Since M=0 there are no other figures
Detailed Analysis 1/5
2. 4th proxy. S=1, M=1, S+M=2
1 figure of 4UEP.
1 more figure as per M
Detailed Analysis  1/1,1/4
3. 3rd proxy. S+M=2+3=5
2 figures of 3 UEP
3 more figures as per M
Detailed Analysis  1/2,2/1,2/3
4. 2nd proxy. S+M=4+8=12
4 2UEPs
8 other figures
Detailed Analysis  1/3,2/2,5/1,4/2
5. 1st proxy S+M=8+20=28
8 1UEP
20 other figures
Detailed Analysis: 1/4,2/3,5/2,12/1,8/1
I have given the Kalita for the 5UP below. Please let me know if this is right. I can then proceed to the next post.
As per Sankhya of 5, there should be a total of 16 permutations.
1. 5th proxy. S=1, M=0 S+M=1
1 figure of 5UEP
Since M=0 there are no other figures
Detailed Analysis 1/5
2. 4th proxy. S=1, M=1, S+M=2
1 figure of 4UEP.
1 more figure as per M
Detailed Analysis  1/1,1/4
3. 3rd proxy. S+M=2+3=5
2 figures of 3 UEP
3 more figures as per M
Detailed Analysis  1/2,2/1,2/3
4. 2nd proxy. S+M=4+8=12
4 2UEPs
8 other figures
Detailed Analysis  1/3,2/2,5/1,4/2
5. 1st proxy S+M=8+20=28
8 1UEP
20 other figures
Detailed Analysis: 1/4,2/3,5/2,12/1,8/1

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#46
Dear members, If you are learning this topic, Talaprastara you can also tally some of these permutations furnished in the 15th post under constructing Kalpana Svaras of the thread, Technical discussions. You can find them along with the rhythmical form in numbers of units, its serial number, jati and Svaras to sing in Svarakalpana. You can apply the same to other compositions of the same Talas and other Talas also. I shall be glad if some of you post here some more varieties of the rhythmical forms of the same Talas or of even other talas in the same manner which obviously reveals that your are really working upon this enthusiastically. Wishing you all the best, amsharma.

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#47
Dear MsAkella Sir,
I have given the details of the panchanga prastara below for 7units. These symbols make me nervous..Numbers seem much more comfortable
0Ś 01
01S 02
10S 03
000S 04
0S1 05
0111 06
1011 07
00011 08
S01 09
1101 10
00101 11
01001 12
10001 13
000001 14
S0 15
1SO 16
00S0 17
S10 18
1110 19
00110 20
01010 21
10010 22
000010 23
0S00 24
01100 25
10100 26
000100 27
S000 28
11000 29
001000 30
010000 31
100000 32
0000000 33
I have given the details of the panchanga prastara below for 7units. These symbols make me nervous..Numbers seem much more comfortable
0Ś 01
01S 02
10S 03
000S 04
0S1 05
0111 06
1011 07
00011 08
S01 09
1101 10
00101 11
01001 12
10001 13
000001 14
S0 15
1SO 16
00S0 17
S10 18
1110 19
00110 20
01010 21
10010 22
000010 23
0S00 24
01100 25
10100 26
000100 27
S000 28
11000 29
001000 30
010000 31
100000 32
0000000 33

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#48
Dear member, sbala, Congrats. You got it. Your progress is very well. Proceed further. OK. Surprisingly enough, even though some of our members started posting about some Talas, no other person, except yourself, seems to have started learning this topic. Very funny! Wishing you all the best, amsharma.

 Posts: 629
 Joined: 30 Jul 2006, 08:56
#49
Dear Msakella Sir,
I have given below the details of Shadanga Prastara Chaturashra jathi 5 units.
U1 01
U00 02
OUO 03
UUUO 04
1U 05
OOU 06
UUOU 07
UOUU 08
OUUU 09
UUUUU 10
I did the Shadanga Prastara for trisram as well for 6 units (25 permutations). I will post the details separately.
I have given below the details of Shadanga Prastara Chaturashra jathi 5 units.
U1 01
U00 02
OUO 03
UUUO 04
1U 05
OOU 06
UUOU 07
UOUU 08
OUUU 09
UUUUU 10
I did the Shadanga Prastara for trisram as well for 6 units (25 permutations). I will post the details separately.
Last edited by sbala on 26 Jan 2007, 20:22, edited 1 time in total.

 Posts: 1939
 Joined: 30 Sep 2006, 21:16
 x 2
#50
Dear member, sbala, Your permutations of Chaturashrajaati are also OK.In the process of the permutation all the prastaras should consist of the right alignment. Even though we are doing the same this alignment is not maintained while pasting these prastaras in these posts (I would like to ask srkrisji about this whether there is any technique to maintain this alignment while pasting the matter in these posts)
If you take the respective figures of the â€˜Table of General Permutationsâ€™, in Panchangaprastara, you will find that the total of the first1st proxy â€˜19â€™ of Drutantyaprastaras, the second2nd proxy â€˜10â€™ of Laghvantaprastaras, the third4th proxy â€˜3â€™ of Gurvantaprastaras and the fourth6th proxy â€˜1â€™ of Plutantaprastaras becomes the respective â€˜Sankhyaâ€™ of (19 + 10 + 3 + 1=) 33 of 7DP. In the same manner you can understand for all other permutations and apply the Kalita also. Do it and inform. amsharma.
If you take the respective figures of the â€˜Table of General Permutationsâ€™, in Panchangaprastara, you will find that the total of the first1st proxy â€˜19â€™ of Drutantyaprastaras, the second2nd proxy â€˜10â€™ of Laghvantaprastaras, the third4th proxy â€˜3â€™ of Gurvantaprastaras and the fourth6th proxy â€˜1â€™ of Plutantaprastaras becomes the respective â€˜Sankhyaâ€™ of (19 + 10 + 3 + 1=) 33 of 7DP. In the same manner you can understand for all other permutations and apply the Kalita also. Do it and inform. amsharma.
Last edited by msakella on 27 Jan 2007, 04:38, edited 1 time in total.