FPGA Central

Hi,

Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
23 kHz tone? I've been arguing with someone about this. He says that
you can, but that goes against what I know about the effect in the
frequency domain of adding signals together in the time domain, not
only that but I tried playing a 19,560 Hz tone together with a 20,000
Hz tone through my PC speakers (which have a decent frequency response
in that range which I tested with a chirp and a mic) and I don't hear
any 440 Hz, unless I push the volume loud enough but in this case for
obvious reasons the actual signal contains low frequency components. It
also goes against my (erronerous?) conception of human hearing which is
pretty much to me a linear system just like a microphone or a loud
speaker.

So it makes me highly sceptical, however when I googled what he pointed
me to it seems that the reality of ultrasonic difference tone hearing
isn't even controversial, so I'd like to know, by what mechanisms could
we hear a 2 kHz tone from a 21 kHz tone added to a 23 kHz tone other
than through distortion, and if we can, why in my experiment did I fail
to hear a 440 Hz tone?

Thanks in advance

Michel Rouzic skrev:
> Hi,
>
> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
> 23 kHz tone? I've been arguing with someone about this. He says that
> you can, but that goes against what I know about the effect in the
> frequency domain of adding signals together in the time domain, not
> only that but I tried playing a 19,560 Hz tone together with a 20,000
> Hz tone through my PC speakers (which have a decent frequency response
> in that range which I tested with a chirp and a mic) and I don't hear
> any 440 Hz, unless I push the volume loud enough but in this case for
> obvious reasons the actual signal contains low frequency components. It
> also goes against my (erronerous?) conception of human hearing which is
> pretty much to me a linear system just like a microphone or a loud
> speaker.
>
> So it makes me highly sceptical, however when I googled what he pointed
> me to it seems that the reality of ultrasonic difference tone hearing
> isn't even controversial, so I'd like to know, by what mechanisms could
> we hear a 2 kHz tone from a 21 kHz tone added to a 23 kHz tone other
> than through distortion, and if we can, why in my experiment did I fail
> to hear a 440 Hz tone?

I know that certain sonar systems generate LF signals as the
difference between two HF source signals. The reason this works
is that the high-power source signals interact nonlinearly generating
strong components at the difference frequencies.

Assuming you are right that this is possible -- I have not heard
of such an effect before -- you ought to look for some nonlinear
interaction somewhere, between the HF source signals.

Rune

Michel Rouzic wrote:

> Hi,
>
> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
> 23 kHz tone?

Yes. If the amplitudes are enough high.

> So it makes me highly sceptical, however when I googled what he pointed
> me to it seems that the reality of ultrasonic difference tone hearing
> isn't even controversial, so I'd like to know, by what mechanisms could
> we hear a 2 kHz tone from a 21 kHz tone added to a 23 kHz tone other
> than through distortion,

The air itself is the nonlinear media. The effects are very considerable
when the sound levels are above 90dB.

and if we can, why in my experiment did I fail
> to hear a 440 Hz tone?

The signal is not strong enough.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

"Michel Rouzic" <[email protected]> wrote in
news:[email protected] oups.com:

> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
> 23 kHz tone? I've been arguing with someone about this.

Well, 'hear' may or may not be accurate, the frequencies may not be right,
and this has nothing to do with physics, but active function of the inner
ear.

It turns out that hair cells can not only sense movement of the membrane in
the inner ear, but they can actually tug on the membrane and cause the ear
to actively emit sound that can be recorded by a microphone in the ear
canal. This is often used to test hearing systems, as it can be seen in
healthy ears. I'm not sure if the test subject actually "hears" the sound
or not.

When you play two tones, the ear plays back other frequencies. I think the
most prominent is 2*f1 - f2, but there is also 2*f2-f1 and 3*f1-f2. The
2Khz number you cite seems to have left the factor of two off the f1. The
real answer would be 2*23-21= 25Kz, or maybe 2*21-23= 19kHz. I can't
remember whether f1 is the higher or lower freq.

You can search up Distortion Product Otoacoustic Emissions for more info.

