Mechanism of Guitar String Tuning: Tension, Length, and Material Properties

Question

I'm trying to understand the precise physical mechanism involved when tuning a guitar string, specifically from the perspective of the string's tension.

When we turn the tuning pegs, we intuitively know we are increasing or decreasing the tension in the string to achieve the desired frequency. However, I'm curious about some of the underlying factors:

Constant Length Assumption: Is it accurate to assume that the vibrating length of the string (between the bridge and the nut) remains perfectly constant during the tuning process? Or does the overall length of the string (including the part wrapped around the tuning peg) change, leading to a change in the vibrating length?

Role of Young's Modulus: Should we consider the Young's modulus of the string material in this scenario? As tension increases, the string will experience a slight elongation. How significant is this elongation in the context of tuning, and how does it relate to the change in frequency?

Linear Mass Density: Does the linear mass density (mass per unit length) of the string change as we increase or decrease the tension? If the string elongates slightly, would this lead to a decrease in the linear mass density? How does this potential change affect the resulting frequency?

I'm looking for a detailed explanation of the interplay between tension, length (and potential elongation), and linear mass density in determining the frequency of a vibrating guitar string during the tuning process. Any insights or relevant formulas would be greatly appreciated.

Answer 1

While these effects are real, they are very small. For example, The length of a fret board is about $2$ feet. The amplitude of a vibrating guitar string is about $0.1$ inch. For simplicity, imagine the shape of the string not curved. The center is displaced sideways $0.1$ inch. The string has two straight sections to the end points. The length of each section is

$$\sqrt{12^2 + 0.1^2} \approx 12.0004 $$

So the elongation is about $0.003$%. A $1^o$ temperature change makes a bigger difference.

Answer 2

Is it accurate to assume that the vibrating length of the string (between the bridge and the nut) remains perfectly constant during the tuning process?

Perfectly constant? No, because when you tune the strings, you are changing the stress on the neck of the instrument, an no matter what the neck is made of, it is not perfectly rigid.

On some instruments, a suitably well trained musician can hear how changing the tuning of one string influences the pitch of the others. But, I'm pretty sure the effect is small—small enough anyway, that tuning a multi-stringed instrument is not an onerous task.

Does the linear mass density (mass per unit length) of the string change?

When you tune a string up, you are stretching it—elongating it. You are pulling part of the string over the instrument's nut, and winding that part around the tuning peg. Meanwhile, at the other end of the instrument, there is not much string between the bridge and the tailpiece, not much movement happening there.

So, if you're pulling mass over the nut, that mass has to come from somewhere, right? When you tighten the string, the distance between the bridge and the nut does not change, but there must be less mass of string in between them—less "linear mass density."

How does this potential change affect the resulting frequency?

Speaking as an amateur* musician, and not as a physicist, I'm pretty sure that the musician does not care. They simply turn the peg one way or the other until they hear the note that they want. I confess, I do not know what physical law relates the mass and the tension and the length of the vibrating part of the string to its fundamental frequency.

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* Ha! That's a joke. I wish I was good enough to call myself "amateur." At least, they're polite enough not to boo me off the stage at open mic night.

Answer 3

Constant length assumption

Theoretically, the neck of the instrument becomes a tad shorter due to the string tension. This is noticeable if you put a single string on the instrument and tune it, and afterwards add the other strings and tune them too. Once you are done, the first string will be significantly flat because the neck just isn't as long as it was before anymore.

Practically, this is a non-concern. It's usually totally dwarved by the effect that string and instrument materials do not fully deform under stress immediately. Instead, they will continue to reorganize internally for a bit, increasing strain while decreasing stress. Especially nylon strings are notorious for this. As such, whenever you put on new strings, you quickly tune the instrument once (quality of the tune does not really matter) and then leave it alone for a bit while the strings lengthen. Depending on the materials, you may not get an instrument that can hold a tune until the day after. Since this effect is most pronounced in nylon strings, it's not a matter of the neck shrinking, the strings can do it without any change in length.

Role of Young's modulus

Young's modulus of the string material only has an effect in how far you need to turn the tuning peg to tension the string. It has no effect on the tuning itself.

That said, Young's modulus has an effect on how the string responds to playing loud notes: When a string is vibrating, it's a bit longer on average than when it's silent. And Young's modulus tells us by how much this extra stretching will increase the tension of the string. A string made from stiff material will tend to sound higher when played loudly, putting the string a bit out-of-tune.

Linear Mass Density

Obviously, when you tension the string, you are moving string matter out of the vibrating zone, which is of virtually constant length. Thus, you are decreasing the linear mass density, which does slightly contribute to making the string vibrate faster. However, the main reason for the faster vibration is the higher tension. And when you turn the tuning peg, you do not really care about how much of the tuning effect is due to which cause. You only measure the vibration frequency that results.

Now, you've forgotten to mention one particularly interesting consideration:

Impact of string flexibility

Most instrument strings don't respond too well to bending. Yes, they can bend, but not to arbitrarily small radius without taking damage, and not without resistance. This straightening force has the effect of increasing wave propagation speed depending on the wavelength. And this is a real problem for the sound of the string: The harmonics of the note that is played are defined by the geometry (1/2 wave on the string, 2/2 waves on the string, 3/2 waves on the string, and so on), i.e. by their wavelength. However, the higher the harmonic, the higher its wave propagation speed, and thus the higher the frequency resulting from the same wavelength. This sharpens the overtones relative to the fundamental, putting them out-of-tune. And we don't like to hear such sounds.

This is why any thick string that you will find on an instrument will be a wound string: A thin, flexible core that carries the tension of the string, and a wire that is wound in a single, long spiral around that core. This outer wire provides most of the string's mass, increasing volume and decreasing pitch, without adding too much stiffness to the string. A solid string of the same thickness would be way too stiff, and not sound well due to the overtones with too high a pitch.

Answer 4

Frequency of an n-th harmonic of a clamped string is (see String vibrations): $$ f_n=\frac{n}{2L}\sqrt{\frac{T}{\mu}}, $$ where

• $L$ is the length of the string
• $T$ is the tension
• $\mu$ is the linear density.