Guitar string tension: effect of total length

[sgb]

I suspect my question will be somewhat unique, please forgive me if it's entirely out of place. (Please also forgive my ignorance when it shows, my knowledge of physics is extremely limited.) You might wonder why I'm asking it here, rather than on a guitar forum. Let me simply say that the guitar community is infused with all manner of voodoo-like beliefs, such that it can be difficult to arrive at a straight answer on such matters. I don't expect that to be a problem here.

I'm interested to learn if the total length of a guitar string should, in theory, have any effect on its tension, i.e., it's bendability. Specifically, my question relates to headstock design. For those unfamiliar with guitars, let me quickly explain a few things.

A guitar string vibrates between two points: (1) the bridge, at the picking end; (2) the nut, at the other end.

The headstock is the material beyond the nut, where the strings are tightened by being wound around tuning pegs. (The strings are anchored at the other end, beyond the bridge.)

Headstocks generally come in two forms: (1) three tuning pegs on each side, called "3+3"; (2) all six pegs on one side, called "in-line."

In-line headstocks themselves come in two forms: (1) standard form, where the tuning pegs are on top, such that the thicker, lower-pitch strings have the least amount of length beyond the nut, since those tuning pegs are closer to the nut; (2) reverse form, where the tuning pegs are on bottom, such that the thinner, higher-pitch strings have the least amount of length beyond the nut.

And now I finally get to my question: Is there any theoretical reason to expect that the amount of string beyond the nut will have any effect on the tension (expressed as bendability) of the string between the nut and the bridge?

If the choice of standard or reverse in-line headstock should in theory have such an effect, I suspect most guitarists would choose the form which would impart greater bendability to the thinner strings and lesser bendability to the thicker ones.

I'd like to know if there's any theoretical reason to expect such an effect, even if in practice it would be too little to expect most players to feel a difference on a typical guitar. Should there be such an effect, palpable to the player, on a ridiculously atypical guitar? Suppose, for example, that the headstock is one meter long, such that there is even more string beyond the nut than between the nut and bridge. Or, for a visual unrelated to guitar, suppose a length of rope, pulled taught across two logs by two persons, such that it could be "plucked" between the logs; should the distance of the persons from the logs, and thus the length of the rope beyond the logs, have any effect on the perceived tension of the rope between the logs?

In order to raise a string of a given gauge to a desired musical pitch, it must be placed under a certain amount of tension. Some guitarists believe that the placement of the bridge and nut along the length of the string will have no effect on its bendability. Others believe that it will, and that the headstock form is therefore relevant.

So, should the total length of a guitar string (or any other wire, cable or rope, for that matter) have any effect on its perceived tension between the terminal points of its vibration?

 

[Danger]

That's a really good question, and well presented. I don't know the answer. Among many reasons, one is that I don't know what effect the bridge has on vibrational nodes. Logic tells me that vibrations should be almost entirely damped out when they hit it, but that obviously isn't the case or messing about with the strings on the neck with your other hand wouldn't do anything. The one thing that I think might be applicable is that a long string (I'm thinking on a large scale here, like a gondola cable or power line) takes a lot more force to be stretched to the same curvature than a short one does. I kinda think that such might be due to gravity rather than an intrinsic property of the string, though.
Good luck with finding an answer from someone else who actually knows about this stuff. There are a lot of them around.

 

[SteamKing]

If the string past the nut is just danglin' in the breeze, it has no effect on the sound made by the taut portion of the string.

The bridge and the nut are node points on the taut string, the points where no vibration takes place.

This article discusses briefly the physics of string instruments:

http://en.wikipedia.org/wiki/String_instrument

The frequency at which a string vibrates depends on the length between nodes, the tension, and the linear density of the string (how much a string weighs for each unit length).

High pitch strings are light and highly tensioned; low pitch strings are heavy and more loosely tensioned. In a guitar, for example, all of the strings are roughly the same length, but in a concert piano, there is much variation in the lengths of the strings, which in turn, influences the shape of the instrument.

 

[Danger]

in a concert piano, there is much variation in the lengths of the strings, which in turn, influences the shape of the instrument.

I always wondered why they look like that. Thanks.

 

[sgb]

If the string past the nut is just danglin' in the breeze, it has no effect on the sound made by the taut portion of the string.

