Halve the length of a string or pipe and its note goes up an octave; close one end of a pipe and it drops an octave.
A string held at both ends, or a pipe open at both, fits whole half-wavelengths, so its frequencies are n v / 2L: twice the length is half the frequency. A pipe closed at one end fits odd quarter-wavelengths, so its lowest frequency is v / 4L, half that of the same pipe open. A guitar string stopped at the twelfth fret is half as long and sounds an octave higher.
It holds for an ideal string and a pipe of uniform bore. Real pipes sound as if they were a little longer than they are, by about 0.6 of the radius at each open end, so short wide pipes sound lower than the rule says. Finger holes and flared bores change the upper harmonics; stiff strings go sharp at the top; and a string’s pitch also depends on its tension (four times the tension doubles it) and its mass per length.
Estimate with the rule, then check it against the calculator that models it properly.
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