# How will the pitch of a string change when I stretch it? (sound)

• kokes
In summary, the linear density (mass per unit length) of a string decreases when it is stretched, however the scale length (the distance from one fret to the next) does not change.
kokes
Is there a way to calculate how frequency will change if I stretch a string by certain amount (0.14 mm in my case)? I found out I can measure its frequency once stretched, but no results as of how to estimate new frequency ahead of the time. I know a speed of string can be calculated from its weight, length and tension, but that doesn't seem to get me anywhere:
- I don't know how to calculate the tension from delta length.
- I don't know how speed of string and its frequency are related.
Any help would be appreciated.
Thanks.

One would need to know the stress versus strain curve of the string.

kokes said:
how frequency will change if I stretch a string by certain amount (0.14 mm in my case)?
The formula requires you to know a lot about the string but, if you already have a known frequency for the given length, you don't need all the data. There are many possible links but this one may help. The grey boxes give the wave velocity and then the fundamental frequency, in terms of Length (L) and Mass per Unit length (m). I assume you can measure the start length L and the extension but how about m?
The problem is that stretching the string will alter both, so length appears twice in the formula. If your string length increases by l (lower case L) the new L is L'=L+l and the new m' = m L/(L+l). If you put these new values into the original formula then you should get the new frequency but with and unknown m in it.
Old and new frequencies will both contain m but, for the ratio, this won't matter because the original m will be top and bottom of the result you get so it will cancel out and give you the value fnew/fold. You never actually needed to know the mass or the modulus of the string.

Give it a try.

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vanhees71
Thank you for your valuable inputs.

Paul Colby:
So I basically need to perform a tensile strength test on the string to be used. That is very clever. Thank you. It will require a specialized instrument.

sophiecentaur:
I don't think stretched string changes its mass. Does it? When I stretch a rubber band its mass won't change. So I need to know at least a tension. That, too, is clever. However it will, once again, require a specialized instrument.

My idea was that I could calculate the length change (angle change of guitar peg [rad]*radius o winch / (2*Pi)) while measuring the frequency. But for that I need to build a measuring guitar first, or specialized instrument.

In summary, regardless of what I do, I cannot go with just calculation. I may need to build a specialized instrument. Is that correct?

If the string stretches then its length has increased (and its cross section shrinks) so the mass per unit length CHANGES. Not obvious at first sight, perhaps. You do not need to measure that nor the elastic modules. All you need is the frequency when it starts and the fractional change in length.
it’s almost a free lunch.

Wow! It is obvious. In my ignorance I thought we were dealing with mass, not mass per unit of length. Frequency when it starts shall I calculate from frequency when elongated then. Fractional change in length is known to me. If my calculations turn out correct, it will be constant throughout the fretboard, which I really hope it will, because what is the point otherwise.

Fanned fretboard with Pythagorean intervals, prior to constant change in pressed string length adjustment and open vs pressed frequency adjustment.

kokes said:
That's very appreciative of you but Gold Member just means I contributed some money to PF. (What am I doing, blowing my own cover? lol)
If it's to represent a real guitar then the actual stretch can be particularly relevant for a badly set up instrument. Every lad has had to deal with a rubbish old starter instrument with a horrid bowed fretboard. Suddenly you get a better guitar with a light action that's been set up right and you can start to play in tune.

Klystron
sophiecentaur said:
If the string stretches then its length has increased (and its cross section shrinks) so the mass per unit length CHANGES. Not obvious at first sight, perhaps. You do not need to measure that nor the elastic modules. All you need is the frequency when it starts and the fractional change in length.
it’s almost a free lunch.

I'm not sure this is true with a guitar since the vibrating part of the string has a constrained length. So when you tune up the tension increases and the linear density decreases but the scale length stays the same.

So I think you do need to know a little bit more...

JT Smith said:
I'm not sure this is true with a guitar since the vibrating part of the string has a constrained length. So when you tune up the tension increases and the linear density decreases but the scale length stays the same.

