Reed physics - harmonics

[Little John]

With, say, a guitar it's easy to understand where the harmonics come from. A sting vibrates at its full length (giving the fundamental) but also at 1/2 its length, 1/3 of its length etc. This can easily be demonstrated by plucking a string and then placing a finger lightly at the centre point. The fundamental is damped and the dominant tone is the first harmonic an octave above. Similarly damping at the seventh fret (1/3 of the length) leaves the second harmonic sounding. You can even see this on some videos because the slow sampling rate aliases all the vibrations down to low frequencies, some of which are easily visible. Halving the length of the string doubles the frequency, dividing the length in three triples the frequency etc. So if f is the fundamental frequency the harmonics have frequencies of 2f, 3f, 4f, etc.

 

But what about a concertina? There is no equivalent mechanism. At a first approximation the reed is a cantilevered beam. As far as I can tell from internet searches the first vibration mode above the fundamental (f) is at 6.25f and the next at 17.5f - distinctly unharmonious frequencies and nothing like the harmonic series of a string or wind column. So it's not the natural vibration modes of the reed that generate the harmonics.

 

We know that if you add sine waves at f, 2f, 3f, 4f etc in the right proportions you can generate any waveform (square wave or sawtoothed waves for example). So I could believe that (in reverse) creating a waveform by chopping the airflow (my earlier question) could generate the harmonics. If it's not that, then what is it?

 

I acknowledge that I'm out of my depth with this. At the end of the day it doesn't really matter - I'm just happy to play the thing - but these questions have been niggling me for years and I'd love to know the answers.

[RAc]

Hi there John -

 

good question! I'm far from being an expert in this area, but I did some searching, and I believe there is an answer here:

 

http://www.sea-acustica.es/fileadmin/publicaciones/Sevilla02_edu01007.pdf

 

On page 5, it reads

"The  accordion  is  rich  in  overtones  of  high  amplitude.  The  overtones  of  the  air-actuated free reed are  not  produced  by  the  different  modes  of  vibration  of  the  reed-tongue,  because  these  are  not  multiples  of  the  fundamental,  but  by  the  non-linear  characteristics  of  the  throttling  action  of  the  tongue  upon  the  air  stream.  This  action  converts  the  steady  air  stream  into  a  pulsating  one  of  the  saw-tooth  type.  Thus,  the  acoustic  spectrum  contains  the  fundamental  and  both  odd  and  even harmonics."

 

I'm not saying I understand this, though...

 

[Richard Mellish]

22 minutes ago, RAc said:

Hi there John -

 

good question! I'm far from being an expert in this area, but I did some searching, and I believe there is an answer here:

 

http://www.sea-acustica.es/fileadmin/publicaciones/Sevilla02_edu01007.pdf

 

On page 5, it reads

"The  accordion  is  rich  in  overtones  of  high  amplitude.  The  overtones  of  the  air-actuated free reed are  not  produced  by  the  different  modes  of  vibration  of  the  reed-tongue,  because  these  are  not  multiples  of  the  fundamental,  but  by  the  non-linear  characteristics  of  the  throttling  action  of  the  tongue  upon  the  air  stream.  This  action  converts  the  steady  air  stream  into  a  pulsating  one  of  the  saw-tooth  type.  Thus,  the  acoustic  spectrum  contains  the  fundamental  and  both  odd  and  even harmonics."

 

I'm not saying I understand this, though...

 

I do understand it and I'd call it spot on.

[Little John]

On 10/5/2020 at 3:01 PM, RAc said:

"... The  overtones  of  the  air-actuated free reed are  not  produced  by  the  different  modes  of  vibration  of  the  reed-tongue,  because  these  are  not  multiples  of  the  fundamental,  but  by  the  non-linear  characteristics  of  the  throttling  action  of  the  tongue  upon  the  air  stream. ..."

 

Thanks, RAc. This is what I was trying to say so it confirms my suggestion.

 

On 10/5/2020 at 3:23 PM, Richard Mellish said:

"... This  action  converts  the  steady  air  stream  into  a  pulsating  one  of  the  saw-tooth  type.  Thus,  the  acoustic  spectrum  contains  the  fundamental  and  both  odd  and  even harmonics. ..."

 

I don't think this bit is right, though. A saw-toothed wave form is generated from even harmonics alone. (And a square wave by odd harmonics alone.) So because the spectrum contains both odd and even harmonics the waveform is not of the saw-tooth type.

 

On 10/5/2020 at 3:23 PM, Richard Mellish said:

I do understand it and I'd call it spot on.

 

Likewise I understand it, but in a hazy intellectual way rather than a nice intuitive way!

 

LJ

[Richard Mellish]

2 hours ago, Little John said:

 

A saw-toothed wave form is generated from even harmonics alone. (And a square wave by odd harmonics alone.) So because the spectrum contains both odd and even harmonics the waveform is not of the saw-tooth type.

I didn't think that was right, but I'm not feeling intelligent enough just now to do a Fourier analysis, so I've used an audio editing program to generate a sawtooth. See the attached screenshots of waveform and spectrum. I chose 110 Hz for the fundamental for a simple relationship to the numbers on the X-axis.

