Swar Prastaar

The word Prastaar means to expand or elaborate. Swar Prastaar is the different ways the seven swars can be combined to give rise to many sequences. For a given a number of unique swars, we can mathematically derive the list of all possible Swar combinations without repetition. 

It was said that one who mastered the Swar-Prastārs is said to have mastered the mysteries of music. Musicianswould only practice the Prastārs for years such that musical ideas would flow so naturally without any difficulties.

According to Ustad Amir Khan Saheb “Memorizing all the 5040 Prastārs is certainly difficult, if not an impossible feat. I now feel that if one memorizes all the four note Paltās alone and does regular Riyāz, that should facilitate Rāgā vistār to a great extent.“ (Singh, 2017)

The concept of Permutation and Combinations is used to get the value of possible patterns for a given set of Swars. the term Factorial (!) will give the number of possible patterns which can be represented with the below formula

n! = n x (n-1) x (n-2) x (n-3) ...

One Swar Prastaar

With only one Swar, say Sa, we can only get one Prastaar. Since there are seven Swars, there can be Seven Swar Prastaars on one Swar each.

Mathematically 1! = 1 x 1 = 1

Three Swar Prastaar

With three Swars, say Sa, Re & Ga, we can get the following possible combinations

Sa Re Ga

Sa Ga Re

Re Sa Ga 

Re Ga Sa

Ga Sa Re 

Ga Re Sa

Hence six Swar Prastaars are possible for any three Swars.

Mathematically 3! = 3 x 2 x 1 = 6

Five Swar Prastaar

With three Swars, say Sa, Re, Ga, Ma & Pa, we can get 120 possible combinations

Mathematically 5! = 5 x 4 x 3 x 2 x 1 = 120

Seven Swar Prastaar

With three Swars, say Sa, Re, Ga, Ma, Pa, Dha & Ni  we can get 504 possible combinations

Mathematically 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040

The above combinations are based on use of the seven Swars in sequential order of Sa Re Ga Ma Pa Dha Ni. If we further change the criteria that the order of the swar in a given pattern need not be sequential. Meaning instead of SRG, we can have SGM or SRM, SMP we get more possible patterns!

Mathematically, we can have the following possible combinations of swars (not appearing consecutively) and without repeating any Swar.

Let us understand the math with an example. Let us take an example to demonstrate the possible combinations that arise out of having two positions.

Here the swars are not fixed to SR alone but use all the seven swars but the number of positions is fixed to two. So, in the pattern of two, the first position can have any of the seven swars. The next position can use seven minus one or rest six swars (as one swar is already occupying the first position). Also, we are not repeating any swar in this. Hence the total combinations are 7X6 = 42 combinations.

Combinations starting with S till N are given in the table below

Indian Music uses 12 Swars in total which are used in the 10 thaats of North Indian Classical Music. 

Carnatic Music uses 72 Thaats called as Melakartas. From the 72, a total of 32 Thaats are recognized by scholars for use in Hindustani Classical Music.

Hence we can get 

1. Total Seven Swar Prastaars for 10 thaats: 10 x 5040 = 50400 Swar Prastaars

2. Total Seven Swar Prastaars for 32 thaats: 32 x 5040 = 161280 Swar Prastaars

3. Total Seven Swar Prastaars for 72 thaats: 32 x 5040 = 362880 Swar Prastaars

In ancient times, the concept of Grama Murchana was used in the way we use Thaat or Mela today to refer to a set of Swars in a Scale. Totally there were three Gramas. From each Grama there were 7 Murchanas possible. Later on, only two Gramas were popular and 14 Murchanas were used. There were 4 types of Murchanas which employed two different types of Vikrit Swars giving rise to 14 x 4 = 56 possible Murchanas. 

Hence, Total Seven Swar Prastaars for 56 Murchanas : 56 x 5040 = 282240Swar Prastaars

While the above table lists the possible combinations of the Swars, scholars have given a specific method of deriving the next pattern in a Swar Prastar. The procedure is detailed below

1. The Krama Rupa of the Swar sequence is always taken in the ascending order – Sa Re Ga Ma Pa Dha Ni. 

2. Sa is the starting swar and no swar lies below it.

3. One has to take the first swar in the starting sequence and below that write the swar which is one note lower. Eg if the first swar is R one has to write S below it. 

   1. The lower swar in this case S, must not be present on the right side of the original sequence

   2. If present, one must choose a swar two notes below.

   3. If the lowest note is S and hence we cannot go further lower, we use a dot below the swar.

4. The remaining swars are used in the sequence as such.

5. In case there are dots, the remaining swars are arranged in ascending order and placed in order in place of each dot.

The Swar Prastaar for three Swars Sa Re Ga is as follows:

1. S R G

2. R S G

3. S G R

4. G S R 

5. R G S

6. G R S

What is today called Merkhand is a term that goes back to the time of Sangita Ratnakara of Sarangadeva. There the word that is used is Khandameru. It refers to a certain diagram that has a a tapering shape. It is a  mathematical/ mnemonic device. In the chapter on Gramas, murchhanas, krama and tana, Sarangadeva attempted a rather exhaustive (and exhausting) analysis of all the note patterns that one could obtained by permuting a set of notes. 

