Origin of 'harmonics' in Helmholts-type resonators

[Robert Hickman]

I'm a maker of concert-tuned transverse ocarinas, which are a kind of Helmholtz resonator, however that does not tell the whole story as they are driven with an air-reed. This set up seems to differ from what is defined by the Helmholtz resonator equation in that it does not have a single resonant frequency. The pitch is highly dependent on air velocity and freely varies up/down by several semitones.

Now the main thing which is causing trouble for me is that of 'harmonic' frequency generation. From what I understand a 'pure' Helmholtz resonator has no harmonics, yet ocarinas can and do generate 'overtones'. This is especially if they are overblown, but they also show up while playing steady state notes. If you look at an FFT graph of the different notes, they all have notable peaks of overtones. The following image shows a particularly bad example:

http://pureocarinas.com/wp-content/uploads/2015/02/ocarina_harmonic.png

The problem I have is that I have no understanding of the physical mechanism by which these originate, it just shows up randomly in my work. Depending on the relation to the fundamental tone, they can give the instrument an unpleasant 'shrieking' tone. So far I've been able to get away with making minor changes to the voicing, but it's unpredictable and time consuming. It shows up unpredictably in ocarinas of different base tunings.

Relating to the above image, this is of an alto C ocarina playing it's top D note, and it has an unpleasant screeching tone. It's played by opening all holes besides right thumb/left pinky. As this appears to show a peak at the octave of the fundamental, one idea that I had was if the chamber is forming a standing wave due to it's length in relation to the wavelength.

The length of the chamber is approximately 12.7CM and the problem note is a D6 which has a wavelength of 29.37CM. The first octave of this is 14.9CM which is close to the length of the chamber, yet not exactly. So I'm uncertain if this is relevant or not.

 

[Quantum Defect]

I'm a maker of concert-tuned transverse ocarinas, which are a kind of Helmholtz resonator, however that does not tell the whole story as they are driven with an air-reed. This set up seems to differ from what is defined by the Helmholtz resonator equation in that it does not have a single resonant frequency. The pitch is highly dependent on air velocity and freely varies up/down by several semitones.

Now the main thing which is causing trouble for me is that of 'harmonic' frequency generation. From what I understand a 'pure' Helmholtz resonator has no harmonics, yet ocarinas can and do generate 'overtones'. This is especially if they are overblown, but they also show up while playing steady state notes. If you look at an FFT graph of the different notes, they all have notable peaks of overtones. The following image shows a particularly bad example:

http://pureocarinas.com/wp-content/uploads/2015/02/ocarina_harmonic.png

The problem I have is that I have no understanding of the physical mechanism by which these originate, it just shows up randomly in my work. Depending on the relation to the fundamental tone, they can give the instrument an unpleasant 'shrieking' tone. So far I've been able to get away with making minor changes to the voicing, but it's unpredictable and time consuming. It shows up unpredictably in ocarinas of different base tunings.

Relating to the above image, this is of an alto C ocarina playing it's top D note, and it has an unpleasant screeching tone. It's played by opening all holes besides right thumb/left pinky. As this appears to show a peak at the octave of the fundamental, one idea that I had was if the chamber is forming a standing wave due to it's length in relation to the wavelength.

The length of the chamber is approximately 12.7CM and the problem note is a D6 which has a wavelength of 29.37CM. The first octave of this is 14.9CM which is close to the length of the chamber, yet not exactly. So I'm uncertain if this is relevant or not.

I found some interesting information here: http://www.campin.me.uk/Music/Ocarina/ ( a "neighbor" of yours in Scotland)

Including a link to this paper which shows excitation of high harmonics (in a simulation) with short pulses of air.

http://arxiv.org/pdf/0911.3567v1.pdf

All of the science references seem to be saying that the Helmholtz resonator is a very crude first approximation of the actual ocarina. In the paper above, it looks like that at short times, the high harmonics are indeed excited, but with time, the only oscillation that is left is the fundamental. With the high notes on your ocarinas, it may be that these higher modes don't decay away, or there may also be some non-linear physics going on, that keeps moving energy from the fundamental to the higher modes -- you see similar non-linear effects in trumpets/trombones played at high volume. There is a very nice BBC video of shock waves being produced from a trombone!