But it isn't just danglin' in the breeze. The string is anchored somewhere behind the bridge (either at the back of the guitar, if it's a string-through-body design, or at the back of the bridge behind the saddle, which is the actual point where the vibrational length of the string begins), and past the nut at the tuning pegs. These points could in theory be any distance from each other; the scale of the guitar is determined by the distance between the bridge saddle and the nut. The string, however, is taught all along its entire length, and, presumably, under equal tension all along its length. Suppose a mile long string, with a standard guitar scale (let's split the difference of the most common scales and call it 25 inches). So we have a bridge and nut 25 inches from each other, with the strings passing over them, at slight angles to maintain pressure at those points. If the tension is constant along the length of the string, should it make any difference how much string there is beyond those points? (Let's eliminate gravity from consideration in the case of my ridiculous mile-long string.)

 

[SteamKing]

But it isn't just danglin' in the breeze. The string is anchored somewhere behind the bridge (either at the back of the guitar, if it's a string-through-body design, or at the back of the bridge behind the saddle, which is the actual point where the vibrational length of the string begins), and past the nut at the tuning pegs. These points could in theory be any distance from each other; the scale of the guitar is determined by the distance between the bridge saddle and the nut. The string, however, is taught all along its entire length, and, presumably, under equal tension all along its length. Suppose a mile long string, with a standard guitar scale (let's split the difference of the most common scales and call it 25 inches). So we have a bridge and nut 25 inches from each other, with the strings passing over them, at slight angles to maintain pressure at those points. If the tension is constant along the length of the string, should it make any difference how much string there is beyond those points? (Let's eliminate gravity from consideration in the case of my ridiculous mile-long string.)

No it shouldn't. When you pluck the guitar string, the part which vibrates and makes the sound is the length between the nodal points, that is, the bridge and where the nut takes up the slack in the string.

I think if you were to measure several different guitars, acoustic, electric, whatever, which are tuned to give the same chords, you'll find that each was made from similar gauge string and each has a similar distance between the bridge and the nut.

If you know any sound engineers, I bet they would have equipment which could tell you at what frequency a given string vibrates when it is plucked. You could round up and test several different instruments, heck, even a 2x4 with some string nailed down on it, and see how the frequency of the sound changes with string length, with excess behind the bridge, the color of the instrument, whatever variables you want to investigate. This is how you learn about something, by making an actual test. The guys who just shoot the breeze will be disappointed, though, because some of their cherished theories may not survive a scientific test. The latter is just as bitter to realize in the musical world as it is in the scientific world.

 

[Bandit127]

Consider a Floyd Rose equipped guitar with clamps on the nut (a locking nut). When the clamps are done up you can cut the string between that and the tuning pegs on the headstock with no change in string tension.

That suggests to me that any string length beyond the nut is irrelevant to string tension.

 

[Danger]

even a 2x4 with some string nailed down on it

Oh, yeah! Bring on the jug band with the washtub bass! :D

 

[sgb]

No it shouldn't. When you pluck the guitar string, the part which vibrates and makes the sound is the length between the nodal points, that is, the bridge and where the nut takes up the slack in the string.

I think if you were to measure several different guitars, acoustic, electric, whatever, which are tuned to give the same chords, you'll find that each was made from similar gauge string and each has a similar distance between the bridge and the nut.

If you know any sound engineers, I bet they would have equipment which could tell you at what frequency a given string vibrates when it is plucked. You could round up and test several different instruments, heck, even a 2x4 with some string nailed down on it, and see how the frequency of the sound changes with string length, with excess behind the bridge, the color of the instrument, whatever variables you want to investigate. This is how you learn about something, by making an actual test. The guys who just shoot the breeze will be disappointed, though, because some of their cherished theories may not survive a scientific test. The latter is just as bitter to realize in the musical world as it is in the scientific world.

What you've said is correct, but it seems that I've failed to convey what I'm trying to determine. Let me try wording it another way.

Suppose you have a 36-gauge (.036 inch) string. In one scenario, you fasten the string at two points exactly 25 inches apart, and tune it to 440 Hz. In another scenario, you fasten the string at two points 50 inches apart, but slide a bridge and nut under the string, 25 inches apart, and tune it to 440 Hz when plucked between those points. In each scenario, the string tension will presumably be the same. The question is whether the string will be more or less bendable in the second scenario.