--
Scott
Reverse name to reply

Scott Seidman wrote:

> "Michel Rouzic" <[email protected]> wrote in
> news:[email protected] oups.com:
>
>
>>Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
>>23 kHz tone? I've been arguing with someone about this.
>
>
> Well, 'hear' may or may not be accurate, the frequencies may not be right,
> and this has nothing to do with physics, but active function of the inner
> ear.
>
> It turns out that hair cells can not only sense movement of the membrane in
> the inner ear, but they can actually tug on the membrane and cause the ear
> to actively emit sound that can be recorded by a microphone in the ear
> canal. This is often used to test hearing systems, as it can be seen in
> healthy ears. I'm not sure if the test subject actually "hears" the sound
> or not.
>
> When you play two tones, the ear plays back other frequencies. I think the
> most prominent is 2*f1 - f2, but there is also 2*f2-f1 and 3*f1-f2. The
> 2Khz number you cite seems to have left the factor of two off the f1. The
> real answer would be 2*23-21= 25Kz, or maybe 2*21-23= 19kHz. I can't
> remember whether f1 is the higher or lower freq.
>
> You can search up Distortion Product Otoacoustic Emissions for more info.
>

Yes, the 2*f1 = f2 is typically the strongest distortion product in
EOAEs. For best results the f2/f1 ratio should be around 1.225.

Paul

> So it makes me highly sceptical, however when I googled what he pointed
> me to it seems that the reality of ultrasonic difference tone hearing
> isn't even controversial, so I'd like to know, by what mechanisms could
> we hear a 2 kHz tone from a 21 kHz tone added to a 23 kHz tone other
> than through distortion, and if we can, why in my experiment did I fail
> to hear a 440 Hz tone?
>
> Thanks in advance

Yes, there are non-linearities that can casue "mixing products" in the
speaker, the air and in your ears...

Mark

Vladimir Vassilevsky wrote:
> Michel Rouzic wrote:
>
> > Hi,
> >
> > Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
> > 23 kHz tone?
>
> Yes. If the amplitudes are enough high.
>
> > So it makes me highly sceptical, however when I googled what he pointed
> > me to it seems that the reality of ultrasonic difference tone hearing
> > isn't even controversial, so I'd like to know, by what mechanisms could
> > we hear a 2 kHz tone from a 21 kHz tone added to a 23 kHz tone other
> > than through distortion,
>
> The air itself is the nonlinear media. The effects are very considerable
> when the sound levels are above 90dB.
>
> and if we can, why in my experiment did I fail
> > to hear a 440 Hz tone?
>
> The signal is not strong enough.

If I understand it correctly, this effects appear when the signal is
strong, so in a way, it's created by some kind of distortion, and from
the other posts in this topic, it has to do with how our ears are made
or the air itselves rather than something in our brain, right?

"Michel Rouzic" schrieb im Newsbeitrag >
> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz
> tone + a 23 kHz tone? I've been arguing with someone about
> this. He says that you can, but that goes against what I know
> about the effect in the frequency domain of adding signals
> together in the time domain,

google for "beat tone", e.g.
http://www.modrec.com/tutorials/beats.html

HTH
Martin

Martin Blume wrote:
> "Michel Rouzic" schrieb im Newsbeitrag >
>> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz
>> tone + a 23 kHz tone? I've been arguing with someone about
>> this. He says that you can, but that goes against what I know
>> about the effect in the frequency domain of adding signals
>> together in the time domain,
>
> google for "beat tone", e.g.
> http://www.modrec.com/tutorials/beats.html

Martin,

Beat tones are not merely perceptual and needn't arise from any
nonlinearity. Beat tones at frequencies below what is audible are
readily perceived, as the examples you provided demonstrate. Plot the
sum of the individual sinusoids and you will see why.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Vladimir Vassilevsky wrote:
> Michel Rouzic wrote:
>
> > Hi,
> >
> > Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
> > 23 kHz tone?
>
> Yes. If the amplitudes are enough high.
>
> > So it makes me highly sceptical, however when I googled what he pointed
> > me to it seems that the reality of ultrasonic difference tone hearing
> > isn't even controversial, so I'd like to know, by what mechanisms could
> > we hear a 2 kHz tone from a 21 kHz tone added to a 23 kHz tone other
> > than through distortion,
>
> The air itself is the nonlinear media. The effects are very considerable
> when the sound levels are above 90dB.
>
> and if we can, why in my experiment did I fail
> > to hear a 440 Hz tone?
>
> The signal is not strong enough.

