Tuning a Violin's Chord (Frequencies)

[Const@ntine]

Homework Statement

A violin's chord has a length of 0.350 meters, and is tuned to the sound of the note Sol, with a frequency of f_G = 392 Hz.

a) How far from the edge of the chord does the violinist need to place his hand, in order to play a note La, with a frequency of f_A= 440 Hz?

b) If the accuracy of that position is +- 0.600 cm, which is the maximum alteration of the stress (T)?

Homework Equations

fn = n*v/2L = n/2L * sqrt(T/μ)

The Attempt at a Solution

a) From what I gather, and from what my book says about tuning chords, f_G is the hamonic. Using that, I can find the speed of the wave in the chord. With n = 1, f_G = 392 Hz & L = 0.350 m I get v = 274.4 m/s. So far, so good.

Initially I figured I'd divide f_A with f_G, find the n and then find the "new" Length, L', and find the difference between that and the initial one, L. Problem is, I don't get an integer result through the division, so I can't go that way.

I looked back to the theory, and it says that you can change the sound of the strings depending on how much you stress them or where you apply it, but it doesn't say anything about whether or not you "tune" it and thus create a "new harmonic" or not.

Since that didn't work, I figure the chord gets tuned again, so I went with f_A as the new harmonic, put in n = 1, v =274,4 Hz & looked for L'. That comes out as 0.31. If you findthe difference between that & L, it comes to 4.0 cm, which is different. The book's answer is 3.8 cm.

So, any help? The book's only examples is just one problem of "put numbers in the formula" and it doesn't explain how tuning and whatnot works.

Every bit of assistance is appreciated!

 

[haruspex]

From what I gather, and from what my book says about tuning chords, fG is the hamonic. Using that, I can find the speed of the wave in the chord.

You are overthinking the problem. "Tuned to" means in both cases this is the fundamental frequency. Since the speed of sound in the string is the same in both cases there is a simple relationship between string length and fundamental frequency.

 

[collinsmark]

Homework Statement

A violin's chord has a length of 0.350 meters, and is tuned to the sound of the note Sol, with a frequency of f_G = 392 Hz.

a) How far from the edge of the chord does the violinist need to place his hand, in order to play a note La, with a frequency of f_A= 440 Hz?

b) If the accuracy of that position is +- 0.600 cm, which is the maximum alteration of the stress (T)?

Homework Equations

fn = n*v/2L = n/2L * sqrt(T/μ)

The Attempt at a Solution

a) From what I gather, and from what my book says about tuning chords, f_G is the hamonic. Using that, I can find the speed of the wave in the chord. With n = 1, f_G = 392 Hz & L = 0.350 m I get v = 274.4 m/s. So far, so good.

Initially I figured I'd divide f_A with f_G, find the n and then find the "new" Length, L', and find the difference between that and the initial one, L. Problem is, I don't get an integer result through the division, so I can't go that way.

I looked back to the theory, and it says that you can change the sound of the strings depending on how much you stress them or where you apply it, but it doesn't say anything about whether or not you "tune" it and thus create a "new harmonic" or not.

Since that didn't work, I figure the chord gets tuned again, so I went with f_A as the new harmonic, put in n = 1, v =274,4 Hz & looked for L'. That comes out as 0.31. If you findthe difference between that & L, it comes to 4.0 cm, which is different. The book's answer is 3.8 cm.

So, any help? The book's only examples is just one problem of "put numbers in the formula" and it doesn't explain how tuning and whatnot works.

Every bit of assistance is appreciated!

For part a), the question is not asking you to find a harmonic. You don't need integer division.

The way a (right handed) violin player chooses notes is by firmly pressing a left-hand finger on a cord/string against the solid "fingerboard." This doesn't necessarily need to be done at the locations of harmonics (it can be, but doesn't necessarily need to be). It can be done anywhere.

The result of doing so is that the cord/string is effectively shortened, producing a higher note than the open string/cord.

