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During a discussion of mELams 56/57, SaNmukhapriya/simhEndramadhyamam, in the Rasika Forum (Thread on  the 72 mELa rAgAmAlika of Maha Vaidyanatha Iyer), the following question was raised by a rasika: "How many mELam pairs exist that differ in exactly one svaram, two swaramsand so forth...?"

In what follows below, I shall give a simple algorithm to answer the one svaram statistics. Similar techniques will yield answers to two, or more svaram cases (but need lengthy analysis covering several cases).

Let us recall how a mELam is defined using the seven notes Sa, Ri, Ga, Ma, Pa, Dha, Ni, in that order, using each note exactly once.  The notes Sa, and Pa are fixed and do not allow any variants. There are two variants of Madhyamam,  namely, shuddha madhyamam (denoted by the mnemonic  phrase “ma”), and prati madhyamam (expressed as “mi”).  There are three varieties of Ri, namely  shuddha Ri (“ra”), catushruti Ri (“ri”), and Shatshruti Ri (“ru”),  three variants of Ga, namely,  shuddha Ga (“ga”), sadharana Ga (“gi”), and antara Ga (“gu”).  Similarly, the Dha has three choices, shuddha Dha (“dha”), catushruti Dha (“dhi”), and Shatshruti Dha (“dhu”).  Finally the note Na also admits three variations, shuddha Ni (“na”), kaishiki Ni (“ni”), and  kakali Ni (“nu”).  But, only six  among the nine possible combinations of Ri – Ga, and six (among the possible nine) combinations of Dha  - Ni are permitted namely, ra ga,  ra gi,  ra gu,  ri gi,  ri gu, ru gu; dha na, dha ni, dha nu, dhi ni, dhi nu, and dhu nu. So the combinations  ri ga,  ru ga,  ru gi,  dhi na,  dhu na and  dhu ni are not permitted.   Hence,  two available choices of Ma, six choices of  Ri-Ga combination, and six choices of Dha-Ni combination result in a total of  2 x 6 x 6 = 72 mELams.  These 72 mELams are arranged in 12 cakrams, each consisting of six mELams.  In each cakram, the Ri – Ga combination is fixed,  and within each cakram, the occurrence of Dha- Ni  is fixed for each mELam  1, 2, 3 , 4 ,5, 6.   The  Ri- Ga combination defined for each cakram is as follows (the cakram numbers appear in parentheses):

               ra ga (1 & 7), ra gi (2 & 8), ra gu (3 & 9), ri gi (4 & 10), ri gu (5 & 11), ru gu (6 & 12).

Also, in each cakram, the Dha-Ni combinations are:

1st mELam (dha na), 2nd mELam (dha ni), 3rd mELam (dha nu), 4th mELam (dhi ni), 5th mELam (dhi nu), and 6th      mELam     (dhu nu).

Finally, the mELams in the first six cakrams (mELams 1 to 36) assume the note ma (shuddha madhyamam), and the mELams in the remaining six cakrams (37 to 72) have the note mi (prati madhyamam).  Thus, the Ri Ga Ma Dha NI combination  fixes a mELam uniquely.  Also, each mELam acquires a unique number from 1 to  72, and  usage of the sanskrit “ka Ta pa yAdi” counting method enables us to obtain the mELam number from the first two letters of  its nomenclature.

For simplicity, let us denote a mELam  X  by a pair (x,y), where “x” denotes the cakram number to which it belongs, and “y” is the rank of the mELam in that particular cakram.  Of course, “x” varies over integers 1 to 12, while “y” ranges over integers 1 to 6.   For  x = 1, 2, 3, 4, 5, 6, the pair (x, y) defines a shuddha madhyama mELam, while  x = 7 onwards, yields a prati madhyama mELam.  From the pair (x,y), the mELam number can easily be recovered by the formula X = 6(x-1)+y = 6x+y-6, since (x, y) is the yth mELam that immediately follows the previous (x-1) cakrams.  For example, VAgadIshvari is (6,4) =6(5)+4 = 34,  and SaNmukhapriya is (10,2) = 6(10-1)+2=56.

Melam pairs differing only in madhyamam

Clearly, if two mELams differ in madhyamam,  one has to take  the shuddha madhyamam (ma) and the other the prati madhyamam (mi). Thus, each is a shuddha/prati madhyamam  counterpart of the other.  Hence, given the mELam (x,y),  the only mELam that differs in madhyamam alone from it is (x+6,y) if  x= 1,2,3,4,5,6 and  (x-6, y)  if x = 7 through 12.

