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Mathematical Fundas in Indian Classical Music

(April 1999)
Ajay Sathyanath

If you have any concerns or ideas regarding this page send mail to : Ajay

Contents

  1. Introduction
    Disclaimer
    Notation
  2. The Algorithm
  3. Some Worked Out Examples
  4. 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.

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.

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)

  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!!!!!!

Some Worked out Examples

  1. 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.

  2. 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.


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