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: Wed, 12 Aug 1992 23:29:03 GMT

In article <1992Aug12.152431.14664@husc3.harvard.edu> elkies@ramanujan.harvard.edu (Noam Elkies) writes:
>In article <1992Aug12.071230.6162@newshost.unh.edu>
>dvf@kepler.unh.edu (David V Feldman) writes:
>:>This brings up a disagreement I have had with some people over when an
>:>interval is "close enough" to a small whole-numbered ratio to be consi-
>:>dered just. Some have claimed that if it lies within the critical band
>:>of the ear's frequency discrimination, the ear can't really tell the
>:>difference. Well of course it can - because of beats. That much is ob-
>:>vious; the point of contention then boils down to the perception of
>:>beats in music. 
>:>effect for me.
>:
>:Before one even plunges into psychoacoustical waters, there is a purely
>:acoustical issue.  Frequency as such makes mathematical sense strictly
>:speaking only for periodic waves of infinite duration.  This is not
>:just a matter of being pedantic.  For short sounds there is an
>:indeterminacy principle (on the same mathematical basis as Heisenberg's
>:in quantum mechanics) that says essentially that the shorter the sound
>:the less definite the frequency.  [...]
>
>I was going to bring this up, but I wonder: in general we're dealing
>not with pure sine waves but also with higher partials; indeed it is the
>higher partials that are thought to be responsible for the resonance of
>intervals with small integer ratios.  So for instance the 5/4 resonance
>of a major third might well be spoiled by a discrepancy too small to
>affect the fundamentals.  Or is there a physical reason to expect that
>overtones must be subject to at least the same indeterminacy as their
>fundamental?  That seems to make sense, but I don't know if it's true.
>
>--Noam D. Elkies (elkies@zariski.harvard.edu)
>  Dept. of Mathematics, Harvard University

Noam is correct, of course: a short note on an oboe will have a more
definite pitch than say a short note on a flute or a french horn, because
of the predominance of upper partials in an oboe's spectrum.  The down
side is that physical systems usually don't produce `exactly harmonic'
harmonics because of non-linearities in the media.  There certainly is
an upper limit to our hearing range also, and I doubt that a lack of
beating between partials that were beyond the auditory range would
contribute to our interval perception, unless by some process that
I don't understand (perhaps we could hear the beats even if we couldn't
here the partials themselves?)  It would be interesting to do an experiment
to see whether the ear would accept unjust intervals as just because
the anharmonic partials were tuned justly.

It is worth pointing out that orchestrators intuitively take account of
the distinction above: staccato writing is considered more idiomatic
for instruments with richer higher partials, or in high pitch instruments.

--David Feldman
Xref: sparky rec.music.classical:11206 rec.music.folk:5593 rec.music.misc:21626 rec.music.dementia:1268