Musical Instrument Project: Frequencies Aren't Right?

(1) My requirement for this project was to build an instrument that can produce all the notes in a C scale. The accuracy of the notes is to be judged through Frequency (Hertz). I decided to build an instrument whose design is comprised of eight electrical conduit tubes (metal) that are about 1.5'' inches in diameter. I strike them with a mallet to produce the sound.

These are the Frequencies my teacher gave me:
C - 261.6 Hz
D - 293.7 Hz
E - 329.6 Hz
F - 349.2 Hz
G - 392.0 Hz
A - 440.0 Hz
B - 493.9 Hz
C - 523.3 Hz​

My problem is that I solved for the Length of the eight pieces of tubing (below), but when I went to go test them, the notes that I produced don't seem to be right.

Together, they definitely do not produce an even scale, and they are not in tune with my piano. If my ear is correct, I believe the notes are all flat. I'll describe the process I went through below. I'd really appreciate any problem-solving advice and suggestions for what could be wrong/how to possibly tune my instrument.

(2) These are the equations I used:
Speed of sound through air:
v = 331 + (.6*Tc) where Tc = the temperature in Celsius.

Frequency (Open-Ended Pipe):
Fn = n * (v/2L) <-- With L being the variable.
Where:
Fn = natural frequency (See the chart above!)
n = 1, 2, 3, etc.
v = speed of sound
L = length of the pipe​

(3) I used 22 degrees Celsius to solve for the speed of sound, and found that my house was actually at 69 degrees F (20.56 degrees C). I adjusted the house temperature to fit my equation (c. 71 degrees F) and tried again but it didn't help much. I also went back to my Frequency equation to solve with the 20.56 degrees C and saw that it was only the difference of 1 Hz.

Using 22 degrees C, I rearranged the Frequency equation to solve for L. I'm not sure if I did this right, but for the first seven notes of the octave, I treated each note as the natural frequency (so n = 1 for the first notes C - B). I was under the impression I did not need to change n = 2 until I reached the second C of the octave, because that would be the second harmonic. If I'm wrong, what do I plug in for n to solve for all the notes?

So for example:
For C (261.6 Hz):
v = 331 + (.6*Tc)
Tc = 22 degrees C
v = 331 (.6*22)
v = 344.2 m/s

Fn = n * (v/2L)
261.6 Hz = 1 * (344.2/2L)
2L = 344.2/261.6
L = (344.2/261.6)/2
L = 0.657874618 meters
0.657874618 meters --> about 66 cm

I rounded because the guy at Home Depot said he couldn't get much more accurate than that, but that might have been the problem too. Would it help to take a few more numbers?

These are the Lengths (L) that I got using my Frequency equation:
C - 66 cm
D - 58.5 cm
E - 52 cm
F - 49 cm
G - 44 cm
A - 39 cm
B - 35 cm
C - 66 cm?

Not sure if this last C note is right! I did the math, and it came out exactly the same, even with n = 2 (because all the 2's cancel out, etc). How do I make this C note an octave higher/lower?

I went and had my lengths of pipe cut, and when I went to play them, they were all off. So, my biggest question is: what's causing the notes to be off? I'm not sure if it's my math, or if I'm using the wrong equation, or if it's something else entirely. Do I need to also account for the speed of sound through the metal tube? If so, how would I do that?

If my idea is an ineffective one for this project, what are some other project suggestions? This project is super important, so I appreciate all the help. Thanks ahead of time! :)

EDIT: I went and got a tuner. The notes are really quite awful! I tested the first five notes: C, D, E, F, G and I got G sharp (the one below middle C), C, E, F sharp, D (above middle C). I have no idea what's going on!

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dynamicsolo
Homework Helper
You describe this as a percussion instrument, for which you are producing the sounds by striking the metal tubes. How have you mounted these tubes? Are they suspended freely or are they clamped in place? Be aware that where they are clamped matters. If you have a tube surface fixed at a point where it is supposed to vibrate, you are creating a vibrational node (static point) there. This means that vibrations will only occur in that tube at frequencies where that point is a node; you could be muting the fundamental frequency and only getting certain overtones.