So what part a) is essentially asking you is, "how much shorter -- how much length needs to be removed -- of the cord/string to produce a 440 Hz note?"

 

[Const@ntine]

You are overthinking the problem. "Tuned to" means in both cases this is the fundamental frequency. Since the speed of sound in the string is the same in both cases there is a simple relationship between string length and fundamental frequency.

For part a), the question is not asking you to find a harmonic. You don't need integer division.

The way a (right handed) violin player chooses notes is by firmly pressing a left-hand finger on a cord/string against the solid "fingerboard." This doesn't necessarily need to be done at the locations of harmonics (it can be, but doesn't necessarily need to be). It can be done anywhere.

The result of doing so is that the cord/string is effectively shortened, producing a higher note than the open string/cord.

So what part a) is essentially asking you is, "how much shorter -- how much length needs to be removed -- of the cord/string to produce a 440 Hz note?"

So every time it's "tuned" or the instrument player touches a chord he essentially changes the fundamental each time? Is there any link where I can read about it? Because the relationship between harmonics & fundementals in instruments is a bit vague in my book and I can't fully grasp it.

Anyway, with all that said, we have: fA = n*v/4L' <=> 440 Hz = 1*274.4 (m/s)/4L' <=> L' = 0.312 m
L = 0.350 m
Therefore, ΔL = L - L' = 3.8 cm, which is the book's answer.

b) Any ideas on this? At first I figured I just needed to find ΔΤ% in regards to T_G & T_A, but that's not it.

 

[collinsmark]

So every time it's "tuned" or the instrument player touches a chord he essentially changes the fundamental each time?

Tuning a violin string is accomplished by twisting a peg and thus changing the string's tension (http://www.violinstudent.com/tuning).

However, when playing a violin, notes are changed by different fingerings on the fingerboard -- these different fingerings change the effective length of the string.

Is there any link where I can read about it?

I suggest looking at videos of people playing the violin. Notice how different notes are accomplished by different fingering positions. Here is one example video.

Because the relationship between harmonics & fundementals in instruments is a bit vague in my book and I can't fully grasp it.

Harmonics are an important and fascinating subject, but they don't apply to this particular problem. This problem deals only with the fundamental.

Anyway, with all that said, we have: fA = n*v/4L' <=> 440 Hz = 1*274.4 (m/s)/4L' <=> L' = 0.312 m
L = 0.350 m
Therefore, ΔL = L - L' = 3.8 cm, which is the book's answer.

That looks about right to me.

b) Any ideas on this? At first I figured I just needed to find ΔΤ% in regards to T_G & T_A, but that's not it.

What answer does the book give? I'm not sure if I understand part b)'s translation very well.

 

[Const@ntine]

Tuning a violin string is accomplished by twisting a peg and thus changing the string's tension (http://www.violinstudent.com/tuning).
However, when playing a violin, notes are changed by different fingerings on the fingerboard -- these different fingerings change the effective length of the string.
I suggest looking at videos of people playing the violin. Notice how different notes are accomplished by different fingering positions. Here is one example video.

Harmonics are an important and fascinating subject, but they don't apply to this particular problem. This problem deals only with the fundamental.
That looks about right to me.

Thanks for all the tips and source! I'll be sure to check it out.

What answer does the book give? I'm not sure if I understand part b)'s translation very well.

The book's answer is "3.85%". To be honest, I don't understand what the book's talking about either. I'll type it here again in case I messed something up the first time:

b) If the accuracy of that position must be equal to half the width of the person's finger (so equal to +- 0.600 cm), which is the maximum relative (or percentage-based) change of the tension of the chord?

