Re: [Phys-l] Exotic harmonies

[John Denker <jsd@av8n.com>]

Bernard Cleyet wrote:
> My friend, the crackpot inventor sent me the msg below.
...
> My device easily shows many mathematical patterns that are not musical 
> harmonies in the western scale, but are inherently harmonic because of 
> the low value integer relationships (like 7:4, 11:8, 13:8, 14:8).

There is some very interesting physics and psychophysics in this.

Your friend is fundamentally wrong, but he's in good company.
Everybody from Pythagoras to Galileo (inclusive) made the
same mistake.

It turns out that integer ratios (and the implied phase relationships)
have little to do (AFAIK nothing to do) with harmony, consonance, or
dissonance ... except very indirectly in special cases.

This should be obvious to anyone who has ever tuned a piano, or
even looked at the frequencies of piano notes.  The "octaves" on
a piano are not 2x apart in frequency;  they are systematically
farther apart than that.  We speak of "stretched" octaves.

Why is that, you ask?  For nondispersive waves, you would
expect the higher partials to be simple integer multiples of
the fundamental frequency.  This is what you get by solving
the wave equation, taking into account only the mass and tension
of the string.

In contrast, wave propagation on a piano string is conspicuously
dispersive.  We observe that the partials have higher frequencies
than mass/tension theory would predict.  A proper description requires
also taking into account the /stiffness/ of the metal strings.
The Green function that describes this involves fourth-order
derivatives, so it is waay different from the usual second-order
wave equation.

Now the psychoacoustic angle is that the octave "sounds" in-tune
when the frequency of the higher note is matched to the second
harmonic of the lower note ... !not! to twice the fundamental of
the lower note.

One example -- the piano -- does not suffice to prove the point, but
Max Matthews did a series of experiments with synthetic instruments
with hugely stretched octaves (and shrunken octaves) and pretty much
removed all doubt.

I heard him play some of these instruments once.  The remarkable thing
is that they didn't sound bad, or even weird.  They sounded fine.  If
you didn't look at the frequency counters, you wouldn't necessarily
know there was anything funny going on.  (I don't have perfect pitch,
in case that wasn't obvious.)  And of course you wouldn't want to play
his instruments in an ensemble with plain old unstretched instruments.

=================

Note:  I absolutely did not say that the ear is insensitive to phase.
There are plenty of cases where phase does matter, including, notably,
the human sonar that blind people use.  (I do know how to do that, at
a very modest level of skill.)   The human auditory system is reeeally
nonlinear.  One of my protégées did her senior thesis on this.

What I said is that integer-ratio relationships are not what control musical
harmony, consonance, or dissonance.