source file: m1542.txt
Date: Fri, 2 Oct 1998 08:11:54 -0500

Subject: beating the beats

From: sethares@eceserv0.ece.wisc.edu (William Sethares)

Bram wrote:

> Might it be possible to find these (Sethares consoannce papers) 
online? 
 
An .html version of "Relating Timbre and Tuning" 
is available online at:

http://eceserv0.ece.wisc.edu/~sethares/consemi.html

and theres a bunch of "supporting stuff" like computer programs to 
do the dissonance calculations branching from my homepage at:

http://eceserv0.ece.wisc.edu/~sethares/

***

As Paul Hahn mentioned, Bram's idea of cancelling beats in 
a "noise cancellation" kind of way is not directly related to the
minimization of beats via a sensory dissonance calculation (which is 
what's going on in "Relating Timbre and Tuning").

In any case, one useful formula from trig is:

sin(x) + sin(y) = 2 cos( (x-y)/2 ) sin( (x+y)/2 )

To apply this to the beat cancellation, use this in the form

sin(w t) + sin( (w + dw)t ) = 2 cos( dw t /2) sin( (w+dw/2)t )

where w = one frequency
w+dw = other frequency
(thus dw = different between the two frequencies)
and t is time.

The "amplitude" is thus the slowly varying wave cos(dw t/2)
while the basic wave if sin( (w+dw/2)t )

***