Guitar String: Newsletter Challenge (03/14/06)

Higher tension is the reason guitar strings as well as rubber bands vibrate at higher frequencies when plucked. The increase in tension of an already tight guitar string results in a noticeably (by humans, that is) higher pitch. A rubber band stretched between thumb and finger vibrates at a low pitch, hence under low tension. A slight increase in length changes the tension very little, because of the rubber band's elasticity, and the result is a very small change in pitch. Ponder this: If a guitar string and a rubber band are put under equal tensions, will they vibrate at equal frequencies?

"If a guitar string and a rubber band are put under equal tensions, will they vibrate at equal frequencies?"

No, because the mass per unit lengths are not the same.

" . . . the mass per unit lengths are not the same." But they could be. e.g. A small diameter guitar string and fat rubber band could easily have the same mass per unit length, but certainly not the same density.

The density is not important. The only variables in the equation are langth, density per unit length, and tension.

f= (1/2L) * (SQRT(T/m),

where L = length (cm), T = tension in Dynes, and M = mass per unit length, in grams.

The elasticity of the rubber band keeps the tension from increasing significantly as the length is decreased. Frequency only goes up as the limits of elasticity are reached, just short of snapping the rubber band.

Well, I didn't say that density was important, now, did I? :-)

YOU asked the question. If you knew the answer,why did you ask?

Well, it fit right in with "Challenge" questions posted by Chris Leonard. I'll bet he knows the answers beforehand too.

Guitar strings are at a much higher tension than the rubber band. At high tension even small changes in tension alter the vibration resonance of the "string". At lower tensions greater percentage of tension changes are required to alter the resonance frequency (see the first formula below and it will become obvious as to why).

The first thing that determines the vibrational pitch of the string is the length of the string and its tension.

Another factor is string mass. The high E string on a guitar is near 0.010" in diameter versus a low E which is usually larger than 0.050, if memory serves from the last set I installed. The tension, string length, and the mass of the string determine its resonance frequency of vibration.

The velocity at which a string vibrates is:

V = SQRT ( Tension / (Mass / unit length ))

The fundamental frequency of vibration is:

F1 = V / 2L

Where 2L is twice the length of the string.

Simply working some examples of the above equations will reveal what the young man has observed.

Oh, also forgot to add, I am not sure, but the length of the rubber band may also be changing. It isn't stated if the rubber band's length is static or if it lengthens as it is tensioned in his hand. If so, then that will further offset the effect of increasing the pitch.

The tension in the steel string is far higher and has very small elasticity and flexion, meaning the guitar string will vibrate far more if plucked the same amount/distance as a rubber band. That's why the frequency is higher. If the string bent more, its frequency (rotations of the string) would be far low 'cause the material would allow it to travel farther. Am I right on this one? i dunno....

you need to ask questions about material properties of guitar string versus rubber. Every body or length of string or rod, etc. has its own natural frequency and its elastic limit. Pitch or frequency is a function of length. When excited, the object oscillates at 'x' times per second, the higher this figure, the higher the pitch. When you tighten the guitar string, you are in fact reducing its oscillating length, thereby increasing its natural frequency. This reduction in oscillating length happens due to the property of the alloy which is high tensile and resisting its increase in length. Notice at higher notes, the up and down vibration of the strings is lower. The rubber band on the other hand does not resists its increase in length - it expands to its new length with comparative ease. Therefore, there is no overall reduction in length leading to its natural frequency being the same. Rubber has a much higher elastic limit compared to the high tensile alloy steel in guitars. If you need to go further, refer to Hooke's Law and Young's Modulus of elasticity that explains material behaviour within its elastic region. I leave it to you to interpret these terms intelligbly to your son (strike the right note, etc..)

I think you are close RiHawaiian, but I believe that the stiffness of the guitar string is very large in comparison with the rubber band. Essentially both are springs where F=kx. The k for the guitar string is orders of magnitude higher than the rubber band. A small change in length (x) is going to give a larger change in tension in the guitar string than for the rubber band, and hence a larger change in the vibration frequency which is driven by tension. The mass per unit length will not change significantly for small changes in length, so that is essentially irrelevant.

There could be one more reason. Metals have very strong electrostatic forces and when under tension (assuming not exceeding the elasticity limit [refer Hooke's Law]), the atoms in the string do not change position that easily. If the elasticity limit is exceeded, the sound will be different. Although rubber is a polymer, when in tension, the macromolecules slide against one another and tend to deform. As a result, rubber will not provide the same pitch as the metal string when plucked. Of course, other reasons illustrated above are the other factors.

Here are a couple of effects to consider. First, the relationship between deformation and tension is drastically different since rubber is non-linear in this respect and assuming you do not exceed Young's modulus the guitar string does not experience a significant change in cross sectional area as the rubber band will. When the rubber band stretches more it becomes thinner. So, higher tension but less mass and the two tend to counter each other.

Rubber has a non-linear modulus and relatively low at low strains therefore it has little relative increase in tension with low strains (which is presumably what's bening dealt with here. Steel on the other hand responds linearly (relatively) and has a high modulus therefore any straining (tightening) causes a sharp increase in tension, therefore a noticeable pitch change. Ultimately you can strain the rubber enough to fully align its molecules at which point the modulus increased sharply and you may start to notice a pitch change.

When you adjust a Guitar string, you pull some of the material over the bridge to make it tighter, make it's vibrating mass lighter, and you maintain the length of the string. When you stretch a rubber band between two fingers, you make it tighter, keep the vibrating mass the same, and you increase it's length quite substantially. The son's problem is that he's not stretching the rubberband the same way he stretches the guitar string. The increase in tension is cancelled by the increase in length. It's interesting that with rubber as the vibrating material band these two variables cancel each other evenly. If he built a sturdy little bridge with a guitar style tension knob and pulled the rubber band over it the pitch would rise properly. ---Chris Cieslewicz

Bingo.

The difference, that counts in this example, between the stretched rubber band and the stretched guitar string is the modulus -- the ratio of stress to strain. If you elongate a rubber band between the thumb and forefinger, you don't have enough range of motion (strain) to increase the tensile stress by a significant multiplier because the rubber is low modulus (low stress to strain ratio). On the other hand, a very small change in length (very small strain) will multiply tensile stress in the steel or even nylon string by a large multiplier because the guitar string is high modulus (high stress to strain ratio). If you change the tensile stress of a plucked member by a larger multiplier, you change the resonant frequency (pitch) by a larger multiplier.

I would say because you couldn't even remotely begin to put the same torque or tension on the rubber band that is on the guitar string

If the rubber band was thick enough, yes you could.

But that wasn't the question .it was between two fingers. so there is no way of getting a rubber band taunt enough or anything else no matter how thick it is or what it is made of.

Interesting question, never thought about it before. The vibration frequency is a function of both the length and the tension. If you shorten a guitar string by touching the middle, it vibrates at twice the frequency (1 octave higher). Or if you increase the tension with the length constant with the tuning peg, the pitch increases. With the rubber band you are increasing both the tension and length. The tension increases proportional to length (hooke's law) so they cancel out.

Keep on tuning

Hry, if you have to keep on tuning your guitar, then maybe you need a new set of strings!