(April 1999)
Ajay Sathyanath
This is common for both Carnatic and Hindustani music, and not suprisingly, because both stem from the same system of music, and are the same in terms of raagas and swaras among several other things.
Relax, for people who don't know what the terminologies are, start reading from the Appendix section onwards. I have also worked out two examples below, so you can get a feel.
|1 2 3| |- 4 5| |- - 6|
It has 6 elements, at vertices given by (1,1), (1,2), (1,3), (2,2), (2,3), and (3,3) the first 3 elements occupy the top row, the next 2 elements occupy the second row, and the last element occupies the third row. the vertices are given by
|(1,1) (1,2) (1,3)| | - (2,2) (2,3)| | - - (3,3)|
Numbering these elements from top down and from left to right as numbers 1 to 6, we get: (1,1) = 1, (1,2) = 2, (1,3) = 3, (2,2) = 4, (2,3) = 5, (3,3) = 6.
Much of the music was kept in its original/pure form, until the advent of the Mughals in the 12th century AD. By virtue of the Mughal grasp of all of North India and many parts of the South, classical music changed to incorporate these influences. However, the South pretty much kept with tradition and was thus called Carnatic music, where the word Carnatic stands for tradition. The North added the Arabic and Persian influence and called its version as Hindustani music, or in other words - the music of India.
Though Indian classical music itself has two forms, each melodious in its own way, the mathematics and the basics remain the same.
Each of the swaras/notes, appear in different frequencies. There are 3 variations of Ri, denoted as R1, R2 and R3; 3 variations of Ga, denoted as G1, G2 and G3; 2 variations of Ma, denoted as M1 and M2; 2 variations of Da, denoted as D1 and D2; and two variations of Ni, denoted as N1 and N2. The swaras Sa and Pa have no variations. Arranging the swaras in order of increasing frequency we get:
S, R1, G1, R2, G2, R3, G3, M1, M2, P, D1, N1, D2, N2, D3, N3.
we now note why R and G are considered together as one group and D and N as another group.
Melakartha raagas are raagas with a pattern, they can be split into two portions, one called the Arohanam and the other called the Avarohanam. The Avarohanam is the mirror image of the Arohanam, and considered as a string of the constituent swaras, the Arohanam plus the Avarohanam makes a palindrome!
Arohanam is a string/sequence of swaras whose frequencies increase from left to right, in a strictly monotonic way. Avarohanam is a string/sequence of swaras whose frequencies decrease from left to right, in a strictly monotonic way. Both the Arohanam and the Avarohanam must consist of all the seven swaras/notes, of course these swaras will have different variations in different Arohanams. For eg:
is one Arohanam andSince G2 > R2, and G1 is not greater than R2. we can have "R2,G2" as a valid substring of an Arohanam, but we CAN'T have "R2,G1" as a valid substring of an Arohanam.
It is called KaTaPaYadi, because all consonants can be partitioned into 4 sets of numbers. consonants from Ka - Ta making one set, Ta - Pa another, Pa - Ya a third, and everything after Ya constituting the fourth.
Ka, kha, ga, gha, nga, (1,2,3,4,5) ca, cha, ja, jha, ngya, (6,7,8,9,0) Ta, tta, da, dda, na, (1,2,3,4,5) tha, ttha, dha, ddha, nna, (6,7,8,9,0) Pa, pha, ba, bha, ma, (1,2,3,4,5) Ya, ra, la, va, sha, (1,2,3,4,5) shha, sa, ha, lla, rra. (6,7,8,9,0)
Because, there are 6 scales to a chakra. ie 6 raagas together are called one chakra. In a given chakra, the swara R and G are always constant.
Because, the first 36 melakartha raagas have the lower Ma (shuddha madhyama) and the next 36 have the higher Ma (prathi madhyama). Besides there is a symmetry between 1 and 37, 2 and 38 etc... they differ only in their Ma values.
The maximum variations for a swara/note is 3. for eg there are 3 types of Ri's R1, R2 and R3 (they differ in their frequencies, with R3 being greater than R2 etc..)
The vertices of an upper triangular matrix A = (R,C). (where R stands for Rows and C for Columns) is such that the rank of R is always greater than the rank of C. this boils down to the fact that the frequency represented by the row is always greater than that of the column. (read the next para below)