Carnatic rāga

Carnatic raga refers to ragas used in Carnatic music. A Carnatic raga has several components - primordial sound (nāda), tonal system (swara), pitch (śruti), scale, ornaments (gamaka) and important tones.

Janaka ragas or Sampurna ragas are parent ragas from which more ragas are derived. Sampurna ragas as the name suggest are those in which all 7 swaras are present. They are also called Melakarta ragas. These ragas have all 7 swaras in their scales (only one of each swara, Sa, Ri, Ga, Ma, Pa, Dha, Ni), following strict ascending and descending scales and are sung in all octaves. Example of melakartha ragas are : Shankarabharana, Kalyani, Natabhairavi, Nattai, Harikambhoji etc.

There are different types of swara sthayis or note pitches that leads to the total number of melakartha ragas. These different variations of swaras leads to more than one combination of musical scale. The different variations in swaras is mentioned below. Using this it is possible to mathematically ascertain the total number of melakartha ragas.
Sa (Shadja) - only one.
Ri (Rishabha) - Shuddha, Chathushruthi and Shatshruti Rishabha in increasing order of pitch.
Ga (Gandhara) - Shuddha, Sadharana, Antara Gandhara in increasing order of pitch.
Ma (Madhyama) - Shuddha, Prathi Madhyama in increasing order of pitch.
Pa(Panchama)- only one.
Dha (Dhaivatha) -Shuddha, Chathushruti, Shatshruthi Dhaivatha in increasing order of pitch.
Ni (Nishadha)- Shuddha, Kaishiki, Kakali Nishadha in increasing order of pitch.
However there are only some combinations that are allowed as for instance Shatshruthi Ri can combine with Antara Gandhara and Chathushruti Rishabha can combine with Sadharana and Antara Gandhara only. Likewise the same principle holds good for Da and Ni. If we label these swaras as Sa, Ra, Ri, Ru, Ga, Gi, Gu, Ma, Mi, Pa, Dha, Dhi, Dhu, Na, Ni, Nu.
The total combinations are:
a) 1 Sa X 1 Ra X 3(Ga, Gi, Gu) Ga + 1 Sa X Ru X 2 Ga (Gu, Gi) + 1 Sa X Ri X Gu = 6.
b) Ma, Mi =2
c) 1 Pa X Dha X 3 Ni (Na, Ni, Nu)+ 1 Pa X Dhu X 2 Ni (Nu, Ni)+ 1 Pa X Dhi X Ni =6
Multiplying these 3 combinations we get 72.

The 72 melakartha ragas are arranged in a cycle called katapayadi sutra which is named so because by the index of the raga we can get the name of the raga and the exact swara combination. The first 36 melakartha ragas have shuddha madhyama whereas, the next 36 ragas have prathi madhyama.

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Wikipedia