Since the helmholtz resonator has a kind of breathing mode, I am unsure what the higher modes look like. They could be transverse, with some nodal planes inside the ocarina, in which case the dimensional argument that you made might be leading to some kind of "truth" or the modes might have radial nodes, in which case, I am guessing that the length might not matter.

The good news is that the ocarina is a simple enough instrument that you might be able to interest some physicists in doing some actual research on the particular problem you would like to solve. There is a very strong and active musical acoustics program at the University of New South Wales that might have someone interested in working on this. They appear to collaborate with many artists. http://newt.phys.unsw.edu.au/music/ Here is an interesting paper where they investigated the unusual problems that tympani can give french horn players! So they are not averse to working on small projects like this when given an interesting puzzle: http://newt.phys.unsw.edu.au/jw/timpani-horn/timpani-horn.html

 

[Robert Hickman]

I found some interesting information here: http://www.campin.me.uk/Music/Ocarina/ ( a "neighbor" of yours in Scotland)

Including a link to this paper which shows excitation of high harmonics (in a simulation) with short pulses of air.

http://arxiv.org/pdf/0911.3567v1.pdf

All of the science references seem to be saying that the Helmholtz resonator is a very crude first approximation of the actual ocarina. In the paper above, it looks like that at short times, the high harmonics are indeed excited, but with time, the only oscillation that is left is the fundamental. With the high notes on your ocarinas, it may be that these higher modes don't decay away, or there may also be some non-linear physics going on, that keeps moving energy from the fundamental to the higher modes -- you see similar non-linear effects in trumpets/trombones played at high volume. There is a very nice BBC video of shock waves being produced from a trombone!

Since the helmholtz resonator has a kind of breathing mode, I am unsure what the higher modes look like. They could be transverse, with some nodal planes inside the ocarina, in which case the dimensional argument that you made might be leading to some kind of "truth" or the modes might have radial nodes, in which case, I am guessing that the length might not matter.

The good news is that the ocarina is a simple enough instrument that you might be able to interest some physicists in doing some actual research on the particular problem you would like to solve. There is a very strong and active musical acoustics program at the University of New South Wales that might have someone interested in working on this. They appear to collaborate with many artists. http://newt.phys.unsw.edu.au/music/ Here is an interesting paper where they investigated the unusual problems that tympani can give french horn players! So they are not averse to working on small projects like this when given an interesting puzzle: http://newt.phys.unsw.edu.au/jw/timpani-horn/timpani-horn.html

Yes, I know Jack and have seen that paper before. Looking through it I can see some issues with the models the author used for the simulations. Fir instance the following model has a substantal curve-in below the voicing which I've known for a long time to cause overtone issues. Also the defined chamber is far wider than high, which is also problematic though for different reasons. Having the chamber 'squat' tends to be too ristrictive and makes the finger holes enormous. The edge should also be more centered with regard to the windway and have a flat area at it's end. Sharp edges are problematic.

As far as I can see, a verry similer model was used in the 3d simulation. Though it is verry crude and nothing like a real transverse ocarina's interior.

Also could you explain what you mean by 'breathing mode'?

I'll contact the UNSW, though working with someone more local would be more practical. Will contact local universities as well.

Regarding OpenFOAM, I have looked at it myself although it seems to be horendiously complicated, and this is coming from someone who is used to working with programming languages. I coulden't make heads or tails of it and couldent find much of any documentation on it eather.

CFD is something I've been wanting to use for a long time, and I know how to 3d model so that part isn't an issue. Though I couldent find any way of getting from the modler (blender) into a format that openfoam understands.

 

[Quantum Defect]

Yes, I know Jack and have seen that paper before. Looking through it I can see some issues with the models the author used for the simulations. Fir instance the following model has a substantal curve-in below the voicing which I've known for a long time to cause overtone issues. Also the defined chamber is far wider than high, which is also problematic though for different reasons. Having the chamber 'squat' tends to be too ristrictive and makes the finger holes enormous.

