From: dvf@kepler.unh.edu (David V Feldman)
Subject: Re: When is close close enough for just intonation
Keywords: intonation, intervals
Organization: University of New Hampshire  -  Durham, NH
Date: Sun, 16 Aug 1992 19:35:10 GMT

In article <l8qjqvINN716@calvin.usc.edu> alves@calvin.usc.edu (William Alves) writes:
>In article <1992Aug15.000215.28004@newshost.unh.edu> dvf@kepler.unh.edu (David V Feldman) writes:
>>In article <l8o3aoINN6c@calvin.usc.edu> alves@calvin.usc.edu (William Alves) writes:
>>>I'm afraid I don't understand the part about decaying amplitude and how
>>>that has anything to do with beats or frequency. A single piano tone
>>>often has beats in it, but that's because most strings are actually
>>>multiple strings which may be slightly out of tune.
>>>
>>Suppose I sound two (sine) tones which begin in phase but differ in frequency by
>>a minute amount.   The beating will be very slow.  For example, if the tones
>>differ in frequency by 1/100 cps (cycles per second) each beat will take
>>100 seconds.  If the process were stopped after only 50 seconds the effect
>>would be that of a single tone decaying in amplitude for 50 seconds.  Indeed,
>>if the sine tones were mixed and produced by a single loud speaker, there
>>would be no mathematical distinction, even in principle, between the two
>>situations.  Turning this backwards, a tone whose amplitude is decaying
>>has a pitch that is no more definite that the interval between the two
>>nearby tones whose simultaneity would produce the same decay.
>
>I assume you are referring to destructive interference. If so, the above
>example only holds for an idealized case. At any rate, it depends on the
>relative phases of the two tones to begin with. If they were to begin
>out of phase, then one's impression would be of a steadily increasing
>loudness. But in any real acoustical environment instead of two tones
>mixed and played over a single loudspeaker, the relative phases of two
>tones would vary depending on the position of your ears. Even turning
>one's head if the frequencies are high enough can give a completely
>different impression.
>
>Also, destructive interference only accounts for beats in unison tones.
>How is it that two tones slightly out of tune from a 5/4 ratio will
>also produce beats in one's brain even though the RMS amplitude remains
>constant?
>
>Bill Alves

Some beating phenomena are purely psychoacoustical and will take place
even if the two tones are heard by different ears.  (There is a Scientific
American article from some years back about this-precise reference anyone?)
I don't think that such phenomena are completely understood.  An out of
tune 5/4 ratio may beat because of destructive interference between the
overtones.

My point is still a point in principle: a decaying tone can be
interpreted as being the result of destructive interference of nearly
unison tones, and therefore cannot be assigned, even mathematically,
a pitch more definite than the difference between those tones.  The longer
the decay, the closer the tones, the more definite the pitch.  I agree
that reversed the situation becomes idealized - that is when two tones
are actually being produced and interpreted as one.  But for the point
I am making, that is irrelevant.

Let me emphasize again that my subject is the mathematical definiteness
of tones of finite duration.  Psychoacoustical phenomena my both help
and hinder pitch perception.  Much work in vision shows that brain 
processing depends on prior assumptions to interpret ambiguous input
(work of Alan Yuille for instance) and it seems reasonable to suppose 
that something similar goes on with various aspects of hearing.  Perhaps
one and the same sound will sound different, at a very fundamental level,
to people brought up on different pitch systems.  (These prior assumptions
account for various optical illusions, making people see genuinely different
things in the same image, depending on how they are conditioned.)   But
I am getting beyond my ken.

David Feldman