Physics class in higher secondary college for the most part was spent working out long-winded derivations and equations that at the time seemed to have no practical application in our teenage lives. So when we got to ‘Simple Harmonic Motion’ and there was a reference to the “oscillations of a violin string” by way of example, I became quite excited. I eagerly suggested to our professor that I could bring my instrument to class to watch these oscillations. The class erupted in laughter, and the professor gave me such a withering look that it killed my curiosity and I dissociated this important link between music and physics for quite some time thereafter. That was a real shame, as it could have been such a wonderful learning experience for us all.

Then I learnt how to play those ‘whistling’ sounds on the violin called harmonics, and I had to revisit this concept. I understood the idea of an ‘open’ string (i.e. a taut string resonating freely between the bridge and the top nut of the instrument, unimpeded by the fingers) vibrating (or ‘oscillating’) at, and therefore producing a sound at a ‘fundamental’ sinusoidal frequency when the bow was drawn across it. And lightly touching the string at ‘nodal’ points that divided it into half, or thirds, or quarters and so on, would produce distinct sounds at multiples of this fundamental frequency.

And the fascinating thing is how mathematical ratios can be exploited to make music. If an open string is lightly touched at its midpoint, the sound produced is an octave (eight notes) above the fundamental. If this is done at a third of its length, you get a fifth (five notes) above this octave; and quarter of its length, you get a tone two octaves above the fundamental. And so on. These are the wonderful whistling pure tones that the famous jazz violinist Stéphane Grappelli used to such dazzling effect in his improvisations.

Harmonics and overtones

An open string being played is in effect ‘vibrating’ simultaneously at its fundamental frequency as well as at these other frequencies, giving it its distinctive ‘fingerprint’.

Musicians and physicists differ in their use of terminology for these sounds. Musicians use the terms harmonics and overtones interchangeably. But for a physicist, the sound produced by an open string ie the fundamental frequency, is the first harmonic. The sound created by lightly dividing the string in half is the second harmonic or the first overtone, and so on.

Back when I was a student, we had to rely on diagrams to understand these concepts but technology has made this so much easier. Here’s an idea for a science project. Take any stringed instrument, and take a video on your camera phone while plucking or bowing a string. When you play back the video clip in slow motion (or even at normal speed if you have a good-quality camera function with decent resolution on your phone) you will see the string vibrate in a typical sinusoidal pattern, with peaks and troughs and nodal points. The lower the pitch of the string, the more visible these oscillations are likely to be.

These principles govern not just vibrating strings, but apply just as well to a vibrating column of air, which means woodwind and brass instruments, panpipes, pipe organ, etc.

This brings us to the concept of ‘ringing’ notes on stringed instruments. So if for example one were to play the note A by stopping the lowermost string on the violin (G) with the first finger on the fingerboard, then you can actually see the A string begin to vibrate ‘sympathetically’, as this is the first overtone of the note being played ie the octave. And the E string, which is pitched at the same frequency as the second overtone, will also vibrate. So although just one note is being played, two other strings join in, and the whole instrument ‘rings’ or resonates sympathetically in a way that would not have been possible had one just had a solitary string stretched across the instrument. This ‘ringing’ quality is one of the secrets of tone production on stringed instruments. John Burton, cellist and professor at the University of Texas, goes so far as to say that “intonation and tone are synonymous”.

These are also considerations that composers consciously or otherwise take into account when deciding in what key their composition will be, and the ‘moods’ ascribed to different keys are in large part a result of this physical aspect of music-making.

And just to make things more interesting, (or more complicated), one can produce harmonics (‘artificial’ harmonics) on a ‘stopped’ string (ie a string shortened by a finger to produce a new ‘fundamental’) with another finger lightly touching the string further away to produce the sound.

Not only this, but each instrument has its own distinctive ‘pattern’ of overtones, governed by the physical qualities of the materials it is constructed from; which is why the same note (example concert pitch A) played on a violin will be noticeably different when it is played on a piano or a clarinet.

This is an area that fascinated thinkers from Pythagoras and Ptolemy to Saint Augustine; Copernicus to Galileo, Descartes, and Kepler; and Leonardo da Vinci to Isaac Newton; and musicians from Josquin des Prez to Gesualdo to Johann Sebastian Bach and his son Carl Philipp Emanuel among so many others. And grappling with the complexities arising from it ie how to ‘divide’ an octave into equal parts without compromising other important internal ratios is what led musicians over time to devise what in western music is known as the twelve-tone or ‘tempered’ scale. The traditional 22 shruti or microtones within an octave in Indian music address this as well.

(An edited version of this article was published on 19 April 2015 in the Navhind Times Goa India)