Is your piano tuned correctly relative to 'A above middle C' at 440 Hz? When and how was it last tuned? I ask because it may sound correct in terms of relative pitch now, but it may not be correct in absolute pitch presently (or maybe never was). You should check your 'A' tube against a standard reference such as a tuning fork or an electronic audio oscillator at 440 Hz, just to make sure it's your tube that's off and not your basis of comparison.

I'll tell you that, in practical matters of homemade instruments, cutting the material can sometimes be just the first step. I have spoken with a member of a musical group that fabricates many of its own instruments; he told me that he did have to work on filing down some components to get them into proper tune. The metal in the tubes you are using may not be sufficiently pure and uniform compositionally and structurally to be properly 'musical', which can complicate matters. Do the tones you get sound reasonably 'pure', or like noise centered near the right frequencies?

[EDIT: It also occurred to me that you're in luck if all of your tubes turn out to sound 'flat' because then you could file them down a little to shorten them and raise their pitches. If they were 'sharp', it would be a bit tougher to lower the pitches...]

As for your question about that top 'C', you should not change the value of n from 1 at all. The symbol 'n' in the frequency equation represents the number of the 'harmonic' for a pipe of a specific length. Since you are interested in setting the 'fundamental frequency' for each tube, you want the 'first harmonic' which is given by n = 1. Using that in your equation will give you

L = [344.2 m/sec]/[2 · 523.3 Hz] = 0.329 m ,

that is, exactly half the length of the 'middle C' tube (as it should be, since the 'high C' tube is supposed to sound one octave higher).

If anything, this project will be a good lesson in the high art of musical instrument making... [If you are interested in pursuing this at all, there are a couple of recent detailed textbooks on the physics of sound and music out there. Good-sounding instruments are not easy to make.]

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I have no comment on your length calculations, but the frequencies that your teacher gave you look correct to me. You have to double the frequency in 12 steps (one octave), so each step is greater than previous one by the factor [the 12th root of 2], that is, 2^(1/12). I believe that the A note 440 Hz is exact. I have heard some people refer to it as the note A 4 on the piano, in a numbering system where C is regarded as the first note of an octave. That gives me:

Code:
       C  2         65.406
C# 2         69.296
D  2         73.416
D# 2         77.782
E  2         82.407
F  2         87.307
F# 2         92.499
G  2         97.999
G# 2         103.83
A  2         110.00
A# 2         116.54
B  2         123.47
C  3         130.81
C# 3         138.59
D  3         146.83
D# 3         155.56
E  3         164.81
F  3         174.61
F# 3         185.00
G  3         196.00
G# 3         207.65
A  3         220.00
A# 3         233.08
B  3         246.94
C  4         261.63
C# 4         227.18
D  4         293.66
D# 4         311.13
E  4         329.63
F  4         349.23
F# 4         369.99
G  4         392.00
G# 4         415.30
A  4         440.00
A# 4         466.16
B  4         493.88
C  5         523.25
C# 5         554.37
D  5         587.33
D# 5         622.25
E  5         659.26
F  5         698.46
F# 5         739.99
G  5         783.99
G# 5         830.61
A  5         880.00
A# 5         932.33
B  5         987.77

Ah, I see about the vibrational nodes... I actually have been trying with them both on the floor and resting on parallel wooden dowels. I was wary of attaching anything permanently while my project is still so out of whack, so that part of the design is not set in stone.

Sorry to ask for more explanation, but I'm a little slow with physics, so I hope you'll bear with me. I used a tuner to check the frequency of the pipes with them on the floor and on the dowels, and both times, the C pipe produced a G sharp. (I'm getting a G sharp, C, E, F sharp, D where I should be getting C, D, E, F, G-- so very, very, very flat notes.) So I'm not seeing the difference in my experimental trials, but I'm not really clamping anything down either, just resting the pipes on the dowels... Does that matter?