I should have stated this before, but the string is free to slide inside its slot in the nut. This is where some guitarists get the idea, which I'm hoping can be confirmed or denied by physics, that a greater or lesser amount of string beyond the nut should have an effect on its bendability. Complicating the issue even further is the lack of consensus on whether more or less string beyond the nut should impart greater bendability! On the one hand, some players believe that a standard in-line headstock should make the thinner strings more bendable, based on the idea that the extra string beyond the nut (on a standard in-line, the tuning pegs for the thinner strings are furthest from the nut) imparts greater elasticity and allows more slide inside the nut slot. On the other hand, some players believe the opposite, that a reverse in-line headstock, having more string beyond the nut for the thicker strings, will make those strings stiffer because of the longer total length!

As you can see, these contradictory beliefs are in need of some solid physics to back up one side or the other, or to disprove the idea altogether and show that total length should have no effect on the bendability of the string between the bridge and nut.

Consider a Floyd Rose equipped guitar with clamps on the nut (a locking nut). When the clamps are done up you can cut the string between that and the tuning pegs on the headstock with no change in string tension.

That suggests to me that any string length beyond the nut is irrelevant to string tension.

Indeed, a locking nut renders the whole discussion moot, but not all guitars use a locking nut. As I hope I've made clear, I do understand that the tension will be the same along the entire length of the string when tuned to pitch between the bridge and nut. The question, as I hope I've now clarified above, is whether more or less extra string beyond the nut will make a string more or less bendable, despite the fact that the tension is the same along the whole string no matter how much extra lies beyond the nut.

 

[SteamKing]

What you've said is correct, but it seems that I've failed to convey what I'm trying to determine. Let me try wording it another way.

Suppose you have a 36-gauge (.036 inch) string. In one scenario, you fasten the string at two points exactly 25 inches apart, and tune it to 440 Hz. In another scenario, you fasten the string at two points 50 inches apart, but slide a bridge and nut under the string, 25 inches apart, and tune it to 440 Hz when plucked between those points. In each scenario, the string tension will presumably be the same. The question is whether the string will be more or less bendable in the second scenario.

I should have stated this before, but the string is free to slide inside its slot in the nut. This is where some guitarists get the idea, which I'm hoping can be confirmed or denied by physics, that a greater or lesser amount of string beyond the nut should have an effect on its bendability. Complicating the issue even further is the lack of consensus on whether more or less string beyond the nut should impart greater bendability! On the one hand, some players believe that a standard in-line headstock should make the thinner strings more bendable, based on the idea that the extra string beyond the nut (on a standard in-line, the tuning pegs for the thinner strings are furthest from the nut) imparts greater elasticity and allows more slide inside the nut slot. On the other hand, some players believe the opposite, that a reverse in-line headstock, having more string beyond the nut for the thicker strings, will make those strings stiffer because of the longer total length!

As you can see, these contradictory beliefs are in need of some solid physics to back up one side or the other, or to disprove the idea altogether and show that total length should have no effect on the bendability of the string between the bridge and nut.
Indeed, a locking nut renders the whole discussion moot, but not all guitars use a locking nut. As I hope I've made clear, I do understand that the tension will be the same along the entire length of the string when tuned to pitch between the bridge and nut. The question, as I hope I've now clarified above, is whether more or less extra string beyond the nut will make a string more or less bendable, despite the fact that the tension is the same along the whole string no matter how much extra lies beyond the nut.

I'm afraid you've lost me when you talk about the 'bendability' of the string. I'm not a musician, so I'm not sure what phenomenon you are trying to describe. All I know is if the string is a certain length, a certain size and at a certain tension, when you pluck it, you should get a certain frequency of vibration. That's what physics tells you about how the string responds.

 

[Bandit127]

I'm afraid you've lost me when you talk about the 'bendability' of the string. I'm not a musician, so I'm not sure what phenomenon you are trying to describe. All I know is if the string is a certain length, a certain size and at a certain tension, when you pluck it, you should get a certain frequency of vibration. That's what physics tells you about how the string responds.

I am a guitarist and I do understand what happens when you bend a string. You increase the tension and therefore the frequency.

However I do not understand "bendability" and I suspect it is entirely subjective anyway. The OP should define bendability in better terms for a physics answer.

 

[sgb]

Agreed, "bendability" is a poor word, but I could think of no other. Let me try another visual.

