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While going through some questions given in my book I came across

Why is the note produced by an open organ pipe sweeter then that produced by a closed organ pipe?

The answer given was:

"The note produced by an open organ pie consists of both odd and even hapmonics but the note produced by a closed organ pipe consists of only odd harmonics. Due to the presence of larger number of overtones or harmonics, the note produced by an open organ pipe is is sweeter."

I disagree with the answer. If we assume two organ pipes of the same length $\ l$ the first one open and the second one closed, the harmonics of both the organ pipes are:

Open: $\frac{v}{2l}$, $\frac{2v}{2l}$,$\frac{3v}{2l}$, etc with a difference of $\frac{v}{2l}$ between successive harmonics

Closed: $\frac{v}{4l}$, $\frac{3v}{4l}$,$\frac{5v}{4l}$, etc with a difference of $\frac{v}{2l}$ between successive harmonics.

In both cases the harmonics form an AP with the same common difference. Just the first term is different. Then how can we say that the open organ pipe would have larger no. of overtones? According to this logic the very statement that the note produced by an open organ pipe is sweeter then that produced by a closed organ pipe seems false. Where am I going wrong?

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If you start with the same fundamental frequency for both open and closed, say $f_0$, you'll notice that for the Open pipe, the harmonics are: $$f_0, 2f_0, 3f_0, ...$$ While for the closed, $$f_0, 3f_0, 5f_0, ...$$ And you'll notice that the closed pipe has greater spacing of harmonics compared to the open pipe.
That is for a given single note. You have to lower the note of the closed pipe by $\frac{1}{2}f_0$ (1 octave) to produce the same spacing of harmonics as the open pipe.

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    $\begingroup$ in order to take the fundamental frequency equal one organ pipe would be double the other. Then the sweetness of the open organ pipe is not because of it being open but because it has a shorter length... $\endgroup$ – oshhh Aug 25 '16 at 7:43
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    $\begingroup$ @OsheenSachdev Yes. You have to take the fundamental frequency equal, because this is the frequency we associate with the "pitch" of a note. Since we are comparing the same note produced by open or closed pipe, then we should use the same "pitch" when comparing, and by doing so, you change the length of either closed or open pipe. $\endgroup$ – philip_0008 Aug 25 '16 at 7:49
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(https://i.stack.imgur.com/HbJQh.jpg) And hence there are more tones in an open organ pipe and it is thus sweeter. But sweetness of a sound or tone is subjective to the listener and hence cannot be calculated or stated. Assumption: The listener feels that the more overtone in a sound more is it sweet.

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