Reposted explanations of beating and quadratic combinational tones from 10/20/05

Part 1 is on quadratic combinational tones
Part 2 (scroll down) is about beating.

Part 1:

Let's say you have two frequencies, x and y.

A general pure tone at frequency x can be written as

a*sin(2pi*xt) + b*cos(2pi*xt)

where a and b are constants, not both zero, and t is time.

For the pure tone at frequency y, we'll write

c*sin(2pi*yt) + d*cos(2pi*yt)

When waves combine in nature, without distortion, they add linearly
(just as forces do). So the combined signal will be:

a*sin(2pi*xt) + b*cos(2pi*xt) + c*sin(2pi*yt) + d*cos(2pi*yt)

If you know your basic Fourier theory, you'll know that there are no
other frequency components besides x and y in this signal. If x and y
are very close, the ear will be unable to resolve them, and this
results in a beating (a separate demonstration).

When there is a quadratic nonlinearity, the output signal (out) can
be written as a function of the input signal (inp) as:

out = f*inp^2 + g*inp + h,

where f, g, and h are constants. Plugging in the combined signal
above, we have

out =

f*(a*sin(2pi*xt) + b*cos(2pi*xt) + c*sin(2pi*yt) + d*cos(2pi*yt))^2
+ g*(a*sin(2pi*xt) + b*cos(2pi*xt) + c*sin(2pi*yt) + d*cos(2pi*yt))
+ h

= f*((a*sin(2pi*xt))^2 + (b*cos(2pi*xt))^2 + (c*sin(2pi*yt))^2 +
(d*cos(2pi*yt))^2 + 2*a*sin(2pi*xt)*b*cos(2pi*xt) + 2*a*sin(2pi*xt)
*c*sin(2pi*yt) + 2*a*sin(2pi*xt)*d*cos(2pi*yt) + 2*b*cos(2pi*xt)*c*sin
(2pi*yt) + 2*b*cos(2pi*xt)*d*cos(2pi*yt) + 2*c*sin(2pi*yt)*d*cos
(2pi*yt))
+ g*(a*sin(2pi*xt) + b*cos(2pi*xt) + c*sin(2pi*yt) + d*cos(2pi*yt))
+ h

Using the following trigonometric identities,

sin(x)*sin(y) = (cos(x - y) - cos(x + y))/2
sin(x)*cos(y) = (sin(x + y) + sin(x – y))/2
cos(x)*cos(y) = (cos(x - y) + cos(x + y))/2

we can derive these additional ones easily:

(sin(x))^2 = 1/2 - cos(2x)/2
(cos(x))^2 = 1/2 + cos(2x)/2
sin(x)*cos(x) = sin(2x)/2

We also note the antisymmetry of the sine function:

sin(-x) = -sin(x)

and using all six, we rewrite the expression for the output as

out = f*(a^2*(1/2 - cos(4pi*xt)) + b^2*(1/2 + cos(4pi*xt)) + c^2*
(1/2 - cos(4pi*yt)) + d^2*(1/2 + cos(4pi*yt)) + ab*sin(4pi*xt) + ac*
(cos(2pi*(x-y)t) - cos(2pi*(x+y)t)) + ad*(sin(2pi*(x+y)t) + sin(2pi*
(x-y)t)) + bc*(sin(2pi*(x+y)t) + sin(2pi*(y-x)t)) + bd*(cos(2pi*(x-y)
t) + cos(2pi*(x+y)t)) + cd*(sin(4pi*yt)))
+ g*(a*sin(2pi*xt) + b*cos(2pi*xt) + c*sin(2pi*yt) + d*cos(2pi*yt))
+ h

= a^2*f/2 + b^2*f/2 + c^2*f/2 + d*2*f/2
+ (b^2 - a^2)*f*cos(4pi*xt) + (d^2 - c^2)*f*cos(4pi*yt)
+ abf*sin(4pi*xt) + cdf*sin(4pi*yt)
+ (bd + ac)*f*cos(2pi*(x-y)t) + (bd - ac)*f*cos(2pi*(x+y)t)
+ (ad - bc)*f*sin(2pi*(x-y)t) + (ad + ab)*f*sin(2pi*(x+y)t)
+ g*(a*sin(2pi*xt) + b*cos(2pi*xt) + c*sin(2pi*yt) + d*cos(2pi*yt))
+ h

