
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 swarams, and 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 RiGa
combination, and six choices of DhaNi 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
DhaNi combinations are:
1^{st} mELam (dha na),
2^{nd} mELam (dha ni), 3^{rd} mELam (dha nu), 4^{th}
mELam (dhi ni), 5^{th} mELam (dhi nu), and 6^{th}
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(x1)+y = 6x+y6,
since (x, y) is the y^{th} mELam that immediately follows the previous
(x1) cakrams. For example, VAgadIshvari is (6,4) =6(5)+4 = 34, and
SaNmukhapriya is (10,2) = 6(101)+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 (x6, 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 (1^{st} 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 (2^{nd}
mELam in each cakram) will admit a dhi ni melam (4^{th} mELams in each
cakram) differing only in Da; also, a dha nu mELam (3^{rd} mELam in
each cakram) admits two more melams (dhi nu, and dhu nu, the 5^{th}
and 6^{th} 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, (4^{th} mELam
in each cakram) we have a dhi nu, melam (5^{th} mELam in each cakram)
differing only in Ni. For the dha na type (1^{st} mELam in each
cakram), we have two possibilties, dha ni, dha nu (namely, the 2^{nd}
and 3^{rd} 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) (2^{nd} mELam in 10^{th} chakram).
So, x =10, and y = 2.
Since x is greater than 6,
the only mELam differing from it in Ma is (106,2) = (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! 