TUNING digest 1541

>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?