If I understand it correctly, this effects appear when the signal is
strong, so in a way, it's created by some kind of distortion, and from
the other posts in this topic, it has to do with how our ears are made
or the air itselves rather than something in our brain, right?

By the way, the context of this argument was a discussion on the
advantages of vinyl as opposed to audio CD, one argument in the favour
of vinyl being that it would record ultrasounds, thus allow those
ultrasonic difference tone to be heard, making it much better for the
audiophile.

I guess that's how audiophiles are supposed to explain the need for a
sampling frequency of 192 kHz.

Michel Rouzic wrote:


> By the way, the context of this argument was a discussion on the
> advantages of vinyl as opposed to audio CD, one argument in the favour
> of vinyl being that it would record ultrasounds, thus allow those
> ultrasonic difference tone to be heard, making it much better for the
> audiophile.

Anything audible produces would be the product of intermodulation, a
type of distortion most audiophiles prefer to avoid.

> I guess that's how audiophiles are supposed to explain the need for a
> sampling frequency of 192 kHz.

Those are probably the same people who insist that speaker wire has to
cost at least $20 a foot to provide decent quality. (Some of them
buy special degaussers for their CDs and vinyl records. Just ignore
those kooks.

Jerry
--
"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
- Chief Justice Charles Evans Hughes, 1927
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Michel Rouzic wrote:

> Vladimir Vassilevsky wrote:
>

>>>Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
>>>23 kHz tone?
>>
>>Yes. If the amplitudes are enough high.
>>
> If I understand it correctly, this effects appear when the signal is
> strong, so in a way, it's created by some kind of distortion, and from
> the other posts in this topic, it has to do with how our ears are made
> or the air itselves rather than something in our brain, right?

This is correct.
The brain is too slow to do the processing of the audio frequencies
directly (except for the very low end of the audible band). The hearing
actually works as the spectrum analyzer. The effects you are asking
about are related to the nonlinear distortions.

> By the way, the context of this argument was a discussion on the
> advantages of vinyl as opposed to audio CD, one argument in the favour
> of vinyl being that it would record ultrasounds, thus allow those
> ultrasonic difference tone to be heard, making it much better for the
> audiophile.

The technical arguments are not applicable in the area of the beliefs
and religions


> I guess that's how audiophiles are supposed to explain the need for a
> sampling frequency of 192 kHz.

This is a different story. There is some value in having the higher
sample rates for the audio. What is commonly referred as the
'quantization noise' and 'the digital processing artifacts' is actually
not a noise, but a mix of the intermod products between the signal
components and the sample rate. At the lower sample rates all of the
intermods are aliased back to the audible band, whereas at the higher
sample rates the big part of the garbage stays in the ultrasound area,
i.e. you can't hear it.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Vladimir Vassilevsky wrote:
> > I guess that's how audiophiles are supposed to explain the need for a
> > sampling frequency of 192 kHz.
>
> This is a different story. There is some value in having the higher
> sample rates for the audio. What is commonly referred as the
> 'quantization noise' and 'the digital processing artifacts' is actually
> not a noise, but a mix of the intermod products between the signal
> components and the sample rate. At the lower sample rates all of the
> intermods are aliased back to the audible band, whereas at the higher
> sample rates the big part of the garbage stays in the ultrasound area,
> i.e. you can't hear it.

Oh, I thought that ideally the signal from a microphone was
analogically low-pass filtered before going to the ADC, thus avoid any
aliasing. Anyways, even if it takes higher sampling rates to minimize
aliasing, it still doesn't justify the need for audio DVD's, as you
just have to filter it, this time digitally, before downsampling it.