Imagine you're standing on the neck of a giant guitar. If you pull on a string, you will clearly see it sliding inside its slot in the nut. The question is whether the amount of string beyond the nut should have any impact on the ease of pulling the string. If the guitar had a locking nut (and if we eliminated any potential movement at the bridge saddles from the equation), then clearly all we'd be doing is stretching the string along its scale length. But with a standard nut, the string is also moving inside its slot, and is stretching along a greater total length. Should this alter the ease of pulling the string (its "bendability")?

As I said before, I'd like to know the theoretical answer to this even if in practice it is unlikely to be felt by the player on a typical guitar. Suppose, if you like, a ridiculously long string beyond the nut; would it have any effect?

Thanks everyone for staying with me so far.

 

[billy_joule]

This is a basic experiment you can do on a lock nut equipped guitar. Bend with the lock not off, then again with it locked down, I think the effect is definitely noticeable.
I found the extra string length would require a greater bend distance to reach a certain pitch.
I concluded it was because you are not only increasing the tension in the vibrating length but also the nut-tuner distance.
Look at your (non locking) nut while bending, you can see the string sliding through the nut while bending - The string is elastic and the nut-tuner length is undergoing extension.
This effect is even more pronounce on certain jazz guitars where the tail piece is some distance from the bridge - One of my guitars (epiphone Joe Pass) has 23cm of non vibrating sting on the G & D strings that undergoes extension while bending.

 

[sgb]

I found the extra string length would require a greater bend distance to reach a certain pitch.

Interesting. So, is it your conclusion that the reverse in-line headstock is preferable to the standard in-line, if the player desires that the treble strings be more "bendable" and the bass strings be less so?

Speaking of experiments, I have an idea for one. It requires: (1) a guitar with an in-line headstock; (2) a firearm trigger-pull scale.

1. String the guitar with only two strings, of the same gauge, at the two outer positions (where the two E strings normally go), tuned to the same pitch.

2. With the scale positioned at the center of a string (12th fret), pull it across the neck to the opposite edge and record the reading on the scale.

3. Repeat with the other string.

If the readings on the scale are different, then it does make a difference how much extra string lies beyond the nut.

Unfortunately I have access to neither of the requirements for the experiment. Can anyone tell me if there is any theoretical reason why the readings for the two strings should or should not be different?

 

[Dr.D]

To speak about "bendability" sounds like you are talking about the bending stiffnress (EI) for the strings. I think all of our theoretical models for vibrating strings assume infinite flexibility.

Another problem with our models is that they assume infinite stiffness for the support structure. In reality, the guitar body and neck have finite stiffness, so the models do not fit in fine detail.

 

[jh0]

I'm sorry to join in so late, but I think what "bendability" is. But in my opinion what is important for the guitarist when he's bending the string is to reach the note he wants to play. It is important to remark this, because in my opinion if there is a lot of string behind the nut (and/or the bridge), it will be easier to bend the string if we think in terms of distance, but the string tension will not rise so easily as with a locking nut or a shorter length of string behind the nut. I believe the effort and the deflection (in terms of length) of the string will be less with a locking nut.
Let's assume the string is attached (locked) at both ends as in the following drawing.http://imageshack.com/a/img674/2648/QHpR67.png
In order to reach the desired note, the guitarist has to bend the string the distance d, so the string is deformed and tension is increased from T to T'. The force he exerts on the string is f.

Now let's make the thought experiment proposed by sgb and think of a guitar with a distance between the tuners and the nut several times the scale length. If we pull the string sideways at the middle of the scale length we want the tension to increase accordingly, from initial tension T to the target tension T'. But as there is a lot of string length to stretch, we will have to deform it (increase its length) more and the string will travel a distance D, in order to reach the same tension T'.

http://imageshack.com/a/img901/7676/1YQz61.png

As D>d, the angle between the two forces T' will be smaller, so the force F that the guitarist has to exert on the string will be considerably higher.
So in this case it will be more difficult to reach the note, the guitarist has to bend further the string and he will need more force.

In my opinion bending should be easier with guitars with locking nuts or with short "free" lengths of string behind the nut and saddle. But as these distances are comparatively small against the scale length the difference will not be much (at least with common guitar designs).

 

[aaddog]

I've been really interested in this question for a few years now, and I'm glad I found this forum post!

