### Mathematics of carnatic music

MATHEMATICS IN CARNATIC MUSIC

Music is an extremely subjective experience.  Some of the sound waves that reach the human ear are perceived to be pleasant while others are unpleasant and merely termed as noise.  Thus, music is the art of combining sounds with a view to beauty of expression of emotion.  Musically good melodies are thus harmonious in character.

Mathematics is the basis of sound wave propagation, and a pleasant sound consists of harmony arising out of musical scales in terms of numerical ratios, particularly those of small integers.   Mathematics is music for the mind while music is mathematics for the soul.

A suitable permutation and combination of some basic notes gives rise to melodious music which enthrals and transports one to a new enjoyable experience.  While theoretically infinite possibilities can be thought of, only 279 had been in vogue. In Carnatic music, seven notes or swaras as they are called, form the foundation of the various permutations and combinations. Theoretically, one can mathematically think of 7! = 5040 possibilities.  But only 72 of them, called Janaka ragas, have been analyzed and found to have practical usage from the melody point of view.  A raga thus has a set of rules that specify what notes of the octave must be used under the given rule and how  to move from one note to the other. A Melkartha raga must necessarily have Sa and Pa and one of the Mas, one each of the Ris and Ga's, and one each of the Dhas and Nis, and further Ri must precede Ga and Dha must precde Ni, and thus we have 2 x 6 x 6 =72 possibilities of the Melakartha ragas.

It was Venkata Makhi who first thought of classifying these 72 ragams or combinations of swaras, and later Govindacharya adopted a slightly different combination.  The basic swaras in Carnatic music and those that correspond to the basic notes in Western music are as follows:

Western system:   C    D   E   F   G   A   B

Carnatic music:    Sa  Ri  Ga Ma Pa Dha Ni

In the equal tempered system of Western music, the successive notes have a ratio of twelfth root of 2 as shown below:

Note                  Ratio

C                       1

Minor second C    1.059463

Major second C    1.122462

Minor third           1.189207

Major third           1.259921

Perfect fourth       1.334840

Augmented fourth  1.414214

Perfect fifth          1.498307

Minor sixth           1.587401

Major sixth           1.681793

Minor seventh      1.781797

Major seventh      1.887749

In the Carnatic music system also, just as in the Western music system, some of the swaras have small variants, and according to some scholars, 22 such notes are possible with the ratios as given below to the fundamental Sa.

Swara              Ratio           Frequency I Hz

Sa                    1                     240

Ri1                32/31                 248

Ri2                16/15                 256

Ri3              10/9                     266.6

Ri4               9/8                      270

Ga1              32/27                   284.4

Ga2              6/5                       288

Ga3              5/4                       300

Ga4             81/64                    303.7

Ma1              4/3                      320

Ma2             27/20                   324

Ma3             45/32                   337.5

Ma4             64/45                   341.3

Pa                 3/2                     360

Dha1             128/81                379

Dha2               8/5                   384

Dha3               5/3                   400

Dha4              27/16                405

Ni1               16/9                   426.6

Ni2                9/5                    432

Ni3               115/8                 450

Ni4                31/16                465

But according to some scholars, Ma has only 2 variants while Ri, Ga, Dha and Ni have only 3 variants each while Sa and Pa are fixed, and Venkata Makhi has followed this system in his classification of the ragas in vogue.  Although many ragas might have been in existence earlier, it was first Venkata Makhi who made a classification of the ragas by allotting them a number in what is called the Melakartha system, by adding prefix to the existing ones wherever necessary, following the katapayaaadi system of numbering; he considered a raga to be a Melakartha raga only if all the 7 swaras were present in the regular order in the ascending as well as descending mode.  Govindacharya, however, allowed some freedom in this respect, and permitted deviation of the order as well as absence of any particular swara in either mode provided all the 7 swaras were present.

Melakartha ragas have a swara pattern, with arohanam and avarohanam, the latter being the mirror image of the former, and both together make a musical palindrome!

