Measured Spectrum of Stopped Wood Organ Pipe Shows ALL Overtones

[greypilgrim]

TL;DR

Why are the even-numbered overtones not suppressed?

Hi.

I have a wooden organ pipe with removable stopper and wanted to confirm the theoretically predicted spectra using a smartphone app. The formulae for the fundamental frequencies for the open and stopped cases work quite well, with the stopped pipe being about an octave lower.

However, theory also predicts that the even-multiple overtones should be absent in the stopped case (or at least clearly weaker than the odd ones). The spectrum does not exhibit this at all, I can clearly see all multiples of the fundamental with no clear pattern for the amplitudes. Same with a clarinet (which should behave like a stopped pipe as well); and I also used different apps and an iPad at some point.

Of course real instruments do not behave ideally, but I observed no indication of suppression of even-multiple overtones at all. Why?

 

[Andrew Mason]

TL;DR: Why are the even-numbered overtones not suppressed?

I have a wooden organ pipe with removable stopper and wanted to confirm the theoretically predicted spectra using a smartphone app. The formulae for the fundamental frequencies for the open and stopped cases work quite well, with the stopped pipe being about an octave lower.

However, theory also predicts that the even-multiple overtones should be absent in the stopped case (or at least clearly weaker than the odd ones). The spectrum does not exhibit this at all, I can clearly see all multiples of the fundamental with no clear pattern for the amplitudes. Same with a clarinet (which should behave like a stopped pipe as well); and I also used different apps and an iPad at some point.

Of course real instruments do not behave ideally, but I observed no indication of suppression of even-multiple overtones at all. Why?

Seems that you may have a leaky pipe stopper. If the stopper is not tight to the wood you will get even and odd harmonics because the pipe is acting as both a closed and open pipe.

AM

 

[Orthoceras]

TL;DR: Why are the even-numbered overtones not suppressed?

However, theory also predicts that the even-multiple overtones should be absent in the stopped case (or at least clearly weaker than the odd ones). The spectrum does not exhibit this at all, I can clearly see all multiples of the fundamental with no clear pattern for the amplitudes.

In support of your point, see below two spectra I recorded just now from a wooden organ pipe in two situations. In the closed pipe the third harmonic is suppressed, while the second harmonic is present. The stopper is not leaky. Apparently the spectrum of a fairly wide, real pipe is more complicated than the ideal one.

 

[Baluncore]

In the closed pipe the third harmonic is suppressed, while the second harmonic is present.

It would appear that the stop is positioned such that it cancels the 2kHz tone rather well.

 

[Andrew Mason]

In support of your point, see below two spectra I recorded just now from a wooden organ pipe in two situations. In the closed pipe the third harmonic is suppressed, while the second harmonic is present. The stopper is not leaky. Apparently the spectrum of a fairly wide, real pipe is more complicated than the ideal one.


View attachment 369682

A closed pipe with a fundamental frequency of 667 Hz should have overtones of 2001, 3335, 4669 Hz etc corresponding to odd multiples of the fundamental. The open pipe at fundamental 667 Hz should have overtones of 1334, 2001, 2668, 3335, 4002, 4669 Hz.

In a wooden pipe, the vibrations in the wood may suppress or enhance overtones. The wooden organ stops are designed (voiced) to absorb some overtones to produce a softer sound. This may be the reason the 2001 Hz overtone is suppressed.

However, resonance of the stopped pipe for the second and fourth overtones means that an antinode is present at the stop position. That means the sound waves for these overtones is not affected by the stopper’s presence. How have you determined that there are no leaks in the stopper?

AM

 

[Orthoceras]

However, resonance of the stopped pipe for the second and fourth overtones means that an antinode is present at the stop position. That means the sound waves for these overtones is not affected by the stopper’s presence. How have you determined that there are no leaks in the stopper?

(Shouldn't that be the second and fourth harmonics, instead of second and fourth overtones?)

When sliding the stopper through the organ pipe, it felt like a snug fit. In addition I tested for leaks in the stopper by blowing through the foot hole, while blocking the labium. No significant amount of air escaped.