The reason I was wondering this is because Jimi Hendrix played a Stratocaster guitar strung backwards. Essentially this means that the strings that get bent more often in blues music (the high E, B and G strings) had LESS string length behind the nut.

According to the ideas described by billy_joule and jh0, this would mean that it would be easier for him to bend the higher 3 strings than on a typical Stratocaster. This, along with the fact that he usually tuned a semitone lower than E standard, might have facilitated his incredible skill and nuance when playing his bluesy bent-note melodies.

In order to gain some practical perspective on this subject, I ordered 2 (essentially) identical necks from warmoth.com. One with a standard headstock, and one with a reverse headstock. In a completely un-scientific study I did a month ago, I compared the "feel" of the two. Personally, I could not feel if the reverse headstock was more "bendable" on the high strings.

In my opinion the difference in string length behind the nut may or may not have helped Jimi's bending, but the fact that he tuned to E flat standard definitely did. As far as my guitar playing is concerned, my conclusion is this: The marginal difference in performance does not outweigh the utter pain in the butt it is to string up and tune a guitar with a reversed headstock.

 

[Chrono G. Xay]

Long story short, sgb, for a couple of years I have been studying some of the equations behind what you asking about, and yes, I believe quite firmly that it does.

If you haven't already, I would recommend looking up the equations for string tension, as well as 'Modulus of Elasticity' and 'Poisson's Ratio'. I have been attempting to write an equation which would allow a person to know the distance they would have to bend any plain string across the same fret to result in the same bend interval, and from there, if they wish, narrow down which plain string gauges they will need soas to better 'set' that bending distance, thereby making it more predictable from string to string, and therefore the learning curve of the instrument that much more steep.

(It would, of course, require a number of measurements of the guitar itself, such as scale length, distance from bridge/tremolo-block to tuner of each string, the string height of each at the 12th fret, the nut-tuner breakover angle of each, each string's diameter, etc.) As a side note, apparently D'Addario (if not also other string manufacturers) use wound strings with a hexagonal core, as the edges of the hexagon make it much each for the wrapping wire to 'bind' to the core wire, so by knowing the area of a hexagon... you could arguably calculate, and by extension better 'set', the bending distance of wound strings!

I would also recommend reading about 'progressive tension'. D'Addario has 'balanced tension' string sets available (and the idea of 'balanced tension' had been out long before they chose to capitalize on it), but some folks felt their bass strings still sounded a little muddy, hence 'progressive tension' came about. (I've tried writing D'Addario about it, but they haven't received enough requests to make seem it worthwhile to them.)

However, with progressive tension comes even MORE progressive friction (meaning progressively worse and worse tuning stability from string to string, which is why a person would be forced to wind more string around the string post (be it up or down). This could be accomplished a little bit easier using string posts whose heights are intentionally 'staggered', and going from there.

 

[Randy Beikmann]

I happened to have done a lot of research in this area, although it was on belt drives in cars. The physics of each is about the same.
Only three things matters for the tuning of a guitar string 1) the amount of tension on the string, 2) the mass per unit length of the string, and 3) the length of the free span of the string. The reason for this is that the bending stiffness of the string is very low - the tension is what makes the string come back to center after you pluck it.
As it vibrates, the string produces forces on the supports at both ends, perpendicular to the supports. These forces vibrate the sounding board of the guitar, which then acts like a speaker and produces the sound.

 

[JBA]

I know nothing about this subject and before scanning through these posts, quantum physics was #1 on my list of complex subjects but now I think I'll drop it to a #3 or so.

 

[Randy Beikmann]

I know nothing about this subject and before scanning through these posts, quantum physics was #1 on my list of complex subjects but now I think I'll drop it to a #3 or so.

That is too funny! I understand you completely.

Here is a quick explanation, devoid of physics jargon:
1) When you pluck it a distance x away from center (away from a straight line), the tension in the string is pulling in a direction towards center.
2) When you release it, the string moves back towards center, speeding up all the while.
3) Moving fast as it reaches center, it keeps moving. The string is straight, and the tension does not speed it up or slow it down.
4) It keeps going until it gets as far to that side as where it started (-x).
5) It repeats 2-4, again and again, with only friction gradually slowing down the motion.

No more to it. The stretching of the string plays no appreciable role. Only its angle of deflection.

