From: alves@calvin.usc.edu (William Alves)
Newsgroups: rec.music.classical
Subject: Re: When is close close enough for just intonation
Date: 14 Aug 1992 12:45:28 -0700
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: calvin.usc.edu
Keywords: intonation, intervals

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. 

>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 don't dispute this, HOWEVER, there are a few important points that
are critical to this issue: First, the effect of beats and frequency
precision depends on hertz, not cents. For example, it takes a tone
of one second duration to be able to identify the frequency to within
a range of one hertz. Now that may make tuning systems inprecise in a
musical context around cello C, but above middle C, where most of the
"action is" in much music, 1 hertz is not very significant.

Secondly, I believe one has to evaluate this issue from "psychoacoustical
waters" because there are a number of complicating factors that the
ear and brain introduce. For example, why is it that one study showed
that people were able to identify the pitch of low frequency sounds
of duration *less than 1 wavelength*? Why is it that I can hear the
change in sonority caused by a beat when the duration of the tones are
less than a quarter of the wavelength of the beat (i.e. 2 notes beating
at 1 hertz played for .25 seconds)? Clearly, the brain identifies pitch
in different ways than a digital tuner. Granted, your "Heisenberg"
principle of pitch uncertainty says that it can only be "right" within
a certain time-dependent range, but that could mean a lot of things in
a musical context.

>In practical term, the is no basis
>to distinguish between a tone whose amplitude is decaying (think of the
>ringing of a piano string) and the the effect of destructive interference
>of two nearby frequencies (which might become beating if the tone(s)
>is long enough compared to the reciprical of the difference of the
>frequencies -- this is just the point.)  

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.

>I am trying to say something
>here in an intuitive way that requires a bit of mathematics to make
>precise; if there is interest, and I haven't been sufficiently precise
>I will try to elaborate.  The upshot:  if you are a composer who wants
>distinction between very similar intervals to be real (not to say
>audible) it behooves you to write (very) slow music.  Actually my
>personal compositional interest is in projecting the identity of these
>intervals by constructing structral functions, extending common practice
>or not, so that the intervals can be recognized by their context.

Because of the two or more complicating factors I outlined above, I don't
think one has to write music with Takemitsu tempos for the differences
between "very similar" intervals to make a difference. La Monte Young did
some pieces along the (very) slow lines that certainly make the differences
in tuning very obvious. But he also did some pieces with very fast notes
where the pitches are in a sense sustained by reverberation. Others have
done similar things with reverberation and sympathetic vibrations.

At any rate, interval differences which take a long time to detect are
really not very common in most justly-tuned pieces. As Noam Elkies pointed
out, the difference between an equally-tempered major third and a 5/4
is quite pronounced. Middle C and the E above it beat about 7 times a
second in equal temperament.

One person, using arguments similar to those above, tried to convince me
that there is really no such thing as just intonation. And in the sense
that the frequency of a tone is only exact when the duration of the tone
is infinitely long, I suppose he was right. But my ears still insist that
Partch's is not the same as tempered music.

Bill Alves