 

[Andrew Mason]

(Shouldn't that be the second and fourth harmonics, instead of second and fourth overtones?)

The terms are interchangeable - at least for our purposes.

When sliding the stopper through the organ pipe, it felt like a snug fit. In addition I tested for leaks in the stopper by blowing through the foot hole, while blocking the labium. No significant amount of air escaped.

This is a fairly short pipe so there may be some end effects, lip anomalies, or pipe shape effects (e.g. different path lengths permitted within a square shaped pipe) that could be causing this. These would be reduced with a longer pipe. If you can find a longer wooden pipe I would suggest you try it with that.

AM

 

[Baluncore]

The terms are interchangeable - at least for our purposes.

After all, they only differ by one.

For this discussion, we should use the term fundamental, then second harmonic at twice the fundamental frequency, (the first of the even harmonics). Then third harmonic at three times the fundamental frequency, (the first of the odd harmonics).

Even harmonics are generated by asymmetrical clipping of the pressure amplitude, like a rectified sinewave.
Odd harmonics are generated by symmetrical clipping about zero, like a sinewave with flat tops and bottoms, or a square wave.

 

[Orthoceras]

After all, they only differ by one.

Yes, for an open organ pipe. For an ideal closed pipe the numbering would again be something else.

If you can find a longer wooden pipe I would suggest you try it with that.

I don't have other pipes. The OP might want to use your suggestion, it's his thread.

 

[Andrew Mason]

After all, they only differ by one.

??

Even harmonics are generated by asymmetrical clipping of the pressure amplitude, like a rectified sinewave.
Odd harmonics are generated by symmetrical clipping about zero, like a sinewave with flat tops and bottoms, or a square wave.

In an organ pipe, the harmonics do not result from clipping sound wave amplitudes. The fundamental and harmonics are generated by resonance within the pipe, which allows standing waves only in discrete multiples of half wave-lengths (open pipe) or discrete odd multiples of quarter wavelengths (closed pipe - using stopper). Resonance occurs when air vibrations created by air passing over the upper lip match a natural frequency of the air column (as determined by its length), creating standing longitudinal waves within the pipe.

AM

 

[Baluncore]

In an organ pipe, the harmonics do not result from clipping sound wave amplitudes.

The blown airflow switches path at the upper lip, (labium), alternately delivering air into and out of the pipe. That is a form of clipping in time.

The fundamental resonance results in pressure variations, which influence the path of air over the labium. The distortion of the fundamental, results in the generation of both odd and even harmonics.

 

[Baluncore]

After all, they only differ by one.

??

In music, for a simple system, the second harmonic of the fundamental, is usually the first overtone. The third harmonic then is often the second overtone.

Counting musical overtones, does not work well with the analysis of Fourier series. There can be independent overtones present that are neither odd, nor even harmonics, of a fundamental. A missing harmonic makes the overtone number meaningless in terms of odd and even harmonics.

When discussing the generation of harmonics, by the distortion of a fundamental, avoid the word overtone, as it leads to confusion.

 

[Andrew Mason]

The blown airflow switches path at the upper lip, (labium), alternately delivering air into and out of the pipe.

Right. That is how the vibrations originate.

That is a form of clipping in time. The fundamental resonance results in pressure variations, which influence the path of air over the labium. The distortion of the fundamental, results in the generation of both odd and even harmonics.

So I am confused by your previous reference to "Even harmonics are generated by asymmetrical clipping of the pressure amplitude, like a rectified sinewave. Odd harmonics are generated by symmetrical clipping about zero, like a sinewave with flat tops and bottoms, or a square wave." It is quite well understood that the tube length and whether it is open or closed determines the resonant frequencies. How does the length of the tube cause this clipping to occur?

AM

 

[Baluncore]

How does the length of the tube cause this clipping to occur?

The length of the tube decides the fundamental. Blown air is switched into the pipe, or out of the pipe, due to pressure changes at the labium, pressure changes due to the resonant fundamental.

