The JI Primer by Doty

Hi fellow tuners

I'm studying Doty's great book called The JI Primer and have a
question on Frequency Beats.

On page 11 he writes...

"The beat frequency (fB), is the difference between the two component
frequencies (f2-f1). It makes no difference which of the two tones is
higher in frequency, the beat frequency will be the same in either
case, occuring at the absolute value of the difference between the
two frequencies." -end-

The part I don't understand is where he says, "occuring at the
absolute value of the difference between the two frequencies."

Say for example I have two frequencies of the same amplitude.
Frequency one is 500Hz and Frequency two is 650Hz. The Beat Frequency
is 150Hz (650Hz - 500Hz).

What is happening at 150Hz ?

Thank you for your time
Walter Lepore NJ

Wally wrote:
>Say for example I have two frequencies of the same amplitude.
>Frequency one is 500Hz and Frequency two is 650Hz. The Beat Frequency
>is 150Hz (650Hz - 500Hz).

>What is happening at 150Hz ?

150 Hz is too high to be perceived as a beat frequency.
The beats in the form of variations in loudness need to be heard
individually, so the range is roughly 0 to 16 Hz. Beyond that it
will be heard as a kind of roughness until you clearly hear the
two tones separately.
Under circumstances a weak difference tone of that frequency can
be heard. This is caused by nonlinearities of the sound transmission
inside the ear. Same for the summation tone. But this aren't beat
frequencies.
If the tones aren't sine waves, then the harmonics might beat. So
then it's a question of the difference in frequency of the tone
components.

Manuel

--- In tuning@yahoogroups.com, manuel.op.de.coul@e... wrote:

> Under circumstances a weak difference tone of that frequency can
> be heard. This is caused by nonlinearities of the sound transmission
> inside the ear. Same for the summation tone. But this aren't beat
> frequencies.

also the cubic difference tone, 2*500 - 650 = 350Hz, can be even
louder in some circumstances.

Manuel wrote:

>150 Hz is too high to be perceived as a beat frequency.
>The beats in the form of variations in loudness need to be heard
>individually, so the range is roughly 0 to 16 Hz. Beyond that it
>will be heard as a kind of roughness until you clearly hear the
>two tones separately. -end-

That's what I'm confused about. I can hear beats but I can't hear a
beat frequency. I understand that the human ear begins to hear
audible sounds at about 20Hz but what exactly is a beat frequency?

I should clarify further what I'm confused about. I understand that
a "Beat" is a periodic variation in amplitude (volume). But what is a
beat frequency and what is it the result of?

It is my understanding that the beat frequency (fB), is the
difference between the two component frequencies (f2-f1). Therefore,
using the example in my previous post, if I have two frequencies of
the same amplitude and frequency number one is 500Hz and frequency
number two is 650Hz. The Beat Frequency is 150Hz (650Hz - 500Hz).

The beat will occur at the absolute value of the difference between
the two frequencies which is 150Hz.

But I thought the beat was simply the periodic rising and falling in
amplitude as a result of two separate tones sounding at the same time
with each having its own distinct frequency. How does the 150Hz beat
frequency relate to the above example??

I hope I'm making sense with this question.

Thank you all for your patience

Walter Lepore

--- In tuning@yahoogroups.com, manuel.op.de.coul@e... wrote:
> Wally wrote:
> >Say for example I have two frequencies of the same amplitude.
> >Frequency one is 500Hz and Frequency two is 650Hz. The Beat
Frequency
> >is 150Hz (650Hz - 500Hz).
>
> >What is happening at 150Hz ?
>
> 150 Hz is too high to be perceived as a beat frequency.
> The beats in the form of variations in loudness need to be heard
> individually, so the range is roughly 0 to 16 Hz. Beyond that it
> will be heard as a kind of roughness until you clearly hear the
> two tones separately.
> Under circumstances a weak difference tone of that frequency can
> be heard. This is caused by nonlinearities of the sound transmission
> inside the ear. Same for the summation tone. But this aren't beat
> frequencies.
> If the tones aren't sine waves, then the harmonics might beat. So
> then it's a question of the difference in frequency of the tone
> components.
>
> Manuel