Problem is, I'm not sure whether "that position" refers to L', or the difference ΔL that I was tasked to find in (a). Apart from that, at first I just went ahead to find T_G & T_A from the formulas (the ones I have written over at the "Relevant Equations" part), and then find ΔΤ% = |Τ_G - T_A|/T_G * 100%, but the result was wildly different, and I didn't even use the "+- 0.6 cm" info. My best guess is that it has something to do with Faults (you know, X +- δx), but I can't think of anything, plus, I don't know how to connect it with the Tension. I'm really stumped here.

 

[collinsmark]

Are you encouraged or even allowed to use calculus in this class? (As in taking derivatives and such.) I ask because if you are there's a somewhat graceful way to solve this problem that involves taking derivatives.

If not, then that's okay too. Calculus is not absolutely necessary to solve this problem. (But it does make it a lot easier.)

Ask yourself, "changing the string length by +/- Δx (where Δx = 0.6 cm and L' = 0.312 m) changes the frequency by how much?"

(It's acceptable to solve for the ratio Δf / f, by the way, if you want [in terms of Δx / L' ]).

Then ask yourself, how does that effect the ratio of the tension, ΔT / T? In other words, solve for ΔT / T in terms of Δf / f.

 

[Const@ntine]

Are you encouraged or even allowed to use calculus in this class? (As in taking derivatives and such.) I ask because if you are there's a somewhat graceful way to solve this problem that involves taking derivatives.

Well, there are no restrictions. As long as the result is correct, go nuts with the explanation.

If not, then that's okay too. Calculus is not absolutely necessary to solve this problem. (But it does make it a lot easier.)

Ask yourself, "changing the string length by +/- Δx (where Δx = 0.6 cm and L' = 0.312 m) changes the frequency by how much?"

(It's acceptable to solve for the ratio Δf / f, by the way, if you want [in terms of Δx / L' ]).

Then ask yourself, how does that effect the ratio of the tension, ΔT / T? In other words, solve for ΔT / T in terms of Δf / f.

So something like this:

L' = 0.312 m
Δx = +- 0.00600 m
f = nv/4L
______________________
For L' + Δx: f₊ = 1*274.4 m/s/4*(0.312 m + 0.006 m) = 431.4 Hz || f - f₊ = 440 Hz - 431.4 Hz = 8.6 Hz
For L' - Δx: f_- = 1*274.4 m/s/4*(0.312 m - 0.006 m) = 448.4 Hz || f_- - f = 448.4 Hz - 440 Hz = 8.4 Hz

So, Δf = (8.6 + 8.4) Hz/2 = 8.5 Hz => Δf = +- 8.5 Hz

Then we have: f = n/2L * sqrt(T/μ), where μ = m/L

The original T, for n = 1, f = 440 Hz & L' = 0.312 m, is: T = 2*L' * f²/m = 120806.4 /m, where m is the mass of the chord (I left out the SI units to make it more streamlined, but it should be N/Kg at the top)

For f + Δf: T₊ = 2*0.312*(431.4)²/m = 116130.1/m || T - T₊ = 4676.3/m
For f - Δf: T_- = 2*0.312 m*(448.4)²/m = 125463.0/m || T - T_- = 4656.6/m

So, Δf = (4676.3/m +4656.0/m)/2 = 4666.4 Hz => Δf = +- 4666.4 Hz

Therefore, ΔΤ/Τ = 4666.4/m / 120806.4/m = 0.0386 or 3.86% (book's answer is 3.85%, so it's just a Significant Digits thing).

And, we're done! Thanks a ton for the help, but I still need to ask one thing: What's the thinking/theory behind this? I mean, I understood how to do it now, but if I come across something similar again, how will I know what to do? Is there something I can read about this?

 

[haruspex]

What's the thinking/theory behind this

Look at part 4 of http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm.
You have, for a given frequency, T∝ L². So ΔT/T = 2ΔL/L.

 

[Const@ntine]

Look at part 4 of http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm.
You have, for a given frequency, T∝ L². So ΔT/T = 2ΔL/L.

Thanks for the link! We did some of the basics there, but not everything. I'll surely give it a look!