Michel Rouzic skrev:
> Vladimir Vassilevsky wrote:
> > > I guess that's how audiophiles are supposed to explain the need for a
> > > sampling frequency of 192 kHz.
> >
> > This is a different story. There is some value in having the higher
> > sample rates for the audio. What is commonly referred as the
> > 'quantization noise' and 'the digital processing artifacts' is actually
> > not a noise, but a mix of the intermod products between the signal
> > components and the sample rate. At the lower sample rates all of the
> > intermods are aliased back to the audible band, whereas at the higher
> > sample rates the big part of the garbage stays in the ultrasound area,
> > i.e. you can't hear it.
>
> Oh, I thought that ideally the signal from a microphone was
> analogically low-pass filtered before going to the ADC, thus avoid any
> aliasing.

The analog signal has to be filtered before sampling, but there
is always some residue. A 60 dB attenuation still leaves 1/1000th
of the power spectrum amplitude. An anti-alias filter does not
*remove* aliasing, it *reduces* it. Hopefully to tolerable levels.

Vladimir's point is that some might find the aliased residue
from 44.1 kHz or 48 kHz sampling intolerable.

Rune

Jerry Avins wrote:
> Martin Blume wrote:
>> "Michel Rouzic" schrieb im Newsbeitrag

>>> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz
>>> tone + a 23 kHz tone?

>> google for "beat tone", e.g.
>> http://www.modrec.com/tutorials/beats.html

> Beat tones are not merely perceptual and needn't arise from any
> nonlinearity.

Though interestingly, a beating percept can result when you feed each
ear a sine of slightly different frequency. I don't know if it works
well in the range Michel mentioned.


Martin

--
Quidquid latine scriptum sit, altum viditur.

Rune Allnor wrote:

>>>>I guess that's how audiophiles are supposed to explain the need for a
>>>>sampling frequency of 192 kHz.
>>>
>>>This is a different story. There is some value in having the higher
>>>sample rates for the audio. What is commonly referred as the
>>>'quantization noise' and 'the digital processing artifacts' is actually
>>>not a noise, but a mix of the intermod products between the signal
>>>components and the sample rate. At the lower sample rates all of the
>>>intermods are aliased back to the audible band, whereas at the higher
>>>sample rates the big part of the garbage stays in the ultrasound area,
>>>i.e. you can't hear it.
>>
>>Oh, I thought that ideally the signal from a microphone was
>>analogically low-pass filtered before going to the ADC, thus avoid any
>>aliasing.
>
>
> The analog signal has to be filtered before sampling, but there
> is always some residue. A 60 dB attenuation still leaves 1/1000th
> of the power spectrum amplitude. An anti-alias filter does not
> *remove* aliasing, it *reduces* it. Hopefully to tolerable levels.
>
> Vladimir's point is that some might find the aliased residue
> from 44.1 kHz or 48 kHz sampling intolerable.

Not quite that.

1. The *real* quantization in the ADC is much less accurate then to the
one msb. There is a significant amount of the additional nonlinearity
associated with it. Therefore the IMD products are the limiting factor
to the ADC performance. Having higher sample rate keeps the IMD in the
ultrasound, whereas at 44.1kHz they are aliased back to the audible
range.

2. The DSP processing of the digital audio requires multiple
requantizations, which results in the noise and the IMD buildup. The
higher sample rate spreads the numeric trash over the wider bandwidth.


Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Martin Eisenberg wrote:
> Jerry Avins wrote:
> > Martin Blume wrote:
> >> "Michel Rouzic" schrieb im Newsbeitrag
>
> >>> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz
> >>> tone + a 23 kHz tone?
>
> >> google for "beat tone", e.g.
> >> http://www.modrec.com/tutorials/beats.html
>
> > Beat tones are not merely perceptual and needn't arise from any
> > nonlinearity.
>
> Though interestingly, a beating percept can result when you feed each
> ear a sine of slightly different frequency. I don't know if it works
> well in the range Michel mentioned.