In a message dated 1/6/03 5:51:38 PM Central Standard Time,
earth7@optonline.net writes:

> But what is a
> beat frequency and what is it the result of?
>
A beat frequency is the # of beats heard in a second. the ET 5th has 2 beats
per second, slow enough to count. It's caused by the two notes not being in
phase. In theoretical, and electronically-created, just intoantion just notes
are in phase, and thus beatless. The intervals and chords thus produced sound
dead and lifless, as such don't occur to any extent in acourstic instruments.

Sincerely,
Pauline W. Phillips, Moderator, <A HREF="/JohannusOrgansSchool ">Johannus Organs eSchool</A>
Johannus Orgelbouw, Holland, builds pipe, pipe-digital, digital-sampled
organs.
Moderator, <A HREF="/JustIntonationOrganSchool/">Just Intonation Organ eSchool</A>

Hello Walter

Back to the basic, beating is based on the exact trigonometric
equation:

sin( F1*t ) + sin( F2*t ) = 2 * sin( (F1+F2)*t/2 ) * cos( (F1-F2)*t/2
).

(I leave appart the 2*pi factor that is not necessary for the
explanation) Mathematically a sum of two sines with different
frequencies equate a sine with a frequency which is the average
between F1 and F2, but multiplied by a modulating enveloppe that has
frequency equating the difference between F1 and F2. If you record the
air pressure in the air (i.e. using a microphone) and trace the curve,
you will clearly see the beating.

The formula shows that the enveloppe frequency is (F1-F2)/2 and does
depends only on the difference, not absolute values of F1 and F2.

The inner ear "usually" select the left form (because the cochlea
works as Fourier transform machine that decompose the sound as sum of
sin of different frequencies). When F1 and F2 are so close that they
cannot be resolved as separate frequencies, the right form is
selected. That is why there no such thing as 150Hz beating is normally
perceived.

The perception of beat in this simple context does not involve any
non-linear phenomena.

yours truly

François Laferrière
--- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
<earth7@o...> wrote:
> Hi fellow tuners
>
> I'm studying Doty's great book called The JI Primer and have a
> question on Frequency Beats.
>
> On page 11 he writes...
>
> "The beat frequency (fB), is the difference between the two
component
> frequencies (f2-f1). It makes no difference which of the two tones
is
> higher in frequency, the beat frequency will be the same in either
> case, occuring at the absolute value of the difference between the
> two frequencies." -end-
>
> The part I don't understand is where he says, "occuring at the
> absolute value of the difference between the two frequencies."
>
> Say for example I have two frequencies of the same amplitude.
> Frequency one is 500Hz and Frequency two is 650Hz. The Beat
Frequency
> is 150Hz (650Hz - 500Hz).
>
> What is happening at 150Hz ?
>
> Thank you for your time
> Walter Lepore NJ

Paul wrote:
>also the cubic difference tone, 2*500 - 650 = 350Hz, can be even
>louder in some circumstances.

Sometimes also called secondary difference tone.

Manuel

Hi Pauline

> A beat frequency is the # of beats heard in a second. the ET 5th
> has 2 beats per second, slow enough to count. It's caused by the
> two notes not being in phase. In theoretical, and
> electronically-created,
> just intoantion just notes are in phase, and thus beatless. The
> intervals and chords thus produced sound dead and lifless, as such
> don't occur to any extent in acourstic instruments.

Beat frequency of ET fifth is not 2 beat across the whole keyboard, it
depends on the root frequency. ET 5th is 1.96 cent below just. This
can be converted in linear percentage as 0.113 % (not much). But it is
not the first harmonics that beat, that is the 3rd harmonic of the
root against the 2 harmonic of the fifth. Thus beat frequency of ET
fifth is (I hope I got the right numbers otherwise I will be
ridiculous again ...)

