Rhythm in logarithm
SUGANTHY KRISHNAMACHARI
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Music is a hidden exercise in arithmetic that involves calculations and numbers.
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Mathematician Whitehead said there was no escape from Mathematics wherever one went. "You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves," he wrote.
The Pythagoreans classified the study of mathematics into four categories — arithmetic, music, geometry and astronomy. In the middle ages, this Quadrivium of four subjects had to be studied for a bachelor's degree. Euclid authored two books on music. Descartes wrote a book called `Compendium Musicae.' Helmholtz and Euler wrote books on music too.
Writing in the Russian Astronomical Calendar in 1919, the physicist Eichenwald spoke of the relationship between piano keys and logarithms. He observed that piano keys are arranged according to the "the logarithms of the numbers of vibrations of the appropriate sounds" the base of the logarithm being two.
In Carnatic music, the system of 22 srutis was arrived at through what are known as the cycles of fourths and fifths. The upward progression of cycle of fifths is the downward progression of fourths and vice versa. We see geometrical progression in the frequency relationship of the octaves.
There is mathematics in our pallavis too. The musician has to sing the same line in different degrees of speed, gati and so on. When a vocalist sings an intricate pallavi, the percussionist has his job cut out for him, because he has to start his tirmanams from the correct point in the avarta.
Even musical instruments bear out the mathematical influence. In the veena, the length of the stem is in a definite ratio to the circumference of the resonator. In the mridangam, the pitch of the two heads will be either identical or will be in the ratio 1:2 or 2:3.
The 72 melakartas or scales have been evolved by permutation. Modal shift of tonic otherwise called grahabedha gave rise to 56 melas, which are called murchanakaraka melas.
The notes of certain ragas (whether melakarta or janya ragas) can be taken as the tonic note or the adhara shadja. Based on this new tonic note, new ragas can be evolved by changing the scale. In some cases deriving the murchanas of ragas is nothing but modular arithmetic in action.
Arijit Mahalanabis, who studied mathematics and computer science at Pennsylvania State University, has demonstrated how it works.
Modular arithmetic
Let us suppose one wants to derive a murchana of a raga. Take the 12 basic notes S r R g G m M P d D n N, and number them from zero to eleven.
Take a raga, and represent it by an integer sequence by replacing its swaras by the corresponding integers. Here is where the modular arithmetic comes into play. While shifting Sa to, say, Ma, to each number in the integer sequence add a constant number and find the remainder when divided by 12, the number added being the shift in swaras that you are considering. Sorting the resulting integer sequence in ascending order yields the integer sequence of the murchana. You can use a fixed formula and apply it to an integer sequence to obtain the integer sequence of the murchana. Mahalnabis also gives the conditions under which his scheme works.
Dr. P. Sriram, an aerospace engineer, has given yet another way of calculating murchanas.
Each of the 12 swaras is either in the sequence of a ragam, or it is not. We denote the presence or absence of a swara in a ragam by a binary digit. So, if a swara is present, we mark its presence by 1, and if a swara is not present, we denote it by 0. Grahabedham using this binary notation is easy, because one just has to move digits from one end of the binary string to the other.
Music then is both an art and a science. Leibniz wrote, music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers. Mathematician Sylvester observed, "May not music be described as the mathematic of sense, mathematic as music of the reason?"
Referring to the fact that mathematicians lived long and healthy lives, Sylvester said this was because "the wings of the soul do not early drop off, nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life."
May the same not be said of musicians too, whose lofty art insulates them from the commonplace? This is probably why quite a few of our musicians have lived well into their eighties and some are still happily with us.
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