 

[Chrono G. Xay]

I think the friction you refer to is also called "drag", and am pretty sure this has *something* to do with the 'damping constant', which, if calculated, could be used to predict the 'sustain' of a string with pre-initial "scale length" L_s experiencing an initial transverse displacement 'd' at a point 'x' along its length. (Preferably at \frac{L_s}{2})

 

[billy_joule]

No more to it. The stretching of the string plays no appreciable role. Only its angle of deflection.

'bending' on a guitar is using your fretting fingers to apply force to increase the overall path of the string to increase tension and pitch.

https://en.wikipedia.org/wiki/String_bending

Factors influencing string bending[edit]
There are numerous mechanical and acoustic properties which heavily influence the resultant pitch from a string bend. Analysis of the physics of string bending [4] suggests that the resultant pitch of a string bend is given by

where L is the length of the vibrating element, T is the tension of the string prior to bending,

is the bend angle, E is the Young's Modulus of the string material, A is the string cross sectional area and

is the linear density of the string material.
Thus, the pitch is not only dependent on the bend angle, but on material properties of the string such as Young's modulus; this may be interpreted as a measure of the stiffness of the string. The force required to bend a string to a given angle

is given by

It is important to note that the resultant pitch from string bending is not linearly correlated with the bending angle, and so a player's experience and intuition is important for accurate pitch modulation.

Wiki's analysis ignores any effect the unplayed portion of the string has so is no help to the discussion at hand, post #16 suggests the length of any unplayed portion does have an effect. My experience agrees with that.

Either way, guitarists only partially rely on force or distance to get to the correct pitch - they mainly rely on a feedback loop where force is increases until the desired pitch is heard.

 

[Chrono G. Xay]

"[F]eedback loop"...

https://en.m.wikipedia.org/wiki/Limit_of_a_function

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html

 

[billy_joule]

"[F]eedback loop"...

https://en.m.wikipedia.org/wiki/Limit_of_a_function

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html

Is this directed at me? Sorry I can't see the relevance.

 

[Hornbein]

Either way, guitarists only partially rely on force or distance to get to the correct pitch - they mainly rely on a feedback loop where force is increases until the desired pitch is heard.

Maybe, but skilled players often bend the string before plucking it. They have to do that by "muscle memory."

 

[Nidum]

(1)

The plucking action for a classical guitar is complex .

Pluck outwards and/or upwards .

Straight across or at an angle

Use of soft edge or hard edge release from nail or pad on finger .

Variation in rate of release - quick snatch to slow draw off .

Variation in amount of string deflection before release .

(2)

One type of vibrato can be generated by applying variable down load or side to side motion on the dead side of a fret .

 

[Chrono G. Xay]

It was directed at you, billy_joule. Like Hornbein corroborated:

Maybe, but skilled players often bend the string before plucking it. They have to do that by "muscle memory."

This is 'pre-bending', and by default requires a limit function moving toward a particular displacement along the fret or particular magnitude of displacing force.

 

[Randy Beikmann]

I think what you're doing by pre-bending, letting the string slide off your fingers, or plucking at the end vs. towards the middle, is changing which harmonics of the string are dominant once the string is released. You excite different string modes. This will definitely change the sound. But as the equation above would show with values plugged in, the vibration of the main span is still affected very little by anything but length density, tension, and spring length.
But the whole idea of a guitar isn't to make a string vibrate, but to make sound. This involves transferring the vibration energy from the main string span through the end supports and the un-played portion of the string, into the main body of the guitar. That's how these elements would "come into play." As you feed the energy into the body it draws energy from the string, as would other forms of drag.

 

[billy_joule]

It was directed at you, billy_joule. Like Hornbein corroborated:

This is 'pre-bending', and by default requires a limit function moving toward a particular displacement along the fret or particular magnitude of displacing force.

I'm not sure if a limit is appropriate here. Human pitch perception is limited* so what most consider an 'in tune bend' is just somewhere close enough to the target pitch. Rather than a limit where Δƒ→0 (ƒ being frequency).

Maybe, but skilled players often bend the string before plucking it. They have to do that by "muscle memory."