Thus for instance if one took just two notes say S and R, one could obtain two patterns S R and R S; with three notes S R G, one has six patterns possible, namely S R G, S G R, R S G, R G S, G S R, G R S. Similarly, four notes give rise to 24 patterns (permutations), five notes to 120, six notes to 720, and seven notes to 5040 permutations. In a given scale, there are of course many ways of selecting subsets of two, three, four, etc. notes, and one begins to appreciate that to literally list all possible patterns systematically would be very difficult in a limited space. So Sarangadeva thought up a mnemonic table which would act like a key that would enable one to quickly arrive at a particular permutation of a subset of a given size. 

This mnemonic table is called Khandameru, and sometimes the whole method is called Khandameru. The process of associating the number to the given note series is called Uddishta. The converse process of figuring out the note series from the number is called Nashta - evidently referring to the problem in which the note series has become obscured.

The following table helps to perform the nasht and uddisht kriya of merukhand

The principle of getting the various numbers in the cells are given below

1. The values in columns from row two onwards are derived by multiplication of numbers 1,2,3,4,5,6

   1. For column two 1 * 1 = 1

   2. For column three 2 * 2 = 4

   3. For column four 6 * 2 = 12, 6 *3 = 18

2. The values in the row two are derived from summation of all the first cells whose count is equal to the position of the value in row two.

   1. The first cell in row two = 1 as there is only one preceeding cell with value 1

   2. The second cell in row two = sum of all first cells in row one and two = 1 + 1

   3. The third cell in row two = sum of all first cells in row one and two and three = 1 + 1 + 4 = 6

The knowledge of Khandmeru helps in learning and identification of Taan patterns. 

1. Uddisht Kriya is the process of identifying the sequence number of a given taan. 

2. Nasht Kriya is the process of identifying the taan pattern given the sequence number.

Let us identify the sequence number for a 4 swar taan M G S R. The original swar prastaar for the given combination is S R G M. To deduce the sequence, we have to use the khandmeru table. Since this is a four swar prastaar, we have to use the first four columns of the khandmeru table.

1. We now need to compare the original swar prastaar and the given prastaar: S R G M | M G S R.

2. Each Swar will be compared based on its position in the Prastaar (M G S R) with the original starting Prastaar (S R G M)

3. The last Swar is R in the given prastaar. It is placed three positions away from the last swar (M) in the original prastaar. In the table, let us refer the third row in the fourth column. This gives us value 12.

4. We are now left with three remaining swars: S G M | M G S.

5. The last swar is S in the given prastaar. It is placed three positions away from the last swar (M) in the original prastaar. In the table, let us refer the third row in the third column. This gives us value 4.

6. We are now left with two remaining swars: G M | M G

7. The last swar is G in the given prastaar and it is placed two positions away from the last swar (M) in the original prastaar. In the table, let us refer the second row in the second column. This gives us value 1 

8. We are now left with one remaining swar M. from the table we get the value 1 from the first column.

9. Now let us sum all the values we identified from the table

10. 12 + 4 + 1 + 1 = 18

Hence the pattern M G S R is the 18th Swar Prastaar of S R G M.

Let us now identify the 21st sequence in the four swar prastaar S R G M. To identify the sequence, we have to use the khandmeru table. Since this is a four swar prastaar, we have to use the first four columns of the khandmeru table.

1. We now need to select four numbers such that the sum of the numbers is 21

2. Also we can only use one number from a column at a time.

3. The selection of numbers will start from right to left.

4. 1 from the first column will always remain constant

1. Starting from the fourth column, let us select the cell whose value is close to 21 which is 18. 

2. This refers to the fourth position from the last swar in the original prastaar S R G M which is S. so the last fourth in the 21st prastaar is X X X S

3. Now 21-18 = 3. The nearest value in the third column is 2. 

4. This refers to the second position from the last swar in the remaining original prastaar R G M which is G. So the third swar in the 21st prastaar is X X G S

5. Now 21 -18 – 2 = 1. Since we have 1 already in the first cell, we have to select 0 from the second column. 

6. This refers to the first position in the remaining original prastaar R M which is M. So the second swar in the 21st prastaar is X M G S.

7. Now the remaining swar is R which can be used to complete the 21st swar prastaar of S R G M, which is now R M G S 

Merukhand Gāyaki trains its practitioners to remember all these combinations by heart and study them and apply it in the Rāgā badat and tāns. 

The patterns are sequenced according to a particular logic and required to be practiced endlessly until they get programmed into the ideation process of the musician. The mastery of these patterns also, obviously, developed the musician‘s technical ability to execute the most complicated melodic passages. When performing a Rāgā, the musician chooses the patterns compatible with Rāgā grammar for exploring the melodic personality of the raga. Merukhand is a technique for doing badhat (progression) in a Rāgā; it is not really a style. (Sharma, 2015) 

1. Singh, P. T. (2017). Ustad Amir Khan Pioneer of Indore Gharana. Kolkata: Thema.

2. Sangeet Visharad

3. https://groups.google.com/g/rec.music.indian.classical/c/uKfYcseLf7Q?pli=1

4. Sharma, A. (2015). Voice cultivation in Hindustani classical and western music a comparative study. Chandigarh, Punjab, India. Retrieved from https://shodhganga.inflibnet.ac.in/handle/10603/200219