[sdiy] Phase shifts and instantaneous frequency

[Ian Fritz]

At 02:26 PM 7/15/2008, Tom Wiltshire wrote:

>I did try it, and as expected I hear a repetitive throbbing in the
>resulting tone. The 1Hz effect is clearly visible in the wave;

Thanks for trying it!


>http://www.electricdruid.com/beatwave.jpg
>
>Since I can see a 1Hz effect in the waveform in front of me, I'm not
>entirely convinced by your explanation that this is only a non-linear
>effect of our hearing. It looks to me like my ears are picking up a
>1Hz variation which exists in this waveform.

The phase difference will, indeed, produce a different summed 
waveform.  But the power spectrum is unchanging, i.e. there is always the 
same power in each component.  So if the ear's response is linear you 
should hear a steady tone.  This is the usual argument that is used to 
claim that relative phase shouldn't matter, even though the waveform itself 
changes.

I convinced myself of the nonlinearity by changing the volume level.  You 
don't hear any beats at low levels and they become quite strong at high 
levels.  You could also try putting each component wave in just one ear 
(using a headset). The beats will disappear if you do that.


>Also, claiming that something that changes at 1Hz is "Quasi-static"
>and therefore says something about waveforms that don't change at all
>seems a bit disingenuous. No-one ever claimed an LFO isn't
>oscillating just because it goes down to 0.1Hz!

No, but consideration of appropriate limits is used all the time in physics 
and math.  Sure, you have to be careful doing this.  The point is that you 
can approach the static limit as closely as you like by making the 
frequencies closer and closer to 2:1. When you do this you will still hear 
the same beats. The quasistatic procedure is used because it is hard to 
compare separated sounds.  Anyway, the effect can't be "pitch bend".  But 
if you don't like the limit argument, you can construct two signals with 
2:1 frequencies but different phase shifts.  According to the original 
research of the 1870's, you will get the largest difference if you use a 
quarter wave phase shift.  You can splice these signals alternately to hear 
the tonal difference.


>I don't understand what I'm hearing exactly, since the sum and
>difference frequencies are so far wrong, but it does seem like I'm
>hearing what I see.

Yes, it may seem non-obvious, but you are not hearing what you 
see.  Because what you hear will vary with amplitude and because if you 
present the signals separately to your ears you will not hear the beats.


>Can you clarify what is going on?

Good questions. Hope this helps!

  Ian