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MUSIC AND MATHEMATICS
Mathematics is found abundantly in music in general, and Carnatic music in particular. We shall now give you in brief, the areas in which it is found.
Mathematics is applied in the tone system. The frequencies of two notes that have an octave difference bear the ratio 1:2. Thus, if the frequency of Shadja (the tonic note) in the middle octave is equal to n vibrations per second, then frequency of the higher Shadja would be 2n, that of the next higher would be 4n and so on. Thus the frequency relationships of the octaves proceed in geometrical progression as, 1, 2, 4, 8, 16 and so on.
The values of different notes were arrived at by the cycle of fourths and fifths.
In the concept of the six degrees of speed (Shatkalas), one finds a regular progression. Here, the length of the note gets progressively reduced from unit time in the first degree of speed to ½, ¼, 1/8, 1/16, 1/32 and so on. In other words, one note is sung to unit time in the first speed, two notes in the second speed, four notes in the third speed and so on.
The arithmetic progression 1, 2, 3, 4, 5, 6, 7, 8, etc., is seen in the frequency relationships of upper partials. The harmonics are heard when a stretched string is sounded.
Using the concept of the 12 notes, the 72 melakarta or scales have been evolved by means of permutation. Other scales and ragas have also been arrived at by the permutation or omission / addition of the 12 notes.
The swara graphs throw light on the contours of a Raga, while the Swarasthana and Srutisthana graphs throw light on the frequencies of the notes used in the Raga.
The Sapta talas give rise to the 35 talas because of the five types of Laghus. Each of these 35-talas again has five varieties because of the five different types of gatis. This gives a total of 175-talas. Apart from this, there is also a set of 108 Talas using certain additional angas.
The Pallavi, of a Ragam-Tanam-Pallavi is highly mathematical, requiring the musician to sing the same line in different degrees of speed, gati etc. It is also usually mounted with a lot of mathematical structures in the kalpanaswara.
The singing of kalpanaswara patterns, in general, requires a reasonable amount of mathematical knowledge.
The permutations of the infinite and finite varieties of Swara or Tala is called Prastara. Prastara enables one to determine the nature and structure of a phrase if the number is given and to determine the serial number if the phrase is given.
Musical / rhythmic patterns like the Yatis (e.g., Gopuccha, Damaru, Mridanga, Srotovaha etc) have specific geometrical patterns.
The Katapayadi formula, an interesting formula that combines language and mathematics, has been used to coin the nomenclature of the 72-Melakartas with their serial numbers.
