source file: m1542.txt Date: Fri, 02 Oct 1998 17:39:26 -0400 Subject: Re: TUNING digest 1541 From: "Andrew Bolce'" >Topic No. 5 > >Date: Thu, 1 Oct 1998 14:17:08 -0700 (PDT) >From: bram >To: tuning@eartha.mills.edu >Subject: Re: Cancelling out beats >Message-ID: > >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. > they're not equal. a*sin(x/a) = a(sq.rt.((1-cosx)/a)) or sq.rt.((a^2-a^2cosx)/(a)) i think. 0 would work however. and so would pi. (or would it be 0 and 1?) >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? > does anyone have any sample keyboard programs out of mild curiousty?