The melakartha raga system is highly mathematical and logical. For the purpose of formulating the melakartha raga system, the great creators of our raga system redefined the 12 tone swara system into a 16 tone system, listed as under:
sa   shadjam 

r1   sudha rishabham 

r2   chatusruthi rishabham  g1 
r3   shatsruthi rishabham  g2 
g1   sudha gandharam  r2 
g2   sadharana gandharam  r3 
g3   andhara gandharam 

m1   sudha madhyamam 

m2   prathi madhyamam 

pa   panchamam 

d1   sudha dhaivatham 

d2   chathusruthi dhaivatham  n1 
d3   shatsruthi dhaivatham  n2 
n1   sudha nishadham  d2 
n2   kaisiki nishadham  d3 
n3   kakali nishadham 

S   Shadjam ( harmonic of the low sa ) 

Thus the basic 12 swaras ( top Sa excluded ) are renamed with the above names for the purpose of construction/ formulation of the 72 melakartha ragas. As mentioned above, the frequency value of r2 is the same as g1, r3 is the same as g2, d2 is the same as n1 and d3 is the same as n2.
1 2 3 concept:
The basic rule for the melakartha ragas is that all ragas will have all seven swaras in the ascent ( aarohanam) and 7 swaras in the descent ( avarohanam) in increasing order of frequency and decreasing order of frequency , respectively. Thus when r2 is used, g1 cannot be used, since its frequency value is the same. Similarly, when r3 is used, g1 and g2 cannot be used since they are smaller or equal in frequency.
To simplify this concept, take the possible combination of double digit numbers using 1, 2 and 3.
They are 1 1, 1 2, 1 3, 2 1, 2 2, 2 3, 3 1, 3 2, 3 3.
Now on this set, apply a rule that the second digit should be equal to or higher than the first digit. This will result in a combination six set combination of 1 1, 1 2, 1 3, 2 2, 2 3 and 3 3.
Now , prefix the first digit with r and the second digit with g in the six set combination, which leads us to a resultant set of  r1 g1, r1 g2, r1 g3, r2 g2, r2 g3 and r3 g3.
Applying the same rule on the six set combination with d and n, we get the following resultant set  d1 n1, d1 n2, d1 n3, d2 n2, d2 n3 and d3 n3.
The 6 combinations of r and g can be combined with the 6 combinations of d and n resulting in 36 possible combinations. For instance the first element of the r and g resultant set  r1, g1 can be combined with d and n resultant set as follows:
r1 g1 d1 n1
r1 g1 d1 n2
r1 g1 d1 n3
r1 g1 d2 n2
r1 g1 d2 n3
r1 g1 d3 n3
Thus there are 36 possible combinations between the resultant set of r and g and the resultant set of d and n. There are two variations of m namely m1 and m2. Hence each variation of m along with the 36 combinations result in the 72 melakartha ragas. In the next blog, we will analyse the classification of these melakartha ragas.