As far as I can see, a verry similer model was used in the 3d simulation. Though it is verry crude and nothing like a real transverse ocarina's interior.

Also could you explain what you mean by 'breathing mode'?

I'll contact the UNSW, though working with someone more local would be more practical. Will contact local universities as well.

With the Helmholtz resonator, the pressure in the body is uniformly changing up and down, and as it does so, the plug of air in the opening is oscillating up and down. There are no nodes, and you do not have a situation where the pressure is rising in one part of the resonator and falling in another (with a node where the pressure is not changing).

With the ocarina, you have an almost Helmholtz oscillator, where I am made to understand that you vary the size of the plug of air, by varying the size of an effective hole. You would like the separate air plugs to be moving synchronously, i.e. the lowest mode being excited. I think that you could investigate whether the problem is with the shape of the ocarina by building a single-hole ocarina that plays only the problematic note -- i.e. with a single hole that gives the same effective area as the real ocarina with the fingering that gives you the problematic note.

You could invesitgate whether there is some transverse node being excited if you measured the pressure oscillations at the opposite ends of the ocarina. If you see component oscillations that are out of phase, this might be indicative of the presence of higher harmonics. You would also see effects of having a transverse node if some of the finger holes are near the node -- the oscillations of the air plug near the node will be smaller, and perturbations to the hole will not change the oscillation as strongly as perturbations to holes near and antinode.

There is a famous demonstration of the placement of a node in the trumpet by the former first trumpet of the Chicago Symphony. Ususally when you play the trumpet, if you open the spit valve, you get a lousy sound -- the spit valve is situated near an antinode. However, on the C trumpet that the player used, there was a fortuitous node at the placement of the hole for the spit valve. He could open and close the spit valve while playing open C, and the sound would change only slightly.

What I meant by a breathing mode is a mode where the nodes are radial. So, rather than having a nodal plane dividing the resonator, you have a closed nodal surface -- a smaller ocarina-shaped surface separating the resonator volume into nested sections.

 

[tech99]

I'm a maker of concert-tuned transverse ocarinas, which are a kind of Helmholtz resonator, however that does not tell the whole story as they are driven with an air-reed. This set up seems to differ from what is defined by the Helmholtz resonator equation in that it does not have a single resonant frequency. The pitch is highly dependent on air velocity and freely varies up/down by several semitones.

Now the main thing which is causing trouble for me is that of 'harmonic' frequency generation. From what I understand a 'pure' Helmholtz resonator has no harmonics, yet ocarinas can and do generate 'overtones'. This is especially if they are overblown, but they also show up while playing steady state notes. If you look at an FFT graph of the different notes, they all have notable peaks of overtones. The following image shows a particularly bad example:

http://pureocarinas.com/wp-content/uploads/2015/02/ocarina_harmonic.png

The problem I have is that I have no understanding of the physical mechanism by which these originate, it just shows up randomly in my work. Depending on the relation to the fundamental tone, they can give the instrument an unpleasant 'shrieking' tone. So far I've been able to get away with making minor changes to the voicing, but it's unpredictable and time consuming. It shows up unpredictably in ocarinas of different base tunings.

Relating to the above image, this is of an alto C ocarina playing it's top D note, and it has an unpleasant screeching tone. It's played by opening all holes besides right thumb/left pinky. As this appears to show a peak at the octave of the fundamental, one idea that I had was if the chamber is forming a standing wave due to it's length in relation to the wavelength.

The length of the chamber is approximately 12.7CM and the problem note is a D6 which has a wavelength of 29.37CM. The first octave of this is 14.9CM which is close to the length of the chamber, yet not exactly. So I'm uncertain if this is relevant or not.

 

[tech99]

Can you explain what the curves mean, and what is the significance of 474 Hz? Are you saying this is the wanted fundamental?

