Distortion created by our ears?

I have an interesting experiment, a nice story and a question for you...

Let's start with the experiment. Load a Hauptwerk sample set (let's stick to the St Annes for now) and draw a single 4' flute on one of the manuals. Play the A and the E in the middle octave. Alternate between playing these 2 notes sequentially and simultaneously.

Play slowly and listen carefully, and maybe you'll hear the same as I am hearing. When playing two notes simultaneously, you may hear a third note. It is a very soft tone, dissonant, a slightly buzzing sound, about two octaves lower. The sound disappears when you play one note only.

Again, play the A and E simultaneously and listen to the buzzing tone. Hold the A, and play a D# instead of the E. You are playing a descending sequence, but you'll notice that the buzzing tone is ascending!

What's going on here? This phenomenon is known as intermodulation distortion. Usually, we hear only multiples of frequencies: the harmonics. But with intermodulation, we hear difference tones.

I am using REW, a great piece of free software for all kinds of audio measurements and analysis, along with a XRef measurement microphone (a good mic for not so much money). In REW, I used the RTA (real time analyzer) while playing the E and A. In the spectrogram, the fundamentals and harmonics of the flute tones are clearly visible. The fundamental of the A is measured at 874 Hz (on St Annes, a0 = 436.5 Hz, so a very close match to twice this frequency), and the fundamental of the E is measured at 654 Hz. The D# measures at approximately 617 Hz. Nothing special here.

Next, I generated a soft sine tone, and I tried to match the frequency of the sine tone to the frequency of the buzzing sound. I got a match at 220 Hz for the A and E, and 256 Hz for the A and D#. Very nice, because this confirms that I am hearing difference tones indeed: 874-654 = 220Hz (spot on!); 874-617 = 257Hz (very close). The difference between A and D# is larger than between A and E, hence the frequency of the difference tone is higher, which explains the ascending tone when going from E-A to D#-A.

So far, so good. Or not so good. Intermodulation is not a nice kind of distortion. It generates spurious dissonant tones. What causes intermodulation distortion? I have to admit I'm far from fully understanding all the causes and physics of audio distortion. But I do know that intermodulation distortion is a significant topic in the design of speakers, amplifiers and DAC's.

So maybe I should improve my audio chain? I have a Focusrite Saffire Audio USB device, and M-Audio DSM3 speakers. Recently, I acquired a second-hand set of Genelec 8020c speakers. Time to put everything to the test...

Now having two pairs of speakers, I have the opportunity to play the A through one pair, and the E through the other. Playing through different speakers should eliminate the intermodulation distortion, if it is caused by the speakers. Moreover, the DSM3's have SPDIF connection (digital) and the 8020's have XLR connection (analog). So different signal paths too!

I absolutely expected that the intermodulation distortion would disappear. But to my suprise, it didn't. Apparently, the speakers are not the source of the distortion, and the signal path probably neither. Anything wrong in the Hauptwerk software? Improbable.

I did another striking observation. The frequencies of the buzzing sound did not show up in the RTA, although I could clearly hear it. Yet, when I played a sine signal at the same frequency, and equally loud as the buzzing tone, it showed very clearly in the RTA.

So apparently, the intermodulation distortion is not caused by my audio chain. It is not even observable in measurements. Only I hear it. Well, my wive hears it too. And you? Do you hear it with digital organs, in Hauptwerk, or even with real pipe organs? Maybe at the same frequencies, maybe you need slightly different notes.

Then where does the intermodulation distortion come from? I'm not sure, but I do have a hypothesis. It's our ears that are not perfect transducers. So perhaps, intermodulation distortion is basically a "defect" of our ears.

I have heard it on pipe organs also. Amplifiers normally have quite a low IM distortion rating, but speakers do not. I suspect it happens in any diaphragm, be it a speaker, or our ears. So it appears if we are pushing both tones through the same speaker, then we hear both the IM from the speaker and our ear drum.

Eric

I have heard it on pipe organs also.
Eric

Many thanks for the observation. I was very curious, but I don't have direct access to a pipe organ. So thanks again for confirming that it is not an audio hardware effect only.

