Re: Physics of Chilly Magnus Chord Organs

["James W. Beauchamp" <jwb@xxxxxxxxxxxxxxxxxxxxx>]

Expanding on Jont's explanation, the frequency of vibration is strictly
proportional to the speed of sound and inversely proportional to the pipe's
effective length (f = c/(2*Leff)). The speed of sound depends on the
properties of the gas contained in the pipe and the temperature of the gas:

        c = const*sqrt(T(kelvin)) = const*sqrt(273+T(celsius))

For dry air, const = 20.1, so at 20 degrees celsius

        c = 20.1*sqrt(273 + 20) = 344 m/s

A useful approximation good for a range of temperatures around zero degrees
C is

        c = 331.3 + 0.6*T(celsius)

If the pipe were cold, at say 50 degrees F (10 degrees C), the frequency
would be 337.3, according to this formula. (338.1 using the more accurate
sqrt formula) When warmed to a room temperature of 20 degrees C we are back
to 344 m/s. So this is an increase of about 1.8% or about 3 tenths of a
semi-tone (which is approx. 5.9%).

You can calculate the "pitch" change in cents using the formula

 cents_change = 3986.3*log10(f2/f1)
              = 3986.3*log10(sqrt((273 + Tc2)/(273 + Tc1)))
             ~= 1993.1*log10(1 + (Tc2-Tc1)/273)
             ~= 865.6*log(1 + (Tc2-Tc1)/273)
             ~= 3.2*(Tc2-Tc1)

This gives 32 cents for a 10 degree C change, again about 0.3 semi-tone.

If someone produces a sound by blowing into a pipe, the air temperature is
presumedly even higher (up to 98 degrees F blood temperature) and more humid.
So this would further affect the value of c, but I'm not sure of the details.

The effective length of the pipe depends on the physical length plus an end
correction that depends on the size of the opening. 0.75*d, where d is the
diameter of the opening is sometimes used. However, it could also depend on
the shape of the pipe. If excited by a reed, a cylindrical pipe sounds an
octave below a conical pipe of the same length.

Jim Beauchamp
University of Illinois at Urbana-Champaign
j-beauch@uiuc.edu

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