RE: Cancelling out beats

Bram, I don't think you're understanding the phenomenon of beats, which
arise as a result of the limited frequency resolution of the ear. You
should do some auditory experiments to convince yourself that you're on
the right (or wrong) track.

Paul H., the nonlinear processing that the ear does actually increases
the frequency resolution over what the standard uncertainty relation
would give for an analyzer with the ear's temporal resolution, thus
decreasing (rather than creating) the incidence of beats. But you're
exactly right about Sethares.

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End of TUNING Digest 1542
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On Sat, 3 Oct 1998, William Sethares wrote:

> There are really two kinds of "beats" being discussed: one is a
> physical wave phenomenon and the other is a psycho-acoustic
> phenomenon.

Yes, I've been thinking about the physical wave phenomenon. I was
previously unaware of the other one.

> I see no theoretical reason why Bram's suggestion shouldn't be do-
> able. But whether you can actually measure the waves, do the
> calculations, and output the appropriate beat-cancelling sound in real
> time -- may be tricky.

Oh, that sort of effect is beyond my ambitions at the moment. :)

Right now I'm just trying to make computer-generated sound files of pure
harmonics with the beats fixed.

I think any ratio can actually have an infinite number of beats,
corresponding to the numbers in it's continued fraction expansion,
although they get much more subtle after the first. An illustration is at

http://gawth.com/~bram/sound/secondary_beat.wav

That file alternates between 6/5 (continued fraction expansion 1 5) and
31/26 (continued fraction expansion 1 5 5) It's playable as a loop (I
*think* I lined up the phases correctly.) The second beat, to my ear,
produces a sound with an obvious relationship to the first one but much
more solemn, possibly corresponding to the adding of a single beat with a
much lower frequency.

-Bram

> Thus (for example) a sine with frequency 100 Hz and another with
> frequency 101 Hz when sounded together appear to be a single wave
> at frequency 100.5 Hz with a beat rate (amplitude modulation) of 1
> Hz. This is the kind of beating that Bram is (I think) trying to cancel
> by adding back in appropriately chosen waves.

> I see no theoretical reason why Bram's suggestion shouldn't be do-
> able.

I probably didn't follow Bram's suggestion then. In the case of Bill's
example, Bram, were you suggesting to add a wave at 100.5Hz whose amplitude
is a constant minus the amplitude of the composite waveform? If so, then
as Bill said, yes, it seems like that ought to work. It would seem pretty
involved when you get into nonsinusoidal waveforms, though.

I was thinking, apparently incorrectly, that (again using Bill's
example) you expected to be able to remove the beating phenomenon by adding
a 1Hz sinewave to the 100Hz & 101Hz composite wave. That didn't seem
likely to work auditorially, and certainly doesn't mean anything in terms
of the objective tones.