psychoacoustic foundations

Paul, mind if I quibble a bit? Yesterday you wrote:

>... the probability that two physical strings are tuned
>to a rational inteval is zero, as Cantor showed...

This is true only if the two strings vibrate independently.
When two strings are mounted on the same instrument, then
there may well be some kinds of interaction between them.
They may, for instance, become "entrained" to each other,
that is, locked into various "simple integer" modes of vibration.
Besides strings, there are other examples of entrainment, for
instance, the orbits of several moons are entrained to simple integer
ratios about their planets. On pianos, it is well known that adjacent
strings interact. Might they, under certain conditions, entrain?

Then, in talking about consonance and dissonance, you said:

>The only definition I care about is one with a psychoacoustic
>correlative.

The sensory consonance theory (including Helmholtz, Plomp and
Levelt, Terhardt, your own work, and my own work) is not the only
theory that has a "psychoacoustic correlate."

Consider for a moment the slightly different question of how the ear
determines pitch. There are two major theories, the "place" and the
"periodicity" theories of pitch perception. In the place theory, pitch is
determined by where on the basilar membrane the maximum
excitation occurs. This classic model, expounded by Von Bekesy in
the 30's was considered the final word for a long time, but careful
measurements of JND's (Just Noticeable Differences) for pitch showed
that the basilar membrane is not sensitive enough to account for all
thats known about the ear's resolution powers.

The ear can also resolve very small temporal differences (as used for
instance, to locate sounds in the environment). The periodicity
theorists suggest that the ear basically "counts up" how many
maxima (or minima or zero crossings, depending on who you read)
occur in a given time interval to determine pitch. There is currently
no experimental evidence that definitely points to one of these
theories over the other, and J Pierce has argued (convincingly, in my
opinion) that the most likely scenario is that both mechanisms are at
work.

Now let's return to the "simple integer ratio" theory of consonance.
One can easily turn the simple integer ratio version into a "length of
waveform" argument. The periodicity pitch theorists already have a
mechanism for determining length of period, so it would be easy to
believe that the this same mechanism could also be used by the
"dissonance determining algorithm" of the brain. The problem with
irrational ratios then becomes the problem of determining the pitch
of "almost periodic waveforms". This is then not a huge problem
since presumably the brain will interpret ambiguous interval ratios
in much the way as it interprets ambiguous pitches.

Thus the "simple integer ratio" theory of consonance has (roughly) as
firm a psychoacoustic foundation as the sensory consonance theory.


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