 

[Quantum Defect]

Can you explain what the curves mean, and what is the significance of 474 Hz? Are you saying this is the wanted fundamental?

http://www.phy.mtu.edu/~suits/notefreqs.html

The fundamental that is desired is D6 (1175 Hz, 29.37 cm). This is the largest yellow peak in the spectrum. Helmholtz resonators nicely oscillate at a single frequency. I.e. you should see only a single peak.

See the spectrum shown here: http://newt.phys.unsw.edu.au/jw/Helmholtz.html

The problem with the OP's ocarina playing this note is that there are additional peaks in the spectrum that are in bad spots. Musically-sounding notes in most instruments will have the fundamental as well as overtones (e.g. 440 Hz, 880 Hz, 1320 Hz ...) If the modes do not line up, the sound is "not good" to our ears. On stringed instruments, the modes automatically line up, and wind instruments go through all sorts of modifications to reproduce this. On looking at the graph some more, it may not be so much the presence of additional peaks that is the problem as it is that the peaks are in the wrong spots. How do these peak positions compare with the actual multiples of the fundamental?

In brass winds, the purpose of the bell flare is to get the higher modes to line up with the fundamental. The modes of oscillation for a straight tube do not give you the nice harmonic relationship. This is why the straight tube sounds "off".



[The explanation of what the funnel does is wrong in this video.]

Benade has an interesting discussion of what the bell flare does in his books, and some of the online material that is out there.

 

[Robert Hickman]

Will read through this in detail in the morning- half asleep right now.

This is the largest yellow peak in the spectrum. Helmholtz resonators nicely oscillate at a single frequency.

In a real life ocarina you actually do want some amount of overtones, as a pure sine sounds verry sterile and is extreamly difficult to blend with other instruments, the slightest pitch error is really obvious. Asian ocarinas tend to go for this 'pure as possible' ideology, and for the most part I don't like listening to them for long. Mine lean more towards the italian tradition with a more 'textured' sound.

It is a good point about getting the overtones to line up in a harmonious way, but first need to know where they come from. You can't control what you don't understand.

The 'pure' sound can be obtained with a verry small voicing, howeaver this tends to 'strangle' the instrument and produce verry poor high notes (a common issue in asian ocarinas).

The ocarina is actually pritty complicated despite it's simple looks. It needs to be shaped in a certain way to work, while also having good ergonomics that will work across a wide range of hand sizes, and also look good astetically. The two can colide, for instance in soprano ocarinas, you must have a long-narrow chamber or the holes end up being really cramped. Yet this is problematic as the chamber itself starts to become a limit on the holes, which cannot be larger than the diamiter of the chamber.

These long chambers are also verry prone to swapping into higher overtones. Yet this itself is not the whole story as the 'bad' note shows up in my alto C ocarina, but does not surface in my alto D which is considerably narrower with respect to its length. In soprano G and higher, the overtones can be used to add 2 extra notes to the top of the range.

-

The spectrum was generated by an EQ plugin of the REAPER DAW, it's not a spectrum analyser 'per-se', but is all I have right now. The frequency showing is just where the mouse cursor is placed and has no relation to the peaks.

This is how my ocarinas generally sound, this recording is of my alto D which does not have any major 'bad' overtones.

https://soundcloud.com/robert-hickman/les-poules-huppees-crested-hens

 

[tech99]

It looks as if the intended note is 1.2 kHz and there are significant harmonics at 2.4kHz and 3.6kHz (approx). With an overtone, I expect the resonator to oscillate close to the frequency of a harmonic, and the fundamental to be absent. Your spectrum does not look like overtone oscillation. A reason the sound is rough may be that the third harmonic is accentuated - the second harmonic will sound smooth. Notice also that harmonics are exact multiples of the fundamental frequency and their frequency will not vary with the tuning of the pipe. Only the amplitude changes due to the response of the pipe. It is abnormal for all harmonics to be accentuated in a uniform transmission line resonator - normally we see odd harmoncs. In your case I think you have a complicated resonator system where a tuned pipe is lump loaded by the mass of air in the various holes. Maybe you could excite the resonator with white noise and study the emerging spectrum?