 

It shows that a sawtooth does contain both odd and even harmonics, though a sawtooth is only a very rough approximation of what a free reed produces.

[Little John]

55 minutes ago, Richard Mellish said:

It shows that a sawtooth does contain both odd and even harmonics, ...

 

 It does indeed. I was thinking of a triangular wave.

 

55 minutes ago, Richard Mellish said:

... though a sawtooth is only a very rough approximation of what a free reed produces.

 

Actually I think the sawtooth bit is a red herring. Here's a waveform produced by a real reed (courtesy of @alex_holden )

 

 

Just over five cycles and nothing like a sawtooth! What is striking about it is the predominance of the fourth harmonic (also evident in the frequency plot).

 

LJ

[DaveM]

To me, I think it's often better to think of these things in the time domain.  "the spectrum is composed of peaks at multiples of a fundamental frequency of 1/T" is equivalent to saying "the waveform (in the time domain) is periodic with period T".   Anything that manages to produce a periodically repeating waveform will be perceived as having a pitch.  Savart wheels (https://en.wikipedia.org/wiki/Savart_wheel) don't have anything that is vibrating à la strings, air columns or reeds, yet still sound pitched.  So as long as the reed manages to make a periodic disturbance in the air, the sound will have harmonic overtones.

 

Even for plucked strings, it's just as good to think of what is going on is something like this:  the string is plucked at a point, two disturbances, one in each direction along the string, propagate away from it, they bounce off the nut/bridge and so on.  The amount of time it takes for the wave pulse to make  a complete round-trip is the fundamental period.  It's that we get this kind of periodic behavior that gives the sound its pitch (of course you won't have the pulses run back and forth along the string without change unless the stationary modes have frequencies that are integer multiples of the fundamental...) Anyway, I find it useful to think this way in order to not get too hung up on the fact that the vibrational modes of reeds don't have harmonic overtones, and, as far as I know, the higher order vibrational modes aren't relevant for the sound production of the reeds.

 

I wonder if anyone has tried exciting a free reed without using airflow.  One could take a steel reed, stimulate it with a magnetic field at the correct frequency and see what a reed sounds like without airflow through the reed frame.  I'm still unclear whether we "hear the reed vibrating" or "hear the reed chopping the airflow through the frame".

[alex_holden]

3 hours ago, DaveM said:

I wonder if anyone has tried exciting a free reed without using airflow.  One could take a steel reed, stimulate it with a magnetic field at the correct frequency and see what a reed sounds like without airflow through the reed frame.  I'm still unclear whether we "hear the reed vibrating" or "hear the reed chopping the airflow through the frame".

 

Other than sustain, I don't see that it would be significantly different from what happens when you pluck a reed in a frame. It's a different tone and much quieter than when the reed is being driven by an air flow, but the pitch is approximately the same.

[Little John]

12 hours ago, DaveM said:

I'm still unclear whether we "hear the reed vibrating" or "hear the reed chopping the airflow through the frame".

 

This is essentially what my other post (Reed physics - frequency) is about. There seems to be a fair consensus that it is the chopping of the airflow we hear*. However, the airflow is chopped twice for each cycle of the reed, and yet we perceive the frequency of the note to be that of the reed; not double it which is the rate of chopping. (I think I have an intuitive answer now.)

 

*It's the same mechanism with beating reed instruments (e.g. saxophone) and brass instruments (e.g. trumpet). In the former the reed hits the mouthpiece and chokes the airflow for an instance; in the latter the lips close together to achieve the same effect. But I rather imagine in both cases the perceived frequency and the frequency of the reed/lips are the same.

[Don Taylor]

3 hours ago, Little John said:

However, the airflow is chopped twice for each cycle of the reed, and yet we perceive the frequency of the note to be that of the reed; not double it which is the rate of chopping

Are you saying that the real frequency of a note played is actually twice the frequency of the note that we perceive? 

 

If so,  when we play what we think and hear as an A4 (440 Hz) then in actuality the reed is playing at A5 (880 Hz)?

 

What would you expect that same note to show up as on a spectrum analyzer?

 

 

Edited October 18, 2020 by Don Taylor

[Richard Mellish]

I'm not at all convinced about the airflow being "chopped twice for each cycle of the reed". Even if the reed bends far enough down for its tip to be more or less clear of the bottom of the frame (which I am dubious about anyway) the air flow will still be more restricted than when the reed bends above the frame, so will be modulated at the frequency of the reed's vibration as well as some harmonics.

 

I posted the screenshots only to confirm that a sawtooth has odd and even harmonics. I agree that the sawtooth wave is not a good approximation of the actual sound from a reed, though it is similarly rich in many harmonics. The rest of the instrument also has a large effect on the proportions of the harmonics that finally emerge from the end.

[David Barnert]

On 10/18/2020 at 3:04 PM, Don Taylor said:

Are you saying that the real frequency of a note played is actually twice the frequency of the note that we perceive? 

 

If so,  when we play what we think and hear as an A4 (440 Hz) then in actuality the reed is playing at A5 (880 Hz)?