It is now obvious that there are 36 mELam pairs that differ only in madhyamam.  In each pair, one belongs to the shuddha madhyama mELam group, and the other to the prati madhyama mELam group.

For melam pairs differing in ia single svaram  other than Ma, we have to look for both members of the pair either in the shuddha madhyama group, or in the prati madhyamam group, as therwise the svaram Ma also will be different. Thus, there are 36 mELam pairs, so that members of each pair differ only in the madhyamam.

Melam pairs differing only in .rSabham

Since the note Ri combines with Ga in only six acceptable ways, we have to be a bit careful. For instance, if a mELam uses the “ra ga” combination (melams 1, 7),  there cannot be another mELam differing from it only in Ri, since  the combinations ri ga, and  ru ga are not available.  But if the mELam is of  ra gi type (mELams 2, 8), then  there is another mELam of ri gi type, namely, melams 4 and 10 (but beware,  not ru gi type) which differs from it only in Ri;  if the mELam is a  ra gu type (melams 3, 9), then there are two mELams, one of ri gu type (mELams 5, 11), and another of ru gu type (mELams 6, 12), differing from it only in Ri.  These observations lead to the following algorithm for any given mELam (x,y):

  • If x= 1, or x = 7, there is no mELam that differs from (x,y) only in .rSabham

  • For  other choices of  x, we have the following  pairs differing only in .rSabham:

  • The pairs (2, y) and (4, y)

  • The pair (8, y) and (10, y) (prati madhyamam counterpart of the above)

  • All three pairs from the triple {(3, y), (5, y), (6 y)}

  • All three pairs from the triple {(9,y), (11,y), 12,y)} (prati madhyamam counterpart of the above)

Here the coordinate  y   has six choices from 1 to 6, so  we have  0 + 6 + 6 +  3(6) + 3(6) = 48 pairs  of mELams such that mELams in each pair differ only in the note Ri.

Melam pairs differeing only in gAndhAram

As in the case of Ri,  we have to consider various possibilities.   MELams in cakrams 6 and 12 employ the   ru gu  combination, so if any mELam has to differ from such a mELam only in the note Ga, the only possibilities are ru ga, or ru gi combination, both not valid.   In this case, we will not obtain any  candidate .   For mELams of ri gi type (mELams 4, 10), there is a mELam of ri gu type (mealms 5, 11) that differs only in Ga. Again, a ra ga type mELam (1, 7) admits two mELams  one of ra gi type (mELams 2,8), and another of ra gu  type (mELams 3, 9), both differing from it only in Ga. These considerations lead to the following  algorithm for Ga differing mELams.

  • For x= 6, or  x = 12, there is no mELam differing from (x , y)  only in gAndhAram. 

  • For  the remaining choices of  x,  we have the following  pairs differing only in gAndhAram:

  • The pair (4 y) and (5,  y)

  • The pair (10, y) and (11, y) (prati madhyamam counterpart of the above)

  • All three pairs from the triple {1, y), (2, y), (3, y)}

  • All three pairs from the triple {7, y), (8, y), (9, y)} (prati madhyamam counterpart of the above)

Here again the coordinate  y  has six choices from 1 to 6, so  we have  0 + 6 + 6 +  3(6) + 3(6) = 48 pairs  of mELams  with the property that mELams in each pair differing only in gAndharam.

Melam pairs differing only in dhaivatam

Here, one has to observe that  if a mELam takes the  dha na  phrase (1st mELam in each cakram), there  cannot be another mELam differing from it only in dhaivatam, since the combinations dhi na, dhu na are not permitted. However, a dha ni melam (2nd mELam in each cakram) will admit a dhi ni melam (4th mELams in each cakram) differing only in Da; also, a dha nu mELam (3rd mELam in each cakram)  admits two more melams (dhi   nu, and dhu nu, the 5th and 6th mELams in each cakram)  ) differing only in Dha.  Thus we have the following algorithm.

  • For  y = 1, there is no mELam that differs  from the mELam (x, y) only in Dhaivatam.. 

  • For the remaining choices  of  y   (that is, y = 2, 3, 4, 5, 6) we have the following pairs differing in Dha

  • The pair (x, 2) and (x, 4)   (dha ni, dhi ni)

  • All three pairs from the triple {(x, 3), (x, 5), (x, 6)}  (dha nu, dhi  nu, dhu nu)

Here, the coordinate  x  has 12 choices (1 to 12), so  we have 0 + 12 + 3(12) = 48 pairs of mELams, with mELams in each pair differing only in Dhaivatam.