Different parts of the fundamental get reinforced, while other parts are reduced. That distorts the fundamental, which produces harmonic distortion, phase locked to the fundamental. That distorted wave pressure variation feeds back on itself, to further distort the resonant wave.

You can excite a pipe at one harmonic of its fundamental, that is, have an integer number of cycles supported in the pipe, (transmission line), but that is not what happens in a blown pipe.

You are thinking of the harmonics, as independent standing waves, somehow excited on the line, but those harmonics are actually due to the distortion of the resonant fundamental.

So I am confused by your previous reference to "Even harmonics are generated by asymmetrical clipping of the pressure amplitude, like a rectified sinewave. Odd harmonics are generated by symmetrical clipping about zero, like a sinewave with flat tops and bottoms, or a square wave."

You need to look at the spectrum of a square wave, symmetrical about zero pressure. It contains only odd harmonics.
https://en.wikipedia.org/wiki/Sawtooth_wave
https://en.wikipedia.org/wiki/Square_wave_(waveform)

 

[Andrew Mason]

You need to look at the spectrum of a square wave, symmetrical about zero pressure. It contains only odd harmonics.

Yes, but the stopped wooden organ pipe does not produce a square wave.

I am not suggesting that the physics of musical instruments is simple. But according to all the literature I have found, the fundamental and harmonics of an organ pipe are determined by the length of the pipe and whether it is open or closed at one end. See, for example, https://newt.phys.unsw.edu.au/music/people/publications/Fletcheretal1983.pdf

There is no mention of clipping. A stopped pipe creates a resonant chamber for only odd harmonics of the fundamental so one could approach a square wave if the amplitudes of the higher odd harmonics were just right (amplitude inversely proportional to harmonic number). But, again, that is not the result of clipping the amplitude, however that might be done.

AM

 

[Baluncore]

Yes, but the stopped wooden organ pipe does not produce a square wave.

True, as that would require an instant step change in pressure. But there are filtered forms of square wave that have attenuated odd harmonics, and that are still missing the even harmonics. (See in depth; the Wikipedia reference at the end of this post).

There are two ways to analyse it. You can look at individual harmonics, or you can look at a distorted fundamental wave. The sum of the fundamental sine wave and the harmonic sine waves, is the distorted fundamental wave.

Fourier analysis involves extracting the amplitude and phase of all the sinusoidal components, that are present in a waveform. Fourier synthesis involves summing all of those sinusoidal components, to make up the waveform.

In the case of an organ pipe, the harmonics are phase locked to the fundamental, so the analysis of the sound generation must involve the harmonics being generated by the non-linear airflow switching at the lip, due to changes in pressure, dominated by the fundamental.

The waveform of the distorted fundamental, is directly related to the harmonic content. It should now be clear, that the organ pipe is a stopped or open transmission line, with a length that supports exactly one cycle of the fundamental, along with any distortion that it has accumulated through its generation, with resonance in the transmission line.

Now is the time to study the Fourier series, and to recognise the relationship between waveform symmetry in time and in amplitude, and the harmonic content.
https://en.wikipedia.org/wiki/Fourier_series

 

[berkeman]

Sorry, I have a question about the driving function of a wooden organ. What happens when the organist presses an organ key? Is there a compressed air system that generates the fundamental sine wave at that frequency and guides it toward the appropriate organ pipe? Or is there a big air chamber that is excited by sinusoidal driven air waveforms that lead to the pipes? How does this system work with chords?

 

[Baluncore]

Compressed air flows to the foot of a pipe while the key is pressed. Each pipe generates a different note, with harmonics, called overtones.

Take a look at page 3 of the reference in post #15.
OCR Extract:
"ORGAN PIPE is excited by air blown in from the bottom. The air is formed into a jet by the flue slit between the lower lip and the languid: a plate running across the pipe. When the air blows across the mouth of the pipe, it interacts with the column of air in the pipe at the upper lip, blowing alternately into and out of the pipe. Waves propagating along the turbulent jet maintain a steady oscillation in the air column, causing the pipe to “speak.” "