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Re: Amplitude and pitch of sound waves



Leigh said that an octave is not necessarily two frequencies that
differ by a factor of two.

I say it depends upon what you want to take as your definition. Do you
want to define an octave by the definition of an equal-tempered scale
(the usual practice) or do you want to define an octave by the
frequency relationship between two notes on a "properly tuned" piano
(like Leigh did)?

The equal-tempered scale, which *is* used by musicians, defines the
relationship between a base frequency and the chromatic scale above it
by f(n) = (12th-root-of-2)^n times f(0). Hence, when n = 12 we have
f(12) = 2 times f(0), and we are one octave above f(0). So the
equal-tempered scale always has an octave frequency ratio of two.

That some musical instruments sound better when tuned slightly
different from this does not necessarily redefine the meaning of an
octave. The reason a piano is tuned "sharp" as we go up in the scale
is because of a "quirk" in the harmonics of the string. Because of the
stiffness of piano wire, the node at the end of the wire tends to move
inward for the higher harmonics. So the first harmonic (music
definition of first) is causing a slightly shorter piece of wire to
oscillate than the fundamental; the string's effective length is
shorter yet for the second harmonic, and so forth.

This means that a stringed instrument with stiff wires under high
tension (i.e. a piano as opposed to a guitar) has a harmonic series
that is not perfect multiples of the fundamental. The harmonic series
goes sharp. That means if you play two notes on a piano that are
supposed to be one octave apart, and their fundamental frequencies are
tuned so the higher fundamental is indeed twice the frequency of the
lower fundamental, then when played simultaneously the first harmonic
of the lower note will beat with the fundamental of the higher note. A
rule of tuning pianos is that octaves are not supposed to beat. So we
have to tune the upper note a bit sharp to keep it from beating with
the first harmonic of the lower note.

Not all instruments are like this (or at least not so severely so), and
I do not think this is justification to redefine an octave. I'll stick
with the definition of an octave as a factor of two in frequency.

Returning to the original statements that initiated this thread. I
would not say that pitch *is* frequency. I would say pitch is *related
to* frequency. So I don't view the pitch statement on the poster as
too much of a travesty. But the amplitude statement is pretty bad.

Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817