I still don't really understand why the notes are so flat, and how to fix it.

I also checked my piano, and it is in tune. :)

dynamicsolo
Homework Helper
I believe that the A note 440 Hz is exact. I have heard some people refer to it as the note A 4 on the piano, in a numbering system where C is regarded as the first note of an octave.
In the current-used system of Western tuning, A4 is indeed defined as exactly 440 Hz. The numbering system you refer to is a survival from how tuning was done before the early 20th Century. 'C' is still considered the first note of an octave; C0 is the first note of the lowest octave that enters the range of human hearing, with A0 at 27.5 Hz. So 'middle C' is C4, four octaves above that. Prior to the introduction of the present system, C4 was exactly 256 Hz, rather than 261.6 as it is now. (This is reminiscent of the change in 1961 from atomic mass being based on oxygen-16 nuclei to using carbon-12 as the basis. I understand that you can hear the difference in tuning if you listen to symphonic performances from the early days of sound recording.) You can read possibly more than you may want to know about this at http://en.wikipedia.org/wiki/Note .

I'd checked the frequencies the teacher provided and they looked fine. The source of the problem is likely something about the fabrication and mounting of the tubes...

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dynamicsolo
Homework Helper
Ah, I see about the vibrational nodes... I actually have been trying with them both on the floor and resting on parallel wooden dowels. I was wary of attaching anything permanently while my project is still so out of whack, so that part of the design is not set in stone.

Sorry to ask for more explanation, but I'm a little slow with physics, so I hope you'll bear with me. I used a tuner to check the frequency of the pipes with them on the floor and on the dowels, and both times, the C pipe produced a G sharp. (I'm getting a G sharp, C, E, F sharp, D where I should be getting C, D, E, F, G-- so very, very, very flat notes.) So I'm not seeing the difference in my experimental trials, but I'm not really clamping anything down either, just resting the pipes on the dowels... Does that matter?
Resting the tubes on dowels has basically the same effect as clamping, in that the point of contact between tube and dowel becomes a vibrational node. If you can find a convenient way to hang your tubes from one end (maybe wrap a string around one end and tape it in place, so you don't have to make the commitment of drilling holes), the tubes will be free of contacts and will vibrate without restriction. Your final arrangement for the instrument will need to be something like wind chimes or tubular bells. This should allow you to judge what your fundamental frequencies actually are.

I'm suspecting your testing method is at least part of the problem because your lengths are supposedly cut close to correctly, but the tones are deviating so far from a musical scale.

The arrangement you have is similar to what is using in tuned percussion instruments such as xylophones, vibraphones, marimbas, etc. It occurs to me that I don't actually know much about how the tuned bars for those instruments are mounted, but I imagine some care must be taken in placing the attachments so that unwanted nodes are not created.

The frequencies you were provided with look to be correct, as is the formula for the frequencies of an open-ended pipe (meaning both ends of the pipe are free to vibrate). The issue with sound speed is not important and is unavoidable in sound perception in any case. Since it is the ratios of the frequencies of notes that determine the musical scale, a change in temperature, air pressure, etc. altering the sound speed simply shifts the entire scale. (In recordings of instruments played in the open air, you can hear the changes in pitch in different seasons, but 'relative pitch' is not affected so you only notice the change by making comparison of the recordings. Differences in thermal expansion of different instruments in varying seasons would also have an effect that would spoil tuning among instruments in an ensemble, but that effect is rather tiny...)
I also checked my piano, and it is in tune. :)
Thank you for checking that -- it eliminates one possible source of error.

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If you can find a convenient way to hang your tubes from one end (maybe wrap a string around one end and tape it in place, so you don't have to make the commitment of drilling holes), the tubes will be free of contacts and will vibrate without restriction. Your final arrangement for the instrument will need to be something like wind chimes or tubular bells. This should allow you to judge what your fundamental frequencies actually are.
That's actually a grand idea... I was so fixated on the idea of a xylophone-like instrument that I hadn't even thought of arranging the thing more like wind chimes. I think I'll go ahead and try that with tape and see if it helps me to get a more accurate sound.

