Beats (was No doubt about Lehman's Bach scale)

Hi Carl,

Thanks for your reply.

On Sun, 16 Oct 2005, you wrote:
>
> > > Wow. Are all these beat rates referring to the beating of upper
> > > partials in a not-quite-JI chord?
> >
> > For example, if I play an A major triad on my piano with the
> > A tuned to 220 Hz, the C# to 275 Hz and the E tuned to 331 Hz,
> > then I'll get beats between the 3rd partial of the A at 660 Hz
> > and the 2nd partial of the E at 662 Hz. These beats will be at
> > 662 - 660 = 2 Hz.
>
> Hi Yahya,
>
> I'll make a few comments. If you want to wait for Paul's reply
> to reply, that's fine by me. I'm just thinking aloud waiting
> for Paul's reply myself.
>
> I'm not sure why you're subtracting these beat rates here.

Look again. I subtracted two near partials to find a slow
audible beat. I did not subtract a beat from anything.

When ANY two frequencies sound simultaneously, so do two
other frequencies - their sum and difference. Since we only
use the term "beats" to refer to low frequencies, I've
focussed on the difference of two tones that lie close to each
other. It's what I listen for when tuning a guitar.

> As I think Paul was trying to say, beats are amplitude
> modulation, which is sort of a second order sound wave.

Combining two pure sinusoidal waves of frequencies A and B
results in sinusoidal waves with components of frequencies
A-B and A+B. IIRC, these components are *multiplied*
rather than *added*, so one (the slower) can be regarded
as an (amplitude) envelope for the other (the faster).

> They represent changes in the amplitude of a sound,

... which our ears respond to ...

> ... not
> a sound in and of themselves.

I'm not sure that this is so.

> ... AM is used for synthesis
> and can create new pitches, called sidebands, but they
> aren't at the frequency of the beats.

Sure? AM radio uses the same "heterodyne" principle
(combining near frequencies) both to create and detect
a modulation of the carrier wave we tune the receiver to.

If the upper and lower sidebands aren't at the
frequencies of the combination tones, what frequencies
are they at?

> ... Perhaps sidebands
> could be evoked from beating in an acoustic musical
> instrument, but I don't think it happens very often (though
> multiphonics on instruments like the bassoon sometimes
> sound like AM to me).
>
> -Carl

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> When ANY two frequencies sound simultaneously, so do two
> other frequencies - their sum and difference.

Yahya, this is not true in general. It's not true when the sound is
amplified or reproduced *linearly*, i.e., with no distortion.
Meanwhile, when there is a quadratic distortion of the signal, you
get not only the sum and difference, but also the 2nd partials of
each of the two frequencies. Shall I show the math?

> > As I think Paul was trying to say, beats are amplitude
> > modulation, which is sort of a second order sound wave.
>
> Combining two pure sinusoidal waves of frequencies A and B
> results in sinusoidal waves with components of frequencies
> A-B and A+B.

This is false, in general, as I wrote above. It's only true in the
presence of distortion. A quadratic nonlinearity is the simplest kind
of distortion, and results in not only A, B, A-B, and A+B, but also
2*A and 2*B as well. Perhaps I should review the math?

> If the upper and lower sidebands aren't at the
> frequencies of the combination tones, what frequencies
> are they at?

They are at the frequencies corresponding to A and B in your example.

i would be curious about this , only the 2nd? or are we leaving timbre out of the equations for the moment.
if you don't mind

>Message: 25 > Date: Mon, 17 Oct 2005 20:52:34 -0000
> From: "wallyesterpaulrus" >Subject: Re: Beats (was No doubt about Lehman's Bach scale)
>
> >
>Yahya, this is not true in general. It's not true when the sound is >amplified or reproduced *linearly*, i.e., with no distortion. >Meanwhile, when there is a quadratic distortion of the signal, you >get not only the sum and difference, but also the 2nd partials of >each of the two frequencies. Shall I show the math?
>
> >
>>>As I think Paul was trying to say, beats are amplitude
>>>modulation, which is sort of a second order sound wave.
>>> >>>
>>Combining two pure sinusoidal waves of frequencies A and B
>>results in sinusoidal waves with components of frequencies
>>A-B and A+B. >> >>
>
>This is false, in general, as I wrote above. It's only true in the >presence of distortion. A quadratic nonlinearity is the simplest kind >of distortion, and results in not only A, B, A-B, and A+B, but also >2*A and 2*B as well. Perhaps I should review the math?
>
> >
>>If the upper and lower sidebands aren't at the >>frequencies of the combination tones, what frequencies >>are they at?
>> >>
>
>They are at the frequencies corresponding to A and B in your example.
>
>
>
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> i would be curious about this , only the 2nd?