Melam pairs differing only in niSAdam

A mELam  taking the note dhu  (last mELam in each cakram) has the only  possible niSAdam  combination nu  (namely,  dhu nu),  so it cannot differ only in Ni from another mELam since dhu na, dhu ni are not allowed.  However, for  dhi ni, (4th mELam in each cakram)  we have a dhi nu, melam (5th mELam in each cakram) differing only in Ni.   For the dha na type (1st mELam in each cakram),  we have two possibilties,  dha ni, dha nu (namely, the 2nd and 3rd mELams in each cakram).  These consideration lead to the following algorithm:

  • For  y = 6, there is no mELam that differs from  the mELam (x  y)  only in Nishadham

  • For the remaining values of   y  (that is, 1,2,3,4,5) we have the following pairs differing only in Ni

  • The pair (x ,4), and (x ,5)  (dhi ni, dhi nu)

  •  All three pairs from the triple {(x ,1), (x ,2), (x ,3)} (dha na, dha ni, dha nu).

Here again, the  coordinate x   has 12 choices ( 1 to 12), so  we have  0 + 12 + 3(12) = 48 pairs of mELams having the property that mELams in each pair differ only in Dhaivatam

Thus, the total number of mELam pairs such that mELam in each pair  differ in exactly one note is  36 + 48 + 48 + 48 + 48 = 228.

Illustrative examples

Our first example is mELam 1, KanakA"ngi = (1, 1). Here x = 1, and  y = 1.

MELam differing only in Ma : (1+ 6, 1) = (7,1) = sAlagam (prati madhyamam counterpart)

MELam differing only in Ri:  no candidate since   x =1.

MELam differing only in Ga: (2, 1) = 7 =  sEnAvati, and (3,1) =13 = gAyakapriya..

Melams differing only in Dha  none, since  y = 1.

MELams differing only in Ni:  (1, 2), and (1,3), that is  RatnA"ngi, and GAnamUrti.

Thus, for KanakA"ngi, we get 5 mELams  that pair with it and differ in exactly one note. 

Our next example is mAyAmALavagaula  = (3, 3) = 2(6)+3 = 15, so a= 3, b=3.

Differing in Ma only:  mELam (9,3) = 51 = KAmavardhani  (prati madhyamam counterpart)

Differing in Ri only:  mELams (5,3) = 27 =SarasA"ngi, and  (6,3) = 33 = GA'ngEyabhUShaNi

Differing in Ga only: mELams (1,3) = GAnamUrti, and (2,3) = 9 = DhEnuka

Differing only in Dha : we have two candidates, (3,5) = 17 = SUryakAntam, and (3,6) = 18 = HAtakAmbari

Differing only in Ni: again two candidates: (3,1) = 13 = GAyakapriya, and (3,2) =14 = VakuLAbharaNam.

Thus, MAyamaLavagaula gives the maximum  possible pairs, a set of nine mELams with which it pairs, and differs from each, exactly in one note.

Finally, let us take SaNmukhapriya :  56 = (10,2)  (2nd mELam in 10th chakram).  So, x =10, and  y = 2.

Since  x is greater than 6, the only mELam differing from it in Ma is (10-6,2) = (4,2) = 6(3)+2=20 = NaTabhairavi (prati madhyamam counterpart).

Since  y = 2 (fixed), the only possibility for Ri difference is the mELam (8,2) =44 = Bhavapriya.

Again,  y = 2, so the only possibility for Ga difference is the mELam (11,2) = 62 = .RSabhapriya

Since  x = 10, the only possibility for the Dha difference is the mELam (10,4) = 58  HEmavati

Finally,  x = 10, and there are now two possible choices for the Ni difference, namely, (10,1) = 55 = ShyAmalA"ngi, and (10,3) = 57 = SimhEndramadhyamam.

Thus, for SaNmukhapriya, we get 6 mELams  that pair with it and differ in exactly one note.

As an easy exercise, one can try mELam (2,3) = 9 = DhEnuka,  and generate 8 pairs; similarly, (4,3) = 21= KIravANi  yields 7  pairs.  Hence the possible pairs for each mELam varies from 5 to 9.

Working in this manner through all mELams in all cakrams, one obtains the following data:

MELams in cakram  1 , 4, 6, 7, 10 and 12 each yield 34 pairs

MELams in cakrams 2 , 5, 8 and 11   each yield  40 pairs

MELams in cakrams 3 and 9  each  yield 46 pairs

This totals 456, but since each pair is counted twice in the above , we have to take half this sum, that is, a grand total of  228 mELam pairs differeing exactly in one note. This number 228 agrees with our previous computation!

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