The difference between difference-tones and beats

The Wikipedia article on Combination Tones
http://en.wikipedia.org/wiki/Combination_tone
contains an unsubstantiated claim for the existence of "binaural difference tones" (4th pragraph). I suspect they are confusing them with "binaural beats". They also seem to be confusing the Bohlen-Pierce scale with Heinz Bohlen's phi scale.

Here's what you have to understand. We have two closely related but different phenomena. Difference tones and Beats. You might be thinking "I know the difference. A beat is a difference tone that's slow enough to be heard as a pulsation". Wrong! Completely and utterly wrong.

Difference tones are real tones. They can be measured with a spectrum analyser. They are subtle, but really there.
Beats are a product of human perception. Perceived as a variation in the volume of a single frequency when really there are two nearby frequencies at constant volume. Beats are obvious, but they are ghosts.

Yes beats must be slower than about 20 Hz to be audible,
and difference tones must be faster than about 20 Hz to be audible,
but a fast beat is not a difference tone, and a slow difference tone is not a beat.

As you speed up a beat, at first it becomes "roughness", and then it becomes inaudible. These are called "critical-band effects".
As you slow down a difference tone it becomes inaudible below about 20 Hz. However it is difficult to produce an infrasonic difference tone without also producing a beat. But if you filter out the two parent tones, leaving just the difference tone, then there will be no beat. Nothing at all will be heard, but a spectrum analyser will still measure the infrasonic difference tone.

You will find that these two different phenomena are often confused in the non-scientific literature and online.

You automatically hear beats if you have two frequencies close enough together.

But you don't automatically get difference tones. Difference tones are only generated by a particular kind of distortion called "non-linearity" or "intermodulation distortion". These are two complementary names for the same thing, one viewing it as a function of time, the other as a function of frequency. Non-linearity is required if you want to turn a beat into a real tone -- a difference tone. Non-linearity gives the ghost solid flesh.

Certain combination-tone chords with four frequencies A:B:C:D as described here /tuning/topicId_105487.html#105487 depend on a difference between two differences, e.g. (D-C) - (B-A). Without non-linearity this would be just the difference between two inaudibly-fast beats. But there's no such thing as a beat between beats -- a ghost of a ghost.

With non-linearity, D-C and B-A become real frequencies, real sounds, and therefore able to be perceived as beating -- as a pulsation in volume of a single frequency midway between the two real frequencies.

Engineers usually go to great pains to make our sound systems as linear as possible, so that they will _not_ create new frequencies that did not exist in the original. Electric guitar amplifiers are an exception.

But if you read the first page here http://books.google.com.au/books?id=eGcfn9ddRhcC&pg=PA277#v=onepage&q&f=false you will learn that I have been over-simplifying for educational purposes, as we teachers often do.

Difference tones, it turns out, can _also_ be a product of human perception. So it's easy to understand why there is confusion of these things in the non-scientific literature. However they remain distinct from beats. Such difference tones are produced by non-linearity in the mechanical components of the middle and inner ear, while beats are produced by higher-level neural processing in the brain.

However, if you rely entirely on your listeners' ears for the required non-linearity, the difference tones will be quite subtle and may require directed attention, and hence any _beating_ of the difference tones may be more subtle again.

On Mon, Dec 3, 2012 at 10:37 AM, dkeenanuqnetau <d.keenan@...>
wrote:
>
> Here's what you have to understand. We have two closely related but
> different phenomena. Difference tones and Beats. You might be thinking "I
> know the difference. A beat is a difference tone that's slow enough to be
> heard as a pulsation". Wrong! Completely and utterly wrong.

Thumbs up to that. This is one of the most common misconceptions out there.

> Difference tones are real tones. They can be measured with a spectrum
> analyser. They are subtle, but really there.

This is also true, unless they're happening in your ears. Then they'd
require us to put a microphone in your ear and record them, but yes,
theoretically possible.

> Beats are a product of human perception. Perceived as a variation in the
> volume of a single frequency when really there are two nearby frequencies at
> constant volume. Beats are obvious, but they are ghosts.