In the range I mentionned you wouldn't hear the beating if you couldn't
hear the two sines in the first place, since they are supposed to be
ultrasonic. But it's not even about hearing a beating in the sound you
hear but rather hearing a whole new low frequency tone.

Martin Eisenberg wrote:
> Jerry Avins wrote:
> > Martin Blume wrote:
> >> "Michel Rouzic" schrieb im Newsbeitrag
>
> >>> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz
> >>> tone + a 23 kHz tone?
>
> >> google for "beat tone", e.g.
> >> http://www.modrec.com/tutorials/beats.html
>
> > Beat tones are not merely perceptual and needn't arise from any
> > nonlinearity.
>
> Though interestingly, a beating percept can result when you feed each
> ear a sine of slightly different frequency. I don't know if it works
> well in the range Michel mentioned.

In the range I mentionned you wouldn't hear the beating if you couldn't
hear the two sines in the first place, since they are supposed to be
ultrasonic. But it's not even about hearing a beating in the sound you
hear but rather hearing a whole new low frequency tone.

Vladimir Vassilevsky wrote:

> 1. The *real* quantization in the ADC is much less accurate then to the
> one msb. There is a significant amount of the additional nonlinearity
> associated with it. Therefore the IMD products are the limiting factor
> to the ADC performance. Having higher sample rate keeps the IMD in the
> ultrasound, whereas at 44.1kHz they are aliased back to the audible
> range.
>
> 2. The DSP processing of the digital audio requires multiple
> requantizations, which results in the noise and the IMD buildup. The
> higher sample rate spreads the numeric trash over the wider bandwidth.

Both is correct, therefore almost all music nowadays is recorded and
processed digitally with at least 24 bits resolution and 96 kHz samplerate.
But there is actually no need to deliver the final result in this format
as well, since nothing hearable is lost when downsampling to 16 bits
at 44.1 kHz. All DACs then use upsampling in one or the other form
to overcome limitiations of the hardware.

I know that there are many CDs which have a bad sound, but that is
not caused by the medium itsself (but by bad recording, processing
and mastering) since there exist CDs which sound almost perfect.

bye
Andreas
--
Andreas Hünnebeck | email: [email protected]
----- privat ---- | www : http://www.huennebeck-online.de
Fax/Anrufbeantworter: 0721/151-284301
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PGP-Key: http://www.huennebeck-online.de/publ...gp_andreas.asc

Michel Rouzic wrote:

> Hi,
>
> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
> 23 kHz tone?

It can happen if both signals pass the same chain and this chain
suffers from intermodulation distortion.

Two japanese scientists performed an experiment where signals
were played back (a) via one driver and (b) via separate drivers.
In case (a) listeners could here something where in (b) they
could not. Also in case (a) the played back signal was recorded with
a microphone, and the FFT analisys revealed the existence of
intermodulation components well within the human hearing range.

Source (german):
http://www.kleinhummel.de/produkte/o...eber_20kHz.pdf
This source refers to paper 5401 by Ashihara Kaoru and Kiryu Shogo, presented
on the 7th. AES Convention.

bye
Andreas
--
Andreas Hünnebeck | email: [email protected]
----- privat ---- | www : http://www.huennebeck-online.de
Fax/Anrufbeantworter: 0721/151-284301
GPG-Key: http://www.huennebeck-online.de/public_keys/andreas.asc
PGP-Key: http://www.huennebeck-online.de/publ...gp_andreas.asc

Michel Rouzic wrote:

(snip)

> If I understand it correctly, this effects appear when the signal is
> strong, so in a way, it's created by some kind of distortion, and from
> the other posts in this topic, it has to do with how our ears are made
> or the air itselves rather than something in our brain, right?

I believe that is true in this case, but there are some cases
where the brain is also involved, especially with musical signals.

If you listen to a combination of, say, 2kHz, 3kHz, and 4kHz,
even in the absence of non-linearities, the brain can
recognize a 1kHz component. I believe this is called something
like the missing fundamental in music terms. The interpreation
of musical signals requires processing in the brain. See, for
example, "The acoustic foundations of music" by John Backus.

