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Interesting Findings And World Unfolding Through My Eyes.

Monday, September 3, 2007

Why is the violin so hard to play?

When you pluck a note on a guitar string, there isn't very much that can go wrong. You may not play the right note at the right time, of course, but a single note will always come out at the expected pitch, and sounding reasonably musical. When a beginner tries to play a violin, things are much more difficult. When a bow is drawn across a string, the result might be a musical note at the desired pitch, but on the other hand it might be an undesirable whistle, screech or graunch. This difference stems from a fundamental distinction between the physics of plucked and bowed strings.

Linear versus nonlinear: plucked versus bowed
A plucked string, like that on a guitar, can be described by linear systems theory. The essential feature of a linear system is that if you can find two different solutions to the governing equations, then the sum of the two is also a solution. In the context of vibration, this idea has a direct physical application.
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The first few vibration modes of a vibrating string...

A vibrating object like a stretched string has certain resonance frequencies, each associated with a particular pattern of vibration called a vibration mode. The corresponding resonance frequencies are the "fundamental" and "harmonics" of the note to which the string is tuned. If the string is set into vibration in the shape of one of these modes it will continue to vibrate in this shape at the corresponding resonance frequency, with an amplitude which gradually dies away as the energy is dissipated into sound and heat.

... and a string vibrating in all three modes at once
Now if the string is vibrated in a way that involves several of the mode shapes at once, then the principle of linearity comes into play. Each mode simply goes its own way, vibrating at its particular resonance frequency, and the total sound is the sum of the contributions from these separate modes (you can read more about adding harmonics in Music of the Primes in Issue 28). The guitar player can vary the mixture of amplitudes of the various modes, by plucking at different points on the string or using a different plectrum, but the set of resonance frequencies is always the same. In musical terms, the pitch of the note is always the same but the tonal quality can be adjusted.

A bowed string is different. A note on a violin can be sustained for as long as your bow-stroke lasts, with a steady amplitude. Although energy is being dissipated into sound and heat, somehow the bow is supplying additional energy at exactly the right rate to compensate. This is one identifying sign of a non-linear system, for which the idea of adding contributions from different vibration modes cannot be applied in the simple way described above. The theory of such systems is always more intricate, and there is scope for very complicated outcomes and chaotic behaviour (read more about chaos in Issue 26). The range of good and bad noises which can be made on a violin string are examples of these complicated outcomes. The same general comments apply equally well to other musical instrument capable of a sustained tone such as the woodwind and brass instruments.

The motion of a bowed string

The string appears to vibrate in a parabola-like shape...
So how does a violin string vibrate? This question was first answered by Hermann von Helmholtz 140 years ago. When a violin is played in a normal way to produce a conventionally acceptable sound, the string can be seen to vibrate. To the naked eye, the string appears to move back and forth in a parabola-like shape, looking rather like the first mode of free vibration of a stretched elastic string.

... but it actually moves in a V-shape.
However, upon closer inspection, Helmholtz observed that it moved in a very unexpected way: the string actually moves in a "V"-shape, i.e. the string gets divided into two straight portions which meet at a sharp corner. The fact that we see a gently curving (parabola-like) outline to the string's motion with the naked eye is because this sharp corner moves back and forth along this curve. Hence we only normally see the "envelope", or outline, of the motion of the string.

This motion, called Helmhotz motion is illustrated in this animation:

Posted by Ajay :: 9:52 AM :: 0 comments

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