--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Chris wrote:
>
> > What makes a "beat" different from a "difference tone"? with
respect the
> > only difference I see between the two is the frequency range in
which they
> > occur.
>
> As Daniel has pointed out, sometimes it is the frequency of the
difference
> tone itself, which we may perceive as beats if it lies outside our
hearing
> range. In some other conditions, it may have to do with the actual
> (exponential) interval size and with the general characteristics
of the
> intervals; and in some of these cases, even a deviation as small
as 27 cents
> can remarcably change them (see my last post).
>
> Petr
>


To my understanding, difference tone is a tone, i.e. periodical
changes of air pressure. For instance, combinational tones (due only
to some nonlinear effect involved in production, transmission or
perception of sound) of two frequencies "a" and "b" are frequencies
a-b and a+b, and they are tones as well. With combinational tones
you actually see the two new frequencies to appear in the spectrum,
a difference tone and a sum tone, and their intensity relative to
the source tones depends on volume.

Instead, beatings are periodical changes of the loudness of some
other (typically faster) tone. Beating is a linear effect due to
wave interference. It does not introduce new frequencies in the
signal, that is, if you make a Fourier spectrum you don't see
additional frequencies.

In my view, beatings are always beatings (linear wave interference)
regardless their frequency. The fact that the ear cannot follow too
rapid beatings does not change the nature of the stimulus. In other
words, can a modulation of the amplitude of a pressure wave (that
has itself a constant average pressure, that is the atmospheric
pressure) turn into a time-dependent average pressure at the
frequency of the beating? I think not, but this topic is actually
what I am trying to understand.

Let's put it in different terms. Take two ultrasonic waves with an
audible frequency difference (say 30000 Hz and 30100 Hz). We should
be able to hear 100 Hz (we know it also from Tartini's third sound).
But we hear that sound only because of nonlinear interaction inside
our ear. If you lower the overall volume such effect should
disappear.

So if you took a (hypothetical) guitar with two strings tuned at
30000 and 30001 Hz, you do not hear any beating at 1 Hz, because you
do not hear the frequency at 30000 Hz and therefore you do not hear
its amplitude modulation! You can say the same with a detuning of
100 Hz, that is in the audible range: if your guitar strings are
tuned at 30000 and 30100 Hz, you do not hear any beating (although
this time it is fast and within the audible range) just because you
can't hear the frequency at 30000 Hz. Or do you? (note: if the
volume is increased enough, as in the previous example of third
sound, combinational tones appear, but in this case we are not
talking about beatings, but of "real" combinational sounds, that you
can see in your Fourier spectra, regardless it is made by a spectrum
analyzer or by your brain)

So, I repute the distinction between pressure (instantaneous)
stimulus and loudness (amplitude of pressure wave) crucial for a
better understanding. I can hear very well 1 Hz beating of two
slightly detuned guitar strings while I try to tune it, but for sure
my ear is not able to hear any 1 Hz (sinewave, pressure change...)
tone.

Max



beating is a change of loudness, while the