[sdiy] Harmonic content of the "sigmoid" half-sine wave

[Jerry Gray-Eskue]

<<The beat frequency is the (a-b)/2 frequency.>>

<<There is no difference frequency produced.  Beats and difference tones are
different thing.>>

For an electronic signal I can see and grasp this as the case, but for us to
hear the signal we must use some form of mechanical transducer, a speaker or
the body of an acoustical guitar for example.

Since the two frequencies are causing physical motion we can get
reinforcement of the (a-b)/2 as the physical motions interact. So do we
create the Beat signal here, or are the physics of the mechanical motion
preserving the two frequencies intact?

Jerry

-----Original Message-----
From: synth-diy-bounces at dropmix.xs4all.nl
[mailto:synth-diy-bounces at dropmix.xs4all.nl]On Behalf Of Ian Fritz
Sent: Thursday, June 04, 2009 9:40 AM
To: Forbes, William ALGLSG-LXES; sdiy DIY
Subject: RE: [sdiy] Harmonic content of the "sigmoid" half-sine wave


At 03:28 AM 6/4/2009, Forbes, William  ALGLSG-LXES wrote:

>Note:
>sin(a) + sin(b) = 2 * ( sin( (a+b)/2 ) * cos( (a-b)/2 ) )
>
>and:
>sin(a) * sin(b) = -( cos(a+b) + cos(a-b) )/2
>
>Thus what happens when we hear the beat frequency between two
>frequencies?
>
>If I play 19khz and 20kHz tone I don't hear the 1kHz until distortion
>causes
>the multiplication terms to be generated.
>My ears don't work at 19kHz so I hear nothing when the system is linear.
>But I can quite easily hear the 1kHz when the system is non-linear.
>
>Yet I can quite easily hear a beat frequency when tuning a guitar.

Easy to get confused by all this.

Your first formula describes the beats you hear when tuning your
guitar.  Look carefully at the right hand side.  The factor with the
(a+b)/2 frequency represents an audio frequency oscillation at the average
frequency of the two pitches.  You can hear this because it is at audio
frequency.  The second factor represents a slow amplitude modulation of the
tone you hear. The amplitude modulation is the beating. The beat frequency
is the (a-b)/2 frequency.

When you repeat this using 19kHz and 20kHz tones you hear nothing, simply
because the average (a+b)/2 frequency is out of your hearing range.  The
physics hasn't changed at all, and the waves will still beat on an
oscilloscope trace, just as in the guitar tuning case.

Note that this is strictly a linear problem.  The Fourier spectrum contains
just the two frequencies "a" and "b". (Left hand side of the
equation.)  There is no difference frequency produced.  Beats and
difference tones are different thing.

Hope this helps.  The hardest thing about teaching physics is overcoming
preconceived misconceptions.

   Ian

_______________________________________________
Synth-diy mailing list
Synth-diy at dropmix.xs4all.nl
http://dropmix.xs4all.nl/mailman/listinfo/synth-diy