Yes. As you can see below, when I wrote that I was only talking about
a quadratic nonlinearity, the simplest kind. You'd need a cubic
nonlinearity to generate 3rd partials, a quartic nonlinearity to
generate 4th partials, etc. Most nonlinear response functions in
nature contain quadratic, cubic, quartic, and higher-order components.

> or are we leaving timbre
> out of the equations for the moment.

We are definitely not leaving timbre out -- see below: Yahya
specified "sinusoidal waves", which of course are pure tones with no
harmonic or inharmonic upper partials.

> if you don't mind

I don't mind at all. Perhaps you could re-read the below and tell me
if there's anything that's still unclear, and/or if I should
explicate the math.

> >Message: 25
> > Date: Mon, 17 Oct 2005 20:52:34 -0000
> > From: "wallyesterpaulrus"
> >Subject: Re: Beats (was No doubt about Lehman's Bach scale)
> >
> >
> >
> >Yahya, this is not true in general. It's not true when the sound
is
> >amplified or reproduced *linearly*, i.e., with no distortion.
> >Meanwhile, when there is a quadratic distortion of the signal, you
> >get not only the sum and difference, but also the 2nd partials of
> >each of the two frequencies. Shall I show the math?
> >
> >
> >
> >>>As I think Paul was trying to say, beats are amplitude
> >>>modulation, which is sort of a second order sound wave.
> >>>
> >>>
> >>Combining two pure sinusoidal waves of frequencies A and B
> >>results in sinusoidal waves with components of frequencies
> >>A-B and A+B.
> >>
> >>
> >
> >This is false, in general, as I wrote above. It's only true in the
> >presence of distortion. A quadratic nonlinearity is the simplest
kind
> >of distortion, and results in not only A, B, A-B, and A+B, but
also
> >2*A and 2*B as well. Perhaps I should review the math?
> >
> >
> >
> >>If the upper and lower sidebands aren't at the
> >>frequencies of the combination tones, what frequencies
> >>are they at?
> >>
> >>
> >
> >They are at the frequencies corresponding to A and B in your
example.
> >
> >
> >
> >
> >
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>

Hi Yahya,

My browser apparently ate my reply, so I'm reconstructing
it from memory.

> > > For example, if I play an A major triad on my piano with the
> > > A tuned to 220 Hz, the C# to 275 Hz and the E tuned to 331 Hz,
> > > then I'll get beats between the 3rd partial of the A at 660 Hz
> > > and the 2nd partial of the E at 662 Hz. These beats will be at
> > > 662 - 660 = 2 Hz.
//
> >
> > I'm not sure why you're subtracting these beat rates here.
>
> Look again. I subtracted two near partials to find a slow
> audible beat. I did not subtract a beat from anything.

Oops... my mistake!

> When ANY two frequencies sound simultaneously, so do two
> other frequencies - their sum and difference. Since we
> only use the term "beats" to refer to low frequencies,

Beats and combination tones (sum and difference) are
different phenomena.

> > As I think Paul was trying to say, beats are amplitude
> > modulation, which is sort of a second order sound wave.
> > They represent changes in the amplitude of a sound,
> > not a sound in and of themselves.
>
> I'm not sure that this is so.

There has been some debate about this -- Bill Sethares
disagreed with Paul once upon a time. IIRC, fast pulses
of white noise can be used to evoke a pitch at the
pulse frequency....

> > ... AM is used for synthesis
> > and can create new pitches, called sidebands, but they
> > aren't at the frequency of the beats.
>
> Sure?

Beats are heard at the average of the two sine waves,
according to Doty's JI Primer.

-Carl

Hi Paul,

On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > When ANY two frequencies sound simultaneously, so do two
> > other frequencies - their sum and difference.
>
> Yahya, this is not true in general.

Think I'll just shut up for a while and listen ! 8-0

> ... It's not true when the sound is
> amplified or reproduced *linearly*, i.e., with no distortion.
> Meanwhile, when there is a quadratic distortion of the signal, you
> get not only the sum and difference, but also the 2nd partials of
> each of the two frequencies. Shall I show the math?

Yes please.

> > > As I think Paul was trying to say, beats are amplitude
> > > modulation, which is sort of a second order sound wave.
> >
> > Combining two pure sinusoidal waves of frequencies A and B
> > results in sinusoidal waves with components of frequencies
> > A-B and A+B.
>
> This is false, in general, as I wrote above.