 

What would you expect that same note to show up as on a spectrum analyzer?

 

If I understand correctly what Alex said in the other thread (and at 3:30 in the morning, I’m too lazy to look for it now), for each cycle of the reed’s vibration there are two puffs of air, but they are not equal: a bigger puff when the reed tongue bends away from the reed shoe and a smaller one when the tongue bends down into the gap in the shoe. The brain perceives this as one long complicated wave having the same frequency as the reed’s vibration rather than two smaller and different waves that alternate.

[alex_holden]

8 minutes ago, David Barnert said:

If I understand correctly what Alex said in the other thread (and at 3:30 in the morning, I’m too lazy to look for it now), for each cycle of the reed’s vibration there are two puffs of air, but they are not equal: a bigger puff when the reed tongue bends away from the reed shoe and a smaller one when the tongue bends down into the gap in the shoe. The brain perceives this as one long complicated wave having the same frequency as the reed’s vibration rather than two smaller and different waves that alternate.

 

That concurs with my understanding. Or perhaps it's more like a sawtooth wave with a big tooth and a little tooth.

[Little John]

5 hours ago, David Barnert said:

... for each cycle of the reed’s vibration there are two puffs of air, but they are not equal: a bigger puff when the reed tongue bends away from the reed shoe and a smaller one when the tongue bends down into the gap in the shoe. The brain perceives this as one long complicated wave having the same frequency as the reed’s vibration rather than two smaller and different waves that alternate.

 

4 hours ago, alex_holden said:

That concurs with my understanding.

 

Mine too. It explains the apparent paradox that we're hearing the chopping of the airflow but at half the frequency.

 

4 hours ago, alex_holden said:

Or perhaps it's more like a sawtooth wave with a big tooth and a little tooth.

 

Except the waveforms you sent to me look nothing like a sawtooth! Here's the close-up version (showing the individual samples):

 

 

I would hazard a guess that the first big (upward) peak corresponds to the "big puff" and that the "noisy" bit a quarter of the way through the cycle is the chopping of airflow as the tongue enters the reed frame.

[Don Taylor]

On 10/18/2020 at 3:04 PM, Don Taylor said:

What would you expect that same note to show up as on a spectrum analyzer?

I do not have a spectrum analyzer but I do have Transcribe! on my PC and it has this handy visual feature for helping you find notes in a recording.

 

I recorded a single, sustained A4 on my New Model Crane and then asked Transcribe! to analyze that note.  Transcribe! shows you a frequency (the piano keyboard) and (I think) the amplitude for each of the notes that it hears.  It 'hears' the fundamental and all of the harmonics overtones - each of the peaks in the lower graph is a harmonic overtone.  The second and fourth are very strong!

 

 

The first peak from the left is A4, the second is A5, the third is E6 and so on.  All of the peaks correspond to successive harmonics overtones of the A4 fundamental.

 

If I understand Alex and John correctly (?) then would I not see something at A3 plus a second set of harmonics overtones starting at A3?

 

Edit added later:  I just realized that I can use Transcribe!'s equalizer to remove bits of the set of harmonics overtones and then get it to play back the result.  Yes, that E6 harmonic overtone is really loud, louder than the fundamental.

Edited October 25, 2020 by Don Taylor
E5 corrected to E6 plus a comment on its volume. Use overtone instead of harmonic (see later discussion about terminology).

[alex_holden]

2 minutes ago, Little John said:

I would hazard a guess that the first big (upward) peak corresponds to the "big puff" and that the "noisy" bit a quarter of the way through the cycle is the chopping of airflow as the tongue enters the reed frame.

 

I'm not so sure about that. We're looking at a sound recording, which is a record of sound pressure waves hitting a microphone diaphragm. I don't know that the speed of air flow through the reed frame directly translates to pressure waves. My guess is that changes in speed may be (more?) significant. So I would expect the instant when the tongue blocks the frame opening on the downswing and the flow suddenly drops to zero to cause the largest pressure spike. Also I would expect to see a much smaller spike at the halfway point between the big spikes, caused by the tongue blocking the frame more gradually on the upswing. What I don't get is why the second spike seems inverted.

 

On a bit of a tangent, I sometimes wonder what part (if any) the bellows play in amplifying the sound. They are essentially a pressurised bag, with the pressure inside pulsating as the reeds sound - would that not cause them to vibrate and act a bit like a loudspeaker cone?

[alex_holden]

15 minutes ago, Don Taylor said:

If I understand Alex and John correctly (?) then would I not see something at A3 plus a second set of harmonics starting at A3?

 

No, I think you're seeing what I would expect. It's an A4 note so the lowest significant peak is at A4. The reed tongue will be vibrating through 440 complete cycles (down and back up) every second.

[Don Taylor]

4 minutes ago, alex_holden said:

 

No, I think you're seeing what I would expect. It's an A4 note so the lowest significant peak is at A4. The reed tongue will be vibrating through 440 complete cycles (down and back up) every second.

In that case shouldn't I see a second set of harmonics starting at A5?

 

BTW.  Although I (and everyone else) hears this note as A4, the loudest sound is from the E6 so our brains are doing some tricks here.