So I think you do need to know a little bit more...
In that event, the calculation is easier because all that has changed would be the mass per unit length. As you say, the length between nut and fret would change very little unless you 'bend' the note sideways.
Edit: When you bend the note, both ends of the string get stretched so you would need to do some trigonometry to take account of the unequal lengths involved. (Too hard guv'.)

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I think you'd need to know the mass of the string. Ideally you'd want to know the initial tension and modulus too so you could determine the initial mass from a weighed (non-tensioned) sample of string, but probably that doesn't matter so much.

(boring anecdote deleted)

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Thank you once again.

I fell asleep, but I will continue with the equal tension calculation when I get a little time today.
I think I can buy set of strings, cut them to their respective lengths and weigh with little difficulty. However I risk measuring error, in my circumstances perhaps significant one. I still have time to think, before I get to my third calculation.

At the age of 13 I bought myself a guitar, because it was beautifully blue. I still play this very guitar quarter of a century later. It has 9 mm action and goes off by 1/2 tone in upper region. It features neck so clumsy that I can bend the notes back and forth pushing or pulling it.

sophiecentaur
kokes said:
I think I can buy set of strings, cut them to their respective lengths and weigh with little difficulty.
No need to cut. Just use a 'stop' at appropriate places on a long string (like guitar frets do). Guitar strings are well made and have uniform linear density so linear density only needs to be measured once for each string.

For practical convenience, you need strings to be as long as possible (in total); you will need to clamp the ends and that is best done with a good length of string held in the clamp. It's little practical details that can add to the time taken to set up a reliable experiment. No problem if you have reasonable metalwork / DIY skills but you will probably need to improvise a lot of this - be prepared.
You may well be able to find out the mechanical properties of a good set of strings from the manufacturer. (Useful, even if just to check). Actually, you would be best off to buy a reel of steel 'piano' wire, which would have specified dimensions which would be to a higher standard than you could measure yourself. All types of metal stock are of higher quality than your average DIYer needs. Use real strings later.

This sounds like a fun investigation. Enjoy it.

kokes said:
It has 9 mm action and goes off by 1/2 tone in upper region. It features neck so clumsy that I can bend the notes back and forth pushing or pulling it.
Ouch, 9mm is pretty large. I assume you’ve added to you guitar collection since.

also as strings age on a well setup guitar they tend to go flat at fret 12 harmonic. Anyone know why this is?

I came up with the second calculation. I modeled it and measured. With my guitar I sense about 1/2 tone flat on overtone vs. pressed string, up 36 Hz from E5 . It looks as with 1.5 mm action (36 times smaller elongation) the tone will change by about 1Hz. I will try it like that, as good as it gets, and will deal with it more in depth if I need to.

Not really. It doesn't hurt after while. Fingers toughen up and forearm learns to help fingers with string pressing. It forces me to play technically. Fast left hand is a no no. I learned right hand as good as I can, and I am looking forward to another level. I miss nothing so I am in no rush. I don't buy musical instruments often, or build them very often, so I feel free to take all the time I like for preparations.

kokes said:
With my guitar I sense about 1/2 tone flat on overtone vs. pressed string,
One reason for that sot of thing can be that the bridge is in the wrong place. What are the measurements? Some guitars have a moveable bridge and other don't. Sometimes the bridge is placed slightly diagonally in order to minimise the overall frequency discrepancy. On electric guitars there is often an adjustment for each string , individually.

sophiecentaur said:
One reason for that sot of thing can be that the bridge is in the wrong place. What are the measurements? Some guitars have a moveable bridge and other don't. Sometimes the bridge is placed slightly diagonally in order to minimise the overall frequency discrepancy. On electric guitars there is often an adjustment for each string , individually.

It's a fixed bridge. I calculated with 610 mm length of e4. Actual length is 620 mm e4. Twelfth fret is 310 mm from nut according to tape measure, verified by strip of paper taped to the guitar, marked and subsequently folded in half. There is no error in the guitar that I can detect. As a matter of fact, I believe I checked it prior to buy, having played on mom's guitar (which is much worse).