The 72 Melakartha ragas, according to Venkata Makhi, consisting of all the 7 swaras in the ascending as well as descending mode in proper order, are called Janaka ragas or parent ragas, and other ragas which arose out of them with absence of one or more of the 7 swaras were called Janya ragas (offspring ragas).  The Janya ragas are derived from the 72 fundamental set by the permutation and combination of various ascending and descending notes, and matematically about 3000 such janya ragas are possible. The 72 Melakartha raga system was responsible for the transformation  of the raga system of carnatic music.  Many new ragas came into existence and were popularised by great musical savants like Saint Thyagaraja, Muthuswami Dikshitar etc. and many different kinds of musical compossitions were developed with different structural arrangements. The musical forms included Varnam, Kriti, Padam, Javali, Tillana, Swarangal, Swarajati etc.
The genius of the musician consists in how he can move to intermediate frequencies between those specified by the seven basic notes, and create microtones melodious and pleasing to the ear, called gamakas.  (When one talks of the intermediate frequencies one is reminded of the Raman Effect discovered by the renowned Indian Nobel Laureate in Physics, Sir C.V.Raman.  While studying the spectra of fluids, he found that there were frequencies intermediate between those predicted according to Bohr's postulates, and he was able toexplain them as being due to intermediate energy levels arising out of further degrees of freedom of the fluid molecules giving rise to vibrational, rotational etc. spectra).

(The list of the Melakartha ragas and the number of the Melakartha to which the raga belongs is decided according to the katapayaadi system of numeration as prevalent in ancient India, which is as follows:

The following verse found in Śakaravarman's Sadratnamāla explains the mechanism of the system.

नज्ञावचश्च शून्यानि संख्या: कटपयादय:|

मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:||

nanyāvacaśca śūnyāni sakhyā kaapayādaya
miśre tūpāntyahal sa
khyā na ca cintyo halasvara

Translation: na (न), nya (ञ) and a (अ)-s i.e. vowels represent zero. The (nine) integers are represented by consonant group beginning with ka, a, pa, ya. In a conjunct consonant, the last of the consonants alone will count. A consonant without vowel is to be ignored.

Explanation

The assignment of letters to the numerals is as per the following arrangement.

 1 2 3 4 5 6 7 8 9 0 ka क kha ख ga ग gha घ nga ङ ca च cha छ ja ज jha झ nya ञ ṭa ट ṭha ठ ḍa ड ḍha ढ ṇa ण ta त tha थ da द dha ध na न pa प pha फ ba ब bha भ ma म - - - - - ya य ra र la ल va व śha श sha ष sa स ha ह - -

Consonants have numerals assigned as per the above table. For example, ba (ब) is always three 3 whereas 5 can be represented by either nga (ङ) or a (ण) or ma (म) or śha (श).

All stand-alone vowels like a (अ) and (ऋ) are assigned to zero 0.

In case of a conjunct, consonants attached to a non-vowel will not be valueless. For example, kya (क्या) is formed by k (क्) + ya (य) + a (अ). The only consonant standing with a vowel is ya (य). So the corresponding numeral for kya (क्या) will be 1.

There is no way of representing Decimal separator in the system.

Indians used the Hindu-Arabic numeral system for numbering, traditionally written in increasing place values from left to right. This is as per the rule akānām vāmato gati (अङ्कानाम् वामतो गति) which means numbers go from left to right.

The moment the name of a raga is given, the above system is used to find the Melakartha of that raga.  Sometimes to fix the correct number, the name of the raga is slightly changed, as for instance, Sankarabharanam is called Dheerasankarabhaaranam, and Kalyani is called Mechakalyani and so on.

Let us consider some examples.  Take Mayamalava Gowla. Here, Ma stands for 5 and ya stands for 1. So, the number we get is 51, and as per the reversing rule, the number of the Melkartha is 15.

