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Problem 43. Two organ pipes, a pipe of fundamental frequency 440 Hz, closed at one end, and a pipe of fundamental frequency 660 Hz, open at both ends, produce overtones. Which choice below correctly describes overtones present in both pipes?

a. After the first overtone of each pipe, every second overtone of the first pipe matches every second overtone of the second pipe.

b. After the first overtone of each pipe, every second overtone of the first pipe matches every third overtone of the second pipe.

c. After the first overtone of each pipe, every third overtone of the first pipe matches every second overtone of the second pipe.

d. After the first overtone of each pipe, every second overtone of the first pipe matches every fourth overtone of the second pipe.

e. After the first overtone of each pipe, every third overtone of the first pipe matches every fourth overtone of the second pipe.

I was trying to solve this problem. However, the concept of overtone is ambiguous. A search on the web gives me that an overtone is any frequency higher than the first harmonic. How can I use that to answer this question?

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    $\begingroup$ Hint: Pipes produce only sounds which are standing waves within them. It's not hard to deduce the frequencies (or "overtones") each pipe will produce from this. $\endgroup$ – ACuriousMind Sep 20 '14 at 16:20
  • $\begingroup$ I did list out the frequencies: for open-closed: 660, 1980, 3300, 4620... for closed-closed: 440, 880, 1320, 1760... I get no match... $\endgroup$ – yolo123 Sep 20 '14 at 16:29
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There's an error in that the type of pipe for each of the two fundamental frequencies as described in your comment don't match the problem description. The pipe with a fundamental frequency of 440Hz is open-closed, and the pipe with a fundamental frequency of 660Hz is open-open. You actually said "closed-closed", which isn't even an option, but even if that's taken to mean "open-open", you've still got the fundamental frequencies backwards from what they are in the problem description.

Assuming you meant to say "open-open" instead of "closed-closed", you've got it right that open-open produces all integer multiples of the fundamental frequency, and open-closed produces only odd multiples; see the Cylinders section of Wikipedia's Acoustic resonance article.

The term "overtone" does indeed mean any pitch higher than the fundamental frequency. But you can assume for this problem that the only overtones involved are harmonic overtones (overtones whose frequency is an integer multiple of the fundamental frequency), and ignore the fact that musical instruments can also produce inharmonic overtones.

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  • $\begingroup$ Hi, this still does not work. $\endgroup$ – yolo123 Sep 20 '14 at 17:24
  • $\begingroup$ Look here: for 440: 440, 1320, 2200, 3080, 3960, 4840... for 660: 660, 1320, 1980, 2640, 3300.... The concordance is not constant. The "right" answer given by teacher is e. But that does not make sense. Sometimes 4 overtones later there is a match, other times only 2, other times only 3. $\endgroup$ – yolo123 Sep 20 '14 at 17:26
  • $\begingroup$ Never mind! It's all good. $\endgroup$ – yolo123 Sep 20 '14 at 17:32

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