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 35talas again has five varieties because of the five different types of gatis. This
gives a total of 175talas. Apart from this, there is also a set of 108 Talas using
certain additional angas.
The Pallavi, of a
RagamTanamPallavi 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 72Melakartas with their serial numbers.
