Help with Building a Xylophone

[piareround]

So since I am a physics major and a help tutor people in Introductory physics (for free), my mom decides to turn everyone who has physics for me to solve. While most of the time I can solve these problems, here is one that I am not sure even where to start. Heck I am not even sure if this is a physics problem or an engineering problem. If I have put this in the wrong place please move this thread for me.

Homework Statement

Son, can you e-mail a student at the following e-mail at [email omitted] regarding any suggestions for her physics project? Her mom [name omitted] works next to me at Tullie. Her note stated that her daughter has to create a musical instrument – she has decided to do a xylophone, but needs to know what length for each electrical conduit would be need for each note. Do you know how to help her or who might be able to assist her?

Basically the problem can be broken up into primary question and a couple of secondary ones.

The main question is as follows:
If you are building a xylophone,what is length for each electrical conduit that would be need for each note? Assume you are starting from the standard 440 Hz tuning note.

Here are the secondary questions, that I had as I read this:

• What are the set of equations related to physics of an acoustic xylophone? Are their correction terms like physics with a flute?
• Is there such thing as an electric xylophone!? If so, what additional or different equations are needed?
• Apparently the person whose doing this project is probably in still in middle school. So does anyone know of through and complete website that could explain the equations behind how a xylophone works?

Homework Equations

None as it is precisely the right set of equations we are looking for.

The Attempt at a Solution

Immediately when I saw this question, I was perplexed as to what they meant by "electrical conduit." I am sending the person an e-mail right now to clarify what they meant and hopefully get more information.

In the mean time, I assumed that it was basically an acoustic xylophone and that xylophones had a similar set of equations that to basic harmonic strings. In other words I tried to answer the first secondary question.
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html#c3"Following this line of logic and a google search, I eventually came across this paper that talks about how to build a copper xylophone...
http://staff.tamhigh.org/lapp/xylophone.pdf"

... as well as I couple of other papers in the NASA archives. I was thinking about sending those papers, to her to help her out when I realized that she was probably in 7th or 8th grade. Thus, she might not be able to fully comprehend the complex jargon we scientists normally use when we write physics papers. As you can see, this lead to my third secondary question.Does anyone have any suggestions?

 

[mhen333]

By electrical conduit I am assuming they are going to buy a length of metal pipe that electricians stuff wires into. I can't offer any help as to the physics, though!

 

[I like Serena]

By electrical conduit I am assuming they are going to buy a length of metal pipe that electricians stuff wires into. I can't offer any help as to the physics, though!

Assuming we're talking about a non-electric xylophone, the pipes need to have length ratios.
For an octave the ratio is 2:1.
For a quint (c-g) the ratio is 3:2.
For a quart (c-f) the ratio is 4:3.

More generally the ratios are powers of 2^(1/12).
The "distance" between c and g is 7 half-notes, yielding a ratio of 2^(7/12) which is approximately 1.5.
The "distance" between c and f is 5 half-notes, yielding a ratio of 2^(5/12) which is approximately 1.33.

 

[Fewmet]

It looks to me like the article about building a copper xylophone gives you what you need. Its Equation 1 says the tube length depends on the desired frequency, the speed of sound in the metal of which the tube is made and the inner and outer diameters. There is a table of speeds in the article. There is also the issue of supporting the tubes in a way that doesn't damp out the sound

You could simply calculate the lengths for the frequencies given in the article and send the student. She could cut them work out where to put the supports by trial and error.

The down side of this is that the student is not really doing any science in this science project. For myself, I'd feel pretty good about a middle schooler understanding that the sound comes the tube vibrating in a transverse wave, how the wavelength and frequency depend on the length of the tube and the idea of locating the supports at the nodes. You might explain to her how to calculate the frequencies: to go from one frequency to the the frequency of the next note in a chromatic scale you just multiply bu the 12th root of two. So if you start with A 440, A3 is 440 x 1.059463094, B is 440 x 1.059463094 x 1.059463094, and so on.

I am not sure how much of this physics will be familiar to you. I know I taught most of it to myself to work with high school students on the physics of music. I'd be happy to clarify anything.

 

[piareround]

Thanks, I have not heard back from the person, but this really helps ^-^. I'll post again when I hear back from them.

 

[Fewmet]

the pipes need to have length ratios.
For an octave the ratio is 2:1.
For a quint (c-g) the ratio is 3:2.
For a quart (c-f) the ratio is 4:3.

More generally the ratios are powers of 2^(1/12).
The "distance" between c and g is 7 half-notes, yielding a ratio of 2^(7/12) which is approximately 1.5.
The "distance" between c and f is 5 half-notes, yielding a ratio of 2^(5/12) which is approximately 1.33.