I remember that Roman army was tuning their ballistae by pinging their strings and listening to the sound. If not in tune, it would shoot off. This might be related in some respect, couldn't it?

Never mind. Roman strings didn't stretch. At all.

Paul Colby said:
also as strings age on a well setup guitar they tend to go flat at fret 12 harmonic. Anyone know why this is?

I haven't noticed that. How much of a difference are you talking about?

Anyway, my guess is the strings get dirty/oxidized and are effectively thicker/stiffer than new strings.

kokes said:
It's a fixed bridge. I calculated with 610 mm length of e4. Actual length is 620 mm e4.

What sort of guitar has a 24 inch scale length? Just curious.

JT Smith said:
I haven't noticed that. How much of a difference are you talking about?

Well, nowhere near a 1/2 tone for certain. The shift is observed by exciting the string octave harmonic by lightly touching the string at the 12'th fret when plucked. Comparing this note to the sound when fully depressed should match. To properly adjust (setup) a guitar, the location of the bridge for each string is adjusted. Once adjusted, the frequency at the 12'th fret is exactly an octave. This takes out the additional tension gained when pressed.

Now as the string ages the 12'th fret goes flat by a few cents. This seems to increase with age. Now, dirt and grim could cause this but single wire strings are easily kept clean and it still happens.

I have an old set on my guitar right now and the intonation is as good as I can detect by ear, clip-on tuner, or using the spectrum plot in Audacity (which only displays to the nearest Hertz). A few cents is only ~1Hz on the lower string.

And my explanation makes no sense anyway. Extra stiffness in the string would effectively reduce the length and make the string go sharp at the harmonic.

I will get myself piezzo pickup once the lock down is released here. It was put into action, you guessed it, today. I have seen Audacity in my sw repo.

The properties of strings change with time. I find it takes about a day for a new set to stop stretching. My guess is they lose stiffness as the age.

Paul Colby said:
The properties of strings change with time. I find it takes about a day for a new set to stop stretching. My guess is they lose stiffness as the age.

They don't really stretch. It's the slack in the whole system, especially the tuning machine windings, that takes time to work out.

I just read one guy's opinion that the strings become uneven over time and that messes up intonation. He didn't explain why though.

JT Smith said:
I just read one guy's opinion that the strings become uneven over time and that messes up intonation. He didn't explain why though.
The strings can get worn by rubbing against the frets and bent by your fingers pressing behind the frets. The mass per unit length will alter for different lengths, in the end. I remember reading that early Lute strings were made of gut and were so non-uniform that the frets had too be formed from loops of gut, tied wound the neck and then placed in the least worst positions for the best tuning of as many of the strings as possible.
I believe that gut strings can be rolled between the fingers to stretch them in parts, to help with the tuning. Nylon and steel are much better behaved.

I was assuming steel strings. It wouldn't surprise me if gut stretches significantly over time.

Guitars are inherently imperfect. There are those True Temperament guitar necks but I've read that many people are so used to imperfect intonation that those improved necks sound funny to them.

sophiecentaur
JT Smith said:
I was assuming steel strings.
Not impervious to sweaty fingers tho'.

I was considering gut strings. But I think I will use steel again. My old strings are old steel strings aged about 20 (I don't use picks any more).

I am looking forward to the new sound. Particularly Adio Cheyenne. But the main plan is to be able to safely recognize notes (including vibratos!) to output a sheet music as I play.

I found a nice illustration of why tension of open string must always differ from tension of pressed string:

This is how ellipse used to be constructed.
Therefore, the string attached between F1 (nut) and F2 (bridge) is always pressed with constant force throughout elliptical fretboard (part of ellipse which is mostly left out, below segment F1F2).
If the string is open, its length must be shorter, regardless of the ellipse used.