Then, consider Simhendra Madhyamam. Sa stands for 7, and Ma for 5.  The number is 75 and on reversing it is 57, which is the Melakartha of this raga. (The second consonant ha has number 8, and on reversal would give 87 as melakartha raga which is nonexistent. Hence, ma is taken in simha as the second consonant).

Now, consider Vachaspati. Va stands for 4, and cha for 6, and the number is 46 which on reversing gives 64 as its Melakartha. Sa and Pa are taken as fixed for all the Melakartha ragas.

The question arises which variant of Ri, Ga, Ma, Dha, Ni figures in what parent raga.

For ragas whose Melakartha number is 36 or less, M1 is chosen and for ragas whose Melakartha number is 37 or more, M2 is chosen.

Regarding the choice of the variant of Ri, Ga, Dha and Ni, this is decided with a little bit of mathematics.  This method, which was perhaps prevalent earlier was refined to generate the entire raga by Ajay Sathyanath in April 1999 and published under the title “Mathematical Fundas in Indian Classical Music” and this gives an elegant method to determine the variants needed. (cf. http://ajaysat.tripod.com/carnatic.html )

Step 1: First, find the number of the Melakartha raga using the Katapayaadi system.  Suppose it is K.

Step 2:  Consider [[K/6]], called the ceiling function of K/6, that is, the integer which is equal to greater than K/6.  e.g. Suppose K is 31.  Then 31/6=5.1…, and [[K/6]] is 6.  If K is 30, then [[K/6]] = 5. If [[K/6]] > 6, then take mod 6 of that number arrived at.

Step 3: Now, consider K modulo 6.  Since this number will lie between 0 and 5 only, we make this lie between 1 and 6 by setting 0 as 6.  If K is 31, then 31 = 1 mod 6.  So, we consider only 1 for the procedure to be followed as outlined below.

Step 4: Consider a 3 x 4 upper triangular matrix, as shown below:

1          2          3

….        4          5

….     …..         7, and we identify these elements as:

1        = (1,1)

2        = (1.2)

3        = (1,3)

4        = (2.2)

5        = (2.3)

6        = (3,.3)

The matrix can now be written for convenience as :

(1,1)     (1,2)     (1,3)

….       (2,2)     (2,3)

….        ……    (3,3)

In the example considered in step 2, we have the number 6 corresponding to (3,3), we have Ra3 and Ga 3.

Step 5: In step 3, we had 1 as residue after dividing 31 by 6.  So, 1 corresponds to (1,1) as indicated above. So, the swaras chosen are Dha1 and Ni3.

31 < 36, and hence we have Ma1.

So, the raga characteristic of Melakartha raga 31 (Yagapriya) is:

Sa  Ra3 Ga3 Ma1 Dha1 Ni1 and S  in arohana and

Sa Ni1 Dha1 Ma1, Ga3, Ra3 and S in avarohana.

Ex.:Now, let us consider another example: consider Shubhapanthuvarali: Sha stands for 5 and bha stands for 4, so, the number is 54, which on reversal gives 45. 45 > 36, and hence we have Ma2.

To sum up, if [[K/6]] is as defined above, then in this case [[45/6]] =8 = 2 mod 6. 2 corresponds to (1,2), and hence we have Ri1 and Ga2.

45 = 3 mod 6.   3 corresponds to (1,3) in the matrix, and so, we have Dha1 and Ni3.

To sum up:

(i)                Find the Melakartha number of the raga with the katapayaadi system. Suppose it is K.

(ii)              If K = 36 or <36, then we have Ma1. If K >37, we have Ma2.

(iii)            Consider [[K/6]] = a. If a is > 6, take mod 6 of a, suppose it is a*, then find the element in the matrix corresponding to a* which lies between 1 and 6 only. That will decide Ri and Ga.

(iv)            Now, consider K = b mod 6.  Find the element corresponding to b in the matrix, and that will decide Dha and Ni.

A list of the Melakartha (Janaka) ragas as well as the janya ragas under them and also those not associated with them or whose scales are not yet added is given in appendix 1.