Amplifiers normally have quite a low IM distortion rating, but speakers do not. I suspect it happens in any diaphragm, be it a speaker, or our ears. So it appears if we are pushing both tones through the same speaker, then we hear both the IM from the speaker and our ear drum.

Eric

Sure. To me it was a revelation that you might try to fight IMD by getting better or more speakers, but diminishing returns kick in earlier than expected because your very own ears spoil the game!

https://en.wikipedia.org/wiki/Combination_tone

Hi Jos

You are a sharp observer.
Could this answer your question. Tartini tones.

Regards Jan

Many thanks Jan! I think I never read about Tartini tones before. I find this article a bit more clear than the Wikipedia article: http://newt.phys.unsw.edu.au/jw/tartini ... ament.html

It seems that I am hearing Tartini tones indeed.

I agree with Jan that you are hearing "Tartani tones" which most organ people simply call "resultant tones." This is exactly how resultant organ stops work - such as creating a pseudo 32' stop from a 16' and rank a 10 2/3' rank.

Intermodulation distortion is something completely different. It is caused by nonlinearities somewhere in the system. Basically, we always assume that our software and audio systems are linear - which is actually a very simply concept. Suppose we have two wave shapes, which I will call A and B. If the result of passing these through your system is C and D respectively, then if we sum the two wave shapes first (A+B) the result of passing this new waveshape through your system should be (C+D).

As long as this relationship is true, we are fine.

As soon as it is not, the result is intermodulation distortion.

The more complex the waveshape (the more different it is than a sine wave), the more pronounced the distortion will be.

All digital systems approximate waveshapes because these systems cannot have infinite resolution - either in amplitude or in time. Hence, there is always some intermodulation distortion in these systems - including Hauptwerk. However, for a well-designed system, it is so minor that it is almost impossible to detect without laboratory instruments.

Even if Hauptwerk were perfect, we would still have some intermodulation distortion from our audio systems (amplifiers, speakers, cables, ...) Once again, these distortions are usually so minor that we cannot hear them. The exception comes when we push these systems to their limits. Playing music too loud for an audio system causes these distortions - which you can hear now as long as you haven't gone deaf first!

None of these effects is due to distortion caused by the human ear (or the brain). This is a whole other topic!

Thank you for this discussion. It confirms the truth that this universe is not totally predictable. How is it that as we tune around the cycle of fifths when we get back around to the original note it is out of tune with our starting note? All temperaments are attempts to reconcile this astounding discrepancy.

And there's the fact, well-known to piano tuners, that our ears hear high notes lower than they actually are when measured by machines. As we go up the scale we have to stretch the octaves progressively in the highest notes for our ears to perceive the notes as being in tune (the opposite as we go towards the low bass).

Human faces are not perfectly symmetrical. That is an essential part of their charm. Mathematical inexactitude is part of the vibrancy of life.

The world is mysterious !

RS

I agree with Jan that you are hearing "Tartani tones"

Intermodulation distortion is something completely different.

Thanks for your comment! Still learning a lot... In this scientific paper https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3578393 I read the following description:

That the human auditory system generates distortion of its own has been known since the 18th century. Armed with little scientific apparatus, probably nothing more sophisticated than a violin, the Italian musician Giuseppe Tartini discovered circa 1754 that when listening to a pair of tones, humans hear additional tones that are not present in the physical stimulus [1]. These additional tones, whose perceived magnitude can exceed 10% of that of the acoustic stimulus (at least 100 times what is common in high-fidelity equipment), have frequencies that consist of simple combinations of the frequencies of the primary tones. For example, for primary-tone frequencies of f1 and f2, where f2 > f1, the resulting ‘combination’, or Tartini, tones, also known as intermodulation-distortion products, will have frequencies of 2f1–f2, f2–fl, 2f2–f1, 3f1–2f2 and so on.

So it seems that intermodulation distortion and Tartini tones can be used interchangeably.

I agree that the causes are totally different (limitations of audio devices versus physiology). But the result is at least in part similar: difference tones are generated/perceived.

The more complex the waveshape (the more different it is than a sine wave), the more pronounced the distortion will be.

