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 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 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 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 Melakartha 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 The following verse found in नज्ञावचश्च शून्यानि संख्या: कटपयादय:|
The assignment of letters to the numerals is as per the following arrangement.
Consonants have numerals assigned as per the above table. For example, ba (ब) is always three 3 whereas 5 can be represented by either All stand-alone vowels like In case of a conjunct, consonants attached to a non-vowel will not be valueless. For example, 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 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 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 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 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 Sa Ni1 Dha1 Ma1, Ga3, Ra3 and S in
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 (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 ( |