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Style as Information in Karnatic Music
Journal of Music Theory (Yale School of Music), 8:2, Winter 1964, pp. 267-72.
Gift Siromoney and  K.R. Rajagopalan

Music can be considered to be a communication system, and information theory has been applied to it by Youngblood*1, Coons*2 and others. Youngblood has explored the usefulness of information theory as a tool for identifying musical styles. He chose melody as stylistic aspect and compared certain works of Schubert, Mendelssohn and Schumann. In this paper we propose to compare the works of three well known Karnatic musicians.

In the Karnatic system of music, seven main notes (svaras) are recognized within the octave and the names of the notes are abbreviated in an early system of sol-fa (sa, ri, ga, ma, pa, dha, ni). The octave is further divided into twenty-two sruthis, popularly called "quartertones". The "quartertones" and "semitones" are not tuned to equal temperament as done in Europe*3.

In this paper, we shall however represent the notes of the Karnatic system by their nearest notes in the western scale.

The basis of Karnatic music is the raga or melody-type*4 and each raga is defined by its particular scale and its melodic pattern. Here we shall be mainly concerned with two common ragas namely Sankarabaranam and Madhyamavathi. The former corresponds to the western major mode with a slight sharpening of the sixth and the latter is a beautiful pentatonic raga.

The system of tala represents the rhythm in Karnatic music. Each tala or time-measure*5 has a fixed number of beats (matras) arranged in characteristic groupings. We shall be mainly concerned here with Athi tala, which corresponds to a four-bar phrase in common time.

Shannon's entropy*6 (H) was chosen by Youngblood. Further advances*7 have made it possible to work out the error in the estimate of H due to chance. The error can be calculated from the variance (D) of H.

We analyse here compositions of three well-known Karnatic musicians -- Tyagaraja (1767-1847), Dikshitar (1776-1835) and Syama Sastrigal (1762-1827).*8  Though contemporaries, they have distinct styles of their own.*9 From each author two compositions were chosen and are analyzed in Table I. All these six compositions are in the raga Sankarabaranam and have the same tala, Athi tala. In Table II we analyse three more compositions, one from each of these three authors. These are in the raga Madhyamavathi and two of these (except D3) are in Athi tala. The quarter note is taken as the unit of time and the notes are given their corresponding weights for calculating the frequencies of all the notes. The distinction between the same note in different octaves is maintained and the proportions of all the different notes are tabulated. All the compositions cover approximately two full octaves.

There seems to be a common pattern running through all the compositions. The notes at the extremities (the high and low) are the least frequent and the most frequent note in all the compositions is the dominant G. In Table I the tonic C in the higher octave is more frequent than the immediately preceding note B and the following note D.

One can define a raga characteristic as a measure which is significantly different for different ragas but the same for all compositions within the same raga. One may also define a style characteristic as a measure which is significantly different between composers but the same for the compositions of a single composer.

Comparing compositions T1 and S3 belonging to different ragas we find that their corresponding entropies are not significantly different; i.e. entropy does not bring out the difference between ragas. Further even within the same raga entropies are significantly different, for example D3 and S3. Therefore entropy is not a raga characteristic. From T1 and T2 it is seen that for Tyagaraja the values of entropy are significantly different. Between these two values lie the entropies of the compositions of the other two composers in Table I. Therefore H is not a style characteristic.

The difference between the entropies of two compositions of Tyagaraja are statistically significant even though they are in the same raga and tala. But the entropies of certain compositions of Dikshitar and Sastrigal are virtually the same even though they are known to have distinct styles of their own. As far as Karnatic music is concerned there seems to be little evidence to support the view that entropy is a style characteristic.

                Raga: Sankarabaranam                                            Tala: Athi tala

              NOTES*TYAGARAJA DIKSHITAR SASTRIGAL 
Tl T2 Dl   D2    Sl

S2

.000 .000 .000 .001 .000 .000
.003 .004 .005 .003 .000 .002
.023 .029 .022 .002 .005 .014
.056 .055 .052 .058 .013 .039
.184 .142 .117 .143 .089 .129
B .092 .098 .074 .101 .066 .071
A .136 .118 .113 .146 .091 .098
G .244 .165 .194 .197 .215 .265
F .139 .130 .133 .125 .162 .153
E .072 .117 .132 .104 .109 .086
D .022 .062 .060 . 041 .067 .052
C .025 .060 .072 .047 .116 .074
.004 .016 .020 .011 .040 .009
. 000 .005 .004 . 002 .021 .008
.000 .000 .002 .000 .009 .000
Total Frequency 256 384 640 480 512 384
Unbiased estimate of H3.023 3.324 3.301 3.217 3.255 3.134
Standard error of H0.0570.0350.0330.0380.0390.050
*A dot above a note denotes the higher octave and a dot below a note denotes the lower octave.

Raga: Madhyamavathi
NOTESTYAGARAJA DIKSHITAR SASTRIGAL 
T3D3S3
.003 .000 .002
.018 .009 .010
.072 .063 .069
.135 .152 .124
Bb .161 .122 .150
G .190 .213 .169
F .171 .155 .116
D .133 .154 .164
C .078 .103 .130
b .029 .022 .042
.011 .006 .023
.000 .000 .002
Total frequency 448 694 512
Unbiased estimate of H2.9832.900 3.051
Standard error of H 0.0400.031 0.037

References:

  1. Joseph E. Youngblood, "Style as Information". In: Journal of Music Theory, II (1958) 24.
  2. Edgar Coons and David Kraehenbuehl, ''Information as a Measure of Structure in Music". In: Journal  of Music Theory, II (1958) 127. 
  3. Abraham Pandither developed the thesis that the octave was in fact divided into 24 alakus or quartertones of equal intervals in the ancient Tamil country (Karunamirtha Sagaram, Tanjore, 1917).
  4. For a detailed description of raga see Herbert A. Popley, The Music of India (Calcutta, 1950) 40.
  5. For a description of tala see Alec Robertson and Denis Stevens, eds., The Pelican History of Music Vol. I (Baltimore, 1960) 32.
  6. C.E. Shannon, "The Mathematical Theory of Communication": In: Bell System Technical Journal, 27 (1948) 379. The entropy (H) is given by the equation
    H = - pr ld pr
    where "Id" stands for the logarithm to the base 2, pr the proportion of the rth note and n the     number of notes considered.
  7. G.P. Basharin (Teoriya veroyatnostei iee Primenenya 4, 361) has proved that H is asymptotically normally distributed and that

    where N is the total frequency.
  8. As given in R. Rangaramanuja Ayyengar's Krithimani Malai (Madras, Vol.2 (1947), Vol. 4(1948), Vol. 5(1953)). The compositions are "Sundaresvaruni" (T1), "Evaraku"(T2), " Ramasamayamu"(T3), "Nagalingam"(D1) "Sadasivamu" (D2), "Dharmasamvardhini(D3), "Devimeena"(S1), "Sarojadala"(S2) and "Palinsugamatchi" (S3).
  9. P.Sambamurthy, Karnatic Music in Tamil, Book I (Madras, 1958) 53.

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