Terms with cos(4pi*xt) and sin(4pi*xt) represent frequency components
at a frequency of 2x. These can't both be zero since, for the first
to be zero, a must equal b, while for the second to be zero, a or b
must equal zero. Since we've already assumed a and b are not both
zero, these conditions can't both be true at once; hence the output
signal a components at frequency 2x.

If I haven't lost you yet (and please let me know if/where I have),
you should be able to continue this line of reasoning and see that
frequency components of 2x, 2y, x+y, and x-y are all in the signal if
f>0; that is, if there is a quadratic nonlinearity.

Part 2:

The hearing apparatus has a finite frequency resolution; this is a
mathematical necessity when you consider that we need to hear
*changes* in the world happening in time (due to the classical
uncertainty principle).

Hence a signal containing two very close frequencies at constant and
equal amplitude will be heard as a single pitch whose amplitude
changes over time.

The trig identities we need are the ones like

sin(x) + sin(y) = 2*sin((x+y)/2)*cos((x-y)/2)

I don't think I need to bring cosines in for full generality but I
welcome you to do so if you wish.

Anyway, what this identity tells us is that adding two equal-
amplitude sine waves very close in frequency is the same thing as
amplitude-modulating a sine wave at the average frequency by a cosine
wave at half the difference frequency. Since the cosine is zero twice
per cycle, and reaches its maximum absolute value twice per cycle,
the average-frequency sine wave will be heard to oscillate in
loudness at twice the modulation frequency, or twice half the
difference frequency, which of course equals the difference frequency.

If the ear is unable to resolve the two frequencies, the right side
of the equation is more directly relevant to how we hear the signal.
On the other hand, if the ear can distinguish them, the right side
becomes completely irrelevant to how we hear. However, it still may
be a relevant representation of a particular method of sound
synthesis; namely amplitude modulation. In that case, you reverse the
direction in which you read the equation: the left side tells you
what "goes in" in terms of how the AM is specified; the right side
tells you what the frequency components are. These are
called 'sidebands' because they normally straddle the modulated
frequency at equal frequency intervals on either side of it. And if
they are different enough, you hear them separately -- this is the
regime where AM results in audible sideband frequencies. When the
sidebands cannot be resolved, though, the original AM specification
on the right side of the equation becomes a direct description of how
we hear the sound -- as a single pitch modulated in amplitude at a
fairly slow rate.

The exact frequency ratios, over the entire spectrum of audible
absolute frequencies, that are just large enough to permit two sine
waves to be resolved separately, have been measured in a large body
of experiments on human subjects. They vary greatly over the
spectrum, but never become narrower than a whole tone (nearing it
only an an optimal "middle" absolute frequency range). Thankfully, we
rarely use a single pair of pure sine waves to make music!

Hi Paul,

On Fri, 20 Jan 2006 "wallyesterpaulrus" wrote:
>
> Part 1 is on quadratic combinational tones
> Part 2 (scroll down) is about beating.
>
And very good explanations I found them, too!
Thanks, Paul.

> Part 1:
[snip]
> When there is a quadratic nonlinearity, the output signal (out) can
> be written as a function of the input signal (inp) as:

And when will we expect to find a quadratic
nonlinearity, and why? Is it also the case
that a quadratic function is also a good
approximation to a higher-order nonlinearity?

[snip]
> Part 2:
> The hearing apparatus has a finite frequency resolution; this is a
> mathematical necessity when you consider that we need to hear
> *changes* in the world happening in time (due to the classical
> uncertainty principle).

When I first read this, I wasn't sure I understood it.
I'm now sure I don't. Could you elaborate, and spell
out the reasoning?