I have a nit to pick about this, however. Beats are very real, and are
just as "real" as the concept of "sounds that change in time."

For instance, if I were to place a tape recorder in your ears and
record everything that you've ever heard across your entire life, I
could take the Fourier transform of the entire tape as one huge
signal, putting it into the frequency domain. When it's in the
frequency domain, the frequency response of the signal will have no
time resolution at all. It'll give you no information about "when"
some frequency appears at some point in your life. It'll just split
the signal up into an infinite summation* of sine waves stretching
from time=the day you're born to time=the day you die that
automagically add back together to reconstruct the original signal,
with the sines interfering with one another in precisely the right way
to recreate what you'd perceive as time information in the signal.

It's a very "dumb" transform in this respect, not much different than
splitting a number up into a prime factorization that automagically
multiplies together to yield the original number. This is, as you
might imagine, not much more useful than just looking at the original
signal in the time domain. This is because we don't just hear
"frequencies," we hear frequencies that change in time. So the
frequency response of a signal isn't any more "real" than the time
domain waveform itself is.

So as you know, what we really want is a time-frequency signal
representation like a spectrogram. And the key is that there's no one
way to do this. You can have a representation which gives you more
time information at the cost of having less frequency information, or
vice versa, and none of these representations are ultimately any more
"real" than any others.

So when your ears hear two sine waves close to one another as just
being one sine wave that's changing in amplitude, this reflects a
certain time-frequency tradeoff being made; in this case, the ear has
sacrificed some frequency resolution for the sake of getting some time
resolution. If you were to drastically increase the frequency
resolution of the ear so that we could perceive 440 and 441 Hz as two
separate pitches ringing out in simultaneity, this would have to come
at the cost of some time resolution in that area. If our ears were set
up like that, we'd probably have 3x as many vowels as we have now, but
we'd all talk a lot more slowly. Maybe 17/16 would be a very
pleasantly harmonic dyad, but triplets would sound chaotic and
disorienting or something.

So any time you have a single frequency "changing in time," whether
it's a sinusoid modulating in amplitude or covered by some envelope,
you can rightly say that the change in time is a "ghost" relative to
what the signal would look like in the frequency domain. However, the
frequency domain itself isn't any more "real" than the time domain;
mixed time-frequency domains correspond more directly to what we
perceive and there's an infinite number of ways to make the tradeoff.
But there's no difference at all between you playing 439 Hz, 440 Hz,
and 441 Hz out of your speaker, vs playing 440 Hz out of your speaker
and paying some guy to sinusoidally modulate your volume knob at 1
cycle per second; the only difference is how you want to think about
it.

TL;DR, whether or not the beats you see are real will depend on the
resolution of the spectrogram.

Other than that point, though, I agree with the rest of the post.

-Mike

PS, some caveats for technical accuracy
- The above is very oversimplified and doesn't handle how the signal
undergoes a wave of severe nonlinear processing after leaving the
cochlea (and particularly during transduction into the auditory nerve,
where it turns into a series of impulses)
- Unlike the spectrogram, the time-frequency resolution of the ear is
not the same across the entire spectrum. As a general rule, the higher
end of the spectrum has better time resolution and the lower end has
better (linear, not logarithmic) frequency resolution (though that
rule may not hold in the lowest registers, not sure)
- There are much more intelligent time-frequency representations of
the signal than your ordinary spectrogram; wavelet analysis is big
business these days and then there's things like the SPEAR algorithm
which Carl's posted on here before. We can safely assume that the
brain is using some such "intelligent" (and adaptive way) to make the
tradeoff.
- Despite the above, the time-frequency tradeoff still has to exist in
some form at the end of the day, and as a very general rule of thumb,
it's true that the phenomenon of beating is really just indicative of
the presence of time resolution in the signal, in general, sacrificing
some frequency resolution.