Fbeat = 3 * Froot * ( 1.00113 - 1 ) / 2

Fbeat = 0.0017 * Froot

Fbeat is 2 Hz for Froot around 1180 Hz (D6). It is 2 Hz between D6 and
A6, 1 Hz between D5 and A5, 0.5 Hz (one beat every two seconds)
between D4 and A4 and so on.

This is slow enough to count for long, sustained and stable tone. For
decaying notes such as those of piano, harpischord etc. the slope
induced by beat is of the order of magnitude of the slope of the
natural decay and both get mixed in such a way that even though the
beat affect the overall decay enveloppe (and the overall perceived
sound quality), the beat cannot be perceived as such.

yours truly

François Laferrière

Walter,
What I believe everyone is getting at, but hasn't said directly, is
that your math in your example is correct, but this phenomenon is
clearly audible only in tones very close to one another. So a better
example would be two frequencies 500 & 505, producing a beat frequency
of 5 beats per second.
Don't be frustrated...it took me three reads to understand it all well
enough to get into 7 limit and I still don't get it all!
Peace,
Chester
--- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
<earth7@o...> wrote:
> Hi fellow tuners
>
> I'm studying Doty's great book called The JI Primer and have a
> question on Frequency Beats.
>
> On page 11 he writes...
>
> "The beat frequency (fB), is the difference between the two
component
> frequencies (f2-f1). It makes no difference which of the two tones
is
> higher in frequency, the beat frequency will be the same in either
> case, occuring at the absolute value of the difference between the
> two frequencies." -end-
>
> The part I don't understand is where he says, "occuring at the
> absolute value of the difference between the two frequencies."
>
> Say for example I have two frequencies of the same amplitude.
> Frequency one is 500Hz and Frequency two is 650Hz. The Beat
Frequency
> is 150Hz (650Hz - 500Hz).
>
> What is happening at 150Hz ?
>
> Thank you for your time
> Walter

In a message dated 1/7/03 4:02:53 AM Central Standard Time,
francois.laferriere@oxymel.com writes:

> Beat frequency of ET fifth is not 2 beat across the whole keyboard, it
> depends on the root frequency.

Roight. I meant at or near Middle C. As the ptich get higher the beats get
higher. However, when other notes are included, as is usually the case, one
doesn't normally notice the difference enough to matter. As beats between
beats tend to level each other out to the ear.

Pauline

>----- Original Message -----
>From: prophecyspirit@aol.com
>
>In a message dated 1/7/03 4:02:53 AM Central Standard Time,
>francois.laferriere@oxymel.com writes:
>
>Beat frequency of ET fifth is not 2 beat across the whole keyboard, it
>depends on the root frequency.
>
>
>Roight. I meant at or near Middle C. As the ptich get higher the beats get
higher.

Wouldn't they get faster?

* David Beardsley
* http://biink.com
* http://mp3.com/davidbeardsley

In a message dated 1/7/03 10:51:29 AM Central Standard Time,
davidbeardsley@biink.com writes:

> Wouldn't they get faster?
>
That's what I meat--higher in number.

Pauline

--- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
<earth7@o...> wrote:
>
>
> Manuel wrote:
>
> >150 Hz is too high to be perceived as a beat frequency.
> >The beats in the form of variations in loudness need to be heard
> >individually, so the range is roughly 0 to 16 Hz. Beyond that it
> >will be heard as a kind of roughness until you clearly hear the
> >two tones separately. -end-
>
> That's what I'm confused about. I can hear beats but I can't hear a
> beat frequency. I understand that the human ear begins to hear
> audible sounds at about 20Hz but what exactly is a beat frequency?

beating is repeated variation in amplitude. if this happens a certain
number of times a second, then the beat frequency is that number (in
Hertz, or cycles per second).
>
> I should clarify further what I'm confused about. I understand that
> a "Beat" is a periodic variation in amplitude (volume). But what is a
> beat frequency and what is it the result of?

it's a result of the fact that your ear is unable to distiguish two
close frequencies as two separate tones. instead you get the
(mathematically equivalent) interpretation of a single, _mean_
frequency varying in amplitude. carl posted a nice website where this
was displayed visually in an applet a little while back . . . carl, you
have that handy?