The integer ratios will give those tones in the Pythagorean scale, whereas the twelfth-root-of-two approach gives them in the modern day well tempered scale. It might be a neat demo to play a xylophone pipe made by each method and compare the sounds. If you based this on A440, the "quarts" based on the 4:3 ratio and on the twelfth-root-of-two approach, there would be a discrepancy that would cause about two beats every three seconds.

I-like-Serena: I can see that quart and quint are what I would call a fourth or a fifth. Do you know if quart and quint are the terms from classical music, or from a European perspective?

 

[I like Serena]

The integer ratios will give those tones in the Pythagorean scale, whereas the twelfth-root-of-two approach gives them in the modern day well tempered scale. It might be a neat demo to play a xylophone pipe made by each method and compare the sounds. If you based this on A440, the "quarts" based on the 4:3 ratio and on the twelfth-root-of-two approach, there would be a discrepancy that would cause about two beats every three seconds.

I-like-Serena: I can see that quart and quint are what I would call a fourth or a fifth. Do you know if quart and quint are the terms from classical music, or from a European perspective?

I thought that most musical instruments were tuned by fifths to be 3:2, so at least those sound pure, at the cost of the purity of other note combinations.

English is not my native language. I guess I have translated incorrectly. In my own language the words come from Latin. Googling seems to suggest that in some contexts these words are used in English as well.

 

[Fewmet]

I thought that most musical instruments were tuned by fifths to be 3:2, so at least those sound pure, at the cost of the purity of other note combinations.

As I understand it that was the goal in the older scales (Mean Tone intonation and Just intonation), but it created clashes when playing in certain keys. (http://www.pyxidium.u-net.com/Acoustics/MusicMaths/MusicMaths.html" are audio files that demonstrate that.) In the well tempered scale the attempt to keep the "perfect fifth" was given up. You can check it through calculation, starting with A440:
440 Hz *3/2 = 660 Hz
440 Hz * 2^(7/12) = 659.2551138 Hz

I have physics students (who are musicians) who can hear that difference, though I cannot. It is enough to cause a noticeable beating.

I have heard (and it makes sense physically) that Barbershop singers use the Pythagorean scale, and that gives that distinctive sound.

English is not my native language. I guess I have translated incorrectly. In my own language the words come from Latin. Googling seems to suggest that in some contexts these words are used in English as well.

Thanks for that clarification. I also checked it on Google and saw a range of kinds of places where those terms were being used. That's what led me to ask.

 

[alpobc]

The speed of sound is different in air than in metals or wood. Of couse a wood xylophone is a marimba
At room temperature ~20°C ~68°F the speed of sound is ~343m/s. I did some experiments with various lengths of EMT (electrical conduit) and a frequency meter in my chill garage and determined the speed of sound to be ~5181m/s of course it will be a bit faster at the warmer room temp of 20°C.
The thickness of the wall of the conduit also plays a factor.
L=sqrt((Pi*v*K*m^2)/(8*f)) where Pi=3.141, v=speed of sound in material, m=0.03112(for the fundamental frequency) 5 for the second, 7 for the third, 2^n+1 etc where n= the mode of resonance. We'll stick with the first mode. f=frequency and K=0.5*sqrt(r^2+R^2) where r=inner radius of the tube and R=outer radius.
Now at the lower frequencies the air column in the tube will play a factor and can cause a noticable 'beating' or wah wah sound. The formula for the air column is f=nv/2(L+0.3d) where n= the mode, in this case 1, v=speed of sound in air (343m/s), L=length and d=inner diameter. Rewriting for L, L=(343/2f)-0.3d. Cut the tubes slightly long and file to tune. You can drill small holes in the pipe at 90° to each other to raise the air frequency to match the metal frequency (a search for tuned wind chimes, found this link http://windworld.com/features/tools-resources/making-self-resonating-chimes/ ). Attach the tubes with elastic cord or rubber bands 0.224L from each end and not resting on anything solid, use nails or something similar and give the bands or cord a couple of twists to keep the tube from touching the nails. Here is an example of piano key frequencies tuned to A440 http://en.wikipedia.org/wiki/Piano_key_frequencies. I suspect the reason for some bamboo chimes to have part of the lower end cut away is not just esthetics, but air column to bamboo tuning, but I haven't researched that to be sure.
Do a search for other scales and tunings and have fun.
PS the formulae and referenced web pages are not mine, I found them using various searches on the internet.
As to the question of an electric xylophone, one could rig up several relay strikers from old doorbell chimes to strike the tubes.