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Re: [Phys-l] Exotic harmonies



Michael Edmiston wrote:

(1) It seems strange to me for someone to say that musical scales and
their temperaments are tangential to the discussion of small
whole-number ratios when small whole-number ratios provide the
historical context for the development of the musical scale.

It seems strange for someone to say that the earth moves and
the sun has spots, when the fixity of the earth and the perfection
of the sun provide the historical context for all of cosmology.

Eppur si muove.

Historically, the theory of continental drift was treated with contempt
by the scientific establishment, long after more-than-sufficient
evidence of drift was available.

If there are facts, tell me the facts. Don't tell me about the
"historical context".

=======

Using temperament to argue against stretching, or against the
importance of higher partials in the perception of harmony, is
like using "green" to argue against "triangular". There exist
objects that are green, or triangular, or both, or neither.
Pianos are stretched for one reason, and tempered for another.

I say again I have no objection to the discussion of temperament,
but it is at best tangential to the discussion of the importance
of higher partials in the perception of harmony.

There exist instruments that are stretched and equal-tempered: ordinary piano.
There exist instruments that are equal-tempered only: ordinary harpsichord.
There exist instruments that are neither stretched nor tempered: violin.
There exist instruments that are stretched but not equal-tempered: piano
optimized for a particular key. The C-major chord on a piano sounds a
lot sweeter if the instrument is *not* equal-tempered.


The
original topic we were asked to comment on was why we don't have ratios
involving seven (8:7, 9:7, 10:7) in our scale. That means the orignal
question was, in fact, a question about scales and temperaments.

I still think the original question is better answered without
reference to equal temperament. The original question could
perfectly well be answered with respect to a non-equal-tempered
piano, one that was optimized to play (say) C major. The original
question did not state any requirement for convenient transposition.

I say again that there are many true and interesting things one
can say about equal temperament ... but they remain tangential
to the topics I was discussing.


(3) Some musicians, perhaps especially violin players and singers, do
not like stretched tuning for a piano. They seem to hear and prefer
octaves as not stetched and this can lead to conflicts when they are
accompanied by a piano that is stretched.

Let's talk first about violins, in particular pizzicato violins. There
is nothing "seeming" about the violin/piano conflict. It is 100% real
and objective. OTOH it has zero cosmic significance. The conflict
does not say that stretching is cosmically good or bad; it just says
stretching is different from no stretching. The dispersion relation
for the piano string is different from the dispersion relation for the
violin string. Not cosmically better or worse, just different.

If you want to argue on religious grounds that an instrument *should*
have integer ratios -- as Pythagoras did -- you are free to do so, but
that is religion, not physics.

The situation is different for the bowed violin, for the voice, and
for a wide range of wind instruments. That's because in those cases,
the sound-production mechanism incorporates a very significant nonlinearity.

Remember, nonlinearity is different from dispersion. Nonlinearity
depends on amplitude, while dispersion depends on frequency. They
could hardly be more different ... except that sometimes partials
reflect the dispersion, and sometimes they reflect the nonlinearity.

Let's start by considering a chopper-type siren. There is a fundamental
frequency, but the chopping process produces all harmonics of that frequency
... in integer ratios. To a good approximation the time-domain waveform
is a Dirac comb, and the Fourier transform thereof is another Dirac comb.
The downstream system is essentially linear, so by Floquet's theorem the
integer-ratio frequencies are preserved and are present at the output.
We can apply Floquet's theorem with confidence, because the rotor in the
siren is massive and is not going to be perturbed by anything downstream.

Now let's consider the human voice. The collision between the vocal chords
is hugely nonlinear. It can be represented by a delta function. The
Fourier transform of a delta function contains all frequencies (integer
ratio frequences as well as others). However, a sequence of equally-
spaced delta functions -- a Dirac comb -- contains only the integer ratio
frequencies. The Fourier transform of a Dirac comb is another Dirac comb.
In this case it is not 100% clear we can apply Floquet's theorem with
confidence, because there is some coupling between the air column and the
vocal chords. However, I suspect the coupling doesn't distort the Dirac
comb very much, and I suspect the coupling is non-systematic, stretching
some intervals and shrinking others, unlike the systematic stretching we
see in the piano.

The clarinet is obviously similar, and I suspect most (perhaps all) wind
instruments are similar. The bowed violin may also be similar, but I
haven't really thought about it.

The point of all this is that there is a wide class of instruments where
the partials come from a nonlinearity, not from a dispersion relation,
and integer ratios are natural for these instruments. So the story about
integer ratios is not entirely dogma; there is also some physics in it.

Yes, integer ratios are relevant for this class of instruments, namely
the instruments where nonlinearity is more significant that dispersion.

But I still don't attribute any /cosmic/ significance to it. It doesn't
have to be that way, as Max Matthews demonstrated. A stretched Dirac
comb -- even quite greatly stretched -- doesn't sound particularly
different from an unstretched one, and harmonizes nicely with others of
the same ilk ... with decidedly non-integer-ratio harmonies.

Integer ratios are not the root of the physics. They are sometimes a
consequence of the physics, but not always even that.