Say we have a string of fixed size on a violin. If I were to play it open (without pressing down on it) it would vibrate at a certain frequency. If I were to play it whilst pressing down on the middle, it would obviously vibrate at a different frequency. Is this because I have changed the length of the string that is allowed to vibrate or that I have changed the tension in the string? My question is primarily about the latter part; I know that decreasing the length of a string that is allowed to vibrate will allow for a higher frequency, but will pressing down on the middle of the string change the tension in the string significantly enough for it to affect pitch? (by "significantly enough" I mean to a similar degree that changing the effective length affects the pitch)
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$\begingroup$ The frequency (or pitch) change is primarily due to the change in length. On a guitar and similar stringed instruments, how hard you fret a string (how hard you push it down onto the fretboard) also has a noticeable effect on the pitch because it affects the string tension. I imagine you might observe the same effect on a violin, although maybe not to the same extent due to the lack of frets. $\endgroup$– PukCommented Apr 22, 2023 at 23:25
2 Answers
When you press a violin string down against the neck/fingerboard, the change in frequency is primarily due to the change in length. Your question is focussed on if the tension does anything.
You'll notice that when you press on the string, above some (pretty small) minimum force, the sound does not change if you press harder. (Side note: when I first started playing the guitar, I would tire my fingers out quickly by pressing too hard.) This is because the wooden handle provides a reaction force so that the net force on the string does not change. To change the tension on a string, you can't press it against something. If you had string that's far way from other objects. If you press down on it with some large force then you would change its tension.
If you want to know how much force you should apply. You can look up the typical tension in violin strings, which is around 15 lbs-f. Then review this post: Tension on a string
When you move the string transversely, the elongation of this one is of the second order compared to the displacement.
More precisely, for a string of half length $l$, the elongation will be $\sqrt{l^2+x^2}-l \cong x^2/2l$
If $x$ = 1 mm and $l$ = 20 cm we find that the elongation will be of the order of a thousandth of a mm.
For an elastic string, tension and elongation are proportional. So, the relative change in tension is related to the relative change in the length of the string. I'm not a musician, and I've never stretched a violin string. But even if the elongation of the string during the tuning is only a few mm, we see that the relative change in tension linked to the transverse displacement of the string will only be of the ordrer of 1/1000th. Surely negligible.
This is an essential point to establish d'Alembert's equation on the vibrating string and thus to obtain the usual formula for the frequency of vibration of the string.
Hope it can help and sorry for my poor english.
