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Klarinet Archive - Posting 000483.txt from 2001/02

From: (Tony Pay)
Subj: [kl] The combination paradox
Date: Tue, 13 Feb 2001 06:21:31 -0500

On Mon, 12 Feb 2001 18:52:20 -0700, said:

> Hi Tony,
> Rather than torture the 90% of the subscribers who aren't interested
> in this, I thought I'd reply off-list. You're welcome to reply
> on-list, and/or forward this to the list if you prefer. I'm still
> trying to understand what the difference is, as it seems to me that
> both beats and difference tones are generated in the same way, but
> below and above the threshold of tone recognition.

This is for the 10%.

> > But, as I said in another post, the added waveform can produce the
> > amplitude modulation only as the *product* of two terms (see below).
> > So the amplitude modulation *isn't* an added waveform. It *is* a
> > beat, though, which is a different thing from a difference tone.
> Actually, the math just shows that you can represent that wave
> *either* as a product (modulation) *or* as a simple sum, that they
> are mathematically equivalent descriptions of the physical process.
> The fact that the wave can be written as a product of sine and cosine
> doesn't make the original sum an invalid representation. If I
> remember correctly, Fourrier's theorem is that *any* periodic wave
> can be represented as a sum of sine and cosine waves of various
> frequencies. Where possible, our hearing system seems to hear
> complex waves as sums of individual fundamentals, with their
> associated harmonics, in effect performing a Fourrier transform.

Yes, I agree with that.

Let me start again, though.

What do I actually mean, when I say that a frequency is present as a
vibration in the air, as opposed to in our middle ear, as a result of
the behaviour of our aural systems?

What I mean is inextricably bound up with the Fourier viewpoint,
according to which, as you say,

> *any* periodic wave can be represented as a sum of sine and cosine
> waves of various frequencies.

In fact,

any periodic wave, of period P, can be represented as a (usually
infinite) sum of sine and cosine waves of frequencies equal to P, 2P,
3P, etc. The proportions of these frequencies are uniquely determined,
which means that no other frequencies can be present in the sum.

With this viewpoint, to say that a given frequency, say 5P, is *present*
means that if you took out all the others, that one would remain,
physically present, and would exert a periodic force on an object it
encountered at the frequency 5P. (If that object had a resonant
frequency of 5P, then it would 'ring'.)

How could we remove all the others? Well, by electronic filtering
perhaps, but because how filtering works isn't at all obvious, we could
just add in a wave that had all the others present, but 180° out of
phase. Then everything would cancel except for the 5P component.

So that's an operational definition of a frequency being actually

Now, if a simple sine wave, of frequency P, excites a medium such as air
that responds linearly to the excitation, then the excitation is
transmitted as a sound wave, and consists entirely of the frequency P on
arrival. And *because of what linearity of response means*, if I have
another frequency Q that excites the air simultaneously, then the
resultant excitation consists of the sum of the excitations P and Q, and
nothing else.

The entire picture is one of summation of sine waves, in a linearly
responding medium of transmission.

If this picture is sufficiently entrenched, the sin + sin = 2sincos
argument for explaining a difference tone looks much more implausible.
You don't get another frequency out of a mathematical recast of the same

> > > Yes, the description above also appear in Benade (which I *can*
> > > locate). It explains why you hear beats between a sine wave
> > > (which has no upper harmonics) at 200 Hz when played against a
> > > tone at 403 Hz.
> >
> > No, it doesn't explain why you hear beats. It explains why you hear
> > difference tones between two frequencies.
> Not what I said.

Yes, I was too hasty. Sorry about that.

But notice:

> If you read Benade's "Fundamentals of Musical Acoustics" (2nd ed.,
> 1990, Dover) at pp. 254-256, he details the experiment. A single
> headphone is supplied with a 400 Hz sine wave (i.e., a single
> component, with no higher harmonics), together with a "search tone"
> that can be adjusted for frequency and amplitude. The search tone
> causes beats to appear when close to (but not on) 400 Hz, and also
> when near 800 Hz, 1200 Hz, etc. You hear beats because the search
> tone is out of tune with harmonics of the 400 Hz sine wave, which are
> generated by the ear's nonlinearity.
> To quote: "We have clear evidence, then, that new, harmonically
> related frequency components have somehow been created from the
> original single-component excitation, and that these new components
> have been called to our attention by their beatings with the search
> tone." (Benade at 255).

...but, in the ear.

I suspect that the reason that Benade doesn't underline this even more
is that he's spent whole chapters generating the picture of summation
and linearity, and doesn't want to confuse the issue.

But here's Roederer, in a book referenced by Benade on p72, note 4:

"In this section we shall focus on a phenomenon belonging to the first
category [originating in the ear, as opposed to the neural system], the
perception of combination tones. These tones are additional pitch
sensations that appear whaen two pure tones of frequencies f1 and f2
are sounded together; they are most easily perceived if the latter are
of high intensity level. These additional pitch sensations correspond to
frequencies that differ from both f1 and f2, as can be established
easily with pitch-matching or pitch-cancellation experiments (Goldstein,
1970). *They are not present in the original sound stimulus* [italics
in the original] -- they appear as the result of a so-called nonlinear
distortion of the acoustical signal in the ear."

> Actually, looking back at the equations, I'm not sure I agree with
> your interpretation:
> > sin Pt + sin Qt = 2 [sin (P + Q)t/2] [cos (P - Q)t/2]
> >
> > ...(which is a standard formula in trigonometry).....
> >
> >
> > ...the sum of (sin Pt) and (sin Qt) can be regarded as the
> > combination of two waves, one of frequency (P+Q)/2, times one of
> > frequency (P-Q)/2.
> >
> > So the eardrum takes (P-Q)/2 as a modulation, and therefore as a
> > tone when large (and at frequency P-Q, because we can't distinguish
> > maxima and minima).
> Am I correct that you are taking P + Q as sounding like (P+Q)/2
> modulated at a frequency of (P-Q)/2 (or P-Q)? That comports with
> what the ear hears in the beat region (e.g., 400 Hz + 404 Hz results
> in 402 Hz beating at 4 Hz), but not for difference tones (e.g., 400
> Hz + 600 Hz does not sound like 500 Hz beating at 200 Hz: it sounds
> like 400 Hz + 600 Hz + a new tone at 200 Hz).

Yes, which is why the 'product' interpretation of sin Pt + sin Qt
doesn't go anywhere.

It's a sort of sleight of hand, but a very convincing one.

You hear the beat frequency, notice that it's of frequency (P-Q)
(actually, (P-Q)/2, but we can't hear the difference between maxima and
minima). Perhaps you do also hear the average frequency (P+Q)/2 as

As we move the tones apart, there comes a discontinuity, a moment when
they separate into two individually perceived tones (which looks nothing
like the waveform at that point, notice). Then we hear the difference
tone rising from below, and forget the fact that there's been a
discontinuity, I suggest partly because the mathematics shows you how
the beat is generated.

> If I'm wrong, I'll happily accept it: but, I want to understand *why*
> I'm wrong if I am. Can you shed any additional light on the matter?

As I said before, I've been where you are! What happened to me was that
I had to disentangle myself gradually from the lure of the
beats=difference tone experience, and from the notion that the
mathematics shows a wave *combination*.

For me, that lure went away when I fully got that 'combination' in
linear vibrations can only be addition, not multiplication.

_________ Tony Pay
|ony:-) 79 Southmoor Rd
| |ay Oxford OX2 6RE GMN artist:
tel/fax 01865 553339

... My economy: Hairline recession; waist inflation; deep depression.

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