[snip]
> The exact frequency ratios, over the entire spectrum of audible
> absolute frequencies, that are just large enough to permit two sine
> waves to be resolved separately, have been measured in a large body
> of experiments on human subjects. They vary greatly over the
> spectrum, but never become narrower than a whole tone (nearing it
> only an an optimal "middle" absolute frequency range). Thankfully, we
> rarely use a single pair of pure sine waves to make music!

Which suggests that the more austere timbres would
probably not be much use in making music with, say,
all notes of 53-EDO. But how about 41-EDO, 31-EDO,
24-EDO, 19-EDO, 15-EDO, 13-EDO ...? Where would
the cutoff lie? Rhetorical questions .... This is food
for thought and experimentation.

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Hi Paul,
>
> On Fri, 20 Jan 2006 "wallyesterpaulrus" wrote:
> >
> > Part 1 is on quadratic combinational tones
> > Part 2 (scroll down) is about beating.
> >
> And very good explanations I found them, too!
> Thanks, Paul.
>
>
> > Part 1:
> [snip]
> > When there is a quadratic nonlinearity, the output signal (out)
can
> > be written as a function of the input signal (inp) as:
>
> And when will we expect to find a quadratic
> nonlinearity, and why?

Obtaining a truly linear response is the holy grail of hi-fi
manufacturers but a nonlinear response is the reality. The ear, for
example, can't respond in a linear way to arbitrarily large
vibrations. At some point, increasing the amplitude of the vibrations
by a given proportion will result in the amplitude of the ear's
response increasing in a smaller proportion. The entire response
function can be expanded (or decomposed) into a series of terms with
different exponents; the first term is the linear terms, and the
second term is the quadratic term. Ultimately, much higher and higher-
order terms must have smaller and smaller coefficients on them so
that the expansion of the response function can converge to a limit
(as well as satisfy conservation of energy, etc.)

> Is it also the case
> that a quadratic function is also a good
> approximation to a higher-order nonlinearity?

This is normally the true if the nonlinearity is symmetrical and the
signal is weak. Then the higher-order terms, which will all be even-
order, will be weaker than the quadratic terms.

The ear and/or brain appear to have some *cubic* nonlinearities,
implying an asymmetrical response function, though. Though the
mechanism is not known, at quiet volumes one of the cubic
combinational tones is often the only audible one.

> [snip]
> > Part 2:
> > The hearing apparatus has a finite frequency resolution; this is a
> > mathematical necessity when you consider that we need to hear
> > *changes* in the world happening in time (due to the classical
> > uncertainty principle).
>
> When I first read this, I wasn't sure I understood it.
> I'm now sure I don't. Could you elaborate, and spell
> out the reasoning?

The classical uncertainty principle? Read the first or all of these:

http://sepwww.stanford.edu/sep/prof/fgdp/c4/paper_html/node2.html
http://www.phys.unsw.edu.au/~jw/uncertainty.html
http://www.ams.org/featurecolumn/archive/uncertainty.html
http://en.wikipedia.org/wiki/Uncertainty_Principle
(and ignore the Heisenberg Uncertainty Principle, which is irrelevant)

> [snip]
> > The exact frequency ratios, over the entire spectrum of audible
> > absolute frequencies, that are just large enough to permit two
sine
> > waves to be resolved separately, have been measured in a large
body
> > of experiments on human subjects. They vary greatly over the
> > spectrum, but never become narrower than a whole tone (nearing it
> > only an an optimal "middle" absolute frequency range).
Thankfully, we
> > rarely use a single pair of pure sine waves to make music!
>
> Which suggests that the more austere timbres would
> probably not be much use in making music with, say,
> all notes of 53-EDO.

If you mean all notes played at once, you're probably right.

> But how about 41-EDO, 31-EDO,
> 24-EDO, 19-EDO, 15-EDO, 13-EDO ...? Where would
> the cutoff lie? Rhetorical questions .... This is food
> for thought and experimentation.

Who makes music using all notes of an EDO played at once?