Yes and pre-bends are generally not as in tune as normal bends. Controlling pitch with muscle memory alone leads to poor results, More so with instruments where the pitch must be controlled at all time like the voice, or the slide guitar but also to regular guitar during bending. eg if a singer cannot hear themselves they must rely on muscle memory alone and their pitch control often suffers dramatically.
this is why good monitoring is so important and has lead to increasingly complex stage monitoring set ups, like in ear monitors where each performer receives a personalised mix through custom fitted isolating ear buds (they often add crowd noise back into add back audience engagement).

I think what you're doing by pre-bending, letting the string slide off your fingers, or plucking at the end vs. towards the middle, is changing which harmonics of the string are dominant once the string is released. You excite different string modes. This will definitely change the sound. But as the equation above would show with values plugged in, the vibration of the main span is still affected very little by anything but length density, tension, and spring length.

Prebending and all other bending most definitely changes the tension and therefore the pitch, not just the harmonic content. Bends up to a tone and a half are common;

(Start at 1.07 for some examples)


*10 cents, or a tenth of a tone according to the study quoted here:
https://en.wikipedia.org/wiki/Just-noticeable_difference#Music_production_applications

 

[Hornbein]

Controlling pitch with muscle memory alone leads to poor results

Tell that to Jimi Hendrix, BB King, Eric Clapton, etc. etc.

 

[billy_joule]

Tell that to Jimi Hendrix, BB King, Eric Clapton, etc. etc.

Surely they would agree. Maybe that's why pre-bends are relatively rare in blues? The fine pitch control required to reach the blue notes (notes that are between the pitches of the 12 equal tempered notes) makes accurate pre-bends difficult as the usual means of pitch control cannot be used.

Pitch control for all musicians of non-fixed pitch instruments relies primarily on their ability to discern pitch.
If a musician cannot recognise when they are in tune then they cannot develop pitch control muscle memory in the first place. Muscle memory pitch control will never be as accurate when the aural feedback loop that was required to develop it is removed. As mentioned, this is a driver for increasing the quality of the feedback signal through better monitoring systems.

 

[Hornbein]

Surely they would agree. Maybe that's why pre-bends are relatively rare in blues?

They do it a lot. Here's my favorite example, where Jimi Hendrix bends down instead of up. //youtu.be/LyqC6_TgLOo?t=40s

BB King does it a ton. Here's an example with a closeup of the fingerboard so it's visible.

 

[Chrono G. Xay]

I think what you're doing by pre-bending, letting the string slide off your fingers, or plucking at the end vs. towards the middle, is changing which harmonics of the string are dominant once the string is released.

That is not the case. You are isolating a smaller mass at roughly the same tension (owing to how the string is depressed slightly at the desired fret, resulting in--hopefully--the smallest of unavoidable bends) and allowing that smaller mass to vibrate instead- a length of string which has its own modes of vibration and fundamental frequency, and whose modes' frequencies follow the harmonic series. Then, that isolated lesser mass has its tension increased further by way of an excursion across the held fret, following a limit function for force/distance needed until the desired ratio of frequency change has been achieved, based upon past experiences with that string at that fret and/or surrounding frets/strings.

Playing 'natural' harmonics, on the other hand, by way of lightly resting the tip of one's finger on a point dividing the string's length by a simple fraction (such as 1/2L, 2/3L, 3/4L, etc.) and then plucking the string, isolates which modes are allowed to vibrate.

 

[billy_joule]

That is not the case. You are isolating a smaller mass at roughly the same tension (owing to how the string is depressed slightly at the desired fret, resulting in--hopefully--the smallest of unavoidable bends) and allowing that smaller mass to vibrate instead- a length of string which has its own modes of vibration and fundamental frequency, and whose modes' frequencies follow the harmonic series. Then, that isolated lesser mass has its tension increased further by way of an excursion across the held fret, following a limit function for force/distance needed until the desired ratio of frequency change has been achieved, based upon past experiences with that string at that fret and/or surrounding frets/strings.

Playing 'natural' harmonics, on the other hand, by way of lightly resting the tip of one's finger on a point dividing the string's length by a simple fraction (such as 1/2L, 2/3L, 3/4L, etc.) and then plucking the string, isolates which modes are allowed to vibrate.

He's not talking about natural harmonics.
The harmonic content of a plucked string is affected by how it's played. That is why a string plucked nearer the fixed end has a different tone than when it is played near the middle of the vibrating section - The former excites more higher order harmonics (more shrill tone) and the latter excites the fundamental more (duller tone).