Which means that if I won't be able to reliably detect vibratos with accuracy I achieve with the fretboard alone, I will need to adjust for tension. Either by MCU modulating output AC wave, or by tuning LC circuits with inputs, or by physically turning the tuner using motor, transistors, external power supply and MCU. The last option has the slowest output, but it appeals to me, should it come this far, because it will, besides shifting fractions of Hz there and forth, keep the guitar in tune any time I play open strings. Either way I will need to monitor which strings are being pressed, perhaps by measuring their impedance. And record them individually.

kokes said:
I was considering gut strings. But I think I will use steel again.
That's a jump of centuries, almost. Nylon is now very good quality and, of course, there are many guitarists (with good ears) who use nylon.
kokes said:
I found a nice illustration of why tension of open string must always differ from tension of pressed string:
Appropriate adjustment of the bridge-nut distance should take care of that. The style of stopping will have an effect, of course - size and squashiness of finger pads. But there seems to be a change of direction in your contributions to the thread - first we were discussing the effect of a stretch and now you are concerned with measuring a frequency change so that you can annotate sheet music.
Afaics, you don't need a highly accurate answer to the OP, you just want a ballpark figure to work with in your software. Why not just record some of your guitar output and see how easily you can work with real sounds? You can see the effect of stretching by using the tuning head and measuring the movement of the string over the nut.
Measuring the frequency of the plucked string is not difficult and nor is it hard to see a resonance when you use a single tone from a nearby loudspeaker to excite the vibration. The output from your guitar pickup will peak at resonance.

kokes said:
...the main plan is to be able to safely recognize notes (including vibratos!) to output a sheet music as I play.

I thought you wanted to know how much the frequency changed when you stretched the string by 0.14mm? If what you really want to do is recognize notes there is likely already software that will do that.
kokes said:
I found a nice illustration of why tension of open string must always differ from tension of pressed string:
View attachment 271421
This is how ellipse used to be constructed.
Therefore, the string attached between F1 (nut) and F2 (bridge) is always pressed with constant force throughout elliptical fretboard (part of ellipse which is mostly left out, below segment F1F2).

I don't believe that's true. The action typically increases as you move down the fret board so there is a larger deflection required to fret the string at the twelfth fret versus the first fret. But the bending force at the lower frets is disproportionately higher due to the geometry and this more than makes up for the difference in the action. You don't have to be a good guitar player to notice that it's harder to bend or even just fret string at the first fret versus the twelfth.

JT Smith said:
I thought you wanted to know how much the frequency changed when you stretched the string by 0.14mm? If what you really want to do is recognize notes there is likely already software that will do that.
There have been apps that can transfer the notes from an instrument to Midi code for years. I haven't understood how the (now revealed) requirement of the OP wouldn't be met by this.
I don't know the error rate of this sort of conversion software but 'proof listening' would be pretty easy.

Thank you. I basically don't need as much as I thought I would. I was unsure whether I could distinguish vibratos and notes. I don't have a pickup around to test, and we are being locked down. So I figured I would work on a hardware a little.

I would rather adjust individual frets then bridge. If I adjust the bridge, I don't have guaranteed known cross section of fret board. The elliptical theory suggests that open string has no tension, which is not true. There is a given ratio of tension open string vs. plucked string. This ratio can be measured and each fret can be moved towards the nut by the ratio times length of vibrating section of the string.

The downsides are that it will only work for particular set of strings.

However I don't necessarily need any of these. I can just choose intervals of notes as needed. Say I have 5 instances of F#4 on my fret board, each a bit different. If I play a vibrato around any of these ranges, it will count as F# with vibrato for the MCU. Is that correct?

My vibratos are subtle. I play three distinct types depending on my mood, but none of them moves the string up and down as in common rock guitar vibrato. The distinction in terms of frequency change may not be too easy.

I take my time to think through resonant frequency idea. I thought I would use whatever the highest frequency MCU can output. I should be big enough a capacitor to shift that frequency reliably when in contact.

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Maybe the discrepancies of my old fret board could be used to my advantage. From frequency alone the guitar could distinguish a position where the melody is being played, no? And output a tabelature.

I need quick release fretboards.

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