Math/Anatomy of Carnatic Melakarthas

There are 7  fundamental notes that are used in both eastern and western system.

Western system:   C    D   E   F   G   A   B 

   Carnatic music:    Sa  Ri  Ga Ma Pa Dha Ni

In order to define a raga, you select a set of notes and define a forward/backward progression of these notes called arohana and avarohana respectively. This particular progression would make musical sense. A sampoorna raga is a raga that uses all the 7 notes. If you try to come up with all sampoorna ragas, there would be of the order of 7! ragas. Obviously not all of them makes musical sense.

Notes

Let us look at the notes before we get to melakarthas. Following are the notes used in melakarthas.

There are 2 Ms, 3 Rs, 3Gs, 3Ds, 3Ns and S/P are fixed. The notes have to follow progression. Let us look at R,G combinations. Since R2=G1 and R3=G2, <R2,G1> and <R3,G2> pairs are omitted. <R3,G1> is omitted because it is not in ascending order. So there are only 6 combinations. Let us try to enumerate  R,G combinations. There are six of them which are

1. R1,G1
2. R1,G2
3. R1,G3
4. R2,G2
5. R2,G3
6. R3,G3

Similarly there are 6 combinations for D,N and there are 2Ms. 

Structure of melakartha

Any melakartha is one of [S] , [ one of the 6 R,G combinations] , [M1 or M2], [P], [one of the six D,G combinations] [S]. Avarohana is just the descending order of aarohana. So we need to count number of ways to make an ascending progression(aarohana) using all 7 notes. That gives us 1(S) X 6(R,G) X 2(M) X 1(P) X 6(D,N) X 1(S) = 72 combinations. Hence there are 72 melakarthas. 

The first half of melakarthas(1-36) uses M1(suddha Madhyamam) and the second half of melakarthas(37-72) uses M2(prati madhyamam).

Now let us try to do a lexicographic ordering in each of these halfs and we get all melakartha ragas. Once we fix S,R,G,M and vary D,N we get 6 ragas and these ragas form a chakra. So the first chakra is S R1 G1 M1 and we vary <D,N> to get 6 combinations and they translate to first 6 ragas in the melakartha system. 

How to get a raga from melakartha number ?

There is a total of 12 chakras and within a chakra there are 6 ways to change <D,N> giving you a total of 12X6 = 72 ragas. Given a raga number N, you can get the chakra number by applying ceil(N/6). Following are the chakras and their S,R,G,M,P notes

1. chakra#1 - Indu - S R1 G1 M1 P
2. chakra#2 - Netra - S R1 G2 M1 P
3. chakra#3 - Agni - S R1 G3 M1 P
4. chakra#4 - Veda - S R2 G2 M1 P
5. chakra#5 - Bana - S R2 G3 M1 P 
6. chakra#6 - Rutu - S R3 G3 M1 P
7. chakra#7 - Rishi - S R1 G1 M2 P
8. chakra#8 - Vasu - S R1 G2 M2 P
9. chakra#9 - Brahm - S R1 G3 M2 P
10. chakra#10 - Disi - S R2 G2 M2 P
11. chakra#11 - Rudra - S R2 G3 M2 P 
12. chakra#12 - Aidtya - S R3 G3 M2 P

If you want a more mathematical method of deriving the aarohana/avarohana from the raga number(N) , do the following.

Any raga is of the form S Ra Gb Mc P Dd Ne S. In order to find a,b,c,d,e do the following

1. c = ceil(N/36) , so N=1-36 will have M1 and 37-72 will have M2
2. To get a,b you find Chakra number = ceil(N/6) and look at the table above
3. To get d,c you do N%6 and you can see the pattern. After every 6 raga <D,N> pattern repeats

From a mathematical perspective, melakarthas are an enumeration of the notes under some constraints. If you sing aarohana, avarohana together you will get a musical palindrome that would make sense and evoke some feeling(rasa).

How to get raga number from raga name?

Apparently there is a way to get raga name from raga number and then aarohana, avarohana from it using the method specified above. This speaks volumes about how methodical and structured indian civilization was back in olden times.

Katapayadi system is basically a way to map words to numbers. It is seen in many places in indian culture. Let us focus on getting the raga number using katapayadi system. In order to do this, we need to take the first two letters and find the number associated with them and then reverse the digits to get the raga number.