> It is my understanding that the beat frequency (fB), is the
> difference between the two component frequencies (f2-f1). Therefore,
> using the example in my previous post, if I have two frequencies of
> the same amplitude and frequency number one is 500Hz and frequency
> number two is 650Hz. The Beat Frequency is 150Hz (650Hz - 500Hz).

again, it's not really a beat frequency, since you can't perceive a
single pitch varying in amplidute 150 times a second!

> The beat will occur at the absolute value of the difference between
> the two frequencies which is 150Hz.
>
> But I thought the beat was simply the periodic rising and falling in
> amplitude as a result of two separate tones sounding at the same time
> with each having its own distinct frequency. How does the 150Hz beat
> frequency relate to the above example??

*if* you could hear beating that fast, what you'd hear is a single tone
of 575 Hz, getting louder and softer 150 times per second.

>
> I hope I'm making sense with this question.

keep 'em comin'!

--- In tuning@yahoogroups.com, prophecyspirit@a... wrote:

> In theoretical, and electronically-created, just intoantion just notes
> are in phase, and thus beatless.

or they can be out of phase, and beatless. only a just interval will
preserve phase over time, be it in phase (which means 0 degree phase
shift), out of phase (which means 180 degree phase shift), or having
some other phase angle.

>carl posted a nice website where this was displayed visually
>in an applet a little while back . . . carl, you have that
>handy?

'fraid not, and can't think of how to search for it just now.

-Carl

Hi Carl

>carl posted a nice website where this
>was displayed visually in an applet a little while back . . . carl,
>you have that handy?

If you can find the time, I'd very much appreciate if you can locate
the website applet you wrote which wallyesterpaulrus so nicely
mentioned.

Thank you for your time and patience.
Walter Lepore

--- In tuning@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >carl posted a nice website where this was displayed visually
> >in an applet a little while back . . . carl, you have that
> >handy?
>
> 'fraid not, and can't think of how to search for it just now.
>
> -Carl

>
> From: "coitmusic <cwinowiecki@hotmail.com>" <cwinowiecki@hotmail.com>

I can think of another example. It is not necessary for the beat to be in close range.
If we took the example up just up one octive played on vibraphone or glockenspeil, it would be unavoidabl to hear a pitch at 300.
Those who have heard reich Drumming live hear like pitches in the section of all glocks. For tyhose who say this only exist in ones hear, one has only to turn down the treble on your stereo and these pitches will not dimish as the uppers tone generators do.
one does not need all these long forulas to say that the differenace between tewo pitch will generate a differance tone that due to various circumstances can be heard.

>

>
>
> Walter,
> What I believe everyone is getting at, but hasn't said directly, is
> that your math in your example is correct, but this phenomenon is
> clearly audible only in tones very close to one another. So a better
> example would be two frequencies 500 & 505, producing a beat frequency
> of 5 beats per second.
> Don't be frustrated...it took me three reads to understand it all well
> enough to get into 7 limit and I still don't get it all!
> Peace,
> Chester
> --- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
> <earth7@o...> wrote:
> > Hi fellow tuners
> >
> > I'm studying Doty's great book called The JI Primer and have a
> > question on Frequency Beats.
> >
> > On page 11 he writes...
> >
> > "The beat frequency (fB), is the difference between the two
> component
> > frequencies (f2-f1). It makes no difference which of the two tones
> is
> > higher in frequency, the beat frequency will be the same in either
> > case, occuring at the absolute value of the difference between the
> > two frequencies." -end-
> >
> > The part I don't understand is where he says, "occuring at the
> > absolute value of the difference between the two frequencies."
> >
> > Say for example I have two frequencies of the same amplitude.
> > Frequency one is 500Hz and Frequency two is 650Hz. The Beat
> Frequency
> > is 150Hz (650Hz - 500Hz).
> >
> > What is happening at 150Hz ?
> >
> > Thank you for your time
> > Walter
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM 8-9PM PST

Hi Tuners

Thanks for all the help with Frequency Beats. I think I understand
now.