In the meantime-- and thank you again for all your detailed help, I really appreciate it-- someone else I talked with about my project suggested that I'm using the wrong equation entirely, and I wanted to double check this statement with someone.

They said: "One formula was presented in a book, "Music, Physics and Engineering" by Olsen, for a tube, free at both ends: f=1.133 pi K v / (l^2)" because they believed the equation I used was only for air blown through an open-ended pipe (like a flute) instead of a pipe being struck, as with my design.

you used the speed of sound in air, but I think the metal will vibrate. This is a much more difficult problem. I found some info here: http://mysite.du.edu/~jcalvert/waves/mechwave.htm" [Broken]

the frequency is proportional to 1/(L^2) with L the length of the bar. This explains why the
distances between the tones that you got were twice as big as desired.

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dynamicsolo
Homework Helper
you used the speed of sound in air, but I think the metal will vibrate. This is a much more difficult problem.
Alas, yes: the formula for the open-ended pipe is for the vibration of the air inside the tube and not the vibration of the metal itself (which is not a problem I'd worked with before).

On the other hand, I am proposing that we work with free-hanging tubes with no clamping at all. It would seem in that case that the fundamental frequency would still be related to standing waves in the length of the tube, so the open-ended pipe formula would still be relevant and the perceived frequency in air would be as given by that. (I could be misguided about this, though...)

Certainly when the metal is fixed at one or more points, the rigidity of the metal and its response to mechanical disturbance is going to be the dominant factor. If this is also very important for the suspended tubes, then the teacher has supplied an equation which is inapplicable. I'm interested to see what happens with the hanging tubes. The chimes I've dealt with don't seem to have a variation like f proportional to 1/(L^2), so I think we may be all right...

If you have the tubes cut to those lengths, you have made palm pipes. Just hit the ends with your palm, and you should be good to go. You may be an octave lower since you will be working with one end of the pipe closed, but you should be fine.

I did manage to somehow by chance get an E pipe that actually produces an E. So here is my question, can I use this equation to solve for the other pipes relative to my E:

L2 = L1 * sqrt(f1 / f2), where L2 is the length I'm solving for, L1 and F1 are the length and frequency of E, and f2 is the frequency that coincides with L2.

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dynamicsolo
Homework Helper
I did manage to somehow by chance get an E pipe that actually produces an E. So here is my question, can I use this equation to solve for the other pipes relative to my E:

L2 = L1 * sqrt(f1 / f2), where L2 is the length I'm solving for, L1 and F1 are the length and frequency of E, and f2 is the frequency that coincides with L2.
This is derived from the f proportional to 1/(L^2) relation, so if that turns out to be applicable, your result here would follow.

Just out of curiosity, how did the length of your E-tube compare with the length you gave in your original post?

The E pipe was actually the exact same one from my original post. :) Lucky coincidence, I guess.

dynamicsolo
Homework Helper
The E pipe was actually the exact same one from my original post.
So it's the 52-cm. tube, yes? We may be able to sort this out empirically before having you cut up any of the other tubes. Can you get a good idea of the frequencies for some of the other tubes? We could check to see if the pieces of conduit seem to follow a 1/L rule, a 1/(L^2) rule, or something else...

My tuner only can tell me what notes I have, give or take 50 Hz. So we could take a really rough estimate. I had G sharp, C, E, F sharp, D-- instead of a regular scale (C, D, E, F, G). So that would be about: 207.65, 261.63, 329.63, 369.99, and 587.33.

A tuner with 50 Hz resolution?! 50 Hz above middle C is higher than D-sharp. you're better off using your ears
The tuners I know give you the closest semitone to what you play and the direction (and sometimes the size) of the error. that means at least 3% accuracy.