Thanks Mike,

I agree with everything you say, but I'm sure you agree that to go into that sort of detail tends to lose the mnemonic value of:

"Beats are obvious ghosts.
Difference tones are subtle flesh.
Distortion enfleshes the ghost."

I guess the point is that the spectrum analyser will always measure the difference tone we hear (with your proviso of a microphone in the ear when the ear is the only cause of non-linearity), but it will only measure the beat we hear if its time-frequency-tradeoff filter is set to be something like the one we have evolved to have, which in terms of pure physics is completely arbitrary. One can imagine that Carl Sagan's enormous slow-moving gas-bag animals in the atmosphere of Jupiter might not hear a beat until the two frequencies get within 0.001 Hz of each other.

Hi,

--- In tuning@yahoogroups.com, "dkeenanuqnetau" <d.keenan@...> wrote:
> I guess the point is that the spectrum analyser will always measure the difference tone we hear (with your proviso of a microphone in the ear when the ear is the only cause of non-linearity), but it will only measure the beat we hear if its time-frequency-tradeoff filter is set to be something like the one we have evolved to have, which in terms of pure physics is completely arbitrary. One can imagine that Carl Sagan's enormous slow-moving gas-bag animals in the atmosphere of Jupiter might not hear a beat until the two frequencies get within 0.001 Hz of each other.

It is true that beats *can* be ghosts; if both ears hear a tone with a slightly different pitch, we hear binaural beats, even if they don't exist on an acoustical level.

It is also true that - from an acoustic perspective - it is arbitrary to say that beats occur if the two frequencies are close enough together that a human would perceive them as beats.

However, it is a mathematical fact that the superposition of two sine waves with identical amplitude and different frequencies f and g can be expressed as a single sine wave with pitch frequency (f+g)/2 (= the arithmetic mean of both frequencies), and beat frequency (f-g)/2, so beats are an acoustical fact as long as we don't premise arbitrary ranges for the difference between f and g.

A fourier transform is designed to represent sounds as superpositions of sine waves, so if you don't recognize beats (or just with arbitrary parameters) when using a spectrum analyzer, this doesn't mean they're not real, it just means you're using a method that is not very efficient to detect them.

If you want to convince yourself that beats are real, just look at the wave form of two superposed sine waves. This works especially well if there's no simple ratio between f and g.

-Gedankenwelt

P.S.: A similar case where the choice of representation matters is FM synthesis, where a tone with a frequency spectrum is identical to a sine wave with a vibrato. Here again, it depends on the parameters how a human would perceive that sound, but independent from the parameters both representations are acoustical facts.

Hi Gedankenwelt,

What you say is true. But it seems we need different terms for the acoustic phenomenon (which can be visualised as a waveform or spectrum display) and the human perception. We do have such terms: "amplitude modulation" vs "beats". But I am aware that the term "beats" is already hopelessly used for amplitude modulation in general, even at radio frequencies that no human can sense.

So yes, I'm well aware that the sum of two sine waves is completely equivalent to the sine wave modulation of the amplitude of a single sine wave mid way in frequency between the two.

My purpose is to help people understand the difference between this phenomenon and the phenomenon of difference tones, which require some nonlinear function to be applied to the sum of sines.

So what does this look like on a waveform display? It is visually indistinguishable unless the nonlinearity is extreme. But on a spectrum display a difference tone will appear as a separate spike while a beat will either not appear at all (only the original two spikes for the two frequencies), or it will appear as a single spike whose height wobbles up and down. This depends on the bandwidth of its filter, as it does with humans.

-- Dave

--- In tuning@yahoogroups.com, "dkeenanuqnetau" <d.keenan@...> wrote:
> But it seems we need different terms for the acoustic phenomenon (which can be visualised as a waveform or spectrum display) and the human perception. We do have such terms: "amplitude modulation" vs "beats". But I am aware that the term "beats" is already hopelessly used for amplitude modulation in general, even at radio frequencies that no human can sense.

Yeah, I think it makes sense to use different terms here, thanks for pointing that out! It seems I talked about something different when I said "beats", but now I understand your point of view much better. ;)