Still pondering... A complex wave shape contains many frequencies. Intermodulation of all these frequencies will generate many more additional frequencies. If audible, I guess we would perceive it as noise. The total power of the distortion signals would be much larger than the distortion originating from two simple sine tones. But noise is not as detrimental as a few dissonant frequencies. So maybe, in the case of audible distortion, the intermodulation distortion of simple waveshapes is "worse".

When I learned about them I was taught the terms "Difference" and "Summation" tones. Difference tones appear to produce the frequency that comes Subtracting the lower frequency from the higher one i.e low C of 16 at 32 would have a G at 48 if perfectly tuned, the difference is 16, low C of 32'. These work better low frequencies where the G is not as far from pure tuning as it is as you get higher.
A 440 would have a perfect fifth of 660, summing them gives 1100 or a pure C# an octave and third above the a. You should hear these to be more pronounced if you use a temperament where the fifth would be pure in tune (Pythagorean would work for Pure 5ths)
If I remember right, It is my understanding that these frequencies aren't actually present but are the result of pyscological/anatomy things.
John

How is it that as we tune around the cycle of fifths when we get back around to the original note it is out of tune with our starting note? All temperaments are attempts to reconcile this astounding discrepancy.

And there's the fact, well-known to piano tuners, that our ears hear high notes lower than they actually are when measured by machines. As we go up the scale we have to stretch the octaves progressively in the highest notes for our ears to perceive the notes as being in tune (the opposite as we go towards the low bass).

RS -- I thought I would pop in here add a little comment in response to the two observations that you made. I'm a retired engineer, and also spent several years as a Registered Piano Technician, so I have some background in these matters.

1) The tuning problem that equal temperament attempts to correct is based purely on mathematics, and applies to all keyboard instruments. A perfect fifth has a frequency ratio of 3/2 between the lower note and the fifth above it. If we walk up the circle of fifths, starting at C and stopping when we return to another C (seven octaves higher), the ratio between those two Cs will be (3/2) raised to the 12th power, or 129.746. On the other hand, if we simply walk up the seven octaves involved, the ratio will be a nice clean 2 to the 7th power, or 128.000. That's where the problem arises -- basically you can't tune both a perfect fifth (or fourth for that matter) and an octave at the same time. The various temperament systems try to partially overcome the difference between pure octaves and fifths.

2) The second item that you mentioned is something that mostly applies only to pianos, and not to organs. Here the problem is that a given piano string has a certain thickness. Because of that thickness, it doesn't vibrate as a perfectly flexible resonator, and the higher harmonics of any given note are just a little bit sharp of where they should be. This is called inharmonicity (you can read the Wikipedia article about it). When you try to tune an octave on any instrument, you are tuning the second harmonic of the lower note against the fundamental of the upper note. But, because the second harmonic of the lower note on a piano is slightly higher than a perfect 2:1 ratio, you need to tune the upper note slightly sharp in order to get it to match and sound in tune. That's where the stretch on a piano arises, and every piano has a slightly different characteristic in this regard. Organ pipes sound well-behaved harmonic series, so the octaves are not stretched. Similarly, with a harpsichord, the strings are so thin that they don't show inharmonicity and the octaves can be tuned without any stretch. This inharmonicity effect is why it is so difficult to get a piano and an organ to sound in tune when played together, especially in the upper octaves.

-- John

Try the original experiment and then vary the spacing between speakers first in the horizontal plane and then in the vertical plane. Of course this is easiest with small speakers that can be held in the hand. You will find that some effects are diminished significantly as the speakers are moved apart in the horizontal plane but not so much when separated vertically where our ears remain at similar distances from the speakers.

This is why it's more effective, in my opinion, to separate multichannel HW speakers by a foot or more and why stacking them vertically is not as effective.

I posted a similar experiment several years ago but had few responses.

Reading some more about Tartini tones.
It is believed now that it’s not created in the ears but that it is neurological.
Experimentation with splitting the tones, the first tone to the left ear and the second tone to the right ear still gives Tartini tones. So the tones must be created in the brains.
Interesting!

Regards Jan

There is an excellent video produced by a student at St. Olaf College on Tartini tones. She gives a good explaination and has a couple of demonstrations.

https://www.youtube.com/watch?v=gPGA2pkrabU