What length to cut pipes for a xylophone?
March 18, 2011 7:14 PM   Subscribe

Need help with some math: figuring out pipe lengths for a xylophone for my daughter's school project.

I wrote out a lot of text which is overwrought so let me just summarize: I failed to find the proper math to use to find good pipe lengths for this project. The pipe we used was electrical conduit from Home Depot, I think it was $2.91 for 12 feet of it. Her total budget for this project is $12.50, aside from donated scraps from her father and me (the frame is built, it's fine, we just need the pipes to be of the proper length).

GOAL: 8 notes, of the C scale. She needs at least one octave to pass this assignment. Something like 15% - 20% of her grade depends on it being actually in tune, or close to it.

References I have referred to:

The table of ratios in this wikipedia article on equal temperament, speaking mostly of the "Decimal value in 12-TET" column.

The list of frequencies for piano keys in this wikipedia article on piano key frequencies

I came up with math that didn't work. I cut the pipes (a very arduous process with a hacksaw, so I would like to minimize any re-cutting necessary) and have the following data to describe the notes we ended up with:
#                   Length (cm)	       Note + cents off
1			30.8		C#6 +10
2			27.2		F6 +25
3			24.1		A6 +26
4			22.8		B6 ----> this one is actually good! (1975.53 hz)
5			20.5		Eb7 -15
6			17		A7 -25
7			15.7		C8 -36
8			15.3		C8 +47

I used AP Tuner to determine the notes. Maybe I would have better luck with something that showed me the actual frequency, but I was unable to find a tuner like that. I do not know how many hz the differing values of cents have at different notes.

I tried a simple relationship, such as: length of pipe * frequency = some constant, but this did not square with the data of the existing pipes and their notes when I ran the numbers in Excel. Even 2 * length = half the frequency didn't seem to work, so I am really at a loss here.

Can anyone lend me a clue here? Or is this too complex and we are pretty much just going to have to cut randomly and hope for the best? B6 is definitely right in tune, can we work some numbers off of that?

This is for a 6th grade magnet program. Ability to show the process used to the devise the pipe lengths is a definite plus.
posted by marble to Education (12 answers total) 10 users marked this as a favorite
Best answer: Experimental Musical Instruments' Free-Bar Length Calculator!
posted by moonmilk at 7:20 PM on March 18, 2011 [3 favorites]

Wikipedia suggests something like:

f = nv / 2 (L+0.8d)


f is the frequency (eg, 440 for A440)
n is the resonant node (I assume you would just leave this at 1 for the fundamental)
v is the speed of sound (wiki has this at 343 meters per second )
L is the length of the pipe
d is the diameter of the pipe
posted by jenkinsEar at 7:33 PM on March 18, 2011

Response by poster: moonmilk I think that is exactly what we need, thank you! I'll cut some tomorrow and see if it is accurate.

odinsdream I am really not feeling like buying yet another tool for this and actually the hacksawing sucks but I have refined my technique a bit after doing it eight times already so it's not *quite* as bad as it was at first.

And this magnet program is crazy with the projects they give as assignments (the last one was a clock). I think they basically assume heavy parental help (for instance, I think she's too young for power tools so I did the drilling for the screws that separate the pipes). She's only eleven, after all. With the reference moonmilk pointed to, that's adequate justification for the pipe lengths that we'll use, so she's covered (she will have to list her references).

I am really quite amazed at the level of acoustic physics they are teaching her at this age. I didn't get any of this information til at least high school. I am lucky she is in the magnet program; the public school system here sucks. The tradeoff is they tend to bury the children with loads more work than they need in order to learn the material, in order to appear "tough", I guess. But I'll stop there and won't go into full rant mode.
posted by marble at 7:43 PM on March 18, 2011

Love to see a picture when you all are done!
posted by Marky at 10:59 PM on March 18, 2011

jenkinsEar, that Wikipedia formula is for resonance of air inside a tube. Example: an organ pipe. You get that kind of resonance by getting the air in the tube to move. A xylophone, however, makes sound by striking the tube, which makes the tube itself vibrate. This is a completely different kind of vibration, so the Wikipedia formula doesn't apply here.
posted by exphysicist345 at 12:37 AM on March 19, 2011 [1 favorite]

exphysicist345: In that case, though, shouldn't the length of the tube be exactly inversely proportional to the frequency? I'd assumed marble's original problem ("I tried a simple relationship, such as: length of pipe * frequency = some constant" — shouldn't that have worked?) was due to end effects (which I additionally assume are significant for air-in-tubes but not for vibrating tubes/bars).

I suppose the other problem might be that electrical conduit isn't perfectly uniform... it's not really precision-manufactured. I'd expect it to be close enough, though.

I do not know how many hz the differing values of cents have at different notes

One cent is 1/100 of the distance between notes; you can probably just approximate linearly but to be precise it'd be an exponential like the rest of the equal-tempered scale. The scale is a compromise musically but it's straightforward mathematically: the ratio of the frequencies of two notes is 2 raised to the power of (1/12 of the number of semitones between the notes). Twelve steps makes exactly an octave; other numbers of steps come out pretty close. (2 to the power of 1/12 is ~1.0594, and you can see that the ratios of adjacent notes in the wikipedia page match that.)

Regarding getting the right lengths, what I'd suggest is cutting all your tubes based on the math but slightly long, then grinding/cutting them down until they reach the right frequency. This is probably only a workable plan if you have some sort of powered grinder (bench grinder, dremel, ...?).
posted by hattifattener at 12:58 AM on March 19, 2011

Sensible eleven year olds are not too young for power tools, especially something like a drill. Not if they know how to use them and the dangers involved.