-- glen

glen herrmannsfeldt wrote:

>[..]
> If you listen to a combination of, say, 2kHz, 3kHz, and 4kHz,
> even in the absence of non-linearities, the brain can
> recognize a 1kHz component. I believe this is called something
> like the missing fundamental in music terms. The interpreation
> of musical signals requires processing in the brain. See, for
> example, "The acoustic foundations of music" by John Backus.

I've heard that before. It is the reason why you can experience a
satisfactory deep bass response from small loudspeakers. The
brain just adds the missing fundamental.

Tschau
Andreas
--
Andreas Hünnebeck | email: [email protected]
----- privat ---- | www : http://www.huennebeck-online.de
Fax/Anrufbeantworter: 0721/151-284301
GPG-Key: http://www.huennebeck-online.de/public_keys/andreas.asc
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Martin Eisenberg wrote:
> Jerry Avins wrote:
>
>>Martin Blume wrote:
>>
>>>"Michel Rouzic" schrieb im Newsbeitrag
>
>
>>>>Is it true that you can hear a 2 kHz tone if you hear a 21 kHz
>>>>tone + a 23 kHz tone?
>
>
>>>google for "beat tone", e.g.
>>>http://www.modrec.com/tutorials/beats.html
>
>
>>Beat tones are not merely perceptual and needn't arise from any
>>nonlinearity.
>
>
> Though interestingly, a beating percept can result when you feed each
> ear a sine of slightly different frequency. I don't know if it works
> well in the range Michel mentioned.
>
>
> Martin
>
Martin,
I am interested in your last comment. Is the beat note similar in
amplitude to what you would get from feeding both the signals into one
ear?

Alternatively, if the beat note is weak, do you think it could be due to
acoustic leakage of signal from one ear to the other?

Please excuse my laziness for not checking this myself, but whenever the
question has occurred to me I have either been at home with only one
oscillator and plenty of time, or at work with plenty of oscillators and
no time.

Regards,
John

glen herrmannsfeldt wrote:

>
> If you listen to a combination of, say, 2kHz, 3kHz, and 4kHz,
> even in the absence of non-linearities, the brain can
> recognize a 1kHz component. I believe this is called something
> like the missing fundamental in music terms.

Hi Glen,
I'm just curious: do you have any experience if this depends on the phase of
the signals, so that either:
- if the phase is related as in real overtones, you "hear" the fundamental
- or if the phase of every tone is arbitrary, you don't "hear" it.
or: if the fundamental tone can be heard even if the frequencies are not
exactly multiples ( e.g. 2kHz,3.001kHz, 4.005kHz,...)

Bernhard

Google for Hypersonic Sound,
see also http://www.ideas21.co.uk/248
There are companies offering speakers based on this principles to be
used at public places, for example museums.

Uli

Michel Rouzic schrieb:

> Hi,
>
> Is it true that you can hear a 2 kHz tone if you hear a 21 kHz tone + a
> 23 kHz tone? I've been arguing with someone about this. He says that
> you can, but that goes against what I know about the effect in the
> frequency domain of adding signals together in the time domain, not
> only that but I tried playing a 19,560 Hz tone together with a 20,000
> Hz tone through my PC speakers (which have a decent frequency response
> in that range which I tested with a chirp and a mic) and I don't hear
> any 440 Hz, unless I push the volume loud enough but in this case for
> obvious reasons the actual signal contains low frequency components. It
> also goes against my (erronerous?) conception of human hearing which is
> pretty much to me a linear system just like a microphone or a loud
> speaker.
>
> So it makes me highly sceptical, however when I googled what he pointed
> me to it seems that the reality of ultrasonic difference tone hearing
> isn't even controversial, so I'd like to know, by what mechanisms could
> we hear a 2 kHz tone from a 21 kHz tone added to a 23 kHz tone other
> than through distortion, and if we can, why in my experiment did I fail
> to hear a 440 Hz tone?
>
> Thanks in advance