Ditto.

> ... It's only true in the
> presence of distortion. A quadratic nonlinearity is the simplest kind
> of distortion, and results in not only A, B, A-B, and A+B, but also
> 2*A and 2*B as well. Perhaps I should review the math?

And ditto.

> > If the upper and lower sidebands aren't at the
> > frequencies of the combination tones, what frequencies
> > are they at?
>
> They are at the frequencies corresponding to A and B in your example.

Whew! Do I get 1 out of 10, then? :-)

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Hi Paul,
>
> On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote:
> >
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:
> >
> > > When ANY two frequencies sound simultaneously, so do two
> > > other frequencies - their sum and difference.
> >
> > Yahya, this is not true in general.
>
> Think I'll just shut up for a while and listen ! 8-0

Well, here's a start, but I need your feedback to know in what manner
I should proceed . . .

> > ... It's not true when the sound is
> > amplified or reproduced *linearly*, i.e., with no distortion.
> > Meanwhile, when there is a quadratic distortion of the signal,
you
> > get not only the sum and difference, but also the 2nd partials of
> > each of the two frequencies. Shall I show the math?
>
> Yes please.

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.

> > > > As I think Paul was trying to say, beats are amplitude
> > > > modulation, which is sort of a second order sound wave.
> > >
> > > Combining two pure sinusoidal waves of frequencies A and B
> > > results in sinusoidal waves with components of frequencies
> > > A-B and A+B.
> >
> > This is false, in general, as I wrote above.
>
> Ditto.

Since 'normal' response is linear, Fourier theory tells us that
combining pure tones yields a signal with frequency components equal
to those of the pure tones, and no other frequency components.

> > ... It's only true in the
> > presence of distortion. A quadratic nonlinearity is the simplest
kind
> > of distortion, and results in not only A, B, A-B, and A+B, but
also
> > 2*A and 2*B as well. Perhaps I should review the math?
>
> And ditto.

Done, above.

> > > If the upper and lower sidebands aren't at the
> > > frequencies of the combination tones, what frequencies
> > > are they at?
> >
> > They are at the frequencies corresponding to A and B in your
example.
>
> Whew! Do I get 1 out of 10, then? :-)

The (quadratic) combinational tones are A+B and A-B, not A and B.

Ouch! Does this not belong to the tuning-math?
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 21 Ekim 2005 Cuma 1:13
Subject: [tuning] Re: Beats (was No doubt about Lehman's Bach scale)

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.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> Ouch! Does this not belong to the tuning-math?\

Right. For tuning, Paul should have used complex numbers and
exponential functions, thereby making the whole derivation easier. :)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> >
> > Ouch! Does this not belong to the tuning-math?\
>
> Right. For tuning, Paul should have used complex numbers and
> exponential functions, thereby making the whole derivation easier. :)

That's funny, I was just going to say that I would have used complex
numbers and exponential functions, which make the derivation far
shorter, had I posted the explanation on tuning-math. But here on
tuning, real numbers are bad enough :) Seriously, I don't know if Yahya
is on tuning-math, and it was hopefully a one-time deal; otherwise, we
can certainly move anything further to tuning-math.

Hi Paul, Gene, Ozan et al,

On Fri, 21 Oct 2005, "wallyesterpaulrus" wrote:
>
>--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>>
>> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>> >
>> > Ouch! Does this not belong to the tuning-math?\
>>
>> Right. For tuning, Paul should have used complex numbers and
>> exponential functions, thereby making the whole derivation easier. :)
>
>That's funny,

It is, isn't it? I would have preferred an e^ix - type formulation,
but should we be introducing anything "imaginary" here - we've
had enough trouble recently with people insisting that _virtual_
ain't real ... :-)

> ...I was just going to say that I would have used complex
>numbers and exponential functions, which make the derivation far
>shorter, had I posted the explanation on tuning-math. But here on
>tuning, real numbers are bad enough :) Seriously, I don't know if Yahya
>is on tuning-math, and it was hopefully a one-time deal; otherwise, we
>can certainly move anything further to tuning-math.

Well, I've been hiding ... I scarce have time to read the tuning list, let
alone respond meaningfully. But if you want to do a mathematical
exposition of beats and other non-linear musical phenomena, just say
the word and I'll follow it there. However, other musicians on _this_
list, not so mathematically inclined, might still appreciate a pratical
summary of said exposition with minimal maths.

Regards,
Yahya

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