For example, MayamalavaGowla [letter1 is ma which maps to "5", letter2 is ya which maps to "1". So we get the raga number as 15. MayamalavaGowla is the 15th raga in melakarthas. ]

Example2: Charukesi , Cha=6 and Ra=2. RagaNumber=26
Example3: Kanakangi, Ka=1, Na=0 , RagaNumber=01

1	2	3	4	5	6	7	8	9	0
ka क క ക	kha ख ఖ ഖ	ga ग గ ഗ	gha घ ఘ ഘ	nga ङ జ్ఞ ങ	ca च చ ച	cha छ ఛ ഛ	ja ज జ ജ	jha झ ఝ ഝ	nya ञ ఞ ഞ
ṭa ट ట ട	ṭha ठ ఠ ഠ	ḍa ड డ ഡ	ḍha ढ ఢ ഢ	ṇa ण ణ ണ	ta त త ത	tha थ థ ഥ	da द ద ദ	dha ध ధ ധ	na न న ന
pa प ప പ	pha फ ఫ ഫ	ba ब బ ബ	bha भ భ ഭ	ma म మ മ	-	-	-	-	-
ya य య യ	ra र ర ര	la ल ల ല	va व వ വ	śha श శ ശ	sha ष ష ഷ	sa स స സ	ha ह హ ഹ	-	-

Rasas

Every raga has some mood/emotions(rasa) associated with it. Following are some of the common rasas associated with ragas.

1. Karuna – sadness, pathos
2. Shringar – love, joy
3. Vira – heroism, valour
4. Hasya – laughter, comedy
5. Raudra – anger
6. Bhayanaka – fear
7. Vibhatsa – disgust
8. Adbhuta – surprise
9. Shantam - peace

Different systems/sources have slightly different classifications. Karuna and bhakti are seen a lot in Carnatic music due to its religious/spiritual connections. Chakras and rasas are not particularly related. You see different rasas on all chakras. This is because ragas in a chakra mostly vary by 1 or 2 notes but that variation can cause huge difference in the emotions it evokes.

It is seen that when notes are evenly paced, it creates smoother and happier emotions. A good example is shankarabharanam. Notes are evenly paced and there is a lot of symmetry. When you start pacing out the notes in an uneven fashion(e.g.:- P D1 N3 S), it would evoke more complex emotions. ShubhaPantuvarali is an example.

Rasas and emotion is a complex topic because most of the time when you pick two ragas, on an average they would vary by 2 notes but the change in feel would be huge.

Time to play a raga

Ancient indians were clearly interested in the connections between medical science(ayurveda), music , arts and so on. It is not surprising to see that vata-pitta-kapha concepts are seen in indian classical  music.

Every day is divided into 6 segments of 4 hours each. The segments are

1. 6-10AM Kapha
2. 10AM-2PM - Pitta
3. 2PM- 6PM - Vata
4. 6PM - 10PM - Kapha
5. 10PM-2AM - Pitta
6. 2AM - 6AM - Vata

A lot of good practices in ayurveda takes this into account very seriously. As far as music is concerned, day is divided into the following 8 segments of 3 hour duration each and they are called prahars. 

1. 6 a.m. - 9 a.m[purvaanha]. first beat of the day. Daybreak; Early Morning; Morning;
2. 9 a.m. - 12 p.m[madhyaanha]. 2nd beat of the day. Late Morning; Noon; Early Afternoon;
3. 12 p.m. - 3 p.m[aparaanha]. 3rd beat of the day. Afternoon; Late Afternoon;
4. 3 p.m. - 6 p.m[saayankaala]. 4th beat of the day. Evening Twilight; Dusk (sunset);
5. 6 p.m. - 9 p.m[pradosha]. first beat of the night. Evening; Late Evening;
6. 9 p.m. - 12 a.m[nishitha]. 2nd beat of the night. Night; Midnight;
7. 12 a.m. - 3 a.m[triyaamaa]. 3rd beat of the night. Late Night
8. 3 a.m. - 6 a.m[ushaa]. 4th beat of the night. Early Dawn; Dawn (before sunrise);

A more fine granular division of time in indian culture is muhurthas. A day has 30 muhurthas of duration 48 min each. Just wanted to point out that all these concepts are related.

So what is the relation between math and melakarthas?

Mathematics is a highly structured and methodical way of looking at things. Carnatic music and many aspects of indian culture was highly organized and methodical and as a result we are able to look at many things from a mathematical perspective. This is really useful in getting a theoretical understanding. On a practical level, music is a divine gift.

Most of the time in mathematics, it is really important to understand the motivation/significance/use case of a mathematical concept/theorem. Personally i feel people should approach ragas also in a similar manner. Understanding the rasas, times/seasons to play the raga, understanding the dosha it balances(e.g. charukesi balances pitta) and understanding the motivation/origin of the raga would be really helpful to know and learn along with the notes/progression.

What is the significance of melakarthas ?

They are the root ragas in south indian classical music. Other ragas like janya ragas, vakra ragas are a derivative of the melakartha ragas. Understanding and practicing the melakartha raga would give you a solid foundation on which you could build upon. Also, most of the melakartas have a hindustani equivalent and hindustani thaat system has a lot of similarities with melakarats. So you could easily pick up hindustani if you learn melakartas properly.