Crazy-irresponsible eleven year olds, sure.
posted by polyglot at 1:27 AM on March 19, 2011

This book might have some ideas. I'd check for you but my copy is lent out.
posted by mecran01 at 2:50 PM on March 19, 2011

Best answer: hattifattener: I agree that the uniformity of electrical conduit ought to be good enough.

From a little web surfing, it looks like the frequency x (length squared) = a constant, and that agrees with the results of the length calculator in moonmilk's link. This is for the fundamental (lowest) frequency of vibration. The particular vibration we're interested in (for a xylophone) is the transverse mode. This is where the middle of the bar vibrates, and the ends wiggle a little. See the nice animation in the first figure on Sarah Tulga's Glockenspiel web page. Further down that page, you can see that the points of the tube which don't move (the anti-nodal points) are located a distance (0.224 x length) measured from each end. This is where you should suspend the tube (using string, rubber bands, whatever) to produce the loudest, clearest note.

There are lots of videos on YouTube about building a xylophone. (It seems to be a popular activity!)
posted by exphysicist345 at 5:34 PM on March 19, 2011

Below are the frequencies and lengths in centimeters I got using freq X length^2 = a constant formula (and calculating the constant from your one right-on note, B6=22.8 cm).

A couple of notes:
  • In the examples you gave above, some lengths are sharp of their predicted values below and some are flat. Maybe your material is a bit inconsistent or maybe the holes you've drilled are affecting the pitch? (The holes will affect the pitch more if they are not right at the anti-nodal points--as exphysicist described above, .224 X length from each end, or if they are different sizes, or if they are different sizes in proportion to the overall size--which of course they are.)
  • Given that, I would second hattifattener in suggesting cutting them a little long, then trimming gradually to tune them. Ideal would be using a grinder, but even a simple file might work. Grind a minute amount, check, grind, check, etc. Make sure you're checking in exact final conditions (ie if it's going to have holes drilled, drill them already, if it's going to lie on some kind of a box, lay it in position before checking, etc--all those things can affect tuning in a minor or (in the case of holes) major way.
  • If you somehow come out sharp of your pitch you can lower the pitch by adding mass to the tube. It might be as simple as putting a few twirls of tape around it--the logical places being either the ends or the middle. (Note that this might change the sound quality some.)
  • If you can afford the extra pipe, I would suggest putting the xylophone in the lower octave listed below (ie, the C4-C5 octave, the octave starting at middle C). Reason: You basically have twice the slack in this octave. That gives you twice the space for mistakes and adjustments and it will make tuning twice as easy. As a bonus, the end result will also be more in your daughter's natural vocal range.
    Note    Freq(hZ)Length(cm)
    C5	523.25	44.30
    C#	554.37	43.04
    D	587.33	41.82
    D#	622.25	40.62
    E	659.26	39.47
    F	698.46	38.34
    F#	739.99	37.25
    G	783.99	36.19
    G#	830.61	35.16
    A	880.00	34.16
    A#	932.33	33.19
    B	987.77	32.24
    C6	1046.50	31.33
    C#	1108.73	30.43
    D	1174.66	29.57
    D#	1244.51	28.73
    E	1318.51	27.91
    F	1396.91	27.11
    F#	1479.98	26.34
    G	1567.98	25.59
    G#	1661.22	24.86
    A	1760.00	24.16
    A#	1864.66	23.47
    B	1975.53	22.80
    C	2093.00	22.15

posted by flug at 7:28 AM on March 20, 2011 [1 favorite]

Response by poster: Thanks for all the help, everyone! I cut the pipes (again) today, re-cutting a few based on the lengths given by that calculator moonmilk linked. I then tested the notes by using AP Tuner, and they all came out to be what they were supposed to be, yay! Not perfectly in tune by any stretch of the imagination, though.

I made sure my daughter did the figuring of which pipes to re-cut and what lengths - I had her put all the numbers in a spreadsheet and so forth, so I feel she has a good grasp of the whole set of procedures. Later in this project she will have to do a big writeup of her design process, stuff she adjusted and why, and all of that. These projects are really involved and they have several grades on different days and have to bring their project to school several times. Luckily this one is a lot sturdier than the clock I helped her with.

We had to go to Home Depot again to get drawer knobs to use for the sticks, and I noticed a cutting tool for electrical conduit pipe but it was $30 so I elected to just stick with the hacksaw, even though it's labor-intensive and not very accurate.

Just extra information on how we did the sticks, if anyone happens to be interested:

We got some small low-profile metal knobs for $1.97 apiece (something like these I think, apparently this style is referred to as a milk bottle shape). I drilled holes in the end of some dowels that were big enough to screw the knob screws into but the threads still got a good bite into the wood. I screwed the screws in pretty deeply, then hacksawed off the screw heads and screwed the knobs on the end. It worked really well, just as I had hoped.

And I'll try to take some pictures the next time she comes over, sorry I forgot to - I was just so glad to be done with the hacksawing. It's not very pretty, but at least it works.
posted by marble at 3:49 PM on March 20, 2011 [1 favorite]

Response by poster: Oh, and I just want to address flug's response: I like the idea of a lower octave (and think it sounds better), however this would make the tubes a lot longer, and she has to take this project back and forth to school on the bus several times, and doesn't have a designated place to store it at school, unfortunately. Right now it fits into a reusable grocery bag.

And now that I think of it, the frame isn't really big enough to support tubes much longer than the ones we used. A bigger frame would make it harder for her to tote the thing around as much as she needs to.
posted by marble at 3:55 PM on March 20, 2011

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