>> Could someone tell me how long to cut a one-inch pipe to get middle C?
I will assume that this pipe is open at the end as you referred to
primitive brass instruments.
The frequency of middle C is about 262 cycles/sec.
The speed of sound (at 68 degrees F) is about 344 meters/sec.
Thus, you want the pipe to resonate at 262 cycles/sec when a blast of
air is sent down it.
This means that the wavelength of the sound wave should be
velocity/frequency or 344/262 or 1.3 meters. My physics book tells me
that the length of an open tube is equal to 1/4 of the wavelength of its
Thus, a tube of .325 meters (i.e., about 12.8 inches) should be a good
start. As you noticed, I have made a number of approximations and
cavalier roundings along the way, so some tuning will be necessary.
Hope this is helpful (and correct :)
Best of Luck.