Mathematical Fundas in Indian Classical Music

(April 1999)
Ajay Sathyanath

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Introduction

Disclaimer
Notation
The Algorithm
Some Worked Out Examples
- Example 1 - HaNuMaToDi
- Example 2 - ViShvamBhaRi
Appendix

History
Notes; Swaras and Raagas
Parent Raagas - Melakartha
The Katapayadi Number
The Sanskrit Consonants
Miscellaneous Information

Introduction

This is a method, part of which I discovered and part of which is already known, to generate the entire raaga given just the name of the Melakartha Raaga. In other words this scheme/algorithm makes it possible to generate the Arohanam and Avarohanam of Melakartha raagas. This is a small 3 step procedure.

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.

Though the Katapayadi number generation is already known in carnatic/hindustani circles, people do not as yet know how to generate the raaga with this number, in ONE CONSOLIDATED FORM. Usually they try remembering the whole table of Melakartha Raagas, or through some other means, like remembering certain properties of this table.

Notation

The seven Swaras or Notes viz. Sa, Ri, Ga, Ma, Pa, Da, and Ni will be denoted by S, R, G, M, P, D, N respectively.

There are 3 variations each for R, G, D, and N. They are represented by R1, R2, R3 and similarly for G, D and N also.
There are two variations for M. and they are M1 and M2.
The notes S and P have no variations.

Algorithm

Heres the Algorithm.

Generate the Katapayadi number 'K' of the Melakartha raaga.

If K > 36 (strictly greater than), then put down Mx as M2,
else put down Mx as M1.
where x simply stands for either 1 or 2.
Let K1, be an integer whose value is given by K1 = ceiling function of K/6.

(for eg., if K/6 = 5.1 then K1 = 6, if K/6 = 5.0 then K1 = 5)
Let K2, be an integer whose value is given by K2 = K1 modulo 6.

since it is a modulo function, we can consider 0 to be equal to 6.
we know that K2 will lie between 0 to 5. We thus make K2 to lie between 1 to 6 by setting 0 to be 6.
Now (don't get afraid !!) consider a (3 x 3) upper triangular matrix. Here is a diagram to help visualise it.
```
       |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.
We group the swaras/notes R and G together ie (R,G) and the notes D, N together as (D,N)
- The first step gives Mx to be either M1 or M2. (ie the lower Ma or higher Ma)
- The second step gives (R,G)
  ie. from the value of K1 find the vertex corresponding to it.
  
  let the vertex be (a,b). Then R is Ra and G is Gb.
  for eg if K1 is 5 then the vertex is (2,3) so we get R2 and G3.
- The third step gives (D,N)
  ie. from the value of K2, find the vertex corresponding to it.
  
  let the vertex be (i,j). Then D is Di and N is Nj.
  for eg if K2 is 4, then the vertex is (2,2). so we get D2 and N2.
Now the Arohanam is written down as
S, Ra, Gb, Mx, P, Di, Nj, S
and the Avarohanam as
S, Nj, Di, P, Mx, Gb, Ra, S.
and together they form the Raaga!!!!!!

Some Worked out Examples

NAME of Melakartha raag := HaNuMaToDi
- the first two syllables in sanskrit are Ha and Nu.
- therefore first two consonants are Ha and Na.
- Ha = 8 ; Na = 0; number = 80; inverting it we get; K = 08 = 8;
- K is not greater than 36, so Mx = M1.
- K1 = ceiling of (8/6) = 2; the 2nd vertex = (1,2) So (R,G) = (1,2); ie, R1, G2.
- K2 = 8 % 6 (8 modulo 6, with 0 considered as 6). ie K2 = 2; and 2nd vertex = (1,2) so (D,N) = (1,2); ie D1, N2.
- Arohanam = S, R1, G2, M1, P, D1, N2, S Avarohanam = S, N2, D1, P, M1, G2, R1, S.
NAME of Melakartha raag := ViShvamBhaRi
- consonants are: Va and Sha.
- Va = 4; Sha = 5; inverted K = 45;
- K = 54; and 54 > 36 is true; so K = 54 - 36 = 18; and Mx = M2;
- K1 = ceiling of (18/6) = 3.0 = 3; 3rd vertex = (1,3) ie (R,G) = (1,3); so R1, G3.
- K2 = 18 % 6 = 0; but 0 is considered as 6. so K2 = 6; 6th vertex = (3,3) so (D,N) = (3,3); ie D3, N3.
- Arohanam = S, R1, G3, M2, P, D3, N3, S. Avarohanam = S, N3, D3, P, M2, G3, R1, S.

Appendix

This section, apart from giving a fair introduction, explains some basics of Indian classical music and certain terminologies concerning it. Certain questions regarding the algorithm given above are also answered in this section.

History
The Origins of Indian classical music can be traced back to the time of the Vedas. The Vedas are four in number viz. Rig, Sama, Yajur and Atharva Veda. The Yajur Veda talks about the accompaniment of the veena for certain recitations. The Sama Veda talks in detail about the seven basic notes - the swara.
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.
Notes; Swaras and Raagas
Just like a language is composed of its alphabet, classical music is composed of notes. There are seven basic notes (just like western classical music) and each of them is called a Swara.
The notes are: Sa, Ri, Ga, Ma, Pa, Da, Ni
pretty much like Do, Re, Mi, Fa, So, La, Ti in the western world !!
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.
Parent Raagas - Melakartha
A Raaga or a tone is a string of Swaras. In other words a combination of the Swaras make a Raaga. There are 72 parent Raagas, combinations and mutations of which result in all known Raagas in the world. These 72 parent Raagas are called Melakartha Raagas.
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:
S, R1, G1, M1, P, D1, N1, S
is one Arohanam and
S, R2, G2, M1, P, D1, N1, S
is another.
Since 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.
The KaTaPaYadi Number
In quite the same light as the ascii coding scheme of modern computer science, the KaTaPaYadi scheme is a mapping of letters in an alphabet to numbers. But unlike ascii, KaTaPaYadi encodes only consonants of the sanskrit alphabet. All sanskrit consonants correspond to a number between 0-9.
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.
The Katapayadi Number of a Melakartha Raaga
take the name of the Melakartha raaga
for eg: DdheeRa Shankarabaranam. (is a melakartha raag)
1. Take the first two syllables as in sanskrit/hindi. in our eg. it is Ddhee and Ra.
2. strip away the vowels, to get only the consonants. in our eg. it becomes Ddha and Ra.
3. Now get the katpayadi number of Ddha and Ra. Ddha is 9. and Ra is 2. if they are N1 and N2 respectively. then write them down as N1N2. and now invert them to get N2N1.
  in our eg., the number is 92 and on inverting, it becomes 29. So the Katapayadi number of DdheeRa Shankarabarnam is 29. (btw, this is the earliest hashing function !!)

The Sanskrit Consonants

       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)

Miscellaneous Information
This sub section answers certain questions that pertain to the algorithm.
- Why does the number 6 appear, in the calculation of k1 and k2 ?
  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.
- Why use the condition "if K > 36, then subtract 36 from it" ?
  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.
- Why arrange them in a 3 x 3 matrix ?
  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..)
- Why arrange them in an upper triangular matrix ?
  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)

Hoping this was informative, and has helped you in anyway. - Ajay Sathyanath