source file: m1541.txt
Date: Thu, 1 Oct 1998 14:17:08 -0700 (PDT)

Subject: Re: Cancelling out beats

From: bram <bram@gawth.com>

On Thu, 1 Oct 1998, Paul Hahn wrote:

> On Thu, 1 Oct 1998, Paul Hahn wrote:
> > On Thu, 1 Oct 1998, bram wrote:
> >>        I know the beats can be essentially calculated using continued
> >> fractions, so I'm guessing there's a rather straightforward way of
> >> calculating, given a*sin(x) + b*sin(y), a and b being volumes and x and y
> >> being wavelengths, there's a rather straightforward way of calculating a
> >> bunch of other sin functions which when added will cancel out all the
> >> beats.
> > 
> > I don't think Bill is canceling out the beats in an antinoise sense;
> 
> Actually, thinking further on it, I don't think that would even be
> possible, considering that the beats don't actually exist physically,
> but arise because of nonlinearities in the auditory system.

Ah, but they do :)

After thinking about it a bit more, I realize that the beats need to be
'augmented' rather than 'cancelled', since they correspond to regions of
decreased volume.

The very simplest example is sin(x/n)+sin(x/(n+1)) since that only has one
beat.

Consider the case where n=4. It starts out at about double the volume of
an individual sin wave, then the volume decreases until the two waves
exactly cancel out at x=10, then come to complement each other and x=20,
and continue to alternate.

If a sin wave is now added which becomes most angular at the place where
the other two waves cancel out, it will tend to make the volume consistent
throughout. Specifically, the function

sin(x/4) + sin(x/5) + 5*sin((x+10)/40)

should have less of a beat.

I say less instead of none because I'm fairly certain I did something
wrong with the volumes. Can someone verify for me whether sin(x) and
a*sin(x/a) are of the same volume? If they are, I know how to make it
exact.

Note that phase information is very important here - simply adding a third
sin wave of the correct wavelength but wrong phase will be unlikely to do
a good job fixing the beat.

Unfortunately, I don't have the tools handy to make neat diagrams or sound
files to illustrate the above - could anybody suggest some good free ones
which work under windows?

I'm still not sure what to do when the phases of the two waves don't line
up exactly, and haven't really looked into how to fix ratios with more
beats (like 21/17) but I think I'll be able to work that out later - it's
an *interesting* problem. 

-Bram