I was also thinking about the actual formula for determining the beat
frequency which is:

Frequecy 1 (500 Hz) minus Frequency 2 (505 Hz) equals the Frequency
beat (5Hz or 5 cycles per second of repeated variation in amplitude).
Thus over the course of one second, there will be a repeated
variation in volume which is defined as the highest audible sound to
the lowest audible sound (1 cycle of amplitude variation) and this
will happen 5 times per second (5 cps or 5 Hz).

I'm typically not the one to just accept a formula at face value. I
like to know how the formula was derived.

Is there an understandable explanation for how the formula was
derived?

Thank you
Walter Lepore

--- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
<earth7@o...> wrote:
> Hi Tuners
>
> Thanks for all the help with Frequency Beats. I think I understand
> now.
>
> I was also thinking about the actual formula for determining the
beat
> frequency which is:
>
> Frequecy 1 (500 Hz) minus Frequency 2 (505 Hz) equals the Frequency
> beat (5Hz or 5 cycles per second of repeated variation in
amplitude).
> Thus over the course of one second, there will be a repeated
> variation in volume which is defined as the highest audible sound
to
> the lowest audible sound (1 cycle of amplitude variation)

that would be 1/2 cycle, actually.

> and this
> will happen 5 times per second (5 cps or 5 Hz).

the sound will reach its loudest (or quietest) 5 times per second,
yes -- thus 5 beats per second. if the 500Hz and 505Hz tones are
equal in amplitude, then it will sound like a 502.5Hz tone beating 5
times per second.

> I'm typically not the one to just accept a formula at face value. I
> like to know how the formula was derived.

francois posted the mathematical identity behind this. it's a bit of
trigonometry, made easier with complex numbers if you wish.

> Is there an understandable explanation for how the formula was
> derived?

the best explanation i've seen is in _the feynman lectures_.
extremely understandable. but in this case, you might want to simply
draw a picture (though pictures sometimes lie when it comes to
hearing, as carl recently saw). if you take the two functions sin
(2pi*t*500) and sin(2pi*t*505) and graphically add them, what you'll
see is a wave that has an "envelope" that varies smoothly between
maximum and minimum amplitude 5 times per second. if you like, i can
create these graphs and post them for you . . .

Hi Wallyesterpaulrus,

>if you like, i can create these graphs and post them for you . . .

Yes, I would very much appreciate if you the graphs so I can see and
understand what you are describing. I'm not sure how to create the
graphs on my own

Thank you for helping
Walter Lepore

--- In tuning@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
> <earth7@o...> wrote:
> > Hi Tuners
> >
> > Thanks for all the help with Frequency Beats. I think I
understand
> > now.
> >
> > I was also thinking about the actual formula for determining the
> beat
> > frequency which is:
> >
> > Frequecy 1 (500 Hz) minus Frequency 2 (505 Hz) equals the
Frequency
> > beat (5Hz or 5 cycles per second of repeated variation in
> amplitude).
> > Thus over the course of one second, there will be a repeated
> > variation in volume which is defined as the highest audible sound
> to
> > the lowest audible sound (1 cycle of amplitude variation)
>
> that would be 1/2 cycle, actually.
>
> > and this
> > will happen 5 times per second (5 cps or 5 Hz).
>
> the sound will reach its loudest (or quietest) 5 times per second,
> yes -- thus 5 beats per second. if the 500Hz and 505Hz tones are
> equal in amplitude, then it will sound like a 502.5Hz tone beating
5
> times per second.
>
> > I'm typically not the one to just accept a formula at face value.
I
> > like to know how the formula was derived.
>
> francois posted the mathematical identity behind this. it's a bit
of
> trigonometry, made easier with complex numbers if you wish.
>
> > Is there an understandable explanation for how the formula was
> > derived?
>
> the best explanation i've seen is in _the feynman lectures_.
> extremely understandable. but in this case, you might want to
simply
> draw a picture (though pictures sometimes lie when it comes to
> hearing, as carl recently saw). if you take the two functions sin
> (2pi*t*500) and sin(2pi*t*505) and graphically add them, what
you'll
> see is a wave that has an "envelope" that varies smoothly between
> maximum and minimum amplitude 5 times per second. if you like, i
can
> create these graphs and post them for you . . .

--- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
<earth7@o...> wrote:
> Hi Wallyesterpaulrus,
>
> >if you like, i can create these graphs and post them for you . . .
>
> Yes, I would very much appreciate if you the graphs so I can see
and
> understand what you are describing. I'm not sure how to create the
> graphs on my own
>
> Thank you for helping
> Walter Lepore

i changed the frequencies to 100 and 105 for easier viewing.

take a look:

/tuning-math/files/Paul/walter.jpg

the horizontal axis is time in seconds and the vertical axis is air
pressure relative to normal or eardrum position or whatever.

at first, the green and blue waves are in phase, so the resultant
(sum), the red wave, is similar in frequency but with twice the
amplitude of either.

after 1/10 of a second, the green and blue waves are out of phase, so
the so the resultant (sum) stays close to zero -- there's almost no
amplitude, so basically silence, at this point in time.

after 1/5 of a second, the green and blue waves are back in phase, so
you're back to maximum amplitude.

Hi Wallyesterpaulrus,

Wow! That's it! Thanks so much!

Now I can actually "see" what's happening while the beat frequency is
taking place.

How did you create this? Its fantastic!

There's allot going on in this picture as I study it. I can finally
see based on your example of two component frequencies (f1 = 105Hz
and F2 = 100Hz) how a single beat occurs in 1/5 of a second for a
total of 5 beats in one second (5Hz).

I also see in the beginning at say 0.01 seconds (at maximum amplitude)
how the peak of component frequencies f1 and f2 are "surrounded" by
the inside walls of the composite frequency. As the component
frequencies (f1 and f2) progress in time and begin the process of
cancelling each other out, they begin to "draw down" or "suck in" the
composite frequency between each other(frequency beat gets draw down
between f1 and f2). Then finally, to completeach component frequency
until (at the moment of minimum amplitude) the component frequency
is "squashed" into silence.

Can you please tell me where the points 0 degrees, 90 degrees, 180
degrees, 270 degrees and 360 degrees are located so I can see where
the complete cycle starts and ends? I understand 0 , 180 and 360
degrees are "silence" where there is no amplitude (assuming both
component frequencies are of the same amplitude to begin with) and 90
and 270 degrees are where maximum amplitude occurs.

In other words, please tell me based on the horizontal chart (time in
seconds)where the points in one complete cycle are located i.e. 0
degrees, 90 degrees, 180 degrees, 270 degrees, and 360 degrees.

Very, very interesting to see the actual mechanics of how all this
works!

Can I (or you) post a new message that says, "Tuners-See Frequency
Beat Chart". I believe it would benefit all tuners to have this graph
in their files for study.

Great work!

