source file: m1543.txt
Date: Sat, 3 Oct 1998 16:56:18 -0700 (PDT)

Subject: Re: beats me

From: bram <bram@gawth.com>

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