 ## Mathematical Fundas in Indian Classical Music

(April 1999)
Ajay Sathyanath

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### Contents

1. Introduction
Disclaimer
Notation
2. The Algorithm
3. Some Worked Out Examples
4. Appendix
History
Notes; Swaras and Raagas
Parent Raagas - Melakartha
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.

#### Disclaimer:

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.
1. 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.

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)

3. 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.

4. 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.

5. 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.

6. 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!!!!!!

### 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.

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