Thank you for your time
Walter Lepore

--- In tuning@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
> <earth7@o...> wrote:
> > Hi Wallyesterpaulrus,
> >
> > >if you like, i can create these graphs and post them for
you . . .
> >
> > Yes, I would very much appreciate if you the graphs so I can see
> and
> > understand what you are describing. I'm not sure how to create
the
> > graphs on my own
> >
> > Thank you for helping
> > Walter Lepore
>
> i changed the frequencies to 100 and 105 for easier viewing.
>
> take a look:
>
> /tuning-math/files/Paul/walter.jpg
>
> the horizontal axis is time in seconds and the vertical axis is air
> pressure relative to normal or eardrum position or whatever.
>
> at first, the green and blue waves are in phase, so the resultant
> (sum), the red wave, is similar in frequency but with twice the
> amplitude of either.
>
> after 1/10 of a second, the green and blue waves are out of phase,
so
> the so the resultant (sum) stays close to zero -- there's almost no
> amplitude, so basically silence, at this point in time.
>
> after 1/5 of a second, the green and blue waves are back in phase,
so
> you're back to maximum amplitude.

--- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
<earth7@o...> wrote:
> Hi Wallyesterpaulrus,
>
> Wow! That's it! Thanks so much!
>
> Now I can actually "see" what's happening while the beat frequency
is
> taking place.
>
> How did you create this? Its fantastic!

a line or two of MATLAB code.

> There's allot going on in this picture as I study it. I can finally
> see based on your example of two component frequencies (f1 = 105Hz
> and F2 = 100Hz) how a single beat occurs in 1/5 of a second for a
> total of 5 beats in one second (5Hz).

right.

> I also see in the beginning at say 0.01 seconds (at maximum
amplitude)
> how the peak of component frequencies f1 and f2 are "surrounded" by
> the inside walls of the composite frequency.

hmm . . . not sure about this terminology, but at least you're seeing
something -- just remember that not everything you see necessarily
corresponds with everything you hear.

> As the component
> frequencies (f1 and f2) progress in time and begin the process of
> cancelling each other out, they begin to "draw down" or "suck in"
the
> composite frequency between each other(frequency beat gets draw
down
> between f1 and f2). Then finally, to completeach component
frequency
> until (at the moment of minimum amplitude) the component

composite?

> frequency
> is "squashed" into silence.

well, you've lost me with your terminology . . .

> Can you please tell me where the points 0 degrees, 90 degrees, 180
> degrees, 270 degrees and 360 degrees are located

0 sec., 0.05 sec., 0.1 sec., 0.15 sec., 0.2 sec., respectively.

> so I can see where
> the complete cycle starts and ends? I understand 0 , 180 and 360
> degrees are "silence" where there is no amplitude (assuming both
> component frequencies are of the same amplitude to begin with) and
90
> and 270 degrees are where maximum amplitude occurs.

not sure where you're getting that. when the phase angle is 0 degrees
= 360 degrees, you're in phase and you have maximum amplitude. when
the phase angle is 180 degrees, you're out of phase and you have zero
amplitude.

> Can I (or you) post a new message that says, "Tuners-See Frequency
> Beat Chart".

you can, if you wish . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_41789.html#41866

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "electricwally77 <earth7@o...>"
> <earth7@o...> wrote:
> > Hi Wallyesterpaulrus,
> >
> > >if you like, i can create these graphs and post them for
you . . .
> >
> > Yes, I would very much appreciate if you the graphs so I can see
> and
> > understand what you are describing. I'm not sure how to create
the
> > graphs on my own
> >
> > Thank you for helping
> > Walter Lepore
>
> i changed the frequencies to 100 and 105 for easier viewing.
>
> take a look:
>
> /tuning-math/files/Paul/walter.jpg
>
> the horizontal axis is time in seconds and the vertical axis is air
> pressure relative to normal or eardrum position or whatever.
>
> at first, the green and blue waves are in phase, so the resultant
> (sum), the red wave, is similar in frequency but with twice the
> amplitude of either.
>
> after 1/10 of a second, the green and blue waves are out of phase,
so
> the so the resultant (sum) stays close to zero -- there's almost no
> amplitude, so basically silence, at this point in time.
>
> after 1/5 of a second, the green and blue waves are back in phase,
so
> you're back to maximum amplitude.

***This is a very